Spectral Measures and Dynamics: Typical Behaviors [1 ed.] 9783031382888, 9783031382895

Table of contents :
Preface
Contents
0 Book's Outline
0.1 Motivation
0.2 Part I
0.3 Part II
1 Spectrum and Dynamics: Some Basic Concepts
1.1 First Basic Concepts
1.1.1 Standard Spectral Classification
1.1.2 Fractal Measures
1.1.3 Dimensions of Measures
1.2 Semigroups
1.2.1 General Definitions
1.2.2 Asymptotics of C0-Semigroups: A Short Account
1.2.3 Joint Resolution of Identity and Normal Semigroups
1.3 Wonderland Theorem
1.3.1 Proof of Wonderland Theorem
Part I Quantum Models
2 Correlation Dimension
2.1 Correlation Dimension and Return Projections
2.2 Gδ Sets For Correlation Dimension
2.3 Generic Correlation Dimension
2.4 Applications
2.4.1 Bounded Operators: Norm Convergence
2.4.2 Bounded Operators: Strong Convergence
2.4.3 Discrete Schrödinger Operators
3 Fractal Measures and Dynamics
3.1 Local Dimensions of Measures and Dynamics
3.2 Fractal Decomposition of Measures
3.3 Dynamics of UαHC and UαHS Measures
3.4 Lower Bounds on Moments
3.4.1 b-adic Intervals
3.4.2 Proof of Theorem 3.26(i)
3.4.3 Proof of Theorem 3.26(iii)
4 Escaping Probabilities and Quasiballistic Dynamics
4.1 Laplace Average Moments
4.2 Escaping Probabilities
4.3 The SULE Condition and Quasilocalization
4.4 Gδ Sets for Dynamical Exponents
4.5 Applications to Schrödinger Operators
4.5.1 Bounded Potentials
4.5.2 Analytic Quasiperiodic Potentials
4.5.3 Unbounded Discrete Schrödinger Operators
5 Generalized Dimensions and Dynamics
5.1 Dynamics of Pure Point Operators
5.2 Box-Counting Dimension
5.3 Thick Point Spectrum and Upper Fractal Dimensions
5.3.1 Proof of Proposition 5.8
5.4 Generic Dimensions for the Hydrogen Atom
5.4.1 Proof of Theorem 5.15
5.5 A Dimensional Heritage
5.6 Dimensions and Moments
5.6.1 Proof of Theorem 5.23
Proof of Theorem 5.34
Proof of Theorem 5.32
5.7 Generic Dimensions for Some Schrödinger Operators
Part II Ergodic Theory and Semigroups
6 Generic Scales of Weak Mixing
6.1 Ergodic Dynamical Systems
6.2 Typical Automorphisms in the Weak Topology
6.2.1 Proof of Rohlin's Lemma
6.2.2 Proof of Halmos' Conjugacy Lemma
6.3 Refined Scales of Weak Mixing
6.3.1 Relation to Correlation Dimensions
6.3.2 Generic Behavior
7 Asymptotics of C0-Semigroups
7.1 Normal Semigroups
7.1.1 Polynomial Decay Rates Spectral Properties
7.1.2 Generic Decay Rates
7.1.3 Stability and Spectrum
7.2 C0-Semigroups in Hilbert Spaces
7.2.1 Generic Decay Rates
7.2.2 Applications
Wave Equation with Localized Viscoelasticity
Thermoelastic Systems of Bresse Type
Damped Wave Equation on the Torus
8 Generic Stability for Self-Adjoint Semigroups
8.1 Generic Stability
8.2 Proof of Theorem 8.1
8.3 A Typical Spectral Property
8.4 Schrödinger Equation Spectrum
Reference
Index

Citation preview

Latin American Mathematics Series UFSCar subseries

Moacir Aloisio Silas L. Carvalho César R. de Oliveira

Spectral Measures and Dynamics: Typical Behaviors

Latin American Mathematics Series

Latin American Mathematics Series – UFSCar subseries Series Editors César R. de Oliveira, Federal University of São Carlos, São Carlos, São Paulo, Brazil Ruy Tojeiro, Universidade de São Paulo, São Carlos, São Paulo, Brazil

Advisory Editors Shiferaw Berhanu, Temple University, Philadelphia, PA, USA Ugo Bruzzo, SISSA - Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy Irene Fonseca, Carnegie Mellon University, Pittsburgh, PA, USA

Published under the Latin American Mathematics Series, which was created to showcase the new, vibrant mathematical output that is emerging from this region, this subseries aims to gather high-quality monographs, graduate textbooks, and contributing volumes based on mathematical research conducted at/with the Federal University of São Carlos, a technological pole located in the State of São Paulo, Brazil. Submissions are evaluated by an international editorial board and undergo a rigorous peer review before acceptance.

Moacir Aloisio ● Silas L. Carvalho ● César R. de Oliveira

Spectral Measures and Dynamics: Typical Behaviors

Moacir Aloisio Mathematics Federal University of Jequitinhonha and Mucuri Valleys Diamantina, Minas Gerais, Brazil

Silas L. Carvalho Mathematics Federal University of Minas Gerais Belo Horizonte, Minas Gerais, Brazil

César R. de Oliveira Mathematics Federal University of São Carlos São Carlos, São Paulo, Brazil

ISSN 2524-6755 ISSN 2524-6763 (electronic) Latin American Mathematics Series ISSN 2524-6755 ISSN 2524-6763 (electronic) Latin American Mathematics Series – UFSCar subseries ISBN 978-3-031-38288-8 ISBN 978-3-031-38289-5 (eBook) https://doi.org/10.1007/978-3-031-38289-5 Mathematics Subject Classification: 81Q10, 35J10, 28A80, 81P10 This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (303503/2018-1), and Fundação de Amparo à Pesquisa do Estado de Minas Gerais (001/17/CEXAPQ00352-17). © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

To the memory of João Batista de Oliveira

Preface

The spectral theory of linear operators in (separable) Hilbert spaces is vast, as well as its field of applications; for instance, such theory plays a fundamental role in the mathematical formulation of quantum mechanics. The spectral measure says a lot about the energy levels and the dynamics of the corresponding quantum state. However, such correspondences are not immediate and many nuances have been discovered in the last 20 years. For instance, during the 1960s and 1970s, the singular continuous spectra were considered exotic and without physical meaning by mathematical physicists. Today the situation is different, since this kind of spectrum is common in models of quasicrystals, including Schrödinger operators with almost periodic potentials, and under some conditions such kind of spectrum is also typical in Baire’s sense (see (ii)). Spectrum and dynamics of linear operators in general (i.e., not necessarily related to quantum systems) have a symbiotic relationship and there are many subtleties among them; for instance: (i) Any self-adjoint operator may be approximated (with respect to the HilbertSchmidt norm) by a pure point operator (this is the Weyl-von Neumann Theorem [53, 137, 139]). (ii) In some topological spaces of self-adjoint operators, the set of elements with purely singular continuous spectra is generic in Baire’s sense (the conclusion of the so-called Wonderland Theorem [125]). (iii) Dense point spectrum implies a form of dynamical instability [1]. (iv) Norm operators with continuous spectrum, whose spectrum has zero distance from the imaginary axis, generate semigroup orbits that display nonuniform behavior [3]. (v) In topological spaces of measure-preserving transformations on Lebesgue spaces with continuous measures, the generic fine properties of weak mixing behavior depend on subsequences of time going to infinity [27]. The aim of this book is to present a study on the generic (in Baire’s sense) behavior of spectral measures and some of their dynamical consequences. A lot of material is not yet found in book form, and part of the material is based on rather recent research by the authors. vii

viii

Preface

The first chapter (Chap. 0) refers to a general view of the book scope and contents. Chapter 1 is dedicated to some basic concepts and classical results on spectral theory and .C0 -semigroups on Hilbert spaces; the description of a normal semigroup is done through the joint resolution of identity and, finally, a proof of Wonderland Theorem is presented. After such extended introduction, Chaps. 2–5 constitute the first part of the text and they address quantum models, which mainly discuss the relations between spectral properties of a self-adjoint operator and the associated unitary evolution group, focusing on generic properties of spectral measures. Chapters 6–8 constitute the second part, and they deal with weak mixing properties in Ergodic Theory and normal semigroups in Hilbert spaces. In Chap. 6, we discuss aspects of time evolution of some discrete dynamical systems (unitary Koopman operators), taking into account extreme scales of variations of (generic) weak mixing; .C0 -semigroups on Hilbert spaces are addressed in Chaps. 7 and 8, also focusing on normal semigroups and generic behavior of spectral measures. The second and third authors of this book have been working in refinements and generalizations of generic (spectral and dynamical) properties of spaces of selfadjoint operators, and recently, the first author has joined the others and published works on variants of the main stream, particularly in the setting of evolution semigroups. Some of the mentioned generalizations are related to additional information about different notions of fractal dimensions of spectral measures; for instance, upper and lower Hausdorff and packing dimensions, Grassberger-Procaccia generalized dimensions, correlation dimensions, and local dimensions. The typical results are of the form “the set of operators with zero lower and one upper correlation dimensions is a generic set.” This book tries to gather results on generic behavior of spectral measures in an expanded and organized form; it also includes some recent results. Sometimes additional information is given in the form of remarks, and if we use a result without proof in the book, a reference is provided. Clearly, there is a bias toward “exotic” spectra and dynamics, following the authors’ taste and research; but it should be clear that this is not a book about singular continuous spectra! The book is directed to graduate students and researchers interested in increasing their knowledge of some advanced aspects of spectral theory and quantum dynamics. Except for the first two and last two chapters, the others can be read (almost) independently. Acknowledgments MA thanks the support of his former institution, the Federal University of Amazonas. SLC acknowledges partial support by Fapemig (Minas Gerais State Brazilian agency). CRdO acknowledges partial support by CNPq (Federal Brazilian agency). Diamantina, MG, Brazil Belo Horizonte, MG, Brazil São Carlos, SP, Brazil March, 2023

Moacir Aloisio Silas L. Carvalho César R. de Oliveira

Contents

0

Book’s Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.2 Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.3 Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 8

1

Spectrum and Dynamics: Some Basic Concepts. . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 First Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Standard Spectral Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Fractal Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Dimensions of Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 General Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Asymptotics of C0 -Semigroups: A Short Account . . . . . . . . . . . . 1.2.3 Joint Resolution of Identity and Normal Semigroups . . . . . . . . . 1.3 Wonderland Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Proof of Wonderland Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 17 23 24 30 31 32 35 37 40

Part I Quantum Models 2

Correlation Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Correlation Dimension and Return Projections. . . . . . . . . . . . . . . . . . . . . . . . 2.2 Gδ Sets For Correlation Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Generic Correlation Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Bounded Operators: Norm Convergence . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Bounded Operators: Strong Convergence . . . . . . . . . . . . . . . . . . . . . 2.4.3 Discrete Schrödinger Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 52 56 57 57 61 63

3

Fractal Measures and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Local Dimensions of Measures and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Fractal Decomposition of Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Dynamics of UαHC and UαHS Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 67 73 80 ix

x

Contents

3.4 Lower Bounds on Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 b-adic Intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Proof of Theorem 3.26(i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Proof of Theorem 3.26(iii) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 85 87 90

4

Escaping Probabilities and Quasiballistic Dynamics . . . . . . . . . . . . . . . . . . . . 4.1 Laplace Average Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Escaping Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The SULE Condition and Quasilocalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Gδ Sets for Dynamical Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Applications to Schrödinger Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Bounded Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Analytic Quasiperiodic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Unbounded Discrete Schrödinger Operators . . . . . . . . . . . . . . . . . .

97 97 106 110 116 119 120 121 122

5

Generalized Dimensions and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Dynamics of Pure Point Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Box-Counting Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Thick Point Spectrum and Upper Fractal Dimensions . . . . . . . . . . . . . . . . 5.3.1 Proof of Proposition 5.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Generic Dimensions for the Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Proof of Theorem 5.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 A Dimensional Heritage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Dimensions and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Proof of Theorem 5.23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Generic Dimensions for Some Schrödinger Operators . . . . . . . . . . . . . . . .

125 125 127 131 136 141 143 149 156 161 172

Part II Ergodic Theory and Semigroups 6

Generic Scales of Weak Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Ergodic Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Typical Automorphisms in the Weak Topology . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Proof of Rohlin’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Proof of Halmos’ Conjugacy Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Refined Scales of Weak Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Relation to Correlation Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Generic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 177 182 186 189 192 193 198

7

Asymptotics of .C0 -Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Normal Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Polynomial Decay Rates × Spectral Properties . . . . . . . . . . . . . . . 7.1.2 Generic Decay Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Stability and Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 C0 -Semigroups in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Generic Decay Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205 205 205 212 218 220 220 222

Contents

8

Generic Stability for Self-Adjoint Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Generic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Proof of Theorem 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 A Typical Spectral Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Schrödinger Equation × Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

229 229 232 234 234

Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Chapter 0

Book’s Outline

0.1 Motivation This book discusses topics related to spectral measures of normal operators, mainly self-adjoint and unitary ones, with emphasis on generic properties in Baire sense; such properties may be related to generic vectors in the Hilbert spaces and/or to generic operators in some complete metric space of operators satisfying suitable properties. This is just an extended version of the book’s title, and more details appear ahead. The subject has applications, since self-adjoint and unitary operators are relevant to many branches of Physics and Mathematics, in particular to quantum mechanics, evolution semigroups, and ergodic theory of dynamical systems. So, this book is naturally divided into two parts, the first one with applications to “quantum models” and the second one to “classical models” (i.e., ergodic theory and semigroups). When the underlying spaces of operators are described by a set of real parameters, it is also natural to think of general properties in terms of Lebesgue measure, and as it is well known, this concept does not necessarily agree with the notion of generic property of Baire; but it is not our intention to elaborate on this in each application. Part of the results presented here is based on rather recent results by the authors, of course supplemented by traditional background and more specific material developed by other researchers; we think the subject is ripe for a book. The discussion of the Hydrogen atom, for example, in the original papers on wave mechanics by Schrödinger, has already indicated that should be a link between spectral types of the Hamiltonian operator and dynamics; indeed, his achievement in describing orbitals was through eigenvectors of the Hamiltonian, whereas the ionized atom was associated with a continuous spectrum (Schrödinger did not use this nomenclature). In order to put the discussion in a broader perspective, consider a self-adjoint operator T , acting in a (infinite dimensional) separable Hilbert space .H , which © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Aloisio et al., Spectral Measures and Dynamics: Typical Behaviors, Latin American Mathematics Series – UFSCar subseries, https://doi.org/10.1007/978-3-031-38289-5_0

1

2

0 Book’s Outline

is the infinitesimal generator of the one-parameter strongly continuous evolution group R  t −→ Ut = e−itT

.

that gives the unique solution .ψ(t) = Ut ψ to the Schrödinger initial value problem i

.

d ψ(t) = T ψ(t), dt

ψ(0) = ψ ∈ dom T ,

where .dom T denotes the domain of T . Sometimes, a unitary operator U is used to generate a discrete time evolution Z  k −→ U k

.

through the k-times composition of U with itself (.U 0 = 1, the identity operator); an important instance is the Koopman operator, which describes the time evolution of some (classical) discrete dynamical systems (a subject of Chap. 6), and the Floquet operator for time-periodic quantum systems (not considered here). The long-time behavior of these systems is characterized by either .t → ∞ or .k → ∞. The spectral measures of such operators will be denote by .μTψ and .μU ψ in case of self-adjoint and unitary operators, respectively. Local and average characteristics of each spectral measure are important to describe the large time evolution of the system, and we will mention its most prominent properties in this book; some properties are rather natural and expected (time evolution of eigenvectors, for example), but others are very subtle (as special values of generalized fractal dimensions of spectral measures). Recall that a basic decomposition of a spectral measure μT = μTp + μTac + μTsc

.

is in its pure point part .μTp (constituted only of atoms and generated by eigenvalues of T ), its absolutely continuous (with respect to Lebesgue measure) part .μTac and its singular continuous part .μTsc (continuous (i.e., no atoms) but singular with respect to Lebesgue measure). In the following, we mainly focus our discussion on the continuous evolution groups and self-adjoint operators, the most frequent case here; more details of the above important decomposition will appear in Chap. 1. Remember that .μTc = μTac + μTsc is the continuous component of the spectral measure. Since the spectral characterization of each operator is given in terms of spectral measures, our interest in this book is in the study of the (Baire) generic behavior of suitable spectral, fractal, and/or dynamical properties of such measures; as already mentioned, sometimes the generic property refers to vectors in the Hilbert space, sometimes to a set of operators. An example of a generic property (values of some generalized dimensions) of a set of vectors in the Hilbert space is the case of dense

0.1 Motivation

3

point spectrum, discussed in Chap. 5; there are also generic results (again about generalized dimensions of spectral measures) for some particular sets of discrete spectra (see Sect. 5.4). Another example is the so-called Wonderland Theorem by Simon [125], which gives sufficient conditions for the set of self-adjoint operators with purely singular continuous spectrum to be generic; we recall this result in Theorem 1.31. So, we see that, although the spectral type may play a role, it is not the last word when talking about generic properties of spectral measures. Based on examples (as the Hydrogen atom mentioned above) and informal discussions, there is a long-standing wisdom that continuous spectrum is related to instabilities and transport, whereas point spectrum is related to stability and localization—Chap. 1 details some suitable quantities to investigate these associations. It is a mathematical challenge to provide specific and complete characterizations of such “popular” wisdom. Maybe the first relevant mathematical contributions in this direction were through the famous Riemann–Lebesgue Lemma (see Lemma 1.4) and Wiener’s Lemma (see Lemma 1.5); both relate the measure types to the asymptotic behavior of its Fourier coefficients, which, but their turn, are related to dynamics. A step further was the so-called RAGE Theorem [40, 53, 121] in scattering theory. However, such mathematical contributions have a qualitative nature (e.g., some quantities vanish but no information on the rate they vanish), and even today one searches for (precise) quantitative descriptions of the link between dynamics and spectra. On top of such qualitative connections, in some cases, one is able to present quantitative descriptions in the Baire generic sense, and some of them are presented in this book—e.g., Theorems 2.6 and 6.31. Although dynamical properties are very appealing, we reinforce that fractal properties of spectral measures and their generic values have an interest per se, and they constitute a large fraction of this text. As mentioned above, the generic properties explored here are not related to a specific spectral type, and usually the results are general, and we do not concentrate on specific models (the Hydrogen atom in Sect. 5.4 is one of the few exceptions), but models will naturally appear in applications of the results (usually to Schrödinger operators). We have found it very curious that the generic behavior has a tendency to be, at first sight, unconventional, with quantities taking extreme values in the possible ranges. For example, as discussed in Chap. 2, the correlation dimensions take values between 0 and 1 and, under suitable hypotheses, the typical behavior of the lower correlation dimension of spectral measures is zero, whereas the upper correlation dimension is 1 (as a spin off, typically the limit defining this dimension does not exist). With respect to the part of spectrum and dynamics of quantum systems, this book is mainly about definitions and estimates involving self-adjoint operators defined in abstract separable Hilbert spaces, with the majority of the applications of these rather abstract results to one-dimensional Schrödinger operators. However, it is worth underlying that the works of Guarneri and Schulz-Baldes [73–77], Combes [33] and Last [97] also concern higher dimensional cases and the dimension d does enter into the basic bounds, but be aware of different expressions for the

4

0 Book’s Outline

p-moments, with the presence of multiplicities in the weights. These modifications of the moments and bounds already show that the dimensions of the spectral measures may not be very relevant for the dynamics in higher dimensions, and this was confirmed in a work by Bellissard and Schulz-Baldes [17]; namely, in this work it was presented a class of hamiltonians in dimension .d = 3, describing a quantum particle in aperiodic medium with absolutely continuous spectrum and subdiffusive behavior. The diffusion exponent .βψ (2), which characterizes the growth of time of the 2-moment (see ahead), can be chosen slightly larger than Guarneri’s inferior limit (see also Remarks 3.29 and 5.24). We close this section with two comments. First, this is not a book on singular continuous or fractal spectrum; the point is that sometimes such features naturally appear related to Baire generic properties; by considering dynamical and dimensional properties, this book goes one step further the statement “exotic spectra are generic.” Second, it is worth quoting Mark Kac [87], page 18: “As is often the case, it is much easier to prove that an overwhelming majority of objects possess a certain property than to exhibit even one such object.” However, soft analysis that leads to generic properties is, at least, existence results and indicates the typical features that should be analyzed in each particular setting. It should be mentioned that there is a recent book, by Damanik and Fillman [43], whose general lines have some overlap with this book. It is about the general theory of one-dimensional discrete Schrödinger operators with ergodic potentials (a second volume, with examples, will appear soon); it is very well written and organized, and we strongly recommend for interested readers. Many of the topics covered in [43] are used in this text, mainly because most of our applications are to one-dimensional operators; however, their interest is not on typical properties, and they do not address different fractal dimensions, neither the classical subject discussed in Part II here.

0.2 Part I Now we (shortly) anticipate some quantities of interest in order to present a general view of the contents of this book. The main dynamical quantities to be considered in the first part of this book are the average (quantum) visit probability of the initial state .ψ, at time t, to .ξ ∈ H , given by T Wξ,ψ (t) = Wξ,ψ (t) =

.

1 t



t

|ξ, ψ(s)|2 ds,

0 ψ

ψ

(see the discussion around equation (1.1)), and the p-moments, .rp (t) = rp (B; t), with respect to a given orthonormal basis .B = {en } of .H , with .p > 0, given by rpψ (t) =



.

n

|n|p Wen ,ψ (t)

1

p

0.2 Part I

5

(see the discussion around (1.3)). Their power-law behavior are described by the exponents defined as − γξ,ψ = lim inf

.

t→∞

ln Wξ,ψ (t) , ln 1/t

+ = lim sup γξ,ψ t→∞

ln Wξ,ψ (t) ln 1/t

(for short .γψ = γψ,ψ ) and βψ− (p) = lim inf

.

t→∞

ψ

ln rp (t) , ln t

βψ+ (p) = lim sup t→∞

ψ

ln rp (t) . ln t

We introduce the first possibility from the fractal point of view; given a (positive) finite Borel measure .μ on .R, one considers its generalized fractal dimensions, parametrized by .q > 0, .q = 1, Dμ− (q) = lim inf

.

ε↓0

ln Iμ (q, ε) , (q − 1) ln ε

Dμ+ (q) = lim sup ε↓0

ln Iμ (q, ε) , (q − 1) ln ε

where  Iμ (q, ε) =

μ((x − ε, x + ε))q−1 dμ(x),

.

with integration restricted to the support of .μ (see around (1.12)). There will also be a discussion on the upper and lower Hausdorff and packing dimensions of .μ; if .dimH (S) and .dimP (S) denote the Hausdorff and packing dimensions of .S ⊂ R, respectively, such upper and lower dimensions are defined as (see Sect. 1.1.3) .

dim+ K (μ) = inf{dimK (S) | μ(R \ S) = 0, S a Borel subset of R},

and the .K-lower dimension of .μ as .

dim− K (μ) = sup{α | μ(S) = 0 if dimK (S) < α, S a Borel subset of R},

with .K = H (for Hausdorff) or .K = P (for packing). Chapter 1 presents more details of the above fractal dimensions, including proofs of the traditional Riemann–Lebesgue Lemma, Wiener’s Lemma, and the Wonderland Theorem. Chapter 2 first discusses an interesting result that precisely links the power-law exponents ruling return probabilities to the correlation dimensions of spectral measures (i.e., .Dμ± (q) with .q = 2; original references are [11, 84, 91]); more precisely, it says that γψ± = Dμ±ψ (2) .

.

6

0 Book’s Outline

Then, sufficient conditions are given so that generically the lower correlation dimension takes the least value (i.e., .Dμ−ψ (2) = 0) and the upper one takes the largest possible value (i.e., .Dμ+ψ (2) = 1). The proofs follow the original source [24]. Applications are presented to sets of general bounded self-adjoint operators with different topologies and discrete Schrödinger operators. Other relevant concepts, related to fractal properties of measures, are the (lower and upper) local dimensions of .μ defined, respectively, as ln μ((x − ε, x + ε)) , ln ε

dμ− (x) = lim inf

.

ε↓0

dμ+ (x) = lim sup ε↓0

ln μ((x − ε, x + ε)) . ln ε

In Sect. 3.1, these local dimensions are related to the .α-derivatives of .μ and it is shown that .

− dim− H (μ) = μ−ess.inf dμ ,

− dim+ H (μ) = μ−ess.sup dμ ,

+ dim− P (μ) = μ−ess.inf dμ ,

+ dim+ P (μ) = μ−ess.sup dμ .

The decompositions of measures are then presented with respect to .α-Hausdorff and .α-packing measures, as well as the concepts of uniform .α-Hölder continuous and uniform .α-Hölder singular measures (see Definition 3.18), along with the wellknown approximation of .α-Hausdorff continuous by uniform .α-Hölder continuous measures [97] and a rather new approximation of .α-packing singular measures by uniform .α-Hölder singular measures [26]; dynamical consequences are discussed in Sect. 3.3. Section 3.4 presents the known inequalities relating the exponents ruling the algebraic growth rate of the p-moments to upper Hausdorff and packing dimensions of the spectral measure, that is, T βψ− (p) ≥ dim+ H (μψ ),

.

T βψ+ (p) ≥ dim+ P (μψ ),

for all .p > 0. Apparently, a seed for the formulations and proofs of such inequalities was a result due to Guarneri [73], in an attempt to quantify RAGE’s Theorem (under an assumption on the spectral measure, basically the condition of uniformly .αHölder); the final versions were obtained in [11, 75, 97]. Chapter 3 ends with some generic results about the values of lower Hausdorff and upper packing dimensions of spectral measures, including some dynamical consequences for limit-periodic Schrödinger operators; this is mainly based on [26]. Chapter 4 begins with an alternative description of the p-moments with respect to orthonormal bases that are dubbed Laplace moments, since their expressions resembles the Laplace transform one; it has some technical advantages over the original definition of moments. The concept of escaping probabilities [45, 70] is discussed, and its relation to the p-moments is proven in Sect. 4.2. Then sufficient conditions, for the presence of generic sets of discrete Schrödinger operators on .l 2 (Zd ), .d ≥ 1, with both quasilocalized (i.e., .βψ− (p) = 0)

0.2 Part I

7

and quasiballistic (i.e., .βψ+ (p) = 1) dynamics are presented, with applications to operators with analytic quasiperiodic potentials and some discrete cases with unbounded potentials. The main reference is [25]. Although for pure point operators T , one usually has .

lim WeTn ,ψ (t) = 0,

t→∞

for members .en of an orthonormal basis B, it is possible to have upper moment exponents .βψ+ (p) ≥ 1 ! The first rigorous example has appeared in [62, 63], and it was considered “a pathological example” by the authors. There is also a different mechanism that implies diffusion in the presence of point spectrum, and it is related to the so-called dimer model, with vanishing of the Lyapunov exponents in some isolated energy values; see additional comments and references at the end of Sect. 5.1. Other examples of diffusion with pure point spectra have been published by different groups, and recently, a step further was given by the present authors in [1], that is, it was concluded that a root for quasiballistic dynamics for self-adjoint operators with pure point spectrum is the presence of thick point spectrum, i.e., pure point spectrum with eigenvalues dense in an interval. More precisely, as discussed in Chap. 5, if a self-adjoint operator has a pure point spectrum dense in a bounded interval, then in the corresponding spectral space, one has .βξ+ (p) ≥ 1, for all .p > 0, and for generic initial conditions .ξ in the Hilbert space. An adaptation of the above result was obtained for some classes of discrete point spectrum [4], for which .βξ+ (p) > 0 and .βξ− (p) = 0 for generic initial conditions .ξ . It is interesting that such classes of operators cover the traditional Hydrogen atom Hamiltonian, with .βξ+ (p) ≥ 1/3 for generic .ξ , and all .p > 0 (see Sect. 5.4). Given .ψ, ϕ ∈ H , consider, for each .k ∈ N, 1 ψk = ψ + ϕ; k

.

for large values of k, it is not clear which properties of .ψk are inherited from .ϕ. Nevertheless, it will be shown in Sect. 5.5 that the spectral measures .μTψk inherit some dimensional .Dμ±T (q) properties from .μTϕ , so that the set of such vectors is ϕ

generic. The precise properties (i.e., Theorem 5.21) depend on if .0 < q < 1 or .q > 1. There are also relations between the power-law exponents for moments and generalized fractal dimensions of spectral measures (the original references are [12, 76, 131]); it was shown that − .β (p, B) ψ

≥

Dμ−T ψ



 1 , 1+p

βψ+ (p, B)

≥

Dμ+T ψ



 1 , 1+p

8

0 Book’s Outline

and Sect. 5.6 covers such results, as well as other relations among fractal dimensions (e.g., Corollary 5.27). Chapter 5 ends with an application to one-dimensional discrete Schrödinger operators T with uniformly bounded potentials, with the topology of pointwise convergence. It is shown that a version of Theorem 2.11 holds in this case, that is, generically one has dimensions .Dμ−T (q) = 0 and .Dμ+T (q) = 1 for all .q > 0, q = 1 f

f

(Theorem 2.11 is for .q = 2). Although in this book we are more interested in results that hold for general classes of Schrödinger operators (which here mainly are one-dimensional and discrete), it is important to stress that for some specific classes of operators, like almost periodic systems with fractal measures, it is possible to obtain upper bounds on growth exponents. Namely, in [77], the authors have shown that for Julia matrices with self-similar spectra, the upper growth rate of the escaping probability (see Sect. 4.2 for the definition) is bounded from above by a Lyapunov exponent-like quantity related to the solutions to the eigenvalue equation. In [46], the authors have obtained a global criteria for the transfer matrix growth (regardless of the specific class of the one-dimensional Schrödinger operator) in order to bound, from above, the growth exponents .β ± of the moments. This method applies specifically to√the so-called Fibonacci operator (where .Vn = λχ[1−θ,1) (nθ ), with .λ ≥ 8 and .θ = ( 5 − 1)/2), as shown in [46]. In [100] this was generalized to Sturmian operators. We also refer to [41, 42] for a more detailed discussion. Such results show, among other things, that at least for such classes of operators, the lower bounds for the growth exponents of the moments discussed in Sect. 3.4 and in Chaps. 4 and 5 are quite tight.

0.3 Part II Ergodic Theory In Chap. 6, we discuss aspects of time evolution of some discrete dynamical systems, in which time is restricted to the integer numbers Z through the iteration of an invertible and measurable mapping ϕ : (Ω, B, m) → (Ω, B, m); Ω is a separable complete metric space and (B, m) is a (σ -algebra, probability measure) pair obtained by the completion under a continuous (nonatomic) Borel probability measure; A, B ∈ B are identified if m(A B) = 0, where A B = (A\B)∪(B \A). It is supposed that m is linked to the dynamics, which is modelled by assuming that m is invariant for ϕ (it is also said that ϕ preserves m), that is, m(ϕ −1 (A)) = m(A) for all A ∈ B. One goal of the Ergodic Theory is to study the long-time evolution, for almost all initial conditions [80, 138]. Celebrated notions in this context are ergodicity, weakmixing, and mixing. Intuitively, ergodicity is a kind of time-average convergence to equilibrium, whereas mixing does not ask for the time average, and weak-mixing

0.3 Part II

9

describes an intermediate behavior—see the discussion in Sect. 6.1. Note that mixing implies weak-mixing, and the latter implies ergodicity (see Proposition 6.8). A possible characterization of such ergodic properties is in terms of the Koopman operator Uϕ : L2m (Ω) → L2m (Ω) acting as Uϕ f (x) = f (ϕ(x)). It is a unitary operator and since its action on constant functions is trivial, what is relevant is its restriction to the orthogonal complement [1]⊥ ⊂ L2m (Ω) of the space of constant functions [1]. With such tool, an automorphism ϕ is weak-mixing if, for all f ∈ [1]⊥ ,

.

L−1 1  |f, Uϕk f |2 = 0. L→∞ L

lim

k=0

By considering the so-called weak topology (see Sect. 6.2) on the space G of automorphisms that preserve m, one obtains a complete metric space; it was a surprise that weak-mixing is generic in G the sense of Baire, and mixing is not. This result is a combination of two classical results, one by Halmos [79] and another by Rohlin [117]. It is worth mentioning that Knill [93] has proven a “Wonderland version” in this setting, that is, under certain conditions, the set of Koopman transformations with purely singular continuous spectrum is generic. In Chap. 6, such results are presented along with recent results [27] on the rate of convergence associated with the weak-mixing property; the main conclusion is that the generic dynamical behavior depends on subsequences of time going to infinity, with extreme cases of rates of convergence. More precisely, it is introduced, for α ∈ [0, 1], f ∈ [1]⊥ and ϕ ∈ G, Mα (ϕ; f ) = lim inf

.

L→∞

M α (ϕ; f ) = lim sup L→∞

L−1 1  |f, Uϕk f |2 , Lα k=0

L−1 

1 L1−α

|f, Uϕk f |2 ,

k=0

and think of small α > 0 in both cases. Such quantities will be related to U the correlation dimension of the spectral measures μf ϕ and it will be shown (Theorem 6.31) that there exists a generic set G ⊂ G of automorphisms such that, for each ϕ ∈ G , the set of f ∈ [1]⊥ with Mα (ϕ; f ) = 0 and

.

M α (ϕ; f ) = ∞,

for all 0 < α < 1, is generic in [1]⊥ . This justifies the above interpretation on the generic rates of convergence, and it is another illustration of generic quantities taking extreme values.

10

0 Book’s Outline

Besides the weak topology, important ingredients in the proof of this result are the density of periodic automorphisms in G (Lemma 6.14) and that the conjugate class of any antiperiodic automorphism is dense in G (Lemma 6.15). Semigroups C0 -semigroups on Hilbert spaces H are addressed in Chaps. 7 and 8. Instead of group of evolutions e−itT , t ∈ R, generated by self-adjoint operators T , in these chapters we discuss the time evolution of semigroups S(t), t ≥ 0, mainly in the setting of normal generators N, i.e., when S(t) = etN , t ≥ 0. Recall the traditional nomenclature that S(t) is strongly stable if lim S(t)u = 0,

.

∀u ∈ H ,

t→∞

and exponentially stable if there exist constants a > 0 and C > 0 such that S(t) ≤ C e−at ,

∀t ≥ 0.

.

Since a normal operator has a decomposition N = NR + iNI (see (1.18)) in terms of commuting self-adjoint operators NR and NI , a first observation is that (see (1.19))  e u =

.

2

tN

0 −∞

R e2ty dμN u (y),

hence the spectral measures of NR that are related to the asymptotic behavior of etN u as t → ∞. Indeed, if one thinks of power-law long time asymptotics, it is shown that they are given by the lower and upper local exponents d −NR (0),

.

μu

d +NR (0), μu

R of the spectral measure μN u at zero (see Definition 3.1 ahead). It is then shown the generic behavior of the decay rates of normal C0 -semigroups of contractions that are stable but not exponentially stable [3]; it is concluded that, generically, their time long asymptotic behaviors depend on sequences of time going to infinity (see Theorem 7.9). Other generic properties relating stability and spectra are also discussed. Generic properties of some nonnormal C0 -semigroups S(t) are also discussed in Chap. 7. Since in this case we do not have a version of the Spectral Theorem for the infinitesimal generators A available, the condition

S(t)A−1  = O(r(t)),

.

with r(t) → 0 as t → ∞, is assumed to hold (among others). The result on generic asymptotic behaviors, discussed for normal semigroups, is partially transferred to

0.3 Part II

11

this (nonnormal) setting; hence, it is also concluded that typically the dynamics depend on sequences of time going to infinity (see Theorem 7.19 ahead). It is worth underlying that although the material of this part of the chapter does not rely on spectral measures, it was included in the book due to the generic properties and many applications to evolution equations whose associated semigroups are (power-law) stable but not exponentially stable, such as systems with localized viscoelasticity, some thermoelastic equations, and a damped wave equation. Chapter 8 is specific for self-adjoints semigroups; for instance, the class of Schrödinger semigroups, whose generators HV have the form (HV u)(x) = u(x) + V (x)u(x),

.

with −a ≤ V (x) ≤ 0 for all x ∈ Rν , ν ≥ 1 and a > 0, which is a complete metric space when endowed with the metric d(HV , HU ) =

∞ 

.

min(2−j , V − U j ),

j =0

where V − U j = supx∈B(0;j ) |V (x) − U (x)|. Denote this space of operators by Xaν . It is concluded that the set of H ∈ Xaν so that etH is stable, but not exponentially stable, is generic in Xaν . By combining this with the generic behavior discussed above, one sees that typically one has nontrivial dynamics for such semigroups [2].

Chapter 1

Spectrum and Dynamics: Some Basic Concepts

1.1 First Basic Concepts Self-adjoint and unitary operators are relevant to many branches of Physics and Mathematics, in particular to quantum mechanics, evolution semigroups, and ergodic theory of dynamical systems. A self-adjoint operator T , acting in a (infinite dimensional) separable Hilbert space .H , is the (unique) infinitesimal generator of the one-parameter strongly continuous evolution group R  t −→ Ut = e−itT

.

giving the unique solution .ψ(t) = Ut ψ to the Schrödinger initial value problem i

.

d ψ(t) = T ψ(t), dt

ψ(0) = ψ ∈ dom T ,

where .dom T denotes the domain of T . Sometimes, a unitary operator U is used to generate a discrete time evolution Z  k −→ U k

.

through the k-times composition of U with itself; an important instance is the Koopman operator, which describes the time evolution of some discrete dynamical systems (a subject of Chap. 6). We are usually interested in the long-time behavior of these systems, characterized by either .t → ∞ or .k → ∞, with different approaches for different applications. A common property of self-adjoint and unitary operators is the presence of spectral measures, which we will denote by .μTψ and .μU ψ in case of selfadjoint and unitary operators, respectively. Characteristics of each spectral measure © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Aloisio et al., Spectral Measures and Dynamics: Typical Behaviors, Latin American Mathematics Series – UFSCar subseries, https://doi.org/10.1007/978-3-031-38289-5_1

13

14

1 Spectrum and Dynamics: Some Basic Concepts

are important to describe the large time evolution of the system, and we will mention its most prominent properties in this book; some properties are rather natural and expected (time evolution of eigenvectors, for example), but others are very subtle (as special values of generalized fractal dimensions of spectral measures): a basic decomposition of a spectral measure μT = μTp + μTac + μTsc

.

is in its pure point part .μTp (constituted only of atoms and generated by eigenvalues of T ), its absolutely continuous (with respect to Lebesgue measure) part .μTac and its singular continuous part .μTsc (continuous (i.e., no atoms) but singular with respect to Lebesgue measure). In the following, we mainly focus our discussion on the continuous evolution groups and self-adjoint operators, the most frequent case here. Recall that .μTc = μTac + μTsc is the continuous component of the spectral measure. The main subject treated in this book is the study of the generic behavior of suitable spectral, fractal, and/or dynamical properties of complete metric spaces of self-adjoint operators. The term generic refers to the Baire category sense; so it may be useful to recall its definition. Definition 1.1 Let X be a complete metric space. A .Gδ set in X is a countable intersection of open sets in X. A set in X is called a generic set, or a residual set, if it contains a countable intersection of dense open sets in X. By Baire’s Theorem, a generic set is also dense in X. Note that countable intersections of generic sets are also generic. An example, which in fact has motivated some of the results presented in this book, is the so-called Wonderland Theorem by Barry Simon [125], which gives sufficient conditions for the set of self-adjoint operators with purely singular continuous spectrum to be generic; we recall this result in Theorem 1.31. Now two important dynamical quantities are described, and in this book they will often be brought into the limelight; they are the average projection (average return probability in quantum mechanics) and the so-called p-moments with respect to an orthonormal basis .B = {en }n of .H (n runs over .N or .Z). Consider the average projection of .ψ(t) = Ut ψ = e−itT ψ, .ψ = 1, .t > 0, onto a fixed .ξ ∈ H : T Wξ,ψ (t) = Wξ,ψ (t) :=

.

1 t



t

|ξ, ψ(s)|2 ds.

(1.1)

0

In case .ψ = ψλ is an eigenvector of T , with .T ψλ = λψλ , then .e−itT ψλ = e−itλ ψλ and Wξ,ψλ (t) = |ξ, ψλ (0)|2 ,

.

1.1 First Basic Concepts

15

and so, the projection does not decay (if it is nonzero). By Wiener’s Lemma 1.5, this nondecay is a result of the point character of the eigenvector spectral measure T = δ , since .W T .μ λ ξ,ψ (t) → 0 as .t → ∞ (and for all .ξ ) if, and only if, .μψ is a ψλ continuous measure; in particular, one is interested in the (algebraic) rate of decay of .Wξ,ψ (t) ∼ (1/t)γ , that is, ln Wξ,ψ (t) , ln 1/t

− γξ,ψ := lim inf

.

t→∞

+ γξ,ψ := lim sup t→∞

ln Wξ,ψ (t) ln 1/t

(1.2)

± ± (of course, .γξ,ψ = γξ,ψ (T )). The larger .γ , the faster the time evolution .ψ(t) is disconnected to .ξ , and so the precise values of these parameters give important information about the dynamics of the system. Such vanishing of the average projection onto any vector is a kind of “instability”; for instance, for an orthonormal basis .B = {en } of .H , since the time evolution is unitary, one has

 1 t ψ(0) ds = ψ(s)2 ds t 0 0   1 t |en , ψ(s)|2 ds = Wen ,ψ (t) , = t 0 n n

1 .ψ(0) = t 2



t

2

and if the spectral measure of .ψ is continuous, then .Wψ,en (t) → 0 as .t → ∞, for each basis vector .en ; so, given .ε > 0, for each .N ∈ N one has 

Wen ,ψ (t) < ε

.

|n| 0, where

rpψ (t) :=

 

.

1

p

|n|p Wen ,ψ (t)

(1.3)

,

n ψ

and if one thinks of an algebraic growth .rp (t) ∼ t β(p) , it is then natural to introduce βψ− (p) := lim inf

.

t→∞

ψ

ln rp (t) , ln t

βψ+ (p) := lim sup t→∞

ψ

ln rp (t) . ln t

Again, the notations were simplified, since .βψ± (p) = βψ± (p, T , B).

(1.4)

16

1 Spectrum and Dynamics: Some Basic Concepts

A particular important case is for .H = l 2 (Z) and for .B = {en } given by the canonical orthonormal basis, that is, .en (j ) = δn,j , since .|ψ(t), en |2 indicates a physical projection onto a distance .|n| from the origin; hence, a positive value of the exponent .βψ represents an effective dislocation of the particle system to large values of .|n| and is interpreted as transport, whereas lack of transport means localization. There are suitable adaptations to the Hilbert spaces .l 2 (Zν ) and .L2 (Rν ) that will be discussed as soon as they are needed. The exponents .βψ± are the standard quantities used to characterize such dynamical behavior, and the system is sometimes also interpreted as conductor (transport) or insulator (localization). The larger p, the more relevant are large values of .|n|, and in case .βψ− (p) < βψ+ (p), we have a dynamics whose behavior is dependent on sequences of time going to infinity! Under suitable conditions (see Theorem 4.5 and [71]), one has .0 ≤ βψ± (p) ≤ 1; note that even if .βψ± (p) = 0, the function .t → rp (t) may be unbounded (e.g., if ψ

ψ

rp (t) ∼ ln t). Based on [25], we introduce the following nomenclature:

.

Definition 1.2 With respect to the orthonormal basis B, we say that the pair (T , ψ) is

.

ψ

• localized, or .(T , ψ) has dynamical localization, if the function .rp (t) is bounded; • • • •

essentially localized if .βψ+ (p) = 0; quasilocalized if .βψ− (p) = 0; quasiballistic if .βψ+ (p) = 1; ballistic if .βψ− (p) = 1;

for all .p > 0. Given a self-adjoint operator T , it is usually a hard question to decide about the dynamical behavior according to Definition 1.2. A common strategy is to try to ± relate the exponents .βψ± and .γψ± := γψ,ψ with the nature of the spectral measure T .μ ψ (that is, if it has an absolutely continuous (w.r.t. the Lebesgue measure), a singular continuous or a pure point component) and also to fractal dimensions of these measures, in particular if .μTψ has any .α-Hausdorff or .α-packing continuous components (which give lower bounds to .βψ− and .βψ+ , respectively). Generalized fractal dimensions .Dμ±ψ (q), with .q = 1/(1 + p), are also very important lower bounds for .βψ± (p). Such relations will be topics of specific chapters of this book. As already mentioned, the main goal of this book is to address recent results describing which properties (as values of .γ ± and .β ± ) give rise to generic sets of operators and/or vectors in the Hilbert space, so describing the typical situation from the topological viewpoint. In other sections of this chapter, we recall the spectral classification of self-adjoint operators, present the definitions of Hausdorff and packing measures on the real line, and define the generalized fractal dimensions of interest. Then we illustrate generic results through the Wonderland Theorem.

1.1 First Basic Concepts

17

1.1.1 Standard Spectral Classification Let us recall details of the standard decomposition of finite Borel measures on .R into point, absolutely continuous and singular continuous components, and the corresponding decompositions for self-adjoint operators; a general reference is [53], Chapter 12. Definition 1.3 Let .μ be a Borel measure on .R. • The support of .μ is the (closed) set supp (μ) := {x ∈ R | μ((x − ε, x + ε)) > 0, ∀ε > 0}.

.

• .μ is supported on a Borel set .S ⊂ R if .μ(R \ S) = 0. For instance, .μ is always supported on .R. Let .A denote the Borel sets of .R and . denote the Lebesgue measure over .A (or over its complement). A Borel measure .μ over .R can be (uniquely) decomposed as .μ = μp + μc , with .μc and .μp denoting, respectively, its continuous (that is, .μc ({t}) = 0, for all .t ∈ R) and point (that is, .μp is supported on a countable set; the points .t ∈ R such that .μp ({t}) = 0 are the atoms of .μ) parts. Note that .μp ⊥ , that is, they are mutually singular measures (that is, there is .A ∈ A so that .μp (A) = (R \ A) = 0). By Lebesgue decomposition, one has (uniquely) .μc = μac + μsc , with .μac   (that is, .μac (A) = 0 if .(A) = 0, .A ∈ A ) and .μsc ⊥ , so that μ = μp + μac + μsc .

.

μac is called the absolutely continuous component of .μ (with respect to Lebesgue measure), whereas .μsc is the singular continuous component of .μ, that is, .μsc is continuous and .μsc ⊥ . Since the spectra of self-adjoint operators are nonempty and real, these decompositions of spectral measures induce corresponding decompositions of self-adjoint operators. If .T : dom T → H is a self-adjoint operator with spectrum .σ (T ) (and so with resolvent set .ρ(T ) = C \ σ (T )), recall that for each pair .ξ, η ∈ H the associated spectral measure .μξ,η = μTξ,η is characterized by

.

 ξ, T η =

x dμξ,η (x),

.

σ (T )

 and we simplify .μξ = μξ,ξ . One has .ξ ∈ dom T if, and only if, . σ (T ) x 2 dμξ (x) < ∞; more generally, for each Borel function .f : R → C,

18

1 Spectrum and Dynamics: Some Basic Concepts

 ξ, f (T )η =

f (x) dμξ,η (x),

.

σ (T )

and the domain of the (normal) operator .f (T ) is given by dom f (T ) = {ξ ∈ H | f ∈ L2μξ (R)}.

.

It tuns out that .μξ,η (σ (T )) = μξ,η (R) = ξ, η (in particular, .μξ (R) = ξ 2 ), .μξ,η  μξ and .μξ,η  μη . An important choice of function is .ft (x) = e−itx , which is directly associated with the time evolution generated by T : ξ, e

.

−itT

 η =

e

−itx

 dμξ,η (x) =

R

σ (T )

e−itx dμξ,η (x) = μˆ ξ,η (t),

(1.5)

that is, it is the Fourier transform .μˆ ξ,η (t) of the spectral measure .μξ,η . The point subspace of T , denoted by .Hp = Hp (T ) ⊂ H , is given by the closure of the linear subspace spanned by the eigenvectors of T ; if .Λ = {λj } is the set of its eigenvalues and .{ξj } are the corresponding (orthonormalized) eigenvectors, that is, .T ξj = λj ξj , then any .ξ ∈ Hp may be written as 

ξ=

.

aj ξj ,

aj = ξj , ξ ,

j

 and its spectral measure, .μTξ = j |aj |2 δλj , is pure point (i.e., it has no continuous component). Its orthogonal complement, denoted by .Hc = Hc (T ) := Hp (T )⊥ , is the continuous subspace of T , since the spectral measure .μξ of any vector .ξ ∈ Hc is continuous; so such spectral measure is decomposed into absolutely continuous and singular continuous components. It turns out that the sets .Hac (T ) := {ξ ∈ H : μξ  } and .Hsc (T ) := {ξ ∈ H | μTξ is singular continuous} are in fact closed subspaces that reduce T ; they are the so-called absolutely continuous subspace of T and the singular continuous subspace of T , respectively. Furthermore, H = Hp (T ) ⊕ Hc (T ) = Hp (T ) ⊕ Hac (T ) ⊕ Hsc (T ),

.

(1.6)

with corresponding operator decomposition T = Tp ⊕ Tac ⊕ Tsc ,

.

(1.7)

where .Tp := T |Hp , and so on. The absolutely continuous spectrum of T is σac (T ) := σ (Tac ) and the singular continuous spectrum of T is .σsc (T ) := σ (Tsc ). The point spectrum of T , .σp (T ) := σ (Tp ), is the closure of the set of its eigenvalues; note that the set of accumulation points of the set of eigenvalues that are not themselves eigenvalues has zero (point) spectral measures.

.

1.1 First Basic Concepts

19

The corresponding orthogonal projection operators will be denote by .PpT , .PcT and so on. Also, the statement “T has purely absolutely continuous spectrum” indicates that only the subspace .Hac (T ) is nontrivial, and so on. With respect to the average projection (1.1), two classical results are now recalled, with pertinent comments. Due to the importance of these results for the subject of this book, proofs will be presented. Lemma 1.4 (Riemann–Lebesgue) If .f ∈ L1 (R), then the Fourier transform .fˆ of f is a continuous function and .lim|t|→∞ fˆ(t) = 0. Proof Write out, for each .s = t ∈ R,      1 −itx −isx  ˆ ˆ (e −e )f (x) dx  .|f (t) − f (s)| = √  2π R  1 ≤√ |(e−itx − e−isx )| |f (x)| dx, 2π R and since .f ∈ L1 (R), it follows from dominated convergence that .fˆ is continuous. ∞ Let, for each .h ∈ R, .fh (t) = f (t + h). Note that for .φ ∈ C∞ 0 (R) (where .C0 (R) stands for the space of infinitely differentiable real functions of compact support), one has .φh − φ1 → 0 as .h → 0 (by the uniform continuity of .φ). Since .C∞ 0 (R) is dense in .L1 (R), given .ε > 0, pick .φ ∈ C∞ (R) with .f −φ1 < ε; by the invariance 0 of the Lebesgue measure under translations one has .fh − φh 1 = f − φ1 < ε. Take .|h| sufficiently small so that .φh − φ1 < ε. Gathering these facts, one gets fh − f 1 ≤ fh − φh 1 + φh − φ1 + φ − f 1 < 3ε,

.

and one concludes that .fh − f 1 → 0 as .h → 0, for any .f ∈ L1 (R). For .t = 0, one has   √ −ixt ˆ e f (x) dx = − e−i(x+π/t)t f (x) dx . 2π f (t) = R



R

π

dx, = − e−ixt f x − t R

and so   √  π

 −ixt  ˆ dx  f (x) − f x − .2 2π |f (t)| =  e t R .

 π    ≤ f (x) − f x −  , t 1

which vanishes as .|t| → ∞.



20

1 Spectrum and Dynamics: Some Basic Concepts

Pick .ψ ∈ Hac (T ) (so, for any .ξ ∈ H it follows that .μξ,ψ  μψ  ) and denote its Radon-Nikodym derivative (with respect to .) by .f (x) = dμξ,ψ /d, an element of .L1 (R). Then,   −itx .ξ, ψ(t) = e dμψ,ξ (x) = e−itx f (x) dx = fˆ(t) R

R

and, by Lemma 1.4, one has lim |ξ, ψ(t)|2 = 0.

.

t→∞

Therefore, for a vector in the absolutely continuous subspace, the projection of its time evolution onto any other vector .ξ vanishes as .t → ∞; consequently, the time average .Wξ,ψ (t) defined in Eq. (1.1) also vanishes. Lemma 1.5 (Wiener) Let .μ be a finite Borel (real or complex) measure on .R and let .μ(t) ˆ be its Fourier transform. Then, 1 . lim t→∞ t



t

 2  μ(s) |μ({λ})|2 , ˆ  ds =

0

(1.8)

λ∈Λ

with .Λ := {λ ∈ R : μ({λ}) = 0}. Proof Since .μ is finite, .Λ is a countable set. From .μ(t) ˆ = by Fubini Theorem (.μ is the complex conjugate of .μ), 1 . t



t 0



Re

−itx

dμ(x) one has,

   1 t −i(u−v)s |μ(s)| ˆ ds = dμ(u) dμ(v) ds e t 0   = g(u, v, t)dμ(u) dμ(v), 2

with .g(u, u, t) = 1 and g(u, v, t) = −i

.

1 − e−i(u−v)t , t (u − v)

u = v.

Since .limt→∞ g(u, v, t) = χ{u} (v) and this function is dominated by the constant function .1 ∈ L1μ×μ¯ (R × R), it follows from dominated convergence that .

1 t→∞ t



t

lim

  2 |μ(s)| ˆ ds =

0

χ{u} (v) dμ(u) dμ(v) 

=

μ({v}) dμ(v) =



|μ({v})|2 .

v∈Λ



1.1 First Basic Concepts

21

For a spectral measure .μξ,ψ one has .μˆ ξ,ψ (t) = ξ, ψ(t), and if .ψ ∈ Hc , then μξ,ψ has no atoms (since .μξ,ψ  μψ ); so, by Eq. (1.8),

.

.

lim Wξ,ψ (t) =

t→∞

  μξ,ψ ({λ})2 = 0 , λ

an important result used above in the motivation for the introduction of moments with respect to orthonormal bases. Remark 1.6 The Riemann–Lebesgue and Wiener’s Lemmas have natural counterparts for unitary operators U on .H . If .S 1 denotes the unit circle in the complex plane, they read, respectively, as: ˆ • If .f ∈ L1 (S 1 ) and if .fˆ(k) denotes its Fourier series, then .lim = 0.  |k|→∞ f (k) 1 • Now, for a Borel measure .μ on .S and .k ∈ Z, let .μ(k) ˆ = S 1 dμ(s) e−iks be the Fourier coefficient and let .Λ = {eiλ ∈ S 1 : μ({eiλ )}) = 0}; then, L−1   2  1   μ({eiλ })2 . μ(k) ˆ  ds = . lim L→∞ L iλ e ∈Λ

k=0



.

In particular, for .0 = ψ ∈ H , WψT (t) =

.

1 t



t

|ψ, ψ(s)|2 ds −→ 0

0

as .t → ∞ if, and only if, .ψ ∈ Hc (T ). In some situations, especially when .ψ ∈ Hac (T ), .ψ, ψ(s) → 0 with no need of time average (as previously shown), and usually the average is needed for vectors .ψ ∈ Hsc (see Section 13.5 of [53]). Next we present a dynamical interpretation of Wiener’s Lemma that is based on Lemma 1.7 (due to Koopman and von Neumann). A subset .∅ = D ⊂ {0, 1, · · · } is said to be of density zero if .

lim

L→∞

#(D ∩ {0, 1, 2, . . . , L − 1}) = 0, L

where .#A denotes the cardinality of the set A. Lemma 1.7 If .(cj ) is a bounded sequence in .[0, ∞) with

.

L−1 1  cj = 0, L→∞ L

lim

j =0

(1.9)

22

1 Spectrum and Dynamics: Some Basic Concepts

then there exists a subset .B ⊂ Z+ of density zero so that .cj → 0 for .j → ∞ and + .j ∈ Z \ B. Proof If .M ⊂ Z+ , let .αM (L) := #({0, 1, . . . , L − 1} ∩ M). We begin by setting the “bad sets” .Bk = {j | cj ≥ 1/k}. Note that .Bk ⊂ Bk+1 for all k, and that each .Bk has density zero, since for .L → ∞,

.

L−1 1  αBk (L) ≤ k cj −→ 0 . L L j =0

Now choose a strictly increasing sequence .0 = n0 < n1 < n2 < · · · of integer numbers so that, for .L ≥ nk , .

1 #(Bk ∩ {0, 1, 2, . . . , L − 1}) < . L k

Set .B := ∪k≥0 (Bk+1 ∩ [nk , nk+1 )); then, one has .cj → 0, provided .j ∈ / B (since if .j > nk and .j ∈ / B, then .j ∈ / Bk+1 , and one has .cj < 1/k). To finish the proof, it is enough to check that B is of density zero. Indeed, given .k ∈ N, pick an integer L so that .nk ≤ L < nk+1 ; since .Bk ⊂ Bk+1 , one has B ∩ [0, L) = [B ∩ [0, nk )] ∪ [B ∩ [nk , L)] ⊂ [Bk ∩ [0, nk )] ∪ [Bk+1 ∩ [0, L)];

.

so .

1 1 αB (L) 1 1 ≤ (αBk (nk ) + αBk+1 (L)) ≤ (αBk (L) + αBk+1 (L)) ≤ + , L L L k k+1

and B is of density zero.   k+1 Now, set .ck = k |ψ, ψ(s)|2 ds and pick .k ≤ sk ≤ k + 1 so that .ck = |ψ, ψ(sk )|2 ; note that, for .L ∈ N, WψT (L) =

.

L−1 1  |ψ, ψ(sk )|2 . L k=0

Hence, if .ψ ∈ Hc (T ), Lemma 1.7 implies .ψ, ψ(sk ) → 0 as .k → ∞, provided that k does not belong to a set of density zero. This opens the possibility of different asymptotic behaviors as time goes to infinity, as discussed, for example, in Chap. 2.

1.1 First Basic Concepts

23

1.1.2 Fractal Measures The classical prototypes of fractal measures on .R (it is enough to restrict ourselves to the real line) are the .α-Hausdorff, .hα , and .α-packing, .P α , measures, for .0 ≤ α ≤ 1. Both families of measures are rather natural generalizations of Lebesgue measure and interpolate the counting measure, for .α = 0, and the Lebesgue measure, for .α = 1. Here, we present these definitions and relevant related concepts necessary for the development of the results presented in this book. For an interval .I ⊂ R, denote by .|I | its diameter (which coincides with .(I )). Definition 1.8 ([82]) For each .α ∈ [0, 1] and each .S ⊂ R, consider the number α .hδ (S)

= inf

 ∞

|Ik | | |Ik | < δ, ∀k; S ⊂ α

k=1

∞ 

 Ik ,

k=1

with the infimum taken over all covers of S by intervals .Ik of diameter at most .δ > 0. The limit hα (S) := lim hαδ (S)

.

δ↓0

is called the .α-dimensional (exterior) Hausdorff measure of S. The Hausdorff dimension of the set S, here denoted by .dimH (S), is defined as the infimum of all .α such that .hα (S) = 0; note that .hα (S) = ∞ if .α < dimH (S). A .δ-packing of an arbitrary set .S ⊂ R is a countable pairwise disjoint collection .(I¯(xk ; rk ))k∈N of closed intervals centered at .xk ∈ S and radii .rk ≤ δ/2, so with diameters at most .δ. Define .Pδα (S), .0 ≤ α ≤ 1, as Pδα (S) = sup

.

∞   (2rk )α | (I¯(xk ; rk ))k is a δ−packing of S , k=1

with the supremum taken over all .δ-packings of S. The decreasing limit P0α (S) = lim Pδα (S)

(1.10)

.

δ↓0

defines a pre-measure. Definition 1.9 ([130, 134]) The .α-packing (exterior) measure .P α (S) of S is given by P α (S) := inf

∞ 

.

k=1

P0α (Sk ) | S ⊂

∞  k=1

 Sk .

24

1 Spectrum and Dynamics: Some Basic Concepts

The packing dimension of the set S, here denoted by .dimP (S), is defined (in analogy with .dimH (S)) as the infimum of all .α such that .P α (S) = 0, which coincides with the supremum of all .α so that .P α (S) = ∞. Both .hα and .P α are (exterior) Borel regular measures, that is, every subset of .R is contained in a Borel set of the same measure, so that in practice one just need to work with Borel sets. They are not .σ -finite for .0 < α < 1 [67, 68, 101]. Furthermore, if .E ⊂ S, then .dimK (E) ≤ dimK (S), for .K = P or .K = H. It is possible to show [67, 68, 101] that the Hausdorff and packing dimensions are related by the inequality .

dimH (S) ≤ dimP (S),

for all .S ⊂ R, and that this inequality may be strict. Nevertheless, the middle third Cantor set .C has .dimH (C ) = dimP (C ) = ln 2/ ln 3. As mentioned before, .P 0 = h0 , 1 1 .P = h and they correspond, respectively, to counting and Lebesgue measures. An important property of Hausdorff and packing dimensions is the countable stability, that is, if .S = ∪∞ j =1 Sj , then, for .K = P or .K = H, .

dimK (S) = sup dimK (Sj ). j

Indeed, first it is clear that, for any j , .dimH (S) ≥ dimH (Sj ), and so .dimH (S) ≥ supj dimH (Sj ). If for all j one has .α > dimH (Sj ), then .hα (Sj ) = 0, so .hα (∪j Sj ) = 0, and one concludes that .α ≥ dimH (S). The same argument applies to the packing dimension. Since a single point has zero Hausdorff and packing dimensions, by the countable stability, all countable sets have zero dimensions as well.

1.1.3 Dimensions of Measures There are different notions of dimensions of a (finite and positive) Borel measure .μ on .R. In what follows, we briefly recall some of them; a specialized monograph was written by Pesin [110]. See also [102] for a discussion about a proposal of the “expected” properties, a dimension of a measure should satisfy. We begin with the extension of the Hausdorff and packing dimensions from sets to measures [101]; let K denote either H (for Hausdorff) or P (for packing). Definition 1.10 Let .μ be as above. The .K-upper dimension of .μ is defined as .

dim+ K (μ) := inf{dimK (S) | μ(R \ S) = 0, S a Borel subset of R},

and the .K-lower dimension of .μ as .

dim− K (μ) := sup{α | μ(S) = 0 if dimK (S) < α, S a Borel subset of R}.

1.1 First Basic Concepts

25

If .A ⊂ R is a Borel set, we shall also consider the dimensions of the restriction of .μ to such set, that is, .μ;A (·) := μ(A ∩ ·). Intuitively, the upper Hausdorff dimension of the measure .μ is the infimum of the Hausdorff dimensions of sets which support .μ, and the lower Hausdorff dimension of .μ is the infimum of the Hausdorff dimensions of sets charged by .μ (similar for packing dimensions). − + + One has .dim− H (μ) ≤ dimP (μ) and .dimH (μ) ≤ dimP (μ), along with the − + expected relations .dimK (μ) ≤ dimK (μ). Example 1.11 If .μ is a finite point measure on .R, then .dim± K (μ) = 0. This is immediate from the fact that any countable set has zero dimensions. . Example 1.12 If .μ is a finite absolutely continuous measure on .R with bounded Radon-Nikodym derivative .f (x) = dμ/d, then .dim± K (μ) = 1. Namely, if .0 < C = −ess.sup f , then for any Borel set .S ⊂ R,  μ(S) =

f (x) dx ≤ C(S).

.

(1.11)

S

It is worth considering a more general setting. Lemma 1.13 Let .ν be a finite Borel measure such that there exist .α ∈ (0, 1) and c > 0 so that .ν(S) ≤ c |S|α for each Borel set S with .|S| < 1. Then, for each Borel set S, one has

.

P α (S) ≥ hα (S) ≥ ν(S)/c,

.

and so, if .ν(S) > 0, then .dimP (S) ≥ dimH (S) ≥ α. Proof Let S be a Borel set such that .ν(S) > 0. If .{Sj } is a (countable) cover of S with .|Sj | < 1 for all j , then ν(S) ≤



.

j

ν(Sj ) ≤ c



|Sj |α .

j

By taking the infimum over covers of this kind, it follows that .ν(S) ≤ c hαδ (S) for  all .0 < δ < 1; so, .ν(S) ≤ c hα (S). Since .hα (S) > 0, one has .α ≤ dimH (S). By Lemma 1.13, it follows that .dim− H (ν) ≥ α; in particular, by relation (1.11), it follows that if .μ(S) > 0, then .dimH (S) = 1, hence .dim− . H (μ) = 1. Now, two families of dimensions of .μ, parametrized by a number .q > 0, .q = 1, are introduced. It is possible to extend the definitions to .q < 0 and .q = 1, but it will not be of interest here. For the accepted values of q and .ε > 0, let  Iμ (q, ε) :=

.

˙ μ(B(x; ε))q−1 dμ(x),

(1.12)

26

1 Spectrum and Dynamics: Some Basic Concepts

˙ with the integrals taken over .supp (μ); .B(x; ε) := [x − ε, x + ε). The idea is to evaluate how such integrals scale for small .ε, that is, 1

Iμ (q, ε) q−1 ∼ εDμ (q) ;

.

the number .Dμ (q) (if it exists) is the so-called q-generalized fractal dimension of μ. Note that for large values of q the main contribution to .Iμ (q, ε) comes from sets for which .μ charges most, whereas for small values of .q < 1 the main contribution is from the most rarefied sets charged by .μ. This interpolation is an advantage of this concept of dimension.

.

Definition 1.14 Let .0 < q, .q = 1, and let .μ be a positive finite Borel measure on .R. The lower and upper q-generalized fractal dimensions of .μ are defined, respectively, as Dμ− (q) := lim inf

.

ε↓0

ln Iμ (q, ε) , (q − 1) ln ε

Dμ+ (q) := lim sup ε↓0

ln Iμ (q, ε) . (q − 1) ln ε

Remark 1.15 ˙ (i) In .Iμ (q, ε), one may use .B(x; ε) = (x − ε, x + ε) instead of .B(x; ε) (check it). (ii) The dimensions .Dμ± (q) have other names, e.g., multifractal dimensions, generalized entropy dimensions, Hentschel-Procaccia dimensions, generalized Rényi dimensions. They seem to have been introduced in the physics literature by Hentschel and Procaccia [83] to characterize invariant measures supported over strange attractors present in some dynamical systems (see also [81]). (iii) .Iμ (q, ε) are well defined since the function .x → μ((x − ε, x + ε)) is Borel measurable; see the proof of Lemma 3.7. 

.

The next result presents some important properties of such dimensions. Proposition 1.16 (i) The functions .q → Dμ± (q) are nonincreasing; (ii) For all .q > 1, one has .0 ≤ Dμ− (q) ≤ Dμ+ (q) ≤ 1. (iii) If there exists .q ∈ (0, 1) such that . k∈Z μ([k, k + 1))q < ∞ (which is particularly true for every .q ∈ (0, 1) if .μ has bounded support), then − + .0 ≤ Dμ (q) ≤ Dμ (q) ≤ 1 Example 1.17 If .μ is a finite Lipschitz (continuous) measure, that is, if there exists an .L > 0 such that .μ((a, b)) ≤ L(b − a) for all intervals .(a, b) with .b − a < 1, then

1.1 First Basic Concepts

27

Dμ± (q) = 1,

.

∀q > 1.

Indeed, for .ε > 0 small enough,  Iμ (q, ε) =

μ(B(x; ε))q−1 dμ(x) ≤ (2L)q−1 2εq−1 μ(R),

.

and so .

ln Iμ (q, ε) ln((2L)q−1 μ(R)) ≥ + 1; (q − 1) ln ε (q − 1) ln ε

the result now follows by taking .ε → 0 and by Proposition 1.16.



.

Example 1.18 If .μ is a finite pure point measure, then Dμ± (q) = 0,

.

∀q > 1.

 Namely, let .{λj } be the atoms of .μ, so that .μ(R) = j μ({λj }) < ∞. For .q > 1 and all .ε > 0, one has    .Iμ (q, ε) ≥ μ({x})q−1 dμ(x) = μ({λj }) μ({λj })q−1 = μ({λj })q = C > 0. R

j

j

Hence, .lim supε→0 ln Iμ (q, ε)/((q − 1) ln ε) ≤ lim supε→0 ln C/((q − 1) ln ε) = 0. 

.

Example 1.19 From some point of view, this example is rather unexpected: a pure point measure with positive upper q-dimension for .0 < q < 1/2 (compare to Example 1.18). This idea is further explored in Chap. 5. Let .(xj )j ≥1 be a sequence in .R such that .xj − xj +1 = 1/j 2 , for each .j ∈ N, and put μ=

.

 1 δx . j2 j j ≥1

Thus, .μ is finite, pure point and .{xj } is the set of its atoms. Given .N ∈ N, let εN = 1/N 2 and .0 < q < 1/2; then,

.

 Iμ (q, εN ) ≥

.

=

R

μ({x})q−1 dμ(x) ≥

N  j =1

N 

μ(xj )μ({xj })q−1

j =1

 N N  1 1 (N 1−2q − 1) , ≥ dy y −2q = μ(xj ) = 1 − 2q j 2q 1 q

j =1

28

1 Spectrum and Dynamics: Some Basic Concepts

and so .

lim sup N →∞

ln Iμ (q, εN ) 1 − 2q . ≥ (q − 1) ln εN 2(1 − q)

Therefore, Dμ+ (q) ≥

.

1 − 2q > 0, 2(1 − q) 

for all .0 < q < 1/2.

.

Proof of Proposition 1.16 We use an alternative expression for the generalized dimensions, presented in Corollary 5.13. (i) We begin recalling Lyapunov’s Inequality for probability measures: for each measurable function .h : X → R, where .X = dom h, and each .0 < s < t, one has 

1 |h(x)|s dμ(x)

.

s



1

≤

|h(x)|t dμ(x)

X

t

;

X

in fact, it is a consequence of Hölder’s Inequality (.1/q + 1/p = 1, p > 1),

1 



 |f (x)g(x)| dμ(x) ≤

.

|f (x)| dμ(x) p

X

p

1 |g(x)| dμ(x) q

q

,

X

X

by taking .f (x) = h(x)s , .g(x) = 1 and .p = t/s. By this inequality, one immediately obtains the result for .0 < q1 < q2 < 1 and .1 < q1 < q2 : namely, for .0 < q1 < q2 < 1, let .h(·) = (μ(B(· ; ε)))−1 in Lyapunov’s Inequality to obtain  |h(x)|1/(1−q2 ) dμ(x)

.

X

⇒ ⇒

ln

ln

 X

 X

1/(1−q2 )

|h(x)|1/(1−q2 ) dμ(x) (q2 − 1) log ε

 ≤

|h(x)|1/(1−q1 ) dμ(x)

1/(1−q1 )

X

 ≤

ln

 X

|h(x)|1/(1−q1 ) dμ(x) (q1 − 1) log ε



   μ(B(x; ε))q2 −1 dμ(x) ln X μ(B(x; ε))q1 −1 dμ(x) ≤ , (q2 − 1) log ε (q1 − 1) log ε

which is valid for .0 < ε < 1 (the proof for .1 < q1 < q2 is completely analogous). Suppose now that .q1 < 1 < q2 . Put .δ := min{1 − q1 , q2 − 1}. Then, for   .q = 1 − δ and .q = 1 + δ, one has

1.1 First Basic Concepts

29

 ln(Iμ (q  , ε)) ln(Iμ (q  , ε)) 1 − = ln . μ(B(x; ε))−δ dμ(x) (q  − 1) (q  − 1) δ   × μ(B(x; ε))δ dμ(x)   2 ln μ(B(x; ε))−δ/2 μ(B(x; ε))δ/2 dμ(x) δ  2 = ln dμ(x) = 0 δ

≥

by Cauchy–Schwarz’s Inequality. Hence, .Dμ± (q  ) ≤ Dμ± (q  ). Finally, by the definition of .δ, one has .q1 ≤ q  < 1 and .1 < q  ≤ q2 . It follows from the first part of the proof that Dμ± (q2 ) ≤ Dμ± (q  ) ≤ Dμ± (q  ) ≤ Dμ± (q1 ) .

.

(ii) It is clear that .0 ≤ Dμ− (q) ≤ Dμ+ (q) for all .q > 0, .q = 1. Let .Ij = [j ε, (j + 1)ε)), .0 < ε < 1/2, .j ∈ Z, and .aj := μ(Ij ). For any .k ∈ Z, one has [k, k + 1) ⊂

r 

.

Ij ⊂ [k − 1, k + 2) ,

(1.13)

j =p

where .p = [k/ε], .r = [(k + 2)/ε] − 1 (.[x] denotes the integer part of x). Note that .r − p + 1 < 2/ε, for .ε small enough. It follows from (1.13) that dk := μ([k, k + 1)) ≤

r 

.

aj ,

j =p

and then, by Hölder’s Inequality for .q > 1 and .k ∈ Z, that ⎛ dk ≤ ⎝

r 

.

⎞1/q ⎛ q aj ⎠

⎝

j =p

r 

⎞(q−1)/q 1⎠

⎛ ≤ (ε/2)

(1−q)/q

j =p

⎝

r 

⎞1/q q aj ⎠

.

j =p

Hence, for each .k ∈ Z, Sμ (q, ε) :=



.

j ∈Z

q

aj ≥

r 

q

q

aj ≥ 21−q εq−1 dk .

j =p

 Since .μ(R) = k∈Z dk = 1, there exists .k ∈ Z such that .dk > 0; so, one has q−1 , so .ln S (q, ε) ≥ ln C(q)+(q −1) ln ε, where .C(q) > 0 .Sμ (q, ε) > C(q)ε μ

30

1 Spectrum and Dynamics: Some Basic Concepts

does not depend on .ε. By Corollary 5.13, since .q > 1, it straightly follows that Dμ+ (q) ≤ 1. (iii) By (1.13), one has .

r 

dk ≤

.

aj ≤ dk−1 + dk + dk+1 .

j =p

 q Now, just apply Hölder’s Inequality to . rj =p aj , with .t = 1/q, .t  = 1/(1 − q), in order to obtain r  .

⎛ q aj ≤ ⎝

j =p

r 

⎞q ⎛ aj ⎠ ⎝

j =p

r 

⎞1−q 1⎠

j =p

≤ 21−q εq−1 (dk−1 + dk + dk+1 )q q

q

q

≤ C(q)εq−1 (dk−1 + dk + dk+1 ) , with .C(q) = 3q · 21−q , where we have used the fact that .p = [k/ε], .r = [(k + 2)/ε] − 1, .r − p + 1 < 2/ε. Thus, Sμ (q, ε) =



.

j ∈Z

q

aj ≤

r 

q

aj ≤ 3C(q)εq−1

k∈Z j =p



q

dk

k∈Z

and .

ln Sμ (q, ε) ≤ ln(3C(q)) + (q − 1) ln ε + ln



q dk ;

k∈Z

 q since .q < 1 and, by hypothesis, . k∈Z dk < ∞, the result follows again from Corollary 5.13. 

1.2 Semigroups Here we recall basic concepts about .C0 -semigroups and put our discussion into perspective. As mentioned previously, regarding the (Baire) generic asymptotic behavior of .C0 -semigroups, in Chaps. 7 and 8 we discuss mainly the case of normal semigroups; with respect to the theory of asymptotics of .C0 -semigroups, at least two questions are well known to be delicate: the situation where the intersection of the spectrum of the generator with the imaginary axis is nonempty and the existence of an adequate

1.2 Semigroups

31

integral representation theory (see [14–16, 21, 119] for a detailed discussion); in order to address these questions, we use mainly the joint resolution of the identity for normal operators [18, 122], and from the technical point of view, this is the main difference between the study presented here and classical texts on the subject.

1.2.1 General Definitions We recall the main basic concepts of the theory of .C0 -semigroups [107, 135]; note that although our discussion is restricted to Hilbert spaces (since the main interest is in normal semigroups), many concepts and results hold true in Banach spaces. Let .B(H ) denote the space of bounded linear operators on the Hilbert space .H . Definition 1.20 A family .(S(t))t≥0 of bounded linear operators acting on .H is called a .C0 -semigroup if the following properties hold: 1. .S(0) = 1 and .S(t + s) = S(t)S(s), .t, s ≥ 0, 2. .lim S(t)u − u = 0, for all .u ∈ H . t↓0

Remark 1.21 It follows that there are constants .ω ≥ 0 and .M ≥ 1 such that S(t) ≤ M eωt ,

.

t ≥ 0;

it also follows that for each .u ∈ H , the mapping .[0, ∞)  t → S(t)u is continuous. . See [107] for a proof of these statements. Definition 1.22 The generator of a .C0 -semigroup .(S(t))t≥0 is the linear operator A given by   S(h)u − u exists , .dom A = u ∈ X | lim h↓0 h Au := lim

.

h↓0

S(h)u − u , h

u ∈ dom A.

A .C0 -semigroup .(S(t))t≥0 is said to be bounded if there exists .C > 0 so that, for each .t ≥ 0, .S(t)B(H ) ≤ C; if .C = 1 is admissible, then it is called a .C0 semigroup of contractions. Recall that a classical solution to the abstract Cauchy problem .

du(t) = Au(t), dt

t ≥ 0,

u(0) = u ∈ H ,

(1.14)

is a continuously differentiable function .u : [0, ∞) −→ H , taking values in dom A, which satisfies (1.14). A continuous function .u : [0, ∞) −→ X is a mild

.

32

1 Spectrum and Dynamics: Some Basic Concepts

solution to (1.14) if there exists a sequence .(un ) ⊂ dom A so that, for each n, the problem (1.14) with initial condition .un has a classical solution .un (t) so that . lim un (t) = u(t) locally uniformly for .t ≥ 0. n→∞

The next theorem is a classical result stating that .S(t)u is a solution to (1.14) for each .u ∈ H (See Chap. 1 in [107] for the proof). Theorem 1.23 Let .(S(t))t≥0 be a .C0 -semigroup and A its generator. Then, (i) A is a closed densely defined linear operator, i.e., .dom A = H . (ii) For each .u ∈ dom A, .t → S(t)u ∈ dom A, and is continuously differentiable, for all .t ≥ 0 and it is the unique solution to (1.14), i.e., .

d S(t)u = AS(t)u = S(t)Au, dt

t ≥ 0.

Remark 1.24 Since .dom A = H , by Theorem 1.23 (ii), each orbit .S(t)u is, at least, a mild solution to (1.14).

1.2.2 Asymptotics of C0 -Semigroups: A Short Account An important question regarding the problem (1.14) is how its solutions behave as t → ∞; in many instances the solutions vanish in this limit, and this is generally reported as stability.

.

Definition 1.25 A .C0 -semigroup .S(t) is called (strongly) stable if, for all .u ∈ H , .

lim S(t)u = 0,

t→∞

and it is called exponentially stable if there exist constants .C > 0, .a > 0 such that for all .t ≥ 0, S(t) ≤ C e−at .

.

In the last decades, there has been a lot of interest in polynomial decay behavior for .C0 -semigroups. This goes back to [14, 15, 21], and polynomial decay rates for semigroups have had extensive applications in the study of PDEs, in particular to damped wave equations (see [7, 109] and references therein). This in turn has led to an investigation of even more subtle asymptotic properties of .C0 -semigroups (see [3, 16, 30, 119]). In order to obtain lower bounds for the decay rates of stable .C0 -semigroups, most authors (see [16, 30, 119, 135] and references therein) usually have related estimates on the norm of the resolvent of the generator A to quantitative decay rates of the form

1.2 Semigroups

33

S(t)A−1  = O(r(t)),

.

t → ∞,

with .limt→∞ r(t) = 0, which implies that all classical solutions to the abstract Cauchy problem (1.14) converge uniformly (on the unit ball of .dom A endowed with the graph norm) to zero at infinity with rate .r(t). Since .dom A is dense in .H , one could argue that such solutions display the typical behavior. In this book, by following [3], for normal semigroups we consider the notion of typical behavior in terms of dense .Gδ subsets of initial conditions .u ∈ H . Namely, we discuss the existence of dense .Gδ sets of initial conditions u such that each of the respective orbits .(S(t)u)t≥0 contains a sequence that decays to zero no faster than a fixed but arbitrarily slow rate and a sequence that decays to zero at a sub-exponential rate. A spectral classification of (strong) stability of these semigroups will also be discussed. Consider now the known Herbst-Howland-Prüss Theorem [113]. Recall that the resolvent of the linear operator A at .z ∈ ρ(T ) is the operator Rz (A) = (A − z1)−1 ∈ B(H ),

.

where .B(H ) stands for the space of bounded linear operators defined on .H . Theorem 1.26 A .C0 -semigroup .(S(t))t≥0 of contractions on a Hilbert space .H , with generator A, is exponentially stable if, and only if, iR ⊂ ρ(A) and

.

lim sup Riλ (A) < ∞; |λ|→∞

In order to better appreciate this result, we complement with the following example. 2 Example 1.27 Let .ϕ : (0, 1) → C be the function .ϕ(y) = y + i 5|y| , and let 2 2 .Mϕ : dom (Mϕ ) ⊂ L ((0, 1)) → L ((0, 1)),

(Mϕ f )(y) = −ϕ(y)f (y),

.

where .f ∈ dom Mϕ = {u ∈ L2 ((0, 1)) | ϕu ∈ L2 ((0, 1))}. 2 , 0 < y < 1}, Mϕ is a normal operator with spectrum .σ (Mϕ ) = {−y − i 5|y| which results in .

.

lim sup Riλ (Mϕ ) ≥ lim sup |λ|→∞

|λ|→∞

1 = ∞, d(iλ, σ (Mϕ ))

where .d(x, A) := inf{|y − x| | y ∈ A}. Hence, by Theorem 1.26, the normal C0 -semigroup of contractions .(etMϕ )t≥0 is not exponentially stable. Our goal in this example is to examine the decay of .etMϕ Mϕ−1 , and we proceed as in [15]. Let .a > 0, let

.

34

1 Spectrum and Dynamics: Some Basic Concepts

γ + = {λ ∈ C | |λ| = a and Re λ ≥ 0},

.

γ − = {λ ∈ C | |λ| = a and Re λ < 0}

.

and let .γ  be a path in .{λ ∈ ρ(A) | Re λ < 0 from ia to .−ia}; these paths are illustrated in Fig. 1.1. By Cauchy’s Theorem, e

.

tMϕ

Mϕ−1

1 =− 2π i −

1 2π i

+

1 2π i

λ2 dλ 1 + 2 Rλ (Mϕ ) T (t) . λ a γ+

 λ2 dλ 1 + 2 Rλ (Mϕ ) etλ . λ a γ

 λ2 dλ 1 + 2 gt (λ) etλ , − λ a γ 



(1.15) (1.16) (1.17)

t where .gt (λ) = 0 e−sλ esMϕ ds is an integer function. Moreover, it is possible to show that the integrals in (1.15) and (1.17) are uniformly bounded in a, independently of t. Therefore, by dominated convergence,  tM −1  e ϕ M  → 0,

.

ϕ

t → ∞;

in particular, such semigroup is stable.   Under this approach, the understanding of the decay rates of .etMϕ Mϕ−1  depends only on (1.16) and therefore, a natural way (at least for specialists) to deal with this case is to build explicitly .γ  , which requires some care since d(iR, σ (Mϕ )) = 0 ,

.

where .d(A, B) := inf{|x − y| | x ∈ A, y ∈ B}. We note that in estimating the length of .γ  , it is expected that estimates on the norm of the resolvent of the generator are involved, since each neighborhood in the open set .ρ(Mϕ ) ⊂ R is built through Neumann series expansions of the resolvent of .Mϕ . . The ideas presented in Example 1.27 are due to Korevaar [95], which were used in [15] to prove a result relating the decay rates of .S(t)A−1  (in a Banach space), in case .iR ⊂ ρ(A), to the arbitrary growth of the resolvent of the generator. Such result due to Batty and Duyckaerts [15] was an important intermediate step between Herbst-Howland-Prüss Theorem (Theorem 1.26) and the results on fine scales of decay of operator semigroups (see, for instance, [3, 16, 30, 119]). Usually, the problem of obtaining lower bounds for the decay rates of stable .C0 -semigroups passes through the details of some theory of integral representation (like, for instance, Cauchy’s theory and the functional calculus of sectorial operators

1.2 Semigroups

35

.σ (Mϕ ), .γ

+

∪ γ − and .γ 

[30, 119]). By employing the joint resolution of the identity for normal operators [18, 122] (see in (1.19) below), in Chaps. 7 and 8 the polynomial decay rates of a normal semigroup is related to the spectral properties of its generator. We also say something about nonnormal semigroups.

1.2.3 Joint Resolution of Identity and Normal Semigroups Let .C+ := {z ∈ C | Re z > 0}; by the Spectral Theorem, each normal operator N on a Hilbert space .H such that .C+ ⊂ ρ(N ) generates a normal .C0 -semigroup of contractions; namely,  etN =

etλ dE N (λ)

.

σ (N )

is such a semigroup, where .E N is the resolution of the identity of N . It is well known that every normal .C0 -semigroup of contractions is of this form [120]. Every normal operator N can be written as .N = NR + iNI , where

36

1 Spectrum and Dynamics: Some Basic Concepts

NR =

.

N + N∗ 2

NI = −i

and

N − N∗ 2

(1.18)

are self-adjoint operators and .NR NI = NI NR . In this case, .E N corresponds to a joint resolution of the identity associated with the operator pair .{NR , NI } [18, 122]. Thus, for every .u ∈ H with .u = 1, etN u2 = et (NR +iNI ) u2  NI R = |et (y+iv) |2 dμN u (y) dμu (v)

.

σ (NR )×σ (NI )



 =

σ (NI )

 =

0

−∞

I dμN u (v)

σ (NR )

R e2ty dμN u (y)

R e2ty dμN u (y) ,

(1.19)

R where .μN u denotes the spectral measure of .NR associated with u; the last equality in (1.19) is a consequence of fact that .(etN )t≥0 is a semigroup of contractions. The identity (1.19) will be important to obtain the main results of Chap. 7 on normal semigroups.

Proposition 1.28 Let N be a normal operator so that .C+ ⊂ ρ(N ). Then, .(etN )t≥0 is exponentially stable if, and only if, .0 ∈ σ (NR ). Proof By (1.19), we may assume, without loss of generality, that N is a self-adjoint operator and .N ≤ 0. Assume that .(etN )t≥0 is exponentially stable. Then, there exist constants .C > 0 and .a > 0 such that for each .t ≥ 0, etN  ≤ Ce−ta ;

.

this implies, for each .u ∈ H , and each .t ≥ 0, etN u2 ≤ C 2 e−2ta u2 .

.

Thus, for each .u ∈ H , each .t ≥ 0, and each .0 < δ < 1, −2ta(1−δ) μN ≤ u ([−a(1 − δ), 0]) e



0

.

 ≤ and therefore,

−a(1−δ) 0 −∞

e2ty dμN u (y)

2 −2ta e2ty dμN u2 , u (y) ≤ C e

1.3 Wonderland Theorem

37

2 −2taδ μN u2 ; u ([−a(1 − δ), 0]) ≤ C e

.

(1.20)

by letting .t → ∞ in (1.20), one obtains μN u ([−a(1 − δ), 0]) = 0,

.

which results in N N μN u ([−a, 0]) = μu (∪0 0 (see Chapter 9 in [53]), that for each .ξ ∈ H ,  Ri (Tn )ξ − Ri (T )ξ  ≤

.

∞

e−s e−isTn ξ − e−isT ξ  ds .

0

Since .e−isTn ξ − e−isT ξ  ≤ 2ξ , it follows from dominated convergence that if −itTn → e−itT strongly, then .T → T in the strong resolvent sense. .e n −1 . So, .r(t) = .(i) ⇔ (v). Let .r : R → C denote the function .r(t) = (t − i) −1 (t + i) , and .Ri (T ) = r(T ), .R−i (T ) = r(T ). Since r is a bounded and continuous function, .(v) → (i) follows. Suppose now that .Tn → T in the strong resolvent sense. Since r separates points of .R and .lim|t|→∞ r(t) = 0, it follows from Stone-Weierstrass Theorem that the set of polynomials .pε (r(t), r(t)) is dense in .C∞ (R) := {ψ ∈ (C(R),  · ∞ ) | ∀ε > 0 there exists .M = M(ε) > 0 so that .|ψ(x)| < ε if .|x| ≥ M}. So, if .φ ∈ C∞ (R), then for each .ε > 0 there exists .pε (r(t), r(t)) such that .φ − ψε ∞ < ε and, by the Spectral Theorem, for each .n ∈ N, both inequalities φ(T ) − pε (Ri (T ), R−i (T )) < ε ,

.

φ(Tn ) − pε (Ri (Tn ), R−i (Tn )) < ε

hold simultaneously. Since .R±i (Tn ) → R±i (T ) strongly, one has, for each .ε > 0, that .pε (Ri (Tn ), R−i (Tn )) → pε (Ri (T ), R−i (T )) also holds strongly. Now, an application of triangle inequality shows that, for each .φ ∈ C∞ (R), .φ(Tn ) → φ(T ) strongly. Finally, let .f : R → C be bounded and continuous. By the Spectral Theorem, .f (T ) ≤ f ∞ and .f (Tn ) ≤ f ∞ , for any .n ∈ N. Let .(φj ) be a monotone increasing sequence in .C0 (R) := {ψ ∈ (C(R), ·∞ ) | ∃M > 0 such that .|ψ(x)| = 0 if .|x| > M} such that .0 ≤ φj ≤ 1 and, for each .t ∈ R, .lim φj (t) = t. Thus, .φj (T ) → 1 and, for each .n ∈ N, .φj (Tn ) → 1, both strongly. Since .f φj ∈ C∞ (R), for each .j ∈ N, .f (Tn )φj (Tn ) → f (T )φj (T ) strongly. If .ξ ∈ H , it follows (after some manipulations) that f (Tn )ξ − f (T )ξ  ≤ f (Tn )ξ − f (Tn )φj (T )ξ 

.

+f (Tn )φj (T )ξ − f (Tn )φj (Tn )ξ  +f (Tn )φj (Tn )ξ − f (T )φj (T )ξ  +f (T )φj (T )ξ − f (T )ξ  ≤ f (Tn )ξ (ξ − φj (T )ξ  + φj (T )ξ − φj (Tn )ξ ) +f (Tn )φj (Tn )ξ − f (T )φj (T )ξ  + f (T )φj (T )ξ − ξ  . Given .ε > 0, let .j ∈ N be large enough so that .ξ − φj (T )ξ  < ε/f ∞ ; then, for large n, one has .φj (T )ξ − φj (Tn )ξ  < ε/f ∞ and .f (Tn )φj (Tn )ξ − f (T )φj (T )ξ  < ε, from which follows that

40

1 Spectrum and Dynamics: Some Basic Concepts

f (Tn )ξ − f (T )ξ  ≤ 4ε,

.

and so .f (Tn ) → f (T ) strongly. −itx is .(i) ⇒ (iii). Given that, for each .t ∈ R, .ft : R → C, .ft (x) := e continuous and bounded, the result follows from .(i) ⇒ (v). .(iii) ⇔ (iv). Since all the operators involved in the convergence are unitary, the weak convergence is equivalent to the strong convergence (see the proof of Proposition 5.1.7 in [53]), and so .(iii) is equivalent to .(iv).  Theorem 1.31 (Wonderland) Let .(X, d) be a regular metric space of self-adjoint operators. If each of the sets • .Cp of .T ∈ X with pure point spectrum; • .Cac of .T ∈ X with purely absolutely continuous spectrum, is dense in X, then the set .Csc of .T ∈ X with purely singular continuous spectrum is generic in X, i.e., .Csc is a dense .Gδ set in X. In applications, one usually proves the existence of dense (and disjoint!) sets of operators with pure point and purely absolutely continuous spectra. Sometimes the Weyl-von Neumann Theorem [53] can be used to get pure point operators, and the absolutely continuous ones from some kind of periodicity (periodic operators have purely absolutely continuous spectrum) or even from perturbations of the free Hamiltonian. Example 1.32 Now we mention an application of the Wonderland Theorem 1.31; see the original paper [125] for other applications. Fix .H and let .X be the subset of .B(H ) (the set of bounded operators on .H ) of self-adjoint operators T with .σ (T ) = [0, 1]. It turns out that the set of operators in with purely singular . continuous spectrum is generic in .X. It is worth mentioning the work by Pearson [108], published in 1978, with explicit examples of potentials .V (x) so that the Schrödinger operator HV = −

.

d2 + V (x), dx 2

acting in .L2 ((0, ∞)), has some singular continuous spectrum.

1.3.1 Proof of Wonderland Theorem The proof consists basically in showing that the sets presented in the statement of the Wonderland Theorem are .Gδ sets; the density comes from the hypotheses, and it is what one usually has to check in the applications. Propositions 1.33 and 1.35 deal with the .Gδ property and have independent interest. Actually, the proof of

1.3 Wonderland Theorem

41

Proposition 1.33 was based on [50]. Recall that .(X, d) is assumed to be a regular metric space of self-adjoint operators. Proposition 1.33 The set .Y := {T ∈ X : σp (T ) = ∅} is a .Gδ in X. Proof By Proposition 1.30, strong resolvent convergence is equivalent to strong dynamical convergence, so for each .ψ ∈ H , .t > 0, the mapping X  T → WψT (t)

.

is continuous and .{T ∈ X : WψT (t) < 1/n} is an open set in X. Let .(ej )j ≥1 be an orthonormal basis of .H . By Wiener’s Lemma 1.5, .σp (T ) = ∅ if, and only if, .limt→∞ WeTj (t) = 0 for each .ej . Since Y =

 

{T ∈ X : WeTj (t) < 1/n},

.

j,n∈N t∈N



it follows that Y is a .Gδ .

Lemma 1.34 A finite (positive) Borel measure .0 = μ in .R and the Lebesgue measure . are mutually singular if, and only if, there exists a sequence of continuous functions .fn : R → [0, 1], .n ≥ 1, so that 



1 .(i) fn d < n 2 R

and

(ii)

R

fn dμ > μ(R) −

1 . 2n

Proof Assume that such sequence of continuous functions exists. Let .Cn := {x ∈ R : fn (x) > 1/2}. Thus,  (Cn ) = 2

.

Cn

1 d ≤ 2 2



1

(i)

R

fn d
, ln 1/tn ln 1/tn

and so .Dμ+T (2) ≥ α + ε. ψ

Suppose now that .Dμ+T (2) ≥ α + ε. Then, there exists a sequence .tn → ∞ so ψ

that .

ln Wψ (tn ) ln 1/tn

>

(α + ε) − 2ε = α − ε ⇒ ln Wψ (tn ) < (α − ε) ln 1/tn 

⇒ Wψ (tn )
0, .lim supt→∞ t α+ε Wψ (t) < ∞, it follows that .lim supt→∞ t α+ε Wψ (t) = 0; the same is true for .lim sup replaced by .lim inf (check this!). Hence, given .c > 0, it follows from (2.10) that ψ

{ψ | lim inf t α+ε Wψ (t) < c} ⊂ C(α+ε)u ⊂ {ψ | lim inf t α−ε Wψ (t) < c}.

.

t→∞

t→∞

Thus, for each fixed .ψ and (small) .ε > 0, one has ψ

{T ∈ X | for each m ∈ N, ∃ t > m with t α+ε Wψ (t) < c} ⊂ C(α+ε)u

.

⊂ {T ∈ X | for each m ∈ N, ∃ t > m with t α−ε Wψ (t) < c} , and by replacing .α with .α + 2ε and by taking .ε = 1/k, for .k ≥ k0 (α) (.k0 (α) is the least integer for which .α + 1/(2k) ≤ 1), one gets ψ Cαu =

.

  

1

{T ∈ X | t α+ k Wψ (t) < c}.

k≥k0 m≥1 t>m

Now, using the same reasoning as before, it follows from (2.9) that  .

n≥1 m≥1

ψ

{T ∈ X | ∃t > m with t α+ε Wψ (t) > n} ⊂ C(α+ε)l

2.2 Gδ Sets For Correlation Dimension

⊂



55

{T ∈ X | ∃t > m with t α−ε Wψ (t) > n};

n≥1 m≥1

by replacing .α with .α − 2ε and by taking .ε = 1/ l, for .p ≥ p0 (α), one gets   

ψ

Cαl =

.

{T ∈ X | t

α− p1

Wψ (t) > n}.

p≥p0 n≥1 m≥1 t>m

If we denote, for each .k ≥ k0 and each .t > 0 Uk (t) = {T ∈ X | t α+1/k Wψ (t) < c},

.

one has ψ Cαu =

  

.

Uk (t).

k≥k0 m≥1 t>m ψ

So, if for each .k ≥ k0 and each .t > 0, .Uk (t) is an open set, then .Cαu is a .Gδ set in X. Fix .k ≥ k0 and .t > 0. If .T0 ∈ Uk (t), it follows from Proposition 1.30 that the mapping .X  T → t α+1/k WψT (t) is continuous, and so if .d(T , T0 ) is small enough, then .T ∈ Uk (t), showing that .Uk (t) is open. ψ For .Cαl , set for each .p ≥ p0 and each .t > 0 Vp (t) = {T ∈ X | t α−1/p Wψ (t) > k};

.

ψ

ψ

one concludes that .Cαl is a .Gδ set in X, in analogy to .Cαu , after noting that ψ

Cαl =

.

  

Vp (t).

p≥p0 n≥1 m≥1 t>m

Now, given that ψ

C1u =

.



ψ

C(1−1/s)u

s≥2

and ψ

C0l =

.



ψ

C(1/s)u ,

s≥2 ψ

ψ

it also follows that .C1u and .C0l are both .Gδ subsets of X.



56

2 Correlation Dimension

2.3 Generic Correlation Dimension As already mentioned, our goal in this chapter is to include dimensional information to generic sets in a regular space .(X, d) of self-adjoint operators. In the following, we present a version that has interesting applications; the original reference is [24]. Recall that .H is a separable infinite dimensional Hilbert space. Theorem 2.6 Let .(X, d) be a regular space of self-adjoint operators acting in .H and let .0 = ψ ∈ H . If each of the sets • .Cp of .T ∈ X with pure point spectrum; ψ • .CL := {T ∈ X | μTψ is Lipschitz.}; is dense in X, then {T ∈ X | Dμ−T (2) = 0 and Dμ+T (2) = 1}

.

ψ

ψ

is generic in X. Proof Since .C0l = {T ∈ X | Dμ−T (2) = 0} and .C1u = {T ∈ X | Dμ+T (2) = 1} are ψ

ψ

ψ

ψ

both .Gδ sets, by Theorem 2.4, one just needs to prove that they are dense in X. By ψ ψ Example 1.17, the hypothesis on .CL implies that .C1u is dense in X, so generic; by ψ Example 1.18, the hypothesis on .Cp implies that .C0l is dense in X, so generic. To ψ ψ  finish the proof, just notice that the set in the statement is .C0l ∩ C1u . Remark 2.7 The proof of Theorem 2.6 can be repeated to include the case in which all spectral measures are restricted to an open interval I , that is, .μTψ;I ; see [24] for details. . We would like to stress that the hypotheses in the statement of Theorem 2.6 could ψ be different (for instance, the result is obviously valid if we assume that both .C0l ψ and .C1u are dense in X), but since it is not obvious how to calculate explicitly the lower and upper correlation dimensions of a measure, we have opted to consider those hypotheses for which there are interesting applications (that is, it is somewhat straightforward to check that they are dense in these examples). In case, if the set ξ

CLn = {T ∈ X | μTξn is Lipschitz}

.

(2.11)

is dense for a countable set .{ξn } of an orthonormal basis of .H , then the conclusions of Theorem 2.6 hold true simultaneously for all vectors .ξn , since one could use the countable intersections

2.4 Applications

57

 .

ξ

C0ln

and

n



ξ

C1un .

n

A natural question refers to the extension of such results for a dense set of vectors in .H ; this is not immediate, since there is no proof that the sets (for each .0 < α < 1) {ξ ∈ H | Dμ−T (2) ≤ α}

.

ξ

and

{ξ ∈ H | Dμ+T (2) ≥ α} ξ

are vector spaces! Think about this problem. Until now, we have discussed a result on correlation dimensions, but there is an immediate and interesting dynamical version of this result. Namely, by Theorem 2.2, if the conclusions of Theorem 2.6 apply to the vector .ψ, then the set {T ∈ X | γψ− (T ) = 0 and γψ+ (T ) = 1}

.

is generic in X and the typical behavior of the average projection .WψT (t), defined on (1.1), depends on sequences of time going to infinity: given a typical operator .T ∈ X, there exist a sequence for which the projection .WψT (t) decays at the fastest powerlaw speed (like a conductor) and other for which .WψT (t) decays at the slowest speed (like an insulator)! This indicates how intricate is the typical dynamical behavior from the topological point of view.

2.4 Applications We are going to discuss three applications of Theorem 2.6. Since the proofs involve many standard results in operator and spectral theory, we present references for the classical results that are employed.

2.4.1 Bounded Operators: Norm Convergence Fix .a < b, and let .Ya,b := {T ∈ B(H ) | T is self-adjoint with spectrum .[a, b]} be equipped with the metric of norm convergence d(T , T  ) = T − T  B(H ) ,

.

which is a complete metric space; here, .B(H ) denotes the set bounded self-adjoint operators on .H . One has the following result. Theorem 2.8 Let .{ej }j ≥1 be an orthonormal basis of .H . Then, the set

58

2 Correlation Dimension

{T ∈ Ya,b | Dμ−T (2) = 0 and Dμ+T (2) = 1 , for all j }

.

j

j

is generic in .Ya,b , where .μTj := μTej . For the proof of Theorem 2.8, we use the following known facts (see also [125]): • F1. Weyl-von Neumann Theorem: Given a self-adjoint operator T acting in .H and .ε > 0, there exists a self-adjoint compact operator .S ∈ B(H ), with .S < ε, so that .T + S is self-adjoint and a pure point operator (see Theorem 12.4.2 in [53]). • F2. Let T and S be self-adjoint operators in .H , with S compact. Then, for the essential spectrum, we have (Corollary 11.3.7 in [53]) σess (T ) = σess (T + S).

.

That is, compact self-adjoint perturbations preserve the essential spectrum of self-adjoint operators; this result is originally due to Weyl. • F3. If .dim H = ∞, given .ξ ∈ H , there is a bounded self-adjoint operator L on .H whose spectral measure .μL ξ is Lipschitz. • F4. If .T = T1 ⊕ T2 (all involved operators are self-adjoint), then (Proposition 11.1.1 in [53]) σ (T ) = σ (T1 ) ∪ σ (T2 ).

.

With respect to F3, one may suppose that .ξ is normalized and pick an orthonormal basis .{ξj } of .H with .ξ0 = ξ . If .M is the multiplication operator by the function .ϕ(x) = x in .L2 ([0, 1]) and .ψ ∈ L2 ([0, 1]) is a bounded and normalized 2 function, then a direct inspection gives .dμM ψ (x) = |ψ(x)| dx, and since .ψ is bounded, say .|ψ(x)|2 ≤ K, it follows that for each Borel set A, .μM ψ (A) ≤ K (A), M and so .μψ is Lipschitz. Let .{ψj } be an orthonormal basis of .L2 ([0, 1]) with 2 .ψ0 = ψ, and let .U : L ([0, 1]) → H be a unitary operator with .U ψj = ξj , L for all j . Then, .μξ is Lipschitz, where .L = U M U −1 . Proof of Theorem 2.8 To simplify the notation, we consider (with no loss of generality) .Y0,1 . Of course, convergence in norm implies strong resolvent convergence (e.g., use the second resolvent identity), and so .Y0,1 is regular. Let .T ∈ Y0,1 ; by F1, given .ε > 0, there exists a compact self-adjoint operator S with .S < ε/2 so that .T + S is pure point. By F2, .σ (T + S) ⊃ [0, 1], with a possible finite set of isolated eigenvalues of finite multiplicity in (−ε/2, 0) ∪ (1, 1 + ε/2).

.

2.4 Applications

59

Such eigenvalues may be shifted to 0 or 1 by a finite rank perturbation of .T + S, say F , with .F  < ε/2. Hence, .T0 := T + S + F is an element of .Y0,1 with pure point spectrum and T − T0  ≤ S + F  ≤ S + F  < ε.

.

ψ

Hence, .Cp is dense in .Y0,1 , given that .ε > 0 is arbitrary. Since .Cp ⊂ C0l for all .ψ ∈ H , it follows from Theorem 2.6 that {T ∈ Y0,1 | Dμ−T (2) = 0}

.

j

is generic for each j , and so {T ∈ Y0,1 | Dμ−T (2) = 0 , for all j }.

.

j

e

ψ

Now we pass to .C1u . The goal is to show that for each j , .CLj (see (2.11)) is dense in .Y0,1 , but by the argument above, it is enough to approximate operators .T ∈ Cp e (with dense point spectrum in .[0, 1]) by operators .An ∈ CLj . Since T is an operator with dense pure point in spectrum in .[0, 1], there exist an orthonormal basis .{ϕk } of .H and eigenvalues .{λk } so that .T ϕk = λk ϕk , for all k. For each .n ∈ N, let

1

I1 = 0, n , 2

I2 =

.

1 2

1  n = 1− , . . . , I , ,1 , 2 2n 2n 2n

and let .ρq = 2q−1 /2n . Define the self-adjoint operator .Bn ∈ B(H ) by Bn ϕk = ρq ϕk ,

.

if λk ∈ Iq

(so, selecting .1 ≤ q(k) ≤ 2n ), and extend it linearly to other vectors. Note that .Bn − T  ≤ 1/2n . Namely, (Bn − T )ϕk  = |ρq(k) − λk |ϕk  ≤

.

and for an arbitrary vector .ξ = (Bn − T )ξ =



.

k



k ak ϕk ,

1 , 2n

one has

ak (B − T )ϕk =



ak (ρq(k) − λk )ϕk ,

k

so (Bn − T )ξ 2 ≤



.

k

|ak |2 |ρq(k) − λk |2 ≤

1  1 |ak |2 = 2n ξ 2 , 2n 2 2 k

60

2 Correlation Dimension

from which follows that .Bn − T  ≤ 1/2n . Fix the basic vector .ej and denote, for each .q = 1, 2, 3, . . . , 2n , the projection of .ej in .Hq := rng χIq (T ) by .ej,q . Since .Hq is infinite dimensional (given that .{λk } is dense in .[0, 1]), it follows from F3 that there exists a self-adjoint operator .Lq , L

L

q is Lipschitz, that is, acting on .Hq , with .σ (Lq ) = [0, 1], such that .μj,qq = μej,q

L

μj,qq (J ) ≤ Cq |J |,

.

for all intervals J such that .|J | < 1 (.|J | denotes the length of J ). Introduce the self-adjoint operators n

L :=

2 

.

Lq

q=1

and 2   n

.

An :=

ρq 1 +

q=1

 1 1 L χIq (T ) = Bn + n L. q n 2 2

The following properties are basic.

1. .σ (An ) = [0, 1], since .σ ρq 1 + 21n Lq = I¯q (the closure of the interval) and, n by F4, .σ (An ) = ∪2 I¯q = [0, 1]. q=1

2. .T − An  ≤ 1/2n−1 . Namely,  1  1 1 T − An  = T − Bn − n L ≤ T − Bn  + n L ≤ 2 n . 2 2 2

.

3. For each q, the spectral measure of the operator .ρq 1 + 21n Lq (in the sum defining .An ) at .ej,q is Lipschitz, since it is just .Lq /2n translated by .ρq . For .η = ej,q ⊕ ej,q  , .ej,q ∈ Hq , ej,q  ∈ Hq  , one has n μA η (J ) = ej,q ⊕ ej,q  , χJ (An )(ej,q ⊕ ej,q  )

.

= ej,q ⊕ ej,q  , χJ (An )ej,q ⊕ χJ (An )ej,q   = ej,q , χJ (An )ej,q H1 + ej,q  , χJ (An )ej,q  H2 An n = μA j,q (J ) + μj,q  (J ) ,

and since this immediately generalizes to .η = ej,1 ⊕ ej,2 ⊕ · · · ⊕ ej,2n , it follows An n that .μA j is Lipschitz, since each .μj,q is Lipschitz, as observed before.

2.4 Applications

61

The above properties ensure that the sequence of operators .An converges to T e e e in .B(H ), and so .CLj is dense in .Y0,1 for each j . Since .CLj ⊂ C1uj , by combining this result with Theorem 2.6, it follows that {T ∈ Y0,1 | Dμ+T (2) = 1}

.

j

is generic for each j , and so {T ∈ Y0,1 | Dμ+T (2) = 1, for all j }.

.

j

The result is now a consequence of the fact that the intersection of two generic  sets is also a generic set.

2.4.2 Bounded Operators: Strong Convergence Basically, the topology of bounded operators is the difference between this application and the one in Sect. 2.4.1; there we had norm convergence of operators and here just strong convergence. Fix .r > 0. Let .Xr = {T ∈ B(H ) | T is self-adjoint, .T  ≤ r}, with the metric d(T , T  ) =

∞ 

.

min(2−j , (T − T  )ej ) ,

j =0

where .{ej }j ≥0 is an orthonormal basis of .H , for which convergence corresponds to strong operator convergence. Theorem 2.9 Fix .r > 0 and an orthonormal basis .(ξj ) of .H . Then, the set {T ∈ Xr | σ (T ) = [−r, r], Dμ−T (2) = 0 and Dμ+T (2) = 1 , for all j }

.

j

j

is generic in .Xr , where .μTj := μTξj . Before we present the proof of Theorem 2.9, we need the following result, which is valid for any regular space X. Proposition 2.10 Fix .[a, b] ⊂ R, .b > a, and .0 = ψ ∈ H . Then, .U ψ := {T ∈ X | T .supp (μ ) ⊃ [a, b]} is a .Gδ set in X. ψ Proof Let .M+ (R) denote the set of positive finite Borel measures on the line endowed with the vague topology. We follow the steps of the proof of Theorem 1.3 in [125]. Fix .0 = ψ ∈ H and let .λn be a countable dense set in .[a, b]. Then,

62

2 Correlation Dimension

        T ∈ X | supp μTψ ⊃ [a, b] = T ∈ X | supp μTψ  λn ,

.

n

so we need only to consider the cases where we replace .[a, b] by .{λ}. Since  .

∞         T ∈ X | supp μTψ ∩ (λ − 1/m, λ + 1/m) = ∅ , T ∈ X | supp μTψ  λ = m=1

the continuity of the mapping .X  T → μTψ ∈ M+ (R) (since strong resolvent convergence results in vague convergence of the spectral measures: namely, by Proposition 1.30, if .Tn → T in the strong resolvent, then, for each .f ∈ Cc (R) (.Cc (R) stands for the set of real continuous functions of compact support), .f (Tn ) → f (T ), which in turn gives, for each .ψ ∈ H , .μTψn → μTψ in the vague sense. Thus, for each .ψ ∈ H , the mapping .X  T → μTψ ∈ M+ (R) is continuous; see also [99]) says that .{T ∈ X | supp (μTψ )  λ} is an .Fσ . Thus, its complement is a .Gδ .  Proof of Theorem 2.9 We begin with the proof that .Cp := {T ∈ Xr | σ (T ) = [−r, r] and it is pure point.} is dense in .Xr . Let .(ϕn )∞ n=−∞ be an orthonormal basis (this way of counting will be convenient) and let .PN be the projection onto .(ϕn )|n|≤N so that .PN → 1 strongly. Let .(αn ) be a counting of the rational numbers in .[−r, r] and let B be the diagonal operator .Bϕn = αn ϕn . Then, .BN := PN APN + (1 − PN )B(1 − PN ) converges strongly to A. Observe that for each .N ∈ N, .σ (BN ) = [−r, r] and it is pure point. By combining this with Theorem 2.6 and Proposition 2.10, it follows that for each j , {T ∈ Xr | σ (T ) = [−r, r], Dμ−T (2) = 0}

.

j

is generic in .Xr , and so {T ∈ Xr | σ (T ) = [−r, r], Dμ−T (2) = 0, for all j }.

.

j

Thus, it remains to show that for each j , .C1uj := {T ∈ Xr | Dμ+T (2) = 1} is ξ

j

ξ

generic; for that, by Theorem 2.6, it is enough to prove that for each j , .CLj := ξj .C L

ξ C1uj .

⊂ The idea is to {T ∈ Xr | μTj is Lipschitz.} is dense in .Xr , given that approximate operators T with dense point spectrum such that .T  ≤ r − ε, where ξj .0 < ε ≤ r, by operators .BN ∈ C . L Suppose, therefore, that T is as above. Let .(γn ) be the normalized eigenvectors of T , say .T γn = λn γn , so that .(λn ) are the corresponding eigenvalues. Take an arbitrary .0 < δ ≤ ε. Pick a sequence of numbers .ηN → 0, .0 < ηN < δ/(2N), and introduce a family of operators .(BN ) by

2.4 Applications

63

BN :=

N −1 

λn 1 + ηN Ln PnN ;

.

n=0

for each .0 ≤ n ≤ N − 1, .PnN is the projection onto the subspace .Hn generated by .(γn+kN )k≥0 ; .Ln is an operator defined on .Hn such that .Ln  ≤ 1 and .μj,n , the spectral measure of .Ln associated with .ξj,n := PnN ξj , is Lipschitz (since .dim Hn = ∞, it follows from condition F3 presented in the proof of Theorem 2.8 that there exists an operator .Ln satisfying the conditions above). Then, for each .N ∈ N, .BN  ≤ r (since . n ηN Ln  < (N δ)/(2N) < ε), and n therefore, .BN ∈ Xr . Note that for each .0 ≤ n ≤ N − 1, .μC j,n , the spectral measure of .Cn := λn 1 + ηN Ln associated with .ξj,n ∈ H , is Lipschitz; therefore, for each BN .N ∈ N, .μ j , the spectral measure of .BN associated with .ξj , is also Lipschitz, since N −1 Cn it equals . n=0 μj,n (see the proof of Theorem 2.8 for more details). At last, in order to prove that .BN → T strongly, as .N → ∞, let .N0 be the least integer such that . j ≥N0 2−j < δ/2; thus, for every .N ≥ N0 , d(BN , T ) =

N −1 

.

min(2−m , ηN Lm γm ) +

min(2−m , (BN − T )γm )

m≥N

m=0


0. Then, the set {hv ∈ Sr | σ (hv ) = [−2 − r, 2 + r], Dμ−v (2) = 0 and Dμ+v (2) = 1}

.

;I

;I

is generic in .Sr . v Remark 2.12 Given that for each .hv ∈ Sr and each .0 = ψ ∈ H , .μhψ;I is v absolutely continuous with respect to .μ (this is a consequence of the fact that .{e0 , e1 } is a set of cyclic vectors for every Schrödinger operator with action (2.12); see [132]), one has:

(i) .supp (μv ) = σ (hv ); v (ii) if .μv purely atomic, then for each .0 = ψ ∈ H , .μhψ;I is also purely atomic. Moreover, one also has the following result. Lemma 2.13 Let .hv be a bounded discrete Schrödinger operator given by a finite rank perturbation of the free operator .h0 , and let .K = (a, b), .−2 < a < b < 2. v Then, .μhψ;K is Lipschitz continuous for every .0 = ψ ∈ H . Therefore, it follows from the previous results and from Theorem 2.11 that {hv ∈ Sr | σ (hv ) = [−2 − r, 2 + r], D −hv (2) = 0 and

.

μψ;I

D +hv (2) μψ;I

= 1, for all ψ = 0}

is generic in .Sr .

.

Proof of Lemma 2.13 Denote the inner product  in .H by .(·, ·). Let J be an interval with .|J | < 1; given .0 = ψ ∈ H , write .ψ = i∈Z ai ei . Then, v μhψ;K (J ) = (ψ, χJ ∩K (hv )ψ) =



.

=



 ai aj

i,j ∈Z

+

 i,j ∈Z

J

ai aj (ei , χJ ∩K (hv )ej )

i,j ∈Z

ui (x)uj (x) dμhe0v;K (x) 

ai aj J

wi (x)wj (x) dμhe1v;K (x) ,

2.4 Applications

65

where .u(x) = (ui (x))i and .w(x) = (wi (x))i represent, respectively, the solution to .(hv ϕ)i = xϕi for .x ∈ R, which satisfies .u0 (x) = 1, .u1 (x) = 0 and .w0 (x) = 0, .w1 (x) = 1 (see Section 2.5 in [132] for details), and .χJ ∩K (hv ) denotes the spectral projection of .hv over .J ∩ K. Since it is known that .u(x) = (ui (x))i and .w(x) = (wi (x))i are continuous functions of .x ∈ K which are uniformly bounded on K (as discussed in Section 5.2 of [70]; namely, one can see .u(x) and .w(x) as sequences related to a sequence of transfer matrices which are, up to a finite number of entries, conjugated to the rotation matrix by .ϕ ∈ [0, π ], where .x = 2 cos ϕ. Therefore, for .x ∈ K, the norms of .u(x) and .w(x) are continuous and uniformly bounded), it follows that 

v μhψ;K (J ) ≤

.

i,j ∈Z

|ai aj | (|Ci Cj | μhe0v;K (J ) + |Di Dj | μve1 ;K (J ))

≤ C 2 μv (J )



|ai aj |

i,j ∈Z

≤ C 2 ψ2 μv;K (J ) , where .Ci :=

sup ui (x) < ∞, .Di :=

x∈J ∩K

sup{|Ci |, |Di |} < ∞.

sup vi (x) < ∞ and .C :=

x∈J ∩K

i∈Z

Thus, since there exists .E > 0 such that .μv;K (J ) ≤ E|J | (.μv;K is Lipschitz, given that its Radon-Nikodym derivative is uniformly bounded on every such .K = (a, b) ⊂ [−2, 2] = σess (hv ) as in the statement of the lemma; see [70]), it follows v that there exists .κ < ∞, which does not depend on J , such that .μhψ;K (J ) ≤ κ |J | v (just take .κ = C 2 Eψ2 ). Hence, .μhψ;K is Lipschitz, by the arbitrariness of J . 

Proof of Theorem 2.11 By following the idea presented in Theorem 4.1 in [125], let .dζ be the product of the Lebesgue measures .(2r)−1 d n , .n ∈ Z, where .supp ( n ) := [−r, r]; thus, .supp (dζ ) = [−r, r]Z . Let .D = {hv ∈ Sr | supp (μv ) = [−2 − r, 2 + r], .hv has pure point spectrum on .[−2 − r, 2 + r]}. Then, .ζ (Sr \ D) = 0, by Anderson localization (see [63] for a discussion about Anderson localization, or even Sect. 4.3 for a discussion about SULE condition, especially the proof of Theorem 4.25) and, therefore, D is dense in .Sr . Now, by Theorem 2.6 and Proposition 2.10, it follows that {hv ∈ Sr | σ (hv ) = [−2 − r, 2 + r], Dμ−v (2) = 0}

.

is generic in .Sr . Consider now the operators .hvk ∈ Sr with potentials .vk = (V˜nk ) given by V˜nk =

.



Vn , n ≤ k . 0, n > k

(2.13)

66

2 Correlation Dimension

Since .vk → v pointwise, .hvk → hv in the strong resolvent sense. Now, given that each .hvk is a finite rank perturbation of .h0 (the free operator), it follows from k Lemma 2.13 that, for each k, .μv;(a,b) is uniformly Lipschitz continuous on .(a, b) ⊂ [−2, 2]. Hence, it follows from Theorem 2.6 that {hv ∈ Sr | Dμ+v

.

;(a,b)

(2) = 1}

is generic, as well as {hv ∈ Sr | Dμ+v (2) = 1},

.

;I

since the argument holds true for all subintervals .(a, b) ⊂ [−2, 2]. Thus, {hv ∈ Sr | σ (hv ) = [−2 − r, 2 + r], Dμ−v (2) = 0 and Dμ+v (2) = 1}

.

;I

is generic, being the intersection of two generic sets.

;I



Chapter 3

Fractal Measures and Dynamics

3.1 Local Dimensions of Measures and Dynamics In Sect. 1.1.3, we have presented the definitions of (upper and lower) Hausdorff, packing and generalized fractal dimensions of a positive finite Borel measure .μ on .R. The basic ingredient present in these definitions is encoded in the concept of (upper and lower) local dimension of a measure, denoted by .dμ± (x). Definition 3.1 Let .μ be a positive finite Borel measure on .R. The upper and lower local dimensions of .μ at the point .x ∈ supp (μ) are defined, respectively, by dμ− (x) := lim inf

.

ε↓0

ln μ(B(x; ε)) , ln ε

dμ+ (x) := lim sup ε↓0

ln μ(B(x; ε)) . ln ε

(3.1)

If .x ∈ / supp (μ), then .dμ− (x) = dμ+ (x) := ∞. The quantities .dμ± (x) are also called the pointwise dimensions or the Hölder exponents of .μ; in [63, 123] one finds possible alternatives of such definitions. It seems that the pioneers in relating such concepts to other dimensions, in particular relating .dμ− to Hausdorff dimensions of measures, were Rogers and Taylor [115, 116], Billingsley [19, 20], Tricot [133], Young [140] and Cutler [35, 36]; the relation between .dμ+ and packing dimensions was proven by Cutler in [38]. Remark 3.2 Note that if .μ =  (the Lebesgue measure on .R), then for every .ε > 0 and every .x ∈ R, one has .

lim ε↓0

ln (B(x; ε)) = 1, ln ε

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Aloisio et al., Spectral Measures and Dynamics: Typical Behaviors, Latin American Mathematics Series – UFSCar subseries, https://doi.org/10.1007/978-3-031-38289-5_3

67

68

3 Fractal Measures and Dynamics

given that .(B(x; ε)) = ((x − ε, x + ε)) = 2ε; if .μ = δx (the pure point measure on .x ∈ R), then for every .ε > 0, one has .

lim ε↓0

ln δx (B(x; ε)) = 0, ln ε

since .δx (B(x; ε)) = 1. The idea is that for other measures, if the limit in Definition 3.1 exists (and so denote it by .dμ (x) = dμ− (x) = dμ+ (x)), then one would have .μ(B(x; ε)) ≈ εdμ (x) for small values of .ε and for .x ∈ supp (μ); in particular, .d (x) = 1 for each .x ∈ supp (μ) and .dδx (x) = 0, so one expects that ± .0 ≤ dμ (x) ≤ 1, for .x ∈ supp (μ) (although this is not exactly true; see Remark 3.5). . Lemma 3.3 The functions .x → dμ± (x) are Borel measurable. Proof The proof follows from the arguments presented in the proof of Lemma 3.7.  The next result illustrates the connections of other dimensions with the local dimensions. Recall that the Hausdorff and packing dimensions of .μ are presented in Definition 1.10. Theorem 3.4 Let .μ be as above. Then, .

− dim− H (μ) = μ − ess.inf dμ ,

− dim+ H (μ) = μ − ess.sup dμ , .

(3.2)

+ dim− P (μ) = μ − ess.inf dμ ,

+ dim+ P (μ) = μ − ess.sup dμ .

(3.3)

Since the proof of Theorem 3.4 requires some preparation, it will be presented after Corollary 3.10. Remark 3.5 It follows from Theorem 3.4 that if .μ is a positive finite Borel measure on .R, then .0 ≤ dμ− (x) ≤ dμ+ (x) ≤ 1 for .μ-a.e. x, confirming in some sense the expectation presented in Remark 3.2. We note that it is not necessarily true that − + .0 ≤ dμ (x) ≤ dμ (x) ≤ 1 if .x ∈ supp (μ). For instance, let .μ be such that .μ(A) =  2 A∩[0,1] x d(x); then, .supp (μ) = [0, 1] and .μ(B(0; ε)) = ε /2 (for .ε < 1), from − + which follows that .dμ (0) = dμ (0) = 2. . Let us rewrite some of the equalities presented in (3.2) and (3.3) in a more explicit way; that is, .

− dim− H (μ) = sup {dμ (x) ≥ t, for μ−a.e. x} , t

and .

+ dim+ P (μ) = inf {dμ (x) ≤ t, for μ−a.e. x} . t

3.1 Local Dimensions of Measures and Dynamics

69

Note that, by Theorem 3.4, .

− dim− H (μ) ≤ dimP (μ),

+ dim+ H (μ) ≤ dimP (μ) .

− + If .dμ− (x) = dμ+ (x) for .μ-a.e. x, then .dim− H (μ) = dimP (μ) and .dimH (μ) = + − + dimP (μ), and in case the measure is exactly scaling, i.e., if .dμ (x) = dμ (x) = d (a constant value) for .μ-a.e. x, as is the case of Lebesgue measure, then, one has − − .dim (μ) = dim (μ) = d. H P Such local dimensions are directly connected to the so-called .α-derivative of measures, defined as follows [101, 115].

Definition 3.6 Let .μ as above and .α ∈ (0, 1]. The lower and upper .α-derivative of .μ at .x ∈ supp (μ) are defined, respectively, as D αμ (x) = lim inf

.

ε↓0

μ(B(x; ε)) , (2ε)α

α

D μ (x) = lim sup ε↓0

μ(B(x; ε)) . (2ε)α

α

(3.4) α

If .x ∈ / supp (μ), then .D αμ (x) = D μ (x) := ∞. The functions .D αμ and .D μ are also called the lower and upper .α-density of the measure .μ, respectively. α

Lemma 3.7 The functions .x → D αμ (x) and .x → D μ (x) are Borel measurable. Proof It is enough to show that, for each .s ≥ 0, the set As = {x | D αμ (x) ≤ s}

.

is a countable intersection of open sets to conclude that .D αμ is a Borel measurable function. ¯ ε)) < s}; since for each .y ∈ R and Set, for each .ε, s > 0, .Osε := {x | μ(B(x; each .ε > 0, ¯ ¯ μ(B(y, ε)) = lim μ(B(y, ε + r)),

.

r↓0

¯ it follows that for each .x ∈ Osε , there exists .r = r(x) > 0 such that .μ(B(x; ε+r)) < ¯ ¯ s, and so if .|y − x| < r, then .B(y, ε) ⊂ B(x; ε + r) and .y ∈ Osε ; that is, .Osε is an ¯ r)) is upper semicontinuous). The same is true for open set (hence, .x → μ(B(s,   ¯ μ(B(x; ε ))

< s, for some ε < ε , Cε = x | (2ε )α

.

since it is the union of suitable sets .Osε (with s depending on .ε ). Now, given that .Cε1 ⊂ Cε2 for .ε1 < ε2 , the set As = ∩∞ k=1 Cεk

.

70

3 Fractal Measures and Dynamics

is a countable intersection of nested open sets, with .εk ↓ 0 (.As is, indeed, a .Gδ set). By considering α

As = {x | D μ (x) ≥ s},

.

α



the measurability of .D μ is obtained in a similar way. Proposition 3.8 Let .μ be as above and .α ∈ (0, 1]. Then, dμ− (x) < α ⇒ D μ (x) = ∞ ⇒ dμ− (x) ≤ α , . α

.

α < dμ− (x) ⇒

α D μ (x)

= 0 ⇒ α ≤ dμ− (x) , .

(3.5) (3.6)

dμ+ (x) < α ⇒ D αμ (x) = ∞ ⇒ dμ+ (x) ≤ α , .

(3.7)

α < dμ+ (x) ⇒ D αμ (x) = 0 ⇒ α ≤ dμ+ (x) , .     α α dμ− (x) = inf α | D μ (x) = ∞ = sup α | D μ (x) = 0 , .     dμ+ (x) = inf α | D αμ (x) = ∞ = sup α | D αμ (x) = 0 .

(3.8) (3.9) (3.10)

Proof The first four statements follow from the definitions; e.g., if .dμ− (x) < α, then there exist .0 < a < α and a sequence .εn → 0 so that .

ln μ(B(x; εn )) μ(B(x; εn )) < α − a ⇒ μ(B(x; εn )) > εnα−a ⇒ > εn−a , ln εn εnα α

α

and so .D μ (x) = ∞. Now, if .D μ (x) = ∞, then for each .m > 0, there exists a subsequence .εn → 0 so that .

μ(B(x; εn )) > m ⇒ ln μ(B(x; εn )) > ln m + ln εnα εnα ⇒

ln m ln μ(B(x; εn )) < + α, ln εn ln εn

and so .dμ− (x) ≤ α, from which (3.5) follows. Finally, (3.9) and (3.10) are directly consequences of the first four relations.  The next result relates the lower and upper .α-derivatives of .μ with the continuity of .μ with respect to .hα and .P α . Proposition 3.9 Let .μ be as above, .E ⊂ R a Borel set and .0 < α ≤ 1, .t > 0. Then, α

(i) If .D μ (x) ≤ t for all .x ∈ E, then .2α hα (E) ≥ μ(E)/t. α (ii) If .D μ (x) ≥ t for all .x ∈ E, then .hα (E) ≤ μ(E)/t. (iii) If .D αμ (x) ≤ t for all .x ∈ E, then .P α (E) ≥ μ(E)/t.

3.1 Local Dimensions of Measures and Dynamics

71

(iv) If .D αμ (x) ≥ t for all .x ∈ E, then .P α (E) ≤ μ(E)/t. Proof ¯ (i) For each .x ∈ E and each .ε > 0, there exists .δ(x) > 0 so that .μ(B(x; r)) ≤ α (t + ε)(2r) if .0 < r ≤ δ(x). Let, for each .k ∈ N, ¯ Ek = {x ∈ E | μ(B(x; r)) ≤ (t + ε)(2r)α

.

if

0 < r ≤ 1/k}.

Then, .E = ∪k Ek , and so .μ(E) = limk→∞ μ(Ek ). j j Fix .k ∈ N. Let .{Ek } be a .δ-cover of .Ek so that .Ek ∩ Ek = ∅ for all j . Then, there exists .δ1 > 0 such that for each .0 < δ < δ1 ,  j . |Ek |α ≤ hα (Ek ) + ε. j j ¯ j ; |E j |) ⊃ E j ∩ Ek . So, for each .0 < δ < Note that if .xj ∈ Ek ∩ Ek , then .B(x k k min{1/k, δ1 },

μ(Ek ) ≤



.

j

μ(Ek ∩ Ek ) ≤

j

≤





¯ j , |E |)) μ(B(x k j

j j

2α (t + ε)(|Ek |)α = 2α (t + ε)

j



j

|Ek |α

j

≤ 2 (t + ε)(h (Ek ) + ε), α

α

and so .μ(E) ≤ 2α t hα (E), given that .ε is arbitrary. (ii) To prove this item, one may use the fact that .hα possesses the centered Vitali covering property (since .hα is a Borel measure on .R): every centered Vitali covering of E contains a countable set of disjoint closed balls .{B¯ j }j such that α ¯ j ) = 0 (a centered Vitali covering of E is a set of closed balls .h (E \ ∪B ¯ containing, for any .x ∈ E and .δ > 0, some closed ball .B(x; r) with .r ≤ δ); see [101] for details. Let .δ > 0, .0 < ε < t, and suppose that E is an open set. If .x ∈ E, there exists .0 < r = r(x) < δ so that .μ(B(x; r)) ≥ (t − ε)(2r)α and .B(x; r) ⊂ E. ¯ Set .U = U (δ) := {B(x; r) | x ∈ E, .r ≤ δ, .μ(B(x; r)) ≥ (t − ε)(2r)α and .B(x; r) ⊂ E}; then, .U is a centered Vitali covering of E, so there exists a ¯ j , rj )}j such that pairwise disjoint subcollection .{B¯ j = B(x .

∪j Bj ⊂ E

and

hα (E \ ∪j B¯ j ) = 0.

Hence, μ(E) ≥ μ(∪j Bj ) =



.

j

μ(Bj ) =

 j

μ(B(xj , rj ))

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3 Fractal Measures and Dynamics

≥

  (t − ε)(2rj )α = (t − ε) hαδ (E ∩ (∪j B¯ j )) j

j

= (t − ε)hαδ (E) (naturally, one can replace open intervals by closed ones in Definition 1.8). By taking .δ ↓ 0, one gets .μ(∪j Bj ) ≥ (t − ε)hα (E). Since .hα (E \ ∪j Bj ) = 0 and .ε > 0 is arbitrary, the result follows. Finally, if E is not an open set, the result follows from the fact that .μ is outer regular; namely, since .μ is a positive and finite Borel measure on .R, it is outer regular (see [101] for details), and therefore, given .ε > 0, there exists an open set U such that .U ⊃ E and .μ(E) > μ(U ) − ε. (iii) One may proceed in analogy to the proof of item (ii). Let .ε > 0 and .δ > 0. For ¯ each .x ∈ E, there exists .r ≤ δ such that .μ(B(x; r)) ≤ (t + ε)(2r)α .

Let .E ⊂ E, and note that ¯ ¯ U = U (δ) := {B(x; r) | x ∈ E , r ≤ δ and μ(B(x; r)) ≥ (t + ε)(2r)α }

.

is a centered Vitali covering of .E ; thus, one may select from .U a countable ¯ i ; ri )}i so that .μ(E \ ∪B¯ i ) = 0 and pairwise disjoint collection .{B¯ i = B(x

¯ .{Bi }i is a .δ-packing of .E . Hence,  .

i

|B¯ i |α =

  μ(B¯ i ) 1 = μ(E ) (2ri )α ≥ t +ε t +ε i

i

1 ⇒ P0α (E ) ≥ μ(E ) t +ε  Let .{Ej } be a countable family of subsets of E such that .E = j Ej . By applying the above inequalities to each .Ej , one obtains from (1.10) that .

  1 1 μ(E) = μ(Ej ) ≤ P α (Ej ), j j 0 t +ε t +ε

and by taking the infimum over all covers of E, it follows from Definition 1.9 that .

1 μ(E) ≤ P α (E). t +ε

Since .ε > 0 is arbitrary, it follows that .μ(E)/t ≤ P α (E). (iv) This proof is very similar to the proof of item (i) and will be omitted. Corollary 3.10 Let .μ be as above, .E ⊂ R a Borel set and .0 ≤ α ≤ 1. Then,



.

3.2 Fractal Decomposition of Measures

(i) (ii) (iii) (iv)

If .dμ− (x) ≤ α for all .x If .dμ− (x) ≥ α for all .x If .dμ+ (x) ≤ α for all .x If .dμ+ (x) ≥ α for all .x

73

∈ E, then .dimH (E) ≤ α. ∈ E and .μ(E) > 0, then .dimH (E) ≥ α. ∈ E, then .dimP (E) ≤ α. ∈ E and .μ(E) > 0, then .dimP (E) ≥ α.

Proof This is a consequence of Proposition 3.9. Let us present the proof of item (i); the others items are similarly proven. If for each .x ∈ E, .dμ− (x) ≤ α, then .dμ− (x) < α + ε for all .ε > 0; so, α+ε

by (3.5), .D μ (x) = ∞ for all .x ∈ E. Now, it follows from Proposition 3.9(ii) that .hα+ε (E) = 0, and so, by the definition of Hausdorff dimension of sets, .dimH (E) ≤ α + ε. Since this holds for all .ε > 0, one has .dimH (E) ≤ α.  We are finally ready to present the proof of Theorem 3.4. Proof of Theorem 3.4 We only discuss two cases, since the others have similar proofs. − − • .dim+ H (μ) = μ − ess.sup dμ . Set .s := μ − ess.sup dμ . Then, there exists a Borel c set .E ⊂ R with .μ(E ) = 0 such that, for each .x ∈ E, .dμ− (x) ≤ s. It follows by Corollary 3.10(i) that .dimH (E) ≤ s, and therefore, by Definition 1.10, that + .dim (μ) ≤ s. H If .s = 0, the above arguments show that .dim+ H (μ) = s = 0, and the equality follows in this case; so, suppose that .s > 0. By contradiction, assume that + + .dim (μ) < s; thus, for each .0 < δ < (s − dim (μ))/2, there exists .Eδ with H H + − .μ(Eδ ) > 0 such that, for each .x ∈ Eδ , .dμ (x) > dim (μ) + δ. It follows H (μ) + δ, a contradiction with the from Corollary 3.10(ii) that .dimH (Eδ ) ≥ dim+ H (μ) (Definition 1.10). definition of .dim+ H + + • .dim− P (μ) = μ − ess.inf dμ . Set .s := μ − ess.inf dμ . First we show that − − .dim (μ) ≥ s. Suppose that there exists .δ > 0 such that .dim (μ) < s − δ; then, P P there exists a Borel set E, with .μ(E) > 0, such that for each .x ∈ E, .dμ+ (x) > + dim− P (μ) + δ. It follows from Corollary 3.10(iv) that .dimP (E) ≥ dimP (μ) + δ, + a contradiction with the definition of .dimP (μ). The result follows from the fact that .δ > 0 is arbitrary.

Now we show that .dim− P (μ) ≤ s. If .s = 1, there is nothing to prove. Then, suppose that .s < 1 and take .t > s. Thus, there exists a Borel set E with .μ(E) > 0 so that, for each .x ∈ E, .dμ+ (x) ≤ t. By Corollary 3.10(iii), .dimP (E) ≤ t, and so − − .dim (μ) ≤ t. Since this holds true for all .t > s, one has .dim (μ) ≤ s.  P P

3.2 Fractal Decomposition of Measures Recall that .hα and .P α are not .σ -finite measures for .0 < α < 1 [101], so there are different notions of continuity and singularity of Borel measures with respect to

74

3 Fractal Measures and Dynamics

them; in the following, we only present some of these notions (see [97, 116] for a broader discussion). Definition 3.11 Let .α ∈ [0, 1] and let K denote either H for Hausdorff or P for packing. A finite positive Borel measure .μ on .R is called: (i) .α−K continuous, denoted .αKc, if .μ(S) = 0 for every Borel set S with .Kα (S) = 0 (for instance, if .K = H, then .μ is called .α-Hausdorff continuous, denoted α .αHc, if .μ(S) = 0 for every Borel set S with .h (S) = 0). (ii) .α −K singular, denoted .αKs, if it is supported on a Borel set S with .Kα (S) = 0. Remark 3.12 It follows from Definition 3.11 that for every .α ∈ (0, 1), if .μ is .αKc, then it is (.α − ε)Kc for every .0 ≤ ε ≤ α. Similarly, if .μ is .αKs, then it is (.α + ε)Ks for every .0 ≤ ε ≤ 1 − α. Note that every finite positive Borel measure is 0Kc (since .K 0 (S) = 0 if, and only if, .S = ∅) and so the only 0Ks measure is the trivial one. . Definition 3.13 A finite positive Borel measure .μ on .R is called: (i) 0-.K dimensional, denoted .0Kd, if it is supported on some Borel set S with .dimK (S) = 0. (ii) 1-.K dimensional, denoted .1Kd, if .μ(S) = 0 for any Borel set S with .dimK (S) < 1. The following result, based on [115, 116] and Theorem 2 in [75], relates the continuity of a Borel measure, with respect to .α-dimensional Hausdorff (packing) measures, with its .α-derivatives. Theorem 3.14 Let .μ be a finite positive Borel measure on .R and denote α

α H∞ := {x ∈ R | D μ (x) = ∞},

.

α P∞ := {x ∈ R | D αμ (x) = ∞} .

α and .P α are Borel sets in .R. Furthermore, Then, for every .α ∈ (0, 1), the sets .H∞ ∞

(i) .μ decomposes uniquely as μ = μαHs + μαHc ,

.

α ∩ ·) is .αHs and .μ α where .μαHs (·) := μ(H∞ αHc (·) := μ((R \ H∞ ) ∩ ·) is .αHc. (ii) .μ decomposes uniquely as

μ = μαPs + μαPc ,

.

α ∩ ·) is .αPs and .μ α where .μαPs (·) := μ(P∞ αPc (·) := μ((R \ P∞ ) ∩ ·) is .αPc.

Proof α ) = 0 and .μ((R \ (i) Due to Definition 3.11, it is enough to show that .hα (H∞ α α H∞ ) ∩ E) = 0, if .h (E) = 0. For .t > 0, let

3.2 Fractal Decomposition of Measures

75

α

Ut = {x ∈ R | D μ (x) ≥ t},

.

α

Lt = {x ∈ R | D μ (x) ≤ t}.

α = ∩∞ U and .R \ H α = H α ∪ H α , where Thus, .H∞ + ∞ k=1 k 0

H0α := {x ∈ R | D μ (x) = 0} = ∩∞ k=1 L1/k α

.

and ∞ H+α := {x ∈ R | 0 < D μ (x) < ∞} = {∪∞ k=1 U1/k } ∩ {∪k=1 Lk }. α

.

By Proposition 3.9(ii), for each .k ∈ N, α hα (H∞ ) ≤ hα (Uk ) ≤ μ(Uk )/k ≤ μ(R)/k,

.

α ) = 0. and so .hα (H∞ α , we deal with .μ;H α and .μ;H α separately. If .F ⊂ For the component .μ;R\H∞ + 0 R is a Borel set with .hα (F ) = 0, one has, by Proposition 3.9(i), α α μ;H0α (F ) = μ(H0α ∩ F ) = μ(∩∞ k=1 (L1/k ∩ F )) ≤ μ(L1 ∩ F ) ≤ 2 h (F ) = 0.

.

Now, it follows again from Proposition 3.9(i) that for each .k ∈ N, μ(Lk ∩ F ) ≤ k2α hα (Lk ∩ F ) = 0,

.

and so μ;H+α (F ) = μ(H+α ∩ F ) ≤ μ(∪∞ k=1 Lk ∩ F ) ≤

∞ 

.

μ(Lk ∩ F ) = 0.

k=1 α is .αHc. Hence, .μ;R\H∞ (ii) By invoking items (iii) and (iv) of Proposition 3.9, the proof of item (ii) is . analogous to the proof of item (i).

Our task now is to relate the notions of .α-continuity and .α-singularity with dimensional properties of measures. Corollary 3.15 Let .μ be as above. If .μ is .αHc, then it is .αPc, and if .μ is .αPs, then it is .αHs. Proof By Theorem 3.14, it is enough to note that α

{x ∈ R | D αμ (x) = ∞} ⊂ {x ∈ R | D μ (x) = ∞}

.

to conclude the proof. Proposition 3.16 Let .μ be as above.



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3 Fractal Measures and Dynamics

− (i) If .μ is .αHc, then .0 ≤ α ≤ dim− H (μ). Conversely, if .dimH (μ) ≥ α is .(α − ε)Hc for each .0 < ε < α. + (ii) If .μ is .βHs, then .dim+ H (μ) ≤ β ≤ 1. Conversely, if .dimH (μ) ≤ β is .(α + ε)Hs for each .0 < ε < 1 − α. − (iii) If .μ is .αPc, then .0 ≤ α ≤ dim− P (μ). Conversely, if .dimP (μ) ≥ α is .(α − ε)Pc for each .0 < ε < α. + (iv) If .μ is .βPs, then .dim+ P (μ) ≤ β ≤ 1. Conversely, if .dimP (μ) ≤ β is .(α + ε)Ps for each .0 < ε < 1 − α.

> 0, then .μ < 1, then .μ > 0, then .μ < 1, then .μ

Proof We prove items (i) and (ii); the other proofs are similar. − (i) By Theorem 3.4, .dim− H (μ) = μ−ess.inf dμ . The case .α = 0 is trivial. Suppose, on the contrary, that .μ is .αHc for some .α > μ − ess.inf dμ− . Then, there exists a Borel set A, with .μ(A) > 0, such that for each .x ∈ A, .dμ− (x) < α. It follows from (3.5) and Theorem 3.14 that .μ(A ∩ ·) is .αHs, a contradiction with the fact that .μ is .αHc. − Conversely, if .dim− H (μ) ≥ α > 0, then for each .0 < ε < α, .μ−ess.inf dμ > α−ε

α − ε, which means, by (3.6), that .μ is supported on .{x ∈ R | D μ (x) = 0}. The result follows from Theorem 3.14. − (ii) By Theorem 3.4, .dim+ H (μ) = μ−ess.sup dμ . The case .β = 1 is trivial. Suppose that .μ is .βHs for some .β < μ − ess.sup dμ− ; hence, there exists a Borel set B, with .μ(B) > 0, such that for each .x ∈ B, .dμ− (x) > β. It follows from (3.6) and Theorem 3.14 that .μ(B ∩ ·) is .βHc, a contradiction with the fact that .μ is .βHs. Conversely, if .dim− H (μ) ≤ β < 1, then for each .0 < ε < 1 − β, .μ − ess.sup dμ− < β + ε, which means by (3.5) that .μ is supported on .{x ∈ R | β+ε

D μ (x) = ∞}. An application of Theorem 3.14 concludes the proof.



.

Remark 3.17 1. One notes that if .dim− H (μ) = α > 0, then it may happen that .μ is .αHs. For instance, let .(Cn ) be a sequence of Cantor-like sets defined as follows. For each n n .n ∈ N, let .[0, 1] = E ⊃ E ⊃ · · · be a decreasing sequence of sets, with each 0 1 k,n n k .E given by the union of .2 disjoint closed intervals .I such that: k i (a) for each fixed .k ≥ 0, .i = 0, . . . , 2k − 1, the .(k + 1)th interval levels .J1 , .J2 contained in .Iik,n are such that the left ends of .J1 and .Iik,n coincide, the right ends of .J2 and .Iik,n coincide, and for each .l ∈ {1, 2}, |Jl |1−1/(n+1) =

.

1 k,n 1−1/(n+1) |I | ; 2 i

(b) for each fixed .k ≥ 0, .Ekn ⊂ Ekn+1 . Then, set .Cn := ∩k≥0 Ekn . It follows that .Cn ⊂ Cn+1 ⊂ [0, 1] and, by Example 4.4 in [68], that .dimH (Cn ) = 1 − 1/(n + 1), with .0 < h1−1/(n+1) (C) < ∞ (and so .(Cn ) = 0).

3.2 Fractal Decomposition of Measures

77

Let .C := ∪n∈N Cn and set .μ as a positive finite measure supported on C such that .μ(S) = 0 if .dimH (S) < 1. Then, .dim− H (μ) = 1, although .μ is 1Hs (since .(C) = 0). One may define such .μ as follows. Let, for each .n ∈ N and each .k ≥ 0, n n k k .cn : [0, 1] → R be given by the law .cn (t) = α χE n (t), where .α is such that k k k 1 k . 0 cn (t) dt = 1; then, if .I ⊂ [0, 1] is an interval, it follows that k →

.

I

cnk (t) dt

is a nonincreasing sequence so that there exists a positive measure .μn , defined on intervals by μn (I ) := lim

.

k→∞ I

cnk (t) dt ;

then, given that n → lim

.

k→∞ I

cnk (t) dt

is a nondecreasing sequence, there exists a positive measure .μ, defined on intervals by μ(I ) := lim μn (I ) ,

.

n→∞

and one can apply the standard procedure for extending .μ from intervals to measurable sets. Such measure .μ is “uniformly distributed over C.” 2. It is also true that if .dim+ H (μ) = β ≤ 1, then it may happen that .μ is .βHc (let .β = 1 and .μ = , for instance). The same is true if we replace the Hausdorff properties by the packing ones in . the previous assertions. In Definition 3.18, two kinds of measures that can be seen as prototypes of measures with “fractal” properties are introduced; see also the original references [26, 129, 134] and also [39] (our nomenclature is adapted from [26, 97]). Definition 3.18 Let .μ be a finite Borel measure on .R and .α ∈ [0, 1]. One says that: (i) .μ is uniformly .α-Hölder continuous, denoted .UαHC, if there exist .C > 0 and α .0 < r0 < 1 such that, for each .0 < r < r0 and each .x ∈ R, .μ(B(x; r)) < C r . (ii) .μ is uniformly .α-Hölder singular, denoted .UαHS, if there exist .C > 0 and α .0 < r0 < 1 such that, for all .0 < r < r0 and for .μ-a.e. x, .μ(B(x; r)) ≥ C r .

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3 Fractal Measures and Dynamics

Remark 3.19 (i) If .μ is U.αHC, .α > 0, then .μ is U.βHC for any .0 ≤ β < α. (ii) If .μ is U.αHS, then .μ is U.βHS for any .β ≥ α. (iii) A finite pure point measure (in particular, .δλ ) is U.αHS for all .α > 0, and it is U.αHC only for .α = 0. (iv) If .μ   with .-a.e. bounded Radon-Nikodym derivative, .f = dμ/d, then .μ is U.αHC for all .0 ≤ α ≤ 1. In fact, if .M =  − ess.sup f , then, for .0 < r ≤ 1, μ(B(x; r)) =

f d ≤ M(B(x; r)) = 2M r.

.

B(x;r)

(v) The Lebesgue measure, ., is U.αHS only for .α = 1. (vi) If .μ is U.αHC and then .dimH A ≥ α. In fact, if .A ⊂ ∪j Uj with  .μ(A)α> 0, .|Uj | < 1, then . |U | > j j j (1/C) μ(Uj ) ≥ (1/C) μ(A) > 0, that is, α .h (A) > 0 and so .dimH A ≥ α. (vii) If .μ is a positive finite Borel U.αHS measure, then there exists a bounded Borel set A such that .μ(Ac ) = 0. Namely, if this is false, then there exist .r > 0 and a countable and pairwise disjoint family i ; r)}i such  of open balls .{B(x that .μ(B(xi ; r)) ≥ Cr α ; then, .μ(R) ≥ i μ(B(xi ; r)) ≥ Cr α i 1 = ∞. Hence, .μ must have compact support. 

.

Proposition 3.20 Let .0 < α < 1. If .μ is U.αHC, then it is .αHc and .dim− K (μ) ≥ α; if .μ is U.αHS, then it is .βPs for all .β > α, and .dim+ (μ) ≤ α. K Proof If .μ is a U.αHC measure, then direct calculations give, for all .x ∈ supp (μ), α

D αμ (x) ≤ D μ (x) ≤ C;

.

thus, by Theorem 3.13, .μ is .αHc; it follows now from Proposition 3.16(i) that .α ≤ dim− K (μ). In a similar way, if .μ is a U.αHS measure, then, for .μ-a.e. x, dμ− (x) ≤ dμ+ (x) ≤ α,

.

so .dim+ K (μ) ≤ α, by (3.3); it follows from Proposition 3.16(iv) that, for each .β > α, .μ is .βPs.  The next result shows that a positive finite Borel measure, which is .αHc, is, in some sense, the limit of a sequence of U.αHC measures, and a positive finite Borel measure which is .αPS is the limit of a sequence of U.αHS measures. These characterizations have dynamical consequences, as discussed in Sect. 3.3. Theorem 3.21 Let .μ be a positive finite Borel measure on .R and .α ∈ (0, 1).

3.2 Fractal Decomposition of Measures

79

(i) If .μ is .αHc (.αPs), then for each .0 < ε ≤ α (.0 < ε ≤ 1 − α) and each .δ > 0, there exist mutually singular measures .μδ1 , μδ2 such that .μδ1 is U.(α − ε)HC (U.(α + ε)HS), .μδ2 (R) < δ, and .μ = μδ1 + μδ2 . (ii) Conversely, if, for each .δ > 0, there exist mutually singular Borel measures δ δ δ δ δ .μ and .μ such that .μ = μ + μ , with .μ a U.αHC (U.αHS) measure and 1 2 1 2 1 δ .μ (R) < δ, then .μ is (.α − ε)Hc ((.α + ε)Ps) for all .0 < ε ≤ α (.0 < ε ≤ 1 − α). 2 Proof We just present the proof of the results involving the U.αHS and .αPs properties; the other ones are completely analogous. + (i) If .μ is .αPs then, by Proposition 3.16(iv), .dim+ P (μ) = μ − ess.sup dμ ≤ α; that is,

.

lim sup r↓0

ln μ(B(x; r)) ≤α ln r

for .μ-a.e. x. Since the sequence .(fr (x))r of measurable functions fr (x) := sup

.

r ≤r

ln μ(B(x; r )) ln r

converges to .dμ+ (x), Egoroff’s Theorem implies that given an arbitrary .δ > 0, there exists a Borel set S with .μ(S c ) < δ such that .fr (x) converges uniformly on S to .dμ+ (x) as .r ↓ 0. But then, given an arbitrary .0 < ε ≤ 1 − α, there exists .0 < r0 < 1 such that, for each .0 < r < r0 and each .x ∈ S, .

ln μ(B(x; r)) < α + ε; ln r

that is, .μ(B(x; r)) > r α+ε for all .x ∈ S. Now, just set .μδ1 = μ;S , .μδ2 = μ;S c , and we are done. (ii) Suppose that for each .δ > 0, .μ = μδ1 + μδ2 , with .μδ1 and .μδ2 satisfying the properties in the statement of the theorem. One must show, for every .0 < ε ≤ 1 − α, that .μ is (.α + ε)Ps; by Proposition 3.16(iv), it is sufficient to prove that .μ − ess.sup dμ+ ≤ α. Let us assume, nonetheless, that .μ − ess.sup dμ+ > α. Thus, there exists a Borel set B, with .μ(B) > 0, such that .dμ+ (x) > α for every .x ∈ B. Fix .0 < s < μ(B). By hypothesis, there exists a Borel set E (which may depend on .s) such that .μ can be decomposed as .μ = μs1 + μs2 , with .μs1 (·) := μ;E (·) = μ(E ∩ ·) U.αHS and .μs2 (·) := μ(E c ∩ ·), with .μs2 (R) < s. By Definition 3.18, there exist constants .C > 0 and .0 < r0 < 1 such that s s α .Cr ≤ μ (B(x; r)) for every .0 < r < r0 and every .x ∈ E \D, where .μ (D) = 1 1 s 0. Now, since .ln μ(·) ≥ ln μ1 (·),

80

3 Fractal Measures and Dynamics

dμ+ (x) = lim sup

.

r↓0

≤ lim sup r↓0

ln μs1 (B(x; r)) ln μ(B(x; r)) ≤ lim sup ln r ln r r↓0 ln C + α = α, ln r

and .dμ+ (x) ≤ α for every .x ∈ E \ D. But then, .(E \ D)c ⊃ B, which results in s > μs2 (E c ∪ D) = μs2 (E c ∪ D) + μs1 (E c ∪ D) = μ(E c ∪ D) ≥ μ(B),

.

a contradiction with .μ(B) > s. Thus, .μ − ess.sup dμ+ ≤ α, and the proof of the theorem is complete. . Remark 3.22 It follows from Remark 3.19(vii) and Theorem 3.21 that if .μ is an αPs finite Borel measure of unbounded support for some .0 < α < 1, then for each δ δ .δ > 0, the support of .μ is bounded, whereas the support of .μ is unbounded. . 1 2 .

3.3 Dynamics of UαHC and UαHS Measures In this section, we are interested in the long time behavior of .ψ(t) = e−itT ψ when the spectral measure .μTψ of the self-adjoint operator T is U.αHC or U.αHS. The results for U.αHC measures were originally presented in [97, 129], whereas the results for U.αHS measures were presented in [26]. Roughly, the dynamics of U.αHC measures is related to decay rates for the average projection onto the initial condition, that is, 1 .Wψ (t) = t =

1 t



t

1 |ψ, ψ(s)| ds = t

t

T 2

μˆ (s) ds; ψ

0



0

2

t

2



e−isx dμTψ (x) ds 0

the dynamics of U.αHS measures, on the other hand, may be related to upper bounds on 

ΞμT (t) :=

.

ψ

dμTψ (x)

dμTψ (y) e−(x−y)

2 t 2 /4

−1/2 .

An important difference to the average return projection is the negative power .−1/2 in the expression of .ΞμT (t). Let us state both results in sequence, so it will be easier ψ to compare them, and then we present their proofs; indeed, the results hold true for more general measures than the spectral ones.

3.3 Dynamics of UαHC and UαHS Measures

81

Theorem 3.23 Let .α ∈ [0, 1] and let .μ be a positive finite U.αHS Borel measure on .R. Then, there exists .D > 0 such that, for sufficiently large t, Ξμ (t) ≤ μ(R) D t α/2 .

.

Theorem 3.24 Let .α ∈ [0, 1], let .μ be a positive finite U.αHC Borel measure on .R and .f ∈ L2μ (R). Then, there exists .D > 0, depending only on .μ, such that for sufficiently large t and each .f ∈ L2μ (R), 1 . t





2



Df 22 −ixs

ds e f (x)dμ(x)

≤ . tα R

t

0

(3.11)

Remark 3.25 (i) For the above discussion, just take .f = 1 and .μ = μTψ in (3.11) to obtain the decay rate .Wψ (t) ≤ C/t α in case .μTψ is U.αHC. (ii) Suppose that .μTψ is U.αHC and let .ξ ∈ H , with .ξ  ≤ 1. By the spectral theorem (see, e.g., Lemma 13.3.4 in [53]) restricted to the cyclic subspace spanned by .ψ, i.e.,   Hψ := f (T )ψ | f ∈ L2μT (σ (T )) ,

.

ψ

the operator T is unitarily equivalent to multiplication by x on .L2μT (R). Then, ψ

there exists .fξ ∈ L2μT (R), with .fξ  ≤ 1 (.fξ = 0 if .ξ ∈ Hψ⊥ ), such that ψ

ξ, ψ(t) =

.

R

e−itx fξ (x) dμTψ (x),

and so, by Theorem 3.24, Wξ,ψ (t) ≤

.

Cfξ 2 C ≤ α, tα t

(3.12)

with the constant C independent of .ξ . 

.

Proof of Theorem 3.23 Since .μ is U.αHS, there exist positive constants C and .r0 , with .r0 < 1, such that for every .0 < r < r0 and .μ-a.e. y, .μ(B(y; r)) ≥ Cr α . Thus, by taking .t0 ≡ 1/r0 , it follows that for .t > t0 and every .x ∈ R, .

e−(x−y)

2 t 2 /4

dμ(y) =

 n≥0 n/t≤|x−y| 0, b ∈ R, .

R

e

Since, for each .0 ≤ s ≤ t, .1 ≤ e1−s .

  |f μ|2 (t) =

1 t

≤

e t

=

e t

=

e t



t



−ax 2 +ibx

dx =

2 /t 2

, it follows that

|f μ(s)|2 ds ≤

0

e

R

−s 2 /t 2

e−s

R

2 /t 2

e t



t

e−s

2 /t 2

|f μ(s)|2 ds

0

|f μ(s)|2 ds ds

R2

f (x)f (y) e−i(x−y)s dμ(x)dμ(y)



R2

π −b2 /(4a) e . a

f (x)f (y)

R

e

−s 2 /t 2 −i(x−y)s

e

ds

dμ(x)dμ(y)

√ e 2 2 f (x)f (y) π t e−t (x−y) /4 dμ(x)dμ(y) t R2 √ 2 2 f (x)f (y) e−t (x−y) /4 dμ(x)dμ(y) = e π =

R2



√ ≤ e π (C−S)

R2



√

≤ e π

√ = e π

2 −t 2 (x−y)2 /4

R2

 ×

   2 2 2 2 |f (x)|e−t (x−y) /8 |f (y)|e−t (x−y) /8 dμ(x)dμ(y)

R2

|f (x)| e

|f (y)|2 e−t

2 (x−y)2 /4



R

|f (x)|2 dμ(x)

R

1 2 dμ(x)dμ(y)

1 2 dμ(x)dμ(y)

e−t

2 (x−y)2 /4

dμ(y)

3.4 Lower Bounds on Moments

83

the symbol C-S (above) stands for Cauchy-Schwarz’s Inequality. Now, for each fixed .x ∈ R, set   n+1 n ≤ |x − y| < . .Ωn := y ∈ R : t t Since .μ is .UαHC and for .t > t0 := 1/r0 , one has .(Ωn ) ≤ 1/t < r0 , it follows that .

R

e−t

2 (x−y)2 /4

dμ(y) =

∞ 

e−t

2 (x−y)2 /4

dμ(y)

n=0 Ωn

≤

∞ 

e

−n2 /4

dμ(y) ≤ Ωn

n=0

∞ 

e−n

2 /4

n=0

C D ≤ α. α t t

By combining the last estimates, one gets for each .t > t0 , .

  D |f μ|2 (t) ≤ α f 22 , t 

and the proof is complete.

3.4 Lower Bounds on Moments The idea now is to use the previous results concerning U.αHC measures in order to obtain lower bounds on the moments of the position operator, defined in (1.3) by rpψ (t) =



.

n

|n|p Wen ,ψ (t)

1

p

,

where .p > 0, .B = {en }n∈Z is an orthonormal basis of .H and .ψ ∈ H . The main goal consists in obtaining lower bounds for the exponents .βψ± (p) = βψ± (p, T , B) (see (1.4)) in terms of the upper Hausdorff and packing dimensions of the spectral measure .μTψ ; T is the self-adjoint operator which is the generator of the unitary evolution group .U (t) = e−itT , .t ≥ 0, so that .ψ(t) = e−itT ψ. Recall that βψ− (p) = lim inf

.

t→∞

ψ

ln rp (t) , ln t

βψ+ (p) = lim sup t→∞

ψ

ln rp (t) . ln t

We refer to [11, 73–75, 97] for the original results. An important point is that if the spectral measure .μTψ is a U.αHC, then there exist .C > 0 and .t0 > 0 so that for each .N ∈ N and .t > t0 ,

84

3 Fractal Measures and Dynamics



Wen ,ψ (t) ≤

.

|n| 0 and each .p > 0, there exist constants .C = C(ψ, δ, p) > 0 and .t0 = t0 (δ) so that, for each .t > t0 , rpψ (t) ≥ C t h−δ .

.

T (ii) .βψ− (p) ≥ dim+ H (μψ ), for all .p > 0. (iii) For each .δ > 0 and .p > 0, there exist a constant .C = C(ψ, δ, p) > 0 and a subsequence .tk → ∞ so that, for sufficiently large k, p−δ

rpψ (tk ) ≥ C tk

.

.

T (iv) .βψ+ (p) ≥ dim+ P (μψ ), for all .p > 0.

Remark 3.27 (i) Note that items (ii) and (iv) of Theorem 3.26 are direct consequences of items (i) and (iii), respectively. For instance, it follows from item (i) that − .β (p) ≥ h − δ, for all .δ > 0, so (ii) follows. ψ (ii) Although the exponents .βψ± (p) depend on the orthonormal basis B and also ψ

on p, the dimensions of the spectral measures don’t; for instance, .rp (t) = ∞ may occur, whereas the dimensions are not greater than 1. This indicates that such lower bounds are far from optimal. A discussion on the special case of operators with dense pure point spectra appear in Chap. 6, for which + T + .dim (μ ) = 0 and .β (p) ≥ 1 for a generic set of initial conditions .ψ; see ψ ψ P Proposition 5.25 and Theorems 5.23 and 5.29. (iii) The lower bound on .βψ+ (p) (.βψ− (p)) stated in item (iv) (item (ii)) is particularly + T T interesting when .dim+ P (μψ ) = 1 (.dimH (μψ ) = 1); in this situation, the pair .(T , ψ) is quasiballistic (ballistic), regardless of the basis B; see Definition 1.2. + T T If, nonetheless, .dim+ P (μψ ) = 0 (.dimH (μψ ) = 0), then the bound is useless. (iv) By Theorem 5.23, it is possible to replace, in items (ii) and (iv), the fractal dimensions of .μTψ by their generalized fractal dimensions, .Dμ±T (q), for .0 < q < 1; this, indeed, improves some of the results exemplified in item (ii) above. 

.

Corollary 3.28 Let T be a self-adjoint operator defined in .H , .B = {en }n∈Z an orthonormal basis of .H and .ψ ∈ H .

3.4 Lower Bounds on Moments

85

(i) If .μTψ is .αHc then, for .p > 0, βψ− (p) ≥ α;

.

(ii) if .μTψ is .αPc then, for .p > 0, βψ+ (p) ≥ α.

.

Proof By combining Proposition 3.16(i) with Theorem 3.26(ii), one obtains item (i). The proof of item (ii) follows from Proposition 3.16(iii) and Theorem 3.26(iv).  Remark 3.29 While the results stated in Theorem 3.26 and Corollary 3.28 are valid in a rather abstract setting (that is, for an arbitrary self-adjoint operator defined in a separable Hilbert space), in specific situations, as the important cases of Schrödinger operators on .l 2 (Zd ) or .l 2 (Nd ), with the usual orthonormal bases B, one has different expressions (physically relevant) for the p-moments and so with different lower bounds [33, 75, 97] βψ− (p) ≥

.

1 T dim+ H (μψ ), d

βψ+ (p) ≥

1 T dim+ P (μψ ). d

In such situations, although for absolutely continuous spectrum one has + T T dim+ H (μψ ) = dimP (μψ ) = 1, it is still possible to have subdiffusive behavior, ± i.e., .βψ (p, B) ≤ 1/2. See an implementation in [17]. The expressions for the p-moments are different from ours (1.3), since the basis is indexed by explicit vectors and there are multiplicities; e.g., in .Z2 , the basis vectors .(2, 0) and .(0, 2) have the same weights in their expressions of p-moments. This difference does not occur in one-dimensional systems. .

.

3.4.1 b-adic Intervals Before we present the proof of Theorem 3.26, some preparation is required. Here, we make a digression on alternatives for calculating the local dimensions of measures; it is possible to restrict the limits to b-adic intervals. Let .b ≥ 2 be an integer number and consider the partition of .R into the so-called b-adic intervals of rank N  N .Ij

=

 j −1 j , , bN bN

j ∈ Z, N ∈ N.

86

3 Fractal Measures and Dynamics

For each .x ∈ R, denote by .IjN(x) the b-adic interval that contains x. Since in this subsection b is kept fixed, we do not include it in the notation .IjN . Given a positive finite Borel measure .μ on .R, consider the b-adic local dimensions − dμ,b (x) := lim inf

.

− ln μ(IjN(x) )

N →∞

N ln b

,

+ dμ,b (x) := lim sup

− ln μ(IjN(x) )

N →∞

N ln b

.

Note that the choice of the base of the logarithm function is immaterial. Proposition 3.30 Let .μ be a positive finite Borel measure on .R. Then, for .μ-a.e. − + x ∈ R, .dμ,b (x) = dμ− (x) and .dμ,b (x) = dμ+ (x).

.

Proof Let .x ∈ supp (μ). Since .IjN(x) ⊂ (x − 1/bN , x + 1/bN ], it follows that for each .x ∈ supp (μ), − dμ,b (x) ≥ dμ− (x)

.

and

+ dμ,b (x) ≥ dμ+ (x).

+ Now we argue by contradiction. If the equality .dμ,b (x) = dμ+ (x) does not hold .μ-a.e., then there exist .s, ε > 0 and a Borel set .A ⊂ R so that .μ(A) > 0 and, for + + all .x ∈ A, .dμ,b (x) > s + 2ε and .dμ+ (x) < s + ε (if .dμ,b (x) = ∞ and .dμ+ (x) < ∞, + (x) > s + 2). Since the then there exists .s > 0 such that .dμ+ (x) < s + 1 and .dμ,b sequence .(fr (x))r of measurable functions

fr (x) := sup

.

b−N ≤r

− ln μ(IjN(x) ) N ln b

+ converges to .dμ,b (x) as .r → 0, by Egoroff’s Theorem, one may suppose that the convergence is uniform on A (by redefining A, if necessary): there exists .r0 > 0 so that for each .0 < r < r0 and each .x ∈ A, there exists .N ∈ N with .b−N < r such that

μ(IjN(x) )
N0 , .ν(B(x; b−N )) ≥ b−N (s+ε) and .2b−N (α+2ε) < b−N (α+ε) (just let .N0 > ln 2/[(α + ε) ln b]). However, for each .x ∈ B and each .N > N0 , ν(B(x; b−N )) ≤

.

  ν(I ) | I of the type IjN and ν (I ∩ B(x; r)) > 0 ,

and this sum contains at most two nonzero terms. Now, by (3.15), there exists .N > N0 such that each of these terms (if there are two of them) has .ν-measure smaller than .b−N (s+2ε) . Thus, b−N (α+ε) ≤ ν(B(x; b−N )) ≤ 2b−N (α+2ε) ,

.

which is an absurd. The proof for .dμ− is completely analogous.



3.4.2 Proof of Theorem 3.26(i) Let .μ = μTψ and recall that .χΛ (T ) denotes the spectral projection of T over the Borel set .Λ. Without loss of generality, we may assume that .ψ = 1 (this does not change the values of the exponents .βψ± and the local dimensions .dμ± ). If .h = 0, there is nothing to prove, so suppose that .h > 0 and let .0 < δ < h. The idea is to use this hypothesis to obtain a vector .φ ∈ H whose spectral measure is then use the decay (3.14) to obtain, for sufficiently large t, lower U(.h − δ)HC, and  bounds on the tail . |n|≥N Wen ,ψ (t), which is actually the responsible for the growth of the moments. It follows from the characterization of .dim− H (μ) given by (3.2) that there exists a Borel set .A ⊂ R, with .μ(A ) > 0, such that for each .x ∈ A , h − δ < dμ− (x) ≤ h.

.

By Egoroff’s Theorem, there exists a Borel subset .A ⊂ A , with .μ(A) > 0, such that the measurable function x →

.

inf

0 bNk 2Nk−1 > 2Nk−1 /Nk2 , it follows that .tk → ∞ as .k → ∞, and −1 (t ), with .f (x) = 2x /(9x 2 ). Therefore, for all k, .Nk < f k 3−α α mtk (bNk /2) > C(α)bN tk > k

.

C(α) f −1 (tk )

tα 6−2α k

It follows from (3.21) that for each .k ∈ N,  .

Wen ,ψ (tk ) < 1 −

|n|≤Fk

bNk , 2

and so  .

|n|>Fk

Wen ,ψ (tk ) >

bNk . 2

=: Fk .

96

3 Fractal Measures and Dynamics

An application of Chebyshev’s Inequality gives ψ p .rp (tk )

1 bN p > k Fk = −1 2 2(f (tk ))2



C(α) f −1 (tk )

p

tα 6−2α k

,

(3.26)

and since .

ln(ln t/ ln 2) ln(f −1 (t)) = lim = 0, t→∞ t→∞ ln t ln t lim

given .ε > 0, there exists .k0 ∈ N so that for each .k > k0 , f −1 (tk ) < tk

ε/(2(3−α+1/p))

.

;

thus, (3.26) can be rewritten, for large .tk , as rpψ (tk ) >

.

C(α) α−ε t . 21/p k

Since this holds for all .0 < α < p and all .ε > 0, item (iii) of Theorem 3.26 follows. Remark 3.34 The proof of item (iii) can be adapted to prove item (i); see the original paper [75]. The proof of item (i) that we have presented is simpler, since it . is not necessary to select subsequences of time going to infinity.

Chapter 4

Escaping Probabilities and Quasiballistic Dynamics

4.1 Laplace Average Moments Let us introduce other mathematical objects that allow us to probe transport (or localization) over (orthonormal) basis vectors .{en } of .H . Recall that the time evolution is given by the initial value problem i

.

d ψ(t) = T ψ(t), dt

ψ(0) = ψ ∈ dom T ,

for a self-adjoint operator T acting in .H ; namely, one has .ψ(t) = e−itT ψ. Before we present them, we emphasize that sometimes, as it will be discussed in this chapter, it is technically convenient to take another type of time average of dynamical quantities; one of these averages is the so-called Laplace average of the projection of .ψ ∈ H over the basis element .en , defined for each .t > 0 as Len ,ψ (t) :=

.

2 t



∞

e−2u/t |en , ψ(u)|2 du ,

(4.1)

0

with the corresponding “Laplace” p-moments lpψ (t) :=

.

 n

|n|p Len ,ψ (t).

(4.2)

Remark 4.1 A motivation for the introduction of the Laplace average is its relation to the resolvent of self-adjoint operators discussed ahead. There is also an adaptation to unitary operators in Theorem 2.3 in [59]. This idea seems to have been introduced in [92], Lemma 3.2. The Laplace average is similar to the Abel average in theory of semigroups. . © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Aloisio et al., Spectral Measures and Dynamics: Typical Behaviors, Latin American Mathematics Series – UFSCar subseries, https://doi.org/10.1007/978-3-031-38289-5_4

97

98

4 Escaping Probabilities and Quasiballistic Dynamics

For each .p > 0, consider the moments without time average ρp (t) = ρpψ (t, B) =



.

|n|p |en , ψ(t)|2 ,

n

so that rpψ (t) =

.



1 t

t

ρp (u) du, 0 ψ

2 t

lpψ (t) =



∞

e−2u/t ρp (u) du.

0

ψ

It turns out that, usually, .rp (t) and .lp (t) have related power-law asymptotic behaviors. In order to state this result precisely, let β˜ψ− (p) := lim inf

.

t→∞

ψ

ln lp (t) , ln t

β˜ψ+ (p) := lim sup t→∞

ψ

ln lp (t) . ln t

(4.3)

Lemma 4.2 Let .ψ ∈ dom T , where T is a self-adjoint operator in .H , and .p > 0. Then, βψ− (p) ≤ β˜ψ− (p),

βψ+ (p) ≤ β˜ψ+ (p),

.

and if there exist parameters .C, m, t0 > 0 (which may depend on p) so that, for ψ every .t ≥ t0 , .ρp (t) ≤ C t m , then βψ− (p) = β˜ψ− (p) ≤ m,

βψ+ (p) = β˜ψ+ (p) ≤ m.

.

Proof Note that rpψ (t) ≤

.

1 t



t

e2 e−2u/t ρp (u) du ≤ e2

0

2 t



∞

e−2u/t ρp (u) du = e2 lpψ (t);

0

thus, .βψ− (p) ≤ β˜ψ− (p) and .βψ+ (p) ≤ β˜ψ+ (p). On the other hand, assuming the hypothesis of the statement of the lemma, it follows that for each .t ≥ t0 and each .ε > 0, 2 ψ .lp (t) = t 2 ≤ t

 

t 1+ε

e−2u/t ρp (u) du +



0 t 1+ε 0

 ρp (u) du + C

t 1+ε

∞

e t 1+ε

∞

 e−2u/t ρp (u) du 

−u/t −u/t m

e

u du ;

for large t, the maximum of the function .u → e−u/t um in the interval .[t 1+ε , ∞) occurs at .t 1+ε , so that for sufficiently large t,

4.1 Laplace Average Moments

lpψ (t) ≤ 2

.

=

tε t 1+ε

99



t 1+ε

ρp (u) du + Ct m(1+ε) e−t

ε

0

ε 2 t ε rpψ (t 1+ε ) + C t m(1+ε)−1 e−2t



∞

e−u/t du

t 1+ε

.

Since .rp (t 1+ε ) is power-law bounded, .C t m(1+ε)−1 e−2t ≤ 2t ε rp (t 1+ε ) holds for sufficiently large t, and so ε

ψ

ψ

ψ

.

ψ

ln rp (t 1+ε ) ln lp (t) ln 4 ≤ + ε + (1 + ε) . ln t ln t ln t 1+ε

Then, ψ

.

lim inf t→∞

ln lp (t) ≤ ε + (1 + ε)βψ− (p), ln t

ψ

lim sup t→∞

ln lp (t) ≤ ε + (1 + ε)βψ+ (p), ln t

and since it holds for all .ε > 0, one gets β˜ψ− (p) ≤ βψ− (p),

.

β˜ψ+ (p) ≤ βψ+ (p).

The condition .ρp (t) ≤ C t m clearly implies .βψ+ (p) ≤ m. ψ

 ψ

The next result presents a sufficient condition for the moments .ρp (t) to be bounded from above by a power of t, for all .p > 0. This condition is a type of Combes-Thomas estimate (given by relation (4.4)); some suitable condition on .ψ, related to the orthonormal basis B, will also be imposed. We denote the distance from .z ∈ C to the spectrum of T by .d(z, σ (T )). Definition 4.3 Let .B = {en } be an orthonormal basis of .H . A vector .ψ =  n an en is B-localized if .|an | decays faster than any power of .|n|, that is, given .r ≥ 1, there exists .D(r) > 0 such that for each .n ∈ Z \ {0}, |an | ≤

.

D(r) . |n|r

Note that one always has .|a0 | ≤ 1, and that this concept is the natural condition for ψ ρp (0, B) < ∞, for all .p > 0.

.

Example 4.4

 (i) Let .B = {en } be an orthonormal basis of .H . A vector .ψ = n an en for which only a finite number of .an ’s are nonzero is B-localized; note that the set of such vectors is dense in .H . (ii) Given a nonzero .ξ ∈ H , then it is B-localized with respect to any orthonormal basis .B = {en } for which .e0 = ξ/ξ . 

.

100

4 Escaping Probabilities and Quasiballistic Dynamics

Theorem 4.5 Let T be a bounded self-adjoint operator on .H , .B = {en } an orthonormal basis and .ψ ∈ H a B-localized vector. Suppose that there exist .C, c > 0 such that, for each .0 < δ ≤ 1 and each .z ∈ ρ(T ) with .d(z, σ (T )) ≥ δ, one has |en , Rz (T )em | ≤

.

C −c δ|n−m| e , δ

∀n, m ∈ Z.

(4.4)

Then, for each .b > 1, there exists .K(b) > 0 so that for large t, ρpψ (t, B) ≤ K(b) t pb .

.

(4.5)

Furthermore, .0 ≤ βψ± (p, B) = β˜ψ± (p, B) ≤ 1 for all .p > 0. ˜ Proof Inequality (4.5) implies, for large t, that .rp (t, B) ≤ K(b) t pb , and so one ± ± has .βψ (p, B) ≤ b; since this holds for all .b > 1, it follows that .βψ (p, B) ≤ 1. By Lemma 4.2, (4.5) implies that .βψ± (p, B) = β˜ψ± (p, B). In order to prove (4.5), one just needs to link the resolvent operator to the time evolution. Since T is bounded, let .a > 0 be such that .σ (T ) ⊂ [−a + 1, a − 1], and consider the contour .Γ given by the concatenation of four curves in the complex plane .Γ = Γ1 ∗ Γ2 ∗ Γ3 ∗ Γ4 , where for .w = x + iy, ψ

Γ1 = {w | x ∈ [−a, a], y = δ},

.

Γ2 = {w | x = −a, y ∈ [−δ, δ]}, Γ3 = {w | x ∈ [−a, a], y = −δ}, Γ4 = {w | x = a, y ∈ [−δ, δ]} , with .δ = min{1, 1/t}. By the functional calculus, if .w ∈ ρ(T ), then .

1 − 2π i



e−itw en , Rw (T )ψ dw Γ

   1 1 dμTen ,ψ (x) dw e−itw 2π i Γ σ (T ) x − w  

1 e−itw = dw dμTen ,ψ (x) σ (T ) 2π i Γ w − x  = e−itx dμTen ,ψ (x) =−

σ (T )

= en , e−itT ψ . 

Note that if .w = x + iy ∈ Γ , then .|y| ≤ δ and . e−itw ≤ etδ ≤ e; hence,

4.1 Laplace Average Moments

.

101



 e

−itT |en , Rw (T )ψ| |dw|. ≤ e , e ψ

n 2π Γ

(4.6)

Now, given that .d(Γ, σ (T )) ≥ δ and by taking into account (4.4), it follows that if ψ = m am em , then

.

.

C



|en , Rz (T )ψ| = |am | e−cδ|n−m| am en , Rz (T )em  ≤ δ m m

(the actual value of C may change throughout the estimates; e.g., in the last term, C has absorbed the constant .e/(2π )); by replacing this into (4.6), one gets .

 C 

|am | e−cδ|n−m| .

en , e−itT ψ ≤ δ m

(4.7)

For each .t ≥ 1 and each .b > 1, write ρpψ (t) =





 2 2 



|n|p en , e−itT ψ + |n|p en , e−itT ψ .

|n|≤t b

|n|>t b

.

It is straightforward to bound .E := namely, E ≤ t pb

.



|n|≤t b

  2 |n|p en , e−itT ψ from above;

2 2 

 



en , e−itT ψ ≤ t pb

en , e−itT ψ = t pb e−itT ψ2 = t pb . n

|n|≤t b

  2  It will be shown that .F := |n|>t b |n|p en , e−itT ψ is bounded for large t (one uses at this point that .b > 1 and that .ψ is B-localized), which completes the proof of the theorem. Since .ψ = 1, one has .|en , e−itT ψ| ≤ 1, and so −itT ψ|2 ≤ |e , e−itT ψ|; hence, by letting .δ = 1/t in (4.7) (.t ≥ 1 if, and .|en , e n only if, .δ ∈ (0, 1]), one gets 

F ≤Ct

.

|n|p

|n|>t b



|am | e−

c|n−m| t

.

m

Since .ψ is B-localized, given .r > 1, there exists .D(r) > 0 such that for each m ∈ Z \ {0}, one has .|am | ≤ D(r)/|m|r , so

.

F ≤Ct



.

|n|>t b ,m

(naturally, .|a0 | ≤ 1).

|n|p |m|−r e−

c|n−m| t

102

4 Escaping Probabilities and Quasiballistic Dynamics

Let .r = p + 3, and write the right hand side of the previous expression as .S = S1 + S2 , with 

S1 = C t

|n|p |m|−r e−

.

c|n−m| t



, S2 = C t

|n|p |m|−r e−

c|n−m| t

.

|n|>t b ,|m|> |n| 2

|n|>t b ,|m|≤ |n| 2

One has(just use the asymptotic behavior of the incomplete Gamma function, ∞ Γ (s, y) = y x s−1 e−x dx ∼ y s−1 e−y , for large .y > 0, .s > 1)

.



S1 ≤ C t

.

|n|p |m|−r e−

c|n| 2t

c b−1

≤ C t pb+1 e− 2 t

|n|>t b ,|m|≤ |n| 2



|m|−(p+3) ,

m

and since .b > 1, one concludes that .S1 is bounded for any .p > 0. For .S2 , first note that .|n|p = |(n − m) + m|p ≤ C(p) (|n − m|p + |m|p ); thus, 

S2 ≤ C t

.

  c|n−m| |n − m|p + |m|p |m|−r e− t ,

|n|>t b ,|m|> |n| 2

and the right-hand side may again be written as the sum of two other terms; the first one is    c|k| c|n−m| .C t |n − m|p |m|−r e− t ≤ C t |m|−r |k|p e− t |n|>t b ,|m|> |n| 2

|m|>t b /2

k

≤ C t · t b(r−1) · t p < C t −(b−1)(p+2) which is bounded, since .b > 1 (the constant C varies throughout the estimates). For the other term, one has 

Ct

.

|m|p |m|−r e−

|n|>t b |m|> |n| 2

c|n−m| t

≤ Ct

 |m|>t b /2

|m|p−r



e−

c|k| t

k

≤ C t · t −b(r−p−1) · t = C t −2(b−1) , and so, it is also bounded.



Remark 4.6 (i) Naturally, if one replaces, in Theorem 4.5, the condition .0 < δ ≤ 1 by the condition .0 < δ ≤ δ0 , with .δ0 > 0, then the conclusions of the theorem still hold true. (ii) Theorem 4.5 holds for unbounded self-adjoint operators T if one projects onto bounded subsets of energy (for instance, by considering .χΛ (T )T , with

4.1 Laplace Average Moments

103

a bounded Borel set .Λ ⊂ R), or if one considers initial conditions of the form f (T )ψ, with f a continuous function of compact support. (iii) It is interesting to remark that since .ψ is B-localized, arguments similar to those used to bound F in the proof of Theorem 4.5 can be used to bound the ψ moments .ρp (t) for small values of .t > 0. .



.

that for the discrete Schrödinger operators .hv , acting in .ψ =  Now, we show 2 (Z) (.B = {e } is the canonical basis of .l 2 (Z)), with action given ψ e ∈ l n n n n n by (2.12), the condition (4.4) holds, and so the conclusions of Theorem 4.5; here, the potential .v = (Vn ) is an arbitrary real sequence, so .dom hv = {ψ ∈ l 2 (Z) | (Vn ψn )n ∈ l 2 (Z)} (.dom hv = l 2 (Z) if v is a bounded sequence). Theorem 4.7 is a version of the so-called Combes-Thomas estimate, originally presented in [34]; see Remark 4.8. Theorem 4.7 (Combes–Thomas Estimate) Let .hv be as above and .z ∈ ρ(hv ) be such that .d(z, σ (hv )) ≥ δ > 0. Then, |en , Rz (hv )em | ≤

.

2 − δ |n−m| e 5 , δ

∀n, m ∈ Z.

Proof For .α ∈ C, let .Uα be the linear operator defined on each .en , .n ∈ Z, by the law .Uα en = enα en , and with domain given by     dom Uα = ψ = ψn en ∈ l 2 (Z) | ψn enα en ∈ l 2 (Z) .

.

n

n

Note that its adjoint is such that .Uα∗ = Uα¯ . A direct calculation gives (U−α hv Uα ψ)n = eα ψn+1 + e−α ψn−1 + Vn ψn = (hv ψ)n + (Q(α)ψ)n ,

.

where .(Q(α)ψ)n = (eα − 1)ψn+1 + (e−α − 1)ψn−1 . Hence, U−α hv Uα = hv + Q(α),

.

and since Q(α) ≤ (e

.

|α|



1

− 1)h0  = 2

|α|e|α|t dt ≤ 2|α| e|α| ,

0

Q(α) is a bounded operator; thus, for each .α ∈ C, .dom U−α hv Uα = dom hv . Note that

.

  hv + Q(α) − z1 = (hv − z1) 1 + Rz (hv )Q(α) ,

.

and if one assumes that

(4.8)

104

4 Escaping Probabilities and Quasiballistic Dynamics

2|α|e|α|
0, there exist constants C and .t0 > 0 so that, for .t > t0 , f (T )e0

lp

.

(t) ≤ C t pm ,

(4.10)

and so .0 ≤ βf±(T )e0 (p, B) = β˜f±(T )e0 (p, B) ≤ 1, for any orthonormal basis B such that .e0 ∈ B; .ψ = f (T )e0 is a kind of “localized initial condition.” See [71] for . details of the continuous case. Due to Remark 4.9, from now on in this chapter, we assume that (4.10) holds for any self-adjoint operator T and any initial condition of the form

106

4 Escaping Probabilities and Quasiballistic Dynamics

ψ = f (T )e0 ,

.

with .f ∈ C∞ 0,+ (R). Since in this situation the initial conditions are then indexed by the functions f , they will usually be indicated by f itself instead of the symbol .ψ; e.g., the average projection (1.1) and p-moments (1.3) will also be denoted by Wen ,f (t),

.

f

lp (t) and

f

rp (t),

respectively; a similar notation will be used for the exponents (4.3) ruling the algebraic growth of the Laplace moments, that is, .β˜f± (p). Note that if T is a bounded operator, then if we take .f ∈ C∞ 0,+ (R) so that .f (x) = 1 for all .x ∈ σ (T ), we get .f (T ) = 1, and so .ψ = f (T )e0 = e0 .

4.2 Escaping Probabilities Let us turn our attention to the escaping probabilities, originally discussed in [45, 70]. Consider the (Laplace) time-average projection onto the “R-tail” of the orthonormal basis, that is,  .PT ;ψ (R, t) := Len ,ψ (t) . |n|≥R

By setting, for .α ≥ 0, .PTα;ψ (t) := PT ;ψ (t α − 1, t), it is natural to introduce its decay exponents as ST−;ψ (α) := − lim inf

.

t→∞

ln PTα;ψ (t)

(4.11)

ln t

and ST+;ψ (α) := − lim sup

.

t→∞

ln PTα;ψ (t) ln t

.

(4.12)

If there exist .α > 0 and .t0 > 0 such that, for every .t ≥ t0 , .PTα;ψ (t) = 0, set ± + − .S T ;ψ (α) = ∞. Thus, for every .α, .0 ≤ ST ;ψ (α) ≤ ST ;ψ (α) ≤ ∞. ± If .ST ;ψ (α) = ∞, then .PTα;ψ (t) = O(t −m ) for every .m > 0, so that for large t, only a negligible part of the wave packet .ψ(t) “escapes” from the basis elements ± α α .en for .|n| ≤ t . On the other hand, if .S T ;ψ (α) = 0, then .PT ;ψ (t) = O(1) and the essential part of the wave packet travels faster than .t α . Thus, it is natural to introduce the dynamical exponents αl± (T , ψ) := sup{α ≥ 0 | ST±;ψ (α) = 0}, .

.

(4.13)

4.2 Escaping Probabilities

107

αu± (T , ψ) := sup{α ≥ 0 | ST±;ψ (α) < ∞}.

(4.14)

Since .ST±;ψ (α) are nondecreasing functions, .0 ≤ αl± (T , ψ) ≤ αu± (T , ψ) ≤ ∞, and .αl± (T , ψ) represent the lower (-) and upper (+) rates of propagation over the basis of the slowest part of the wave packet, whereas .αu± (T , ψ) quantify the rates of propagation of the fastest part of the wave packet. The dynamical and transport exponents are not completely independent, as we will discuss now. Theorem 4.10 Let T be a self-adjoint operator, .ψ ∈ H , and suppose that there exists .m > 0 so that for each .p > 0, there exist constants C and .t0 > 0 such that lpψ (t) ≤ C t pm ,

∀t > t0 .

.

(4.15)

Then, for every .p > 0, one has αu− (T , ψ) ≥ β˜ψ− (p, T ) ≥ αl− (T , ψ) ,

.

αu+ (T , ψ) ≥ β˜ψ+ (p, T ) ≥ αl+ (T , ψ) . Furthermore, .αu± (T , ψ) = limp→∞ β˜ψ± (p, T ) and .αl± (T , ψ) = limp→0 β˜ψ± (p, T ). Proof Let .α ≥ 0 be such that .S − (α) = ST−;ψ (α) < ∞; then, for each .ε > 0, there exists .t0 > 0 such that for .t > t0 , PTα;ψ (t) ≥ t −S

.

− (α)−ε

,

and so, a constant .C > 0 such that  − ψ .lp (t) ≥ |n|p Len ,ψ (t) ≥ (t α − 1)p PTα;ψ (t) ≥ C t αp−S (α)−ε ; |n|≥t α −1

therefore, β˜ψ− (p, T ) ≥ α −

.

S − (α) . p

Given that the relation above follows trivially for .S − (α) = ∞, it is valid for every .α ≥ 0, and so   S − (α) = sup{α ≥ 0 | S − (α) = 0} = αl− . β˜ψ− (p, T ) ≥ sup α − p α≥0

.

108

4 Escaping Probabilities and Quasiballistic Dynamics

The same argument, but now for time subsequences, gives the inequality .αl+ ≤ β˜ψ+ (p, T ). As a spin-off, note that αl± ≤ β˜ψ± (0, T ),

(4.16)

.

where .β˜ψ± (0, T ) := limp↓0 β˜ψ± (p, T ). Moreover, for every .α < αu± , one gets, by using the same reasoning as above, that β˜ψ± (p, T ) ≥ α −

.

S ± (α) , p

and so, by taking .p → ∞, one gets β˜ψ± (∞, T ) ≥ αu± ,

(4.17)

.

where .β˜ψ± (∞, T ) := limp→∞ β˜ψ± (p, T ). If .αu+ (T , ψ) = ∞, then .β˜ψ+ (p, T ) ≤ αu+ (T , ψ) holds trivially for any .p > 0. So, assume that .αu+ = αu+ (T , ψ) < ∞, let .m > 0 be as in (4.15) and let .δ, ε > 0 be so that .αu+ + ε ≤ m + δ. If .α ∈ (αl+ , αu+ ), write lpψ (t) = l1 (t) + l2 (t) + l3 (t) + l4 (t),

.

where l1 =



.

|n|p Len ,ψ (t),

t α −1≤|n|≤t αu



|n|p Len ,ψ (t),

.

+ +ε

t αu

|n|p Len ,ψ (t), + +ε

|n|t m+δ

≤|n|≤t m+δ

One has to estimate each of these terms; we will use C to denote different constants throughout the proof. Since . n |en , ψ(t)|2 = ψ(t)2 = ψ2 , one has l1 ≤ (t α − 1)p

.

 n

Len ,ψ (t) ≤ C t pα .

Now, for sufficiently large t, +

+

l2 ≤ t p(αu +ε) PTα;ψ (t) ≤ C t p(αu +ε)−(S

.

For .l3 , one has

+ (α)−ε

).

4.2 Escaping Probabilities

109 α + +ε

u l3 ≤ t p(m+δ) PT ;ψ (t),

.

and since .S + (αu+ + ε) = ∞, given .a > 0, there exists .t0 > 0 such that for each .t > α + +ε

u t0 , one has .PT ;ψ (t) ≤ t −a ; hence, .l3 ≤ 1 for large t (just pick .a > p(m + δ)). By the definition of m, for any .s > 0, there exist .C > 0 and .t0 > 0 such that, for each .t > t0 ,



l4 ≤ t −s(m+δ)

|n|p+s Len ,ψ (t) ≤ t −s(m+δ) lp+s (t) ≤ C t (p+s)m−s(m+δ) ψ

.

|n|>t m+δ

= C t pm−sδ , and so, by letting .s > pm/δ, one has .l4 ≤ 1 for large t. Thus, the above estimates give (in fact, the estimates on .l1 and .l2 ), for every .ε > 0 and every .α ∈ (αl+ , αu+ ),  ST+;ψ (α)  β˜ψ+ (p) ≤ max α, αu+ + ε − . p

.

(4.18)

In a similar way, one gets, for every .ε > 0 and every .α ∈ (αl− , αu− ),  ST−;ψ (α)  . β˜ψ− (p) ≤ max α, αu− + ε − p

.

(4.19)

Since .p → β˜ψ± (p) are increasing functions and since .ε > 0 is arbitrary, if one takes .p → ∞ on the right-hand side of (4.18) and (4.19), it follows that β˜ψ± (p) ≤

.

inf

αl± 0 if, and only if, .αl− (T , ψ) = αu− (T , ψ). Accordingly, .β˜ + (p, T ) = β + for every .p > 0 if, and only if, .α + (T , ψ) =

.

αu+ (T , ψ).

ψ

l

In other words, quantum systems with a homogeneous dynamical behavior are characterized by the fact that their wave packets travel at a unique speed (but not necessarily constant, since .αl− (T , ψ) and .αl+ (T , ψ) may differ) and wave packets do not spread out.

4.3 The SULE Condition and Quasilocalization We discuss how the presence of a complete set of specially localized eigenfunctions of a self-adjoint operator T affects its dynamical exponents. We begin with the definition of SULE (semi-uniformly localized eigenfunctions), originally proposed in [63]. We will see that this concept is strong enough to result in dynamical localization (see Definition 1.2).

4.3 The SULE Condition and Quasilocalization

111

Definition 4.15 Let T be a self-adjoint operator in .H = l 2 (Zd ) and J some subset of .R. One says that T has SULE on J (or just has SULE, in case .J ⊃ σ (T )) if: (i) T has a complete set .{ϕn }n≥1 , in .χJ (T )H , of orthonormal eigenfunctions (say, .T ϕn = λn ϕn , with .λn ∈ J ); (ii) there exist .α > 0 and .mn ∈ Zd (the “localization centers”) such that for every d .δ > 0, there exists a constant .Cδ so that, for every .m ∈ Z , .n ∈ N, |ϕn (m)| ≤ Cδ eδ|mn |−α|m−mn | .

.

(4.20)

Proposition 4.16 Let .B = {em }m∈Zd be any orthonormal basis of .l 2 (Zd ), J be a subset of .R, and suppose that the self-adjoint operator T has SULE on J . Then, for each .f ∈ C∞ 0,+ (J ), .(T , f (T )e0 ) has dynamical localization. Proof First, we will show that there exists .τ > 0 so that for each .δ > 0, there exists Dδ such that for .q, m ∈ Zd ,

.

.

sup |eq , e−itT em | ≤ Dδ eδ|m|−τ |q−m| .

(4.21)

t

Namely, let .{ϕk }k≥1 be the eigenfunctions in .χJ (T )H and .{λk }k≥1 the corresponding eigenvalues, with .λk ∈ J ; then, eq , e−itT em  =



.

eq , ϕn  ϕn , e−itT em  =



n

ϕn (q)eitλn ϕn (m) ,

n

and since T has SULE on J , one gets .

sup |eq , e−itT em | ≤ t



|ϕn (q)ϕn (m)| ≤ Cδ2



n

e2δ|mn | e−α(|q−mn |+|m−mn |) .

n

Since |q − mn | + |m − mn | ≥ max{|q − m|, |mn | − |m|} ,

.

it follows that for each .ε ∈ (0, min{α, δ}), e−α(|q−mn |+|m−mn |) ≤ e−(α−ε)(|q−mn |+|m−mn |) e−ε(|q−mn |+|m−mn |)

.

≤ e−ε|mn | eδ|m| e−(α−ε)|m−q| . So, .

sup |eq , e−itT em | ≤ Cδ2 eδ|m| e−(α−ε)|m−q| t

 n

e−(2δ+ε)|mn | .

(4.22)

112

4 Escaping Probabilities and Quasiballistic Dynamics

It remains to show that .A0 :=



ne

−(2δ+ε)|mn |

is finite.

Claim For each .η > 0, there exists .Fη so that for each .n ∈ Zd and each .L > 0, 

|ϕn (m)|2 ≤ Fη e−αηL e−αη|mn |/2 .

.

|m−mn |≥η(|mn |+L)

Namely, by (4.20) (taking .δ = αη/2), there exists .Cη1 so that |ϕn (m)| ≤ Cη1 eα(η|mn |/2−|m−mn |) .

.

If .|m − mn | ≥ η(|mn | + L), then .|m − mn | ≥ |m − mn |/2 + η(|mn | + L)/2, and so |ϕn (m)| ≤ Cη1 e−ηα L/2 e−α|m−mn |/2 ,

.

from which it follows that  . |ϕn (m)|2 ≤ (Cη1 )2 e−ηα L



e−α|k| ≤ Fη e−ηα L/2 e−αη|mn |/2 ,

|k|≥η|mn |

|m−mn |≥η(|mn |+L)

proving the claim. By the Claim, if .|mn | ≤ L, then 



|ϕn (m)|2 ≤

.

|m|≥(1+2η)L

|ϕn (m)|2 ≤ Fη e−ηα L/2

|m−mn |≥η(|mn |+L)

(namely, for each .η > 0, .(1 + 2η)L ≥ η(|mn | + L) + L, so .{m ∈ Z | |m| ≥ (1 + 2η)L} ⊂ {m ∈ Z | |m| − L ≥η(|mn | + L)} ⊂ {m ∈ Z | |m − mn | ≥ η(|mn | + L)}) and so, since for each .n ∈ N . m |ϕn (m)|2 = 1, one has 

|ϕn (m)|2 ≥ 1 − Fη e−ηα L/2 .

.

|m|≤(1+2η)L

 Now, since for each .m ∈ Zd , . n |ϕn (m)|2 = 1, one has (2(1 + 2η)L + 1)d ≥



.

|ϕn (m)|2 ≥

n,|m|≤(1+2η)L



|ϕn (m)|2

n∈CL ,|m|≤(1+2η)L

≥ #CL (1 − Fη e−ηα L/2 ) , where .CL := {n | |mn | ≤ L}. Thus, the cardinality .#CL is finite and .lim sup(2L + 1)−d #CL ≤ 1; in particular, there exists .c0 such that, for each .L ≥ 1, #CL ≤ c0 Ld .

.

4.3 The SULE Condition and Quasilocalization

113

 This shows, in particular, that for .ε, δ > 0, .A0 = n e−(2δ+ε)|mn | is finite. Now, let 2 .ε = α/2 in (4.22) and set .τ = α/2, .Dδ = (Cδ ) A0 in order to get (4.21). ∞ Finally, let .f ∈ C0,+ (I ), where .I ⊃ J is open, and set .ψ = f (T )e0 . Then, it follows from (4.21) that there exists .C = C(δ, f ) such that, for each .q ∈ Zd , .

sup |eq , ψ(t)| ≤ C e−τ |q| ; t

namely, given that .f ∈ C∞ 0,+ (J ), there exists .m0 with .

sup |eq , ψ(t)| ≤ sup t

t



|em , ψ||eq , e−itT em |

|m|≤m0

⎛ ≤ Dδ ⎝



⎞ |em , ψ|e(τ +δ)|m| ⎠ e−τ |q| = C e−τ |q| .

|m|≤m0 f

Now, it is straightforward to show that for each .p > 0, .supt rp (t) < ∞ and f .supt lp (t) < ∞. This concludes the proof of the proposition.  Corollary 4.17 Let J be an open subset of .R and suppose that the self-adjoint + operator T has SULE on J . Then, for each .f ∈ C∞ 0,+ (J ), .αu (T , f ) = 0, and so, + for every .p > 0, .β˜f (p, T ) = 0. An important example of operator that has SULE is the so-called onedimensional Anderson operator .Tω : l 2 (Z) → l 2 (Z), given by the action (Tω ψ)n = (h0 ψ)n + Vn (ω)ψn

.

whose potential .v(ω) = (Vn (ω))n∈Z is a compactly supported sequence of independent and identically distributed (real) random variables with probability distribution .g(x)d(x), .0 ≤ g ∈ L∞ (R); think of .ω as an element of .(Ω, τ ), where .Ω = RZ and .τ is the probability measure obtained as the product of such distributions. Indeed, the next result (which combines Theorems 7.6 and 7.7 in [63] for the case .d = 1) shows that .Tω has SULE for .τ -a.e. .ω. Proposition 4.18 Let .Tω be the one-dimensional Anderson operator defined above. Then, for .τ -a.e. .ω, .Tω has SULE. The next result is an immediate consequence of Proposition 4.18 and Corollary 4.17. Corollary 4.19 Let .Tω be the one-dimensional Anderson operator defined above. + Then, for .τ -a.e. .ω and each .f ∈ C∞ 0,+ (R), .αu (Tω , f ) = 0. Depending on the technique, the proof of pure point spectrum may also imply the SULE conditions for selected initial conditions. This is the case, for instance,

114

4 Escaping Probabilities and Quasiballistic Dynamics

with certain bounded perturbations .v : Z → R of the one-dimensional discrete Schrödinger operator .hv,E with uniform electric field of intensity .E = 0, given by the law .

  hv,E ψ n = (hv ψ)n + En ψn

with domain  2  

n |ψn |2 < ∞ . dom hv,E = ψ ∈ 2 (Z) |

.

n

By applying the so-called (quantum) KAM technique [55], pure point spectrum and SULE (here, the initial conditions are the elements of the canonical basis of .l 2 (Z)) were obtained if .V ∞ ≤ 1/40, and so dynamical localization. There is a version of such results for time-periodic perturbations [54]. It is worth mentioning that dynamical localization follows from a variable energy multi-scale analysis [44], a technique used to prove almost everywhere pure point spectrum of random Schrödinger operators (Anderson models) and also applied to wave propagation in random media. The next result will be used later (more specifically, in the proof of Theorem 4.25). Lemma 4.20 Let J be a bounded self-adjoint operator acting on .l 2 (Z) such that 2 d .σ (J ) = [a, b] and J has SULE. Consider, in .l (Z ), the basis .{en }n∈Zd , the Kronecker sum .T = J ⊗ 1 ⊗ · · · ⊗ 1 + · · · + 1 ⊗ 1 ⊗ · · · ⊗ J , and write   .T = T + vj ej , ·ej , |j |≤l

with .l ∈ N (that is, a finite-rank perturbation of T ). Then, for every .f ∈ C∞ 0,+ ((a · d, b · d)), one has .αu+ (T  , f ) = 0. Proof We initially show that T has SULE (on .[a · d, b · d]). Denote by .{ϕj } the complete set, in .l 2 (Z), of orthonormal eigenfunctions of J (such that .J ϕj = λj ϕj ). Since .{Ψk }k , with .Ψk (m) := ϕk1 (m1 ) ⊗ · · · ⊗ ϕkd (md ), is a complete set, in 2 d .χ[a·d,b·d] (T )(l (Z )), of orthonormal eigenfunctions of T (such that .T Ψk = Λk Ψk , with .Λk = λk1 + · · · + λkd ∈ [a · d, b · d]), one needs to prove that there exist .α > 0 and .mn ∈ Zd so that, for each .δ > 0, there exists .Cδ such that |Ψn (m)| ≤ Cδ eδ|mn |−α|m−mn | ,

.

for each .m ∈ Zd and each .n ∈ N. But then, since there exist, for every .p = p 1, . . . , d, .α p > 0 and .mnp ∈ Z so that, for each .δ > 0, there exists .Cδ such that |ϕnp (mp )| ≤ Cδ eδ|mnp |−α

.

p

p |m −m | p np

,

4.3 The SULE Condition and Quasilocalization

115

d p p the result follows by setting .Cδ := p=1 Cδ , .α := min α and .mn := (mn1 , . . . mnp ). Now, fix .q ∈ Zd , .|q| > l and .u > 0. By Duhamel’s formula, one obtains 

eq , e−iuT e0  = eq , e−iuT e0  − i



u



.



vj eq , e−isT ej ej , e−i(u−s)T e0  ds.

0 |j |≤l

Since T has SULE, it satisfies condition (4.21) (see the proof of Proposition 4.16), that is, that there exists .β > 0 such that for every .δ > 0, there exists .Dδ so that, for every .q, r ∈ Zd , .

sup |eq , e−iuT er | ≤ Dδ eδ|r|−β|r−q| . u

Combining the previous relations with the fact that, for every .y ∈ R,  |ej , e−iyT e0 | ≤ 1, one has

.

|eq , e

.

−iuT 

e0 | ≤ |eq , e

−iuT

e0 | +



 |vj |

|j |≤l

≤ Dδ e−β|q| + Dδ u



u

|eq , e−isT ej | ds

0

|vj | eδ|j |−β|j −q|

|j |≤l

≤ Dδ e where .A := PTr  ;e0 (t) ≤



|j |≤l

(1 + u A e(δ+β)l ),

|vj |. Now, for every .t > 0 and every .r > 0, it follows that



.

=

−β|q|

|q|≥t r −1



2 t



∞

0



2 e−2u/t Dδ2 e−2β|q| 1 + uAe(δ+β)l du



2  Dδ2 e−2β|q| 1 + Ae(δ+β)l t + A e(δ+β)l t / 2

|q|≥t r −1

≤ Eδ e

−2βt r

 d

2   (d − 1)! t r(d−p) (δ+β)l (δ+β)l 1 + Ae , t + Ae t /2 (d − p)! (2β)p p=1

with .Eδ some finite constant, and consequently, + .S  T ;e0 (r)

  tr − 2 − r(d − 1) = ∞. ≥ lim sup 2β ln t t→∞

Since .r > 0 is arbitrary, it follows from (4.14) that .αu+ (T  , e0 ) = 0.

116

4 Escaping Probabilities and Quasiballistic Dynamics

It remains to prove that .αu+ (T  , f ) = 0 for every .f ∈ C∞ 0,+ (I ). So, let .f ∈ ∞ f,I 2 := sup{(f (x)) | x ∈ I }. Since .0 < C f,I < ∞, one has, C0,+ (I ) and define .C for every .R, t > 0, PT ;f (R, t) =



.

Len ,f ≤ C f,I PT ;e0 (R, t).

|n|≥R

Now, fix .β > 0; it follows from definition (4.12) that 

β

+ .S T ;f (β)

= − lim sup

ln PT ;f (t) ln t

t→∞

≥ − lim sup

β

ln PT ;e0 (t) ln t

t→∞

ln C f,I + ln t

 = ST+;e0 (β). (4.23)

that .ST+;e0 (β)

Since .αu+ (T , e0 )

= 0, it follows ∞ for every .β > 0, and so .αu+ (T , f ) = 0.

= ∞. Hence, by

(4.23), .ST+;f (β)

= 

4.4 Gδ Sets for Dynamical Exponents Let X be a regular metric space of self-adjoint operators (see Definition 1.29), so convergence in its metric implies convergence in the strong resolvent sense. Fix an interval .I = (a, b) ⊂ R, .f ∈ C∞ 0,+ (I ) and, as before, given .T ∈ X, let .ψ = f (T )e0 , where .e0 is an element of the orthonormal basis .{en } of .H . Introduce the sets   U f = T ∈ X | supp (μTf ) ⊃ I ,   f C0− = T ∈ X | αu− (T , f ) = 0 ,   f C1+ = T ∈ X | αl+ (T , f ) = 1 . .

f

f

Proposition 4.21 Each of the sets .U f , .C0− and .C1+ is a .Gδ set in X. Hence, in each application, in order to get generic sets of operators in X, one f f has to argue that the corresponding set (i.e., .U f , .C0− and .C1+ ) is dense; in general, each application has its own strategy. We will present some applications to explicitly metric spaces of Schrödinger operators in Sect. 4.5. The following results are required in the proof of Proposition 4.21. Lemma 4.22 Let T , f be as above, and .α, γ > 0. Then, for every .0 < ε ≤ γ , one has {f | lim inf t γ PTα;f (t) < ∞} ⊂ {f | ST−;f (α) ≥ γ }

.

t→∞

4.4 Gδ Sets for Dynamical Exponents

117

⊂ {f | lim inf t γ −ε PTα;f (t) < ∞} , t→∞

{f | lim sup t

γ

t→∞

PTα;f (t)

< ∞} ⊂ {f | ST+;f (α) ≥ γ } ⊂ {f | lim sup t γ −ε PTα;f (t) < ∞} . t→∞

Proof If, for a fixed .α > 0, .PTα;f (t) = 0 for every .t > t0 > 0, then .STL ;f (α) = ∞, and the above inclusions are trivially satisfied. Otherwise, the result follows directly from definitions (4.11) and (4.12).  Lemma 4.23 Let f be as above. Then, ∞

f

∞

∞ ∞

1/r+1/p

C0− =

.

(4.24)

Qk−1/ l r=1 p=1 k=1 l=2

and ∞

f

∞

∞ ∞ ∞

C1+ =

.

1−1/r−1/p

Y1/k+1/ l

(n) ,

(4.25)

r=2 p=2 k=1 l=1 n=1

where, for .α > 0, .β > 0,   Qαγ := T ∈ X| for each m > 0, ∃t > m with t γ PTα;f (t) < 1   Yγα (n) := T ∈ X| for each m > 0, ∃t > m with t γ PTα;f (t) > n . .

Proof Only the proof of relation (4.24) will be presented, since the proof of (4.25) is similar. Let .f ∈ C∞ 0,+ ((a, b)) be such that, for every .T ∈ X and every .t > t0 > α 0, .PT ;f (t) = 0 (which is the case when, for every .T ∈ X, .supp (μTψ ) = ∅); f

then, .Qαk−1/ l = X = C0l . Otherwise, note that for every .α, γ > 0 and .0 < ε ≤ γ , the first set of inclusions in Lemma 4.22 is equivalent to  .

  f | lim inf t γ PTα;f (t) = 0 ⊂ f | ST−;f (α) ≥ γ } t→∞   ⊂ f | lim inf t γ −ε PTα;f (t) = 0 ; t→∞

thus, it follows that for each .f ∈ C∞ 0,+ ((a, b)), ∞

Qαγ =

!

.

m=0 t>m

  T ∈ X | t γ PTα;f (t) < 1 ⊂ T ∈ X | ST−;f (α) ≥ γ }

118

4 Escaping Probabilities and Quasiballistic Dynamics ∞

⊂

!

 T ∈ X | t γ −ε PTα;f (t) < 1 = Qαγ −ε

m=0 t>m

(actually, one may replace 1 with any .c > 0 on the definition of .Qαγ ). Now, if one replaces .γ by .γ − ε and sets .γ = k, .k ≥ 1, .ε = 1/ l, .l ≥ 2, in the previous relation, one gets ∞ ∞

∞ ∞

{T ∈ X | ST−;f (α) ≥ k − 1/ l} ⊂

Qαk−1/ l ⊂

.

k=1 l=2

k=1 l=2

∞ ∞

Qαk−2/ l , k=1 l=2

and consequently, {T ∈ X | ST−;f (α) = ∞} =

∞ ∞

Qαk−1/ l .

.

(4.26)

k=1 l=2

Finally, for each .δ > 0, the set inclusions ∞ ∞

Qαk−1/ l

.



⊂ T ∈X|

αu− (T , f )



∞ ∞

Qα+δ k−1/ l ,