Sturm-Liouville Operators and Applications : Revised Edition [Revised] 0821853163, 978-0-8218-5316-0, 9781470415808, 1470415801

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STURM-LIOUVILLE OPERATORS AND APPLICATIONS REVISED EDITION

VLADIMIR A. MARCHENKO

AMS CHELSEA PUBLISHING American Mathematical Society • Providence, Rhode Island

Sturm-LiouviLLe operatorS and appLicationS reviSed edition

Sturm-LiouviLLe operatorS and appLicationS reviSed edition

vLadimir a. marchenko

AMS CHELSEA PUBLISHING

American Mathematical Society • Providence, Rhode Island

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UN

8 DED 1

SOCIETY

Α Γ ΕΩ ΜΕ

ΤΡΗΤΟΣ ΜΗ

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HEMATIC AT A M

88

2000 Mathematics Subject Classification. Primary 34A55, 34B24, 35Q51, 47E05, 47J35.

For additional information and updates on this book, visit www.ams.org/bookpages/chel-373

Library of Congress Cataloging-in-Publication Data Marchenko, V. A. (Vladimir Aleksandrovich), 1922– Sturm-Liouville operators and applications / Vladimir A. Marchenko. — Rev. ed. p. cm. Rev. ed. of: Sturm-Liouville operators and applications. 1986. Includes bibliographical references. ISBN 978-0-8218-5316-0 (alk. paper) 1. Spectral theory (Mathematics) 2. Transformations (Mathematics) 3. Operator theory. I. Title. QA320.M286 2011 515.7222—dc22 2010051019

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 1986 held by the American Mathematical Society. All rights reserved.  c 2011 by the American Mathematical Society. Revised Edition  Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines 

established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

16 15 14 13 12 11

CONTENTS

PREFACE TO THE REVISED EDITION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Chapter 1 THE STURM-LIOUVILLE EQUATION AND TRANSFORMATION OPERATORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Riemann’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Transformation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3. The Sturm-Liouville Boundary Value Problem on a Bounded Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4. Asymptotic Formulas for Solutions of the Sturm-Liouville Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5. Asymptotic Formulas for Eigenvalues and Trace Formulas . . . . . . . . . . . . . . 67 Chapter 2 THE STURM-LIOUVILLE BOUNDARY VALUE PROBLEM ON THE HALF LINE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 1. Some Information on Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2. Distribution-Valued Spectral Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3. The Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4. The Asymptotic Formula for the Spectral Functions of Symmetric Boundary Value Problems and the Equiconvergence Theorem . . . . . . . . . 153 Chapter 3 1. 2. 3. 4. 5.

THE BOUNDARY VALUE PROBLEM OF SCATTERING THEORY . . 173 Auxiliary Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 The Parseval Equality and the Fundamental Equation . . . . . . . . . . . . . . . . 200 The Inverse Problem of Quantum Scattering Theory . . . . . . . . . . . . . . . . . 216 Inverse Sturm-Liouville Problems on a Bounded Interval . . . . . . . . . . . . . . 240 The Inverse Problem of Scattering Theory on the Full Line . . . . . . . . . . . 284

Chapter 4 1. 2. 3. 4.

NONLINEAR EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Transformation Operators of a Special Form . . . . . . . . . . . . . . . . . . . . . . . . . 307 Rapidly Decreasing Solutions of the Korteweg-de Vries Equation . . . . . . 322 Periodic Solutions of the Korteweg-de Vries Equation . . . . . . . . . . . . . . . . 332 Explicit Formulas for Periodic Solutions of the Korteweg-de Vries Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

Chapter 5 STABILITY OF INVERSE PROBLEMS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .363 1. Problem Formulation and Derivation of Main Formulas . . . . . . . . . . . . . . . 363 2. Stability of the Inverse Scattering Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 v

vi

CONTENTS

3. Error Estimate for the Reconstruction of a Boundary Value Problem from its Spectral Function Given on the Set (−∞, N 2 ) Only. . . . . . . . . . 380 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

Preface to the revised edition In the first edition of this book the main attention was focused on the methods of solving the inverse problem of spectral analysis and on the conditions (necessary and sufficient) which the spectral data must satisfy in order to make it possible to reconstruct the potential of the corresponding Sturm-Liouville operator. These conditions imply that the spectral data (e.g. spectral function or scattering data) must be known for all values of spectral parameter which belong to the spectrum of the operator. But from the physical meaning of the inverse problem it is obvious that the values of spectral data on the whole spectrum are impossible to obtain from any observations. For example, in the inverse problem of quantum scattering theory the energy of the particles acts as the spectral parameter, and in order to find the values of scattering data on the whole spectrum one has to conduct an experiment with the particles of infinitely large energy. But for big enough values of energy the scattering process is not any more described by Schr¨ odinger equation with potential q(x). Therefore, even allowing, ideally, the possibility to experiment with particles of arbitrarily large energies, we would obtain, starting from a certain energy, data relevant to process, which has certainly nothing to do with the equation that we want to reconstruct. Hence, a principal question is as follows: what information about the potential q(x) can be obtained, if the spectral function or scattering data are known (generally speaking, approximately) only on a finite interval of values of the spectral parameter? The new Chapter 5, devoted to solving this problem, was added to this edition. The convenient formulae are obtained, which allow to estimate the precision with which the eigenfunctions and potentials of Schr¨odinger operator can be restored when the scattering data or spectral function are known only on a finite interval of values of spectral parameter. V. Marchenko

vii

x

xi

xii

xiii

CHAPTER 5 STABILITY OF INVERSE PROBLEMS

1. PROBLEM FORMULATION AND DERIVATION OF MAIN FORMULAS It was shown in the previous chapters that the Sturm-Liouville boundary-value problem can be completely reconstructed either from its spectral function or from the scattering data and that the reconstruction procedures are quite efficient. In particular, they allowed us to find necessary and sufficient conditions for spectral functions and scattering data of the boundary-value problems under consideration. These conditions show that the symmetric boundary-value problem is uniquely reconstructed from its spectral function ρ(µ) given for all µ. The same is valid for the reconstruction of the boundary-value problem from its scattering data. At the same time, the physical sense of the inverse problems indicates that neither the spectral function nor the spectral data can be completely known. This is clearly seen in the case of the inverse quantum scattering problem. Indeed, in this problem the parameter λ2 is proportional to the system energy (see (3.3.2)), so in order to know the scattering data for all values of λ one has to conduct experiments with particles of arbitrarily large energy. But for sufficiently large, though finite, values of energy, the scattering process is not described anymore by the Schr¨odinger equation (3.3.1) with the ~2 q(x). Therefore, even allowing, ideally, the possibility to experiment potential V (x) = 2M with particles of arbitrarily large energies, we would obtain, starting from a certain energy, data relevant to a process, which has certainly nothing to do with the equation that we want to reconstruct. Hence, a principal question is as follows: what information about the function q(x) or the boundary-value problem in general can be obtained, if the spectral function or the scattering data are known (generally speaking, approximately) only on a finite interval of values of the spectral parameter? To answer this question, one has to know to what extend can two boundary-value problems differ from each other, if it is known that their spectral functions or scattering data differ slightly for λ2 varying on a finite interval. It is evident that if nothing is known a priori about these problems, then

364

STABILITY OF INVERSE PROBLEMS

they can differ as much as you want. For example, for any boundary-value problems with q(x) ≥ N and h > 0, the corresponding spectral functions vanish for all µ < N and, therefore, they coincide on the interval (−∞, N ). Therefore, a meaningful (and natural, from the physical point of view) question about the stability of inverse spectral problems is the following: how much can two boundary-value problems differ from each other, if their spectral functions differ a little on a given interval of values of the spectral parameter Rx λ2 , under the condition that certain estimates for |h| + 0 |q(t)|dt are known a priori? In a similar way one can formulate the stability problem for the inverse problem of the quantum scattering theory. Notice the similarity of these questions with the question that is typical for the approximation theory: what can one say about a function, for which a finite part of its Fourier series is known? Or (which is actually the same): how can one estimate a function whose few first Fourier coefficients are equal to 0? It is known that one can answer this question (with the help of the G. Bohr inequality) only assuming that the function under consideration belongs to a certain functional class (for example, its derivative is bounded by a prescribed number, etc.) Now let us introduce definitions and notations, which will be used in what follows. Denote the symmetric boundary-value problems (2.2.1), (2.2.2) by {h, q(x)} and the boundary-value problem (3.1.1), (3.1.2), that satisfies condition (3.1.3), by {q(x)}. The function q(x) is called the potentials. Let a(x) be an arbitrary nondecreasing continuous function with a(0) = 0 and let A be an arbitrary nonnegative number. Denote by V {A, a(x)} the set of all boundary-value problems {h, q(x)} such that Z x |q(t)|dt ≤ a(x) (0 ≤ x < ∞), (5.1.1) |h| ≤ A, 0

and denote by V {α(x)} the set of boundary-value problems {q(x)} such that Z ∞ |q(t)|dt ≤ α(x) (0 < x < ∞),

(5.1.2)

x

where α(x) is a continuous nonincreasing function integrable on (0, ∞). We will study the accuracy of the reconstruction of the boundary-value problem {h, q(x)} from a part of its spectral function in the set V {A, a(x)} while the accuracy of the reconstruction of the boundary-value problem {q(x)} from a part of its scattering data will be studied in the set V {α(x)}. Our primary interest is the accuracy of reconstruction of the solutions ω(λ, x; h) or e(λ, x) of the corresponding equations; this is because their reconstruction is more stable.

365

PROBLEM FORMULATION

In this section we derive convenient representations for the differences of such solutions in terms of the differences of the corresponding spectral functions (or the scattering data). Consider two boundary-value problems {q1 (x)} and {q2 (x)} from the set V {α(x)}. Subtracting the main integral equations for the corresponding inverse problems gives the equation Z

∞

F1 (x + y) − F2 (x + y) + K1 (x, y) − K2 (x, y) +

F1 (t + y)× x

Z

∞

×{K1 (x, t) − K2 (x, t)}dt +

{F1 (t + y) − F2 (t + y)}K2 (x, t)dt = 0, x

where K1 (x, y), and K2 (x, y) are the kernels of the corresponding transformation operators and the functions F1 (x) and F2 (x) are constructed from the scattering data by (3.2.7). To shorten notations, define K1,2 (x, y) = K1 (x, y) − K2 (x, y),

F1,2 (x) = F1 (x) − F2 (x).

Then the equation above can be written as Z ∞ Z K1,2 (x, y) + F1 (t + y)K1,2 (x, t)dt = −{F1,2 (x + y) + x

∞

F1,2 (t + y)K2 (x, t)dt}.

x

For each fixed x ≥ 0, this equality is an equation with respect to the function K1,2 (x, y). Solving it we find K1,2 (x, y) = (I + F1x )

−1

Z

∞

{F1,2 (x, y) +

F1,2 (t + y)K(x, t)dt},

(5.1.3)

x

where, in view of (3.2.16), (I + F1x )−1 = (I + K∗1x )(I + K1x )

(5.1.4)

and the operators K∗1x , K1x are defined by (3.2.16’). Let {Sj (λ), λk , mk (j)} (j = 1, 2) be the scattering data and let ej (λ, x) be the solutions of the considering problems, defined in Lemma 3.1.1. To shorten notations we omit the index j at λk and assume that mk (j) = 0 if iλk is not a zero of ej (λ, 0). Then Z ∞ X 1 −λk x 2 2 e {mk (1) − mk (2)} + {S2 (λ) − S1 (λ)}eiλx dλ. F1,2 (x) = 2π −∞ k

Thus Z

∞

F1,2 (x + y) +

K2 (x, t)F1,2 (t + y)dt = x

X k

e−λk y e2 (iλk , x)×

366

STABILITY OF INVERSE PROBLEMS

×{m2k (1)

−

m2k (2)}

Z∞

1 + 2π

{S2 (λ) − S1 (λ)}eiλy e2 (λ, x)dλ,

−∞

Z

∞

φ(x, y) = (I + K1x ){F1,2 (x, y) +

K2 (x, t)F1,2 (t + y)dt} =

X

x

×e2 (iλk , x){m2k (1)

−

m2k (2)}

1 + 2π

e1 (iλk , y)×

k

Z∞ {S2 (λ) − S1 (λ)}e1 (λy)e2 (λ, x)dλ. (5.1.5) −∞

Equations (5.1.3) and (5.1.4) imply that K1,2 (x, y) = −(I + K∗1x )φ(x, y)

(5.1.6)

and, therefore, for Im µ > 0 we have Z ∞ K1,2 (x, y)eiµy dy = − ({I + K∗1x }φ(x, y), exp(−iµy))x = e1 (µ, x) − e2 (µ, x) = x

  = − (φ(x, y), {I + K1x } exp(−iµy))x = − φ(x, y), e1 (µ, y) , x

2

where (f, g)x denotes the scalar product in the space L (x, ∞): Z ∞ (f, g)x = f (t)g(t)dt. x

Thus we obtain the equation Z

∞

e1 (µ, x) − e2 (µ, x) = −

φ(x, y)e1 (µ, y)dy,

(5.1.7)

x

where Im µ > 0 and the function φ(x, y) is defined by the r.h.s. of (5.1.5). From the equations for e1 (λ, y) it follows that Z ∞ e0 (λ, x)e1 (µ, x) − e1 (λ, x)e01 (µ, x) e1 (λ, y)e1 (µ, y)dy = 1 . λ2 − µ2 x Using this equality and (5.1.5), we have e1 (µ, x) − e2 (µ, x) =

X

E1,2 (iλk , µ, x){m2k (1) − m2k (2)}+

k

+

1 2π

Z

∞

{S2 (λ) − S1 (λ)}E1,2 (λ, µ, x)dλ, −∞

(5.1.8)

PROBLEM FORMULATION

367

where Ej,i (λ, µ, x) =

ei (λ, x){ej (λ, x)e0j (µ, x) − e0j (λ, x)ej (µ, x)} . λ2 − µ2

Now we notice that one can interchange the indices 1 and 2 in (5.1.8), which gives e2 (µ, x) − e1 (µ, x) =

X

E2,1 (iλk , µ, x){m2k (2) − m2k (1)}+

k

+

1 2π

Z

∞

{S1 (λ) − S2 (λ)}E2,1 (λ, µ, x)dλ.

(5.1.9)

−∞

Multiply the both sides of (5.1.8) by e2 (µ, x) and the both sides of (5.1.9) by e1 (µ, x) and sum them up. As a result, in the l.h.s. we obtain −{e1 (µ, x) − e2 (µ, x)}2 . In order to represent the r.h.s. in a form convenient for further considerations, we perform the following transformations: (λ2 − µ2 ){E1,2 (λ, µ, x)e2 (µ, x) − E2,1 (λ, µ, x)e1 (µ, x)} = = e2 (µ, x)e2 (λ, x)e1 (λ, x)e01 (µ, x) − e2 (µ, x)e2 (λ, x)e01 (λ, x)e1 (µ, x)− −e1 (µ, x)e1 (λ, x)e2 (λ, x)e02 (µ, x) + e1 (µ, x)e1 (λ, x)e02 (λ, x)e2 (µ, x) = = e1 (λ, x)e2 (λ, x){e01 (µ, x)e2 (µ, x) − e1 (µ, x)e02 (µ, x)}− −e1 (µ, x)e2 (µ, x){e01 (λ, x)e2 (λ, x) − e1 (λ, x)e02 (λ, x)} = Z∞ = {q1 (t) − q2 (t)}{e1 (µ, x)e2 (µ, x)e1 (λ, t)e2 (λ, t)− x

−e1 (λ, x)e2 (λ, x)e1 (µ, t)e2 (µ, t)}dt. Here we have used the identities Z ∞ {q1 (t) − q2 (t)}e1 (ν, t)e2 (ν, t)dt = −{e01 (ν, x)e2 (ν, x) − e1 (ν, x)e02 (ν, x)}, x

that follow from the Sturm-Liouville equations for e1 (ν, x) and e2 (ν, x). Hence, the r.h.s. of the resulting equation can be written as Z ∞ {q1 (t) − q2 (t)}{A1,2 (µ, x, t) − A1,2 (µ, t, x)}dt, x

368

STABILITY OF INVERSE PROBLEMS

where X m2 (2) − m2 (1) k

A1,2 (µ, x, t) = e1 (µ, x)e2 (µ, x)

k

e1 (µ, x)e2 (µ, x) + 2π

Z∞

k

λ2k − µ2

e1 (iλk , t)e2 (iλk , t)+

S2 (λ) − S1 (λ) e1 (λ, t)e2 (λ, t)dt. λ2 − µ2

(5.1.10)

−∞

Thus we have proved the following LEMMA 5.1.1. For all µ from the open upper half plane, for which Imµ2 = 6 0, the following identity holds: {e1 (µ, x) − e2 (µ, x)}2 =

Z∞ {q1 (t) − q2 (t)}{A1,2 (µ, t, x) − A1,2 (µ, x, t)}dt, (5.1.11) x

where the function A1,2 (µ, x, t) is defined by the r.h.s. of (5.1.10). Now consider two boundary-value problems, {hj , qj (x)} (j = 1, 2) , from the set √ V {A, a(x)} and denote by ωj ( λ, x) the solutions of the following equations: −y 00 + qi (x)y = λy,

y 0 (0) = hi

y(0) = 1,

(i = 1, 2).

(5.1.12)

LEMMA 5.1.2. For all µ with Im µ 6= 0, the following identity holds: √ √ {ω1 ( µ, x) − ω2 ( µ, x)}2 = (h1 − h2 ){B1,2 (µ, 0, x) − B1,2 (µ, x, 0)}+ Z ∞ + {q1 (t) − q2 (t)}{B1,2 (µ, t, x) − B1,2 (µ, x, t)}dt, (5.1.13) x

where √ √ B1,2 (µ, x, t) = ω1 ( µ, x)ω2 ( µ, x)

Z

∞

−∞

√ √ ω1 ( λ, t)ω2 ( λ, t) d{ρ1 (λ)−ρ2 (λ)}, (5.1.14) λ−µ

and ρj (λ) (j = 1, 2) are the spectral functions of the boundary-value problems {hj , qj (x)}. √ PROOF. Let I+Klj be the transformation operators which transform the solution ωj ( λ, x) √ to the solution ωl ( λ, x). Consider the integrals Z ∞ Z x √ √ √ ω2 ( λ, x)dρj (λ) ω1 ( λ, t)ω1 ( µ, t)dt, Ij (µ, x) = −∞

0

PROBLEM FORMULATION

369

where ρj (λ) (j = 1, 2) are the spectral functions of the associated boundary-value problems. Then we have Z ∞ Z x √ √ √ I1 (µ, x) = ω1 ( λ, x)dρ1 (λ) ω1 ( λ, t)ω1 ( µ, t)dt+ −∞

Z

∞

+ −∞

0

 Z x √ 2 K1 (x, t)ω1 ( λ, t)dt

Z 0

x

 √ √ ω1 ( λ, t)ω1 ( µ, t)dt dρ1 (λ),

0

K12 (x, t)

is the kernel of the operator K21 . By the equiconvergence theorem, the where √ first integral in this equality exists and is equal to 12 ω1 ( µ, x). From the Parceval equality we find the second summand: Zx

√ √ √ K12 (x, t)ω1 ( µ, t)dt = ω2 ( µ, x) − ω1 ( µ, x).

0

This implies the existence of I1 (µ, x) and the equality 1 √ √ I1 (µ, x) = ω2 ( µ, x) − ω1 ( µ, x). 2 Applying the equiconvergence theorem to I2 (µ, x), we get Z∞ I2 (µ, x) =

x

Z √ √ √ ω2 ( λ, x)dρ2 (λ) ω2 ( λ, t)ω1 ( µ, t)dt+

−∞

Z∞ +

0

Zx Zt √ √ 1 √ √ ω2 ( λ, x)dρ2 (λ) K21 (t, ξ)ω2 ( λ, ξ)ω1 ( µ, t)dξdt = ω1 ( µ, x)+ 2

−∞

Z

∞

+

√ ω2 ( λ, x)

−∞

=

x

Z

0

0

Z

x

0

1 1 √ ω1 ( µ, x) + 2 2

√ K21 (t, ξ)ω1 ( µ, t)ω2 (

 √ λ, ξ)dtdξdρ2 (λ) =

ξ

Z ξ

x

 1 √ √ K21 (t, ξ)ω1 ( µ, t)dt ξ=x = ω1 ( µ, x). 2

Therefore Z∞ −∞

x

Z √ √ √ ω2 ( λ, x) ω1 ( λ, t)ω1 ( µ, t)dtd{ρ1 (λ) − ρ2 (λ)} = 0

√ √ = I1 (µ, x) − I2 (µ, x) = ω2 ( µ, x) − ω1 ( µ, x),

370

STABILITY OF INVERSE PROBLEMS

and, using the equality Zx

√ √ √ √ √ ω10 ( λ, x)ω1 ( µ, x) − ω1 ( λ, x)ω10 ( µ, x) √ , ω1 ( λ, t)ω1 ( µ, t)dt = − λ−µ

0

we arrive at the formula √ √ ω1 ( µ, x) − ω2 ( µ, x) =

Z∞

√ √ √ √ ω10 ( λ, x)ω1 ( µ, x) − ω1 ( λ, x)ω10 ( µ, x) × λ−µ

−∞

√ ×ω2 ( λ, x)d{ρ1 (λ) − ρ2 (λ)}, which is similar to (5.1.8). Further transformations, which are in complete analogy with those given above, are based on the parity of indices 1 and 2.



2. STABILITY OF THE INVERSE SCATTERING PROBLEM Let the scattering data of a boundary value problem {q(x)} ∈ V {α(x)} be defined for all λ2 < N 2 only, with an error δ. How accurately can one reconstruct this boundary-value problem? To fix ideas, we will assume that δ = 0; the general case can be treated in a completely similar way, but the resulting formulas turn out to be much more cumbersome. To answer the question above, one has to understand how much can differ two boundary-value problems {qj (x)} ∈ V {α(x)} (j = 1, 2) having the scattering data {Sj (λ; λk (j); mk (j)} coinciding for λ2 ∈ (−∞, N 2 ), that is, S1 (λ) = S2 (λ), −N < λ < N,

λk (1) = λk (2), mk (1) = mk (2) (k = 1, .., n).

THEOREM 5.2.1. If the scattering data of two boundary-value problems {qj (x)} ∈ V {α(x)} (j = 1, 2) coincide for all λ2 ∈ (−∞, N 2 ), then for all µ2 ∈ (−∞, N 2 ) the following inequalities hold: |e1 (µ, x) − e2 (µ, x)|2 ≤ Z

8α(x) exp{4α1 (x)} ,  2 2 πN 1 − |µ|2N+µ 2

∞

|e1 (µ, x) − e2 (µ, x)|2 dx ≤

0

4e2α1 (0) sinh{2α1 (0)}   , 2 2 πN 1 − |µ|2N+µ 2

(5.2.1)

(5.2.2)

where Z

∞

α1 (x) =

α(t)dt. x

(5.2.3)

INVERSE SCATTERING PROBLEM

371

PROOF. Let us first suppose that µ belongs to the upper half plane and that Im µ2 6= 0. Then one can use formula (5.1.11), where, since the scattering data of the considered problems coincide for all λ2 ∈ (−∞, N 2 ), one has A1,2 (µ, x, t) =

e1 (µ, x)e2 (µ, x) 2π

Z

S2 (λ) − S1 (λ) e1 (λ, t)e2 (λ, t)dλ. λ2 − µ2

(5.2.4)

|λ|>N

Hence, formulas (5.1.11) and (5.2.4) are valid also for µ2 ∈ (−∞, N 2 ), which can be verified by passing to the limit. From the estimate (3.1.19) and the inequality (5.1.2) it follows that for all real ν and x ≥ 0 one has |ej (ν, x)| ≤ exp α1 (x), j = 1, 2, where α1 (x) is defined by formula (5.2.3). Thus for all N 2 > 0 and µ2 < N 2 we have | A1,2 (µ, x, t) |≤

exp{2(α1 (x) + α1 (t))} 2π

Z |λ|≥N

≤

|S1 (λ) − S2 (λ)|   dλ ≤ 2 2 λ2 1 − |µ|2N+µ 2

2 exp{2(α1 (x) + α1 (t))}  ,  2 2 πN 1 − |µ|2N+µ 2

and hence, in view of (5.1.11), we obtain the estimate 4 exp{2α1 (x)}   |e1 (µ, x) − e2 (µ, x)| ≤ 2 2 πN 1 − |µ|2N+µ 2 2

≤

Z∞ |q1 (t) − q2 (t)| exp{2α1 (t)}dt ≤ x

8α(x) exp{4α1 (x)} .  2 2 πN 1 − |µ|2N+µ 2

Integrating now the inequality (5.2.1) along the whole positive semiaxis and taking into account that α(x) = −α10 (x), we obtain (5.2.2).  The estimate (5.2.1) holds for all positive N , x, and µ2 < N 2 . Nevertheless, an evident estimate |e1 (µ, x)−e2 (µ, x)|2 ≤ 4 exp{2α1 (x)}, that follows from the inequalities (3.1.9), shows that the estimate (5.2.1) is nontrivial for −1  |µ|2 + µ2 N > 2π −1 α(x) exp{2α1 (x)} 1 − 2N 2 only. Therefore, it is desirable to get more precise estimates for large N , which are valid for all x.

372

STABILITY OF INVERSE PROBLEMS For the further convenience, let us introduce the function ∆(x, N ) = α1 (x) − α1 (x + N −1 ) =

−1 x+N Z

α(t)dt. x

It is defined for all x ≥ 0 and N > 0, is positive and nonincreasing with respect to the both variables. Evidently, the potentials of the considered boundary-value problems satisfy the inequalities Z∞

Z∞ |q(t)|dt = σ(x) ≤ α(x),

σ(t)dt ≤

σ1 (x) =

x

Z∞

x

σ1 (x) − σ1 (x + N

−1

α(t)dt = α1 (x), x

−1 N Z

) ≤ ∆(x, N ) ≤ ∆(0, N ) =

α(t)dt, 0

from which the following inequalities follow: Z∞ sin λ(t − x) |q(t)|dt ≤ σ1 (x) − σ1 (x + |λ|−1 ) ≤ ∆(x, |λ|), λ

(5.2.5)

x

Z

Z

∞

x

|λ|≥N

 sin2 λ(t − x) |q(t)|dt dλ ≤ 2π(σ1 (x)−σ1 (x+N −1 )) ≤ 2π∆(x, N ). (5.2.6) λ2

Namely, since for all real values of λ and y, |λ−1 sin λy| ≤ |y|,

|λ−1 sin λy| ≤ |λ|−1 ,

(5.2.7)

it follows that Z∞ sin λ(t − x) |q(t)|dt ≤ λ x

−1 x+|λ| Z

(t − x)|q(t)|dt + |λ| x

−1

Z∞ |q(t)|dt =

x+|λ|−1

−1 x+|λ| Z

(t − x)dσ(t) + |λ|−1 σ(x + |λ|−1 ) = −|λ|−1 σ(x + |λ|−1 )+

=− x −1 x+|λ| Z

σ(t)dt + |λ|−1 σ(x + |λ|−1 ) = σ1 (x) − σ1 (x + |λ|−1 ) ≤ ∆(x, |λ|),

+ x

373

INVERSE SCATTERING PROBLEM and  Z

Z∞





2

sin λ(t − x) |q(t)|dt dλ = λ2

x

|λ|≥N



Z∞

 |q(t)|  x

Z

 (t − x)|q(t)|  x

2

sin λ(t − x)  dλ dt ≤ λ2

|λ|≥N





−1 x+N Z

 Z

2

sin λ(t − x)  dλ dt + λ2 (t − x)

Z∞

Z |q(t)|dt

x+N −1

|λ|≥0

|λ|−2 dλ =

|λ|≥N

−1 x+N Z

(t − x)dσ(t) + 2N −1 σ(x + N −1 ) = −2πN −1 σ(x + N −1 )+

= −2π x −1 x+N Z

σ(t)dt + 2N −1 σ(x + N −1 ) ≤ 2π(σ1 (x) − σ1 (x + N −1 )) ≤ 2π∆(x, N ).

+2π x

Now introduce the functions φ(λ, x) = e(λ, x) − eiλx ,

(5.2.8)

ψ(λ, x) = φ(λ, x) − φ(−λ, x) = e(λ, x) − e(−λ, x) − 2i sin λx. In view of (3.1.20), these functions satisfy the inequalities |φ(λ, x)| ≤ (σ1 (0) − σ1 (|λ|−1 ))eσ1 (0) ≤ ∆(0, |λ|)eα1 (0) , |ψ(λ, x)| ≤ 2∆(0, |λ|)eα1 (0) , which implies that for all x ≥ 0 and λ ∈ (−∞, ∞), m(λ, x) = sup |φ(λ, t)| < ∞, t≥x

and for all N1 > N > 0, Z J(N, N1 ) = sup t≥0

ψ(λ, t) λ dλ < ∞.

N N

For |λ| > N and h > N −1 , in view of (5.2.22) we have |A(y, λ)| ≤

1 |λ|h 1 −

1 4λ2 h2


≤

4 3|λ|h

and, according to (5.2.21) and (5.2.23), Z Z π 6α(y) π 12α(y) . |r(y + hξ, λ)|| cos ξ|dξ ≤ | cos ξ|dξ = |B(y, λ)| ≤ |λ| |λ| 0 0

(5.2.24)

380

STABILITY OF INVERSE PROBLEMS

Therefore, |A(y, λ)| + |B(y, λ)| ≤

4 (1 + 9hα(y)) , 3|λ|h

and formula (5.2.14) implies that Z 2 1 |S1 (λ) − S2 (λ)|(|A(y, λ)| + |B(y, λ)|)dλ ≤ (1 + 9hα(y)) × 2πh 3πh2 |λ|>N

  S1 (λ) − 1 S2 (λ) − 1 + dλ ≤ 16∆(0, N ) (1 + 9hα(y)) . λ λ 3h2 (1 − 2∆(0, N ))

Z ×

(5.2.25)

|λ|>N

Next, by the mean-value theorem, we have   ˜ − q 0 (y + hξ) ˜ , Q1,2 (y, λ, ξ) = hξ q10 (y + hξ) 2 where 0 ≤ ξ˜ ≤ ξ, and formula (5.2.19) implies that Q1,2 (y, h, ξ)| ≤ 2hπD(y, h) and 1 2

Z

π

π

Z |Q1,2 (y, h, ξ)| sin ξdξ ≤ hπD(y, h)

sin ξdξ = 2hπD(y, h).

0

(5.2.26)

0

Now the statement of the theorem follows directly from the inequalities (5.2.24), (5.2.25), and (5.2.26).  3. ERROR ESTIMATE FOR THE RECONSTRUCTION OF A BOUNDARY VALUE PROBLEM FROM ITS SPECTRAL FUNCTION GIVEN ON THE SET (−∞, N 2 ) ONLY. √ We will need estimates for the spectral functions ρ(λ) and the solutions ω( λ, x) = √ ω( λ, x, h) of boundary-value problems {h, q(x)} ∈ V {A, a(x)}. Let us introduce the following notations: Z x Z x a(t)dt, b(x) = A + a(x), b1 (x) = b(t)dt = Ax + a1 (x), a1 (x) = 0

0

and note, that the boundary value problems {h, q(x)} ∈ V {A, a(x)} satisfy inequalities x

Z 0

x

Z |q(t)|dt ≤ a(x),

σ0 (x) =

σ0 (t)dt ≤ a1 (x),

σ1 (x) =

|h| ≤ A.

0

It is convenient to normalize the spectral functions of these problems by ρ(−∞) = 0.

RECONSTRUCTION FROM SPECTRAL FUNCTION

381

LEMMA 5.3.1. The spectral functions of the boundary value problems {h, q(x)} ∈ V {A, a(x)} satisfy the inequality 3√ λ exp 2b1 (λ−1/2 ), 2

ρ(λ) ≤

λ > 0.

(5.3.1)

√ √ PROOF. It follows from formula (2.4.1) and equality 1 − cos 2 λx = 2 sin2 λx, that for all x > 0 √ !2 Z∞ Z2x sin λx 2 √ dρ(λ) = 2x + (2x − t)L(t, 0, h)dt. 2x λx −∞

0

2

Since for all t ∈ (−∞, 1) 2  2  sin t 2 t2 25 ≥ , ≥ 1− ≥ t 6 36 3 then −2

x Z2x Z 2 2x+ (2x−t)L(t, 0, h)dt ≥ 2x

−∞

0

x−2 √ !2 Z sin λx 4 2 4 √ dρ(λ) ≥ x dρ(λ) = x2 ρ(x−2 ). 3 3 λx −∞

So, ρ(x−2 ) ≤

     Z 2x  Z 2x t 3 3 −1 x L(t, 0, h)dt ≤ x−1 1 + 1+ 1− L(t, 0, h)dt , 2 2x 2 0 0

and since by (1.2.34’)       t t t exp 2 |h| + σ1 ≤ |L(t, 0, h)| ≤ |h| + σ0 2 2 2       d t t t exp 2b1 = exp 2b1 , ≤b 2 2 dt 2 then
3 ρ x−2 ≤ x−1 exp 2b1 (x). 2  Inequality (5.3.1) follows by putting here λ = x−2 . COROLLARY. Z∞ Z∞ Z∞ dρ(λ) ρ(λ) dλ ρ(N 2 ) 3 −1 =− + dλ ≤ exp 2b1 (N ) = 3N −1 exp 2b1 (N −1 ). λ N λ2 2 λ3/2 N2

N2

N2

382

STABILITY OF INVERSE PROBLEMS

√ √ LEMMA 5.3.2. The solutions ω( λ, x) = ω( λ, x, h) of the boundary value problems {h, q(x)} ∈ V {A, a(x)} satisfy inequalities √  √    √ |ω( λ, x)| ≤ (1 + |h|x) exp σ1 (x) + Im λ x ≤ exp b1 (x) + Im λ x , √ moreover, in the domain a(x) < λ they also satisfy inequalities √ √  √   √  √ λ + |h| λ+A |ω( λ, x)| ≤ √ exp Im λ x ≤ √ exp Im λ x . λ − σ0 (x) λ − a(x)   √ √ PROOF. The functions φ(λ, x) = ω( λ, x) exp − Im λ x satisfy equations √ ! √ √ sin λx e(−|Im λ|x) + cos λx + h √ λ

φ(λ, x) =

√ sin λ(x − t) (−|Im √λ|(x−t)) √ + e q(t)φ(λ, t)dt. λ 0 The latter can be solved by the method of successive approximations: x

Z

φ(λ, x) =

∞ X

(5.3.2)

φk (λ, x),

k=0

where √ ! √ sin λx cos λx + h √ e(−|Im λ|x) , λ √

φ0 (λ, x) =

√ sin λ(x − t) (−|Im √λ|(x−t)) √ φk (λ, x) = e q(t)φk−1 (λ, t)dt. λ 0 Putting mk (x) = max |φk (t)| we find, that m0 (x) ≤ 1 + |h|x and Z

x

0 0   max |φ(λ, t)| Z x |h| 0 N 2 Zy  δ −1   √ √ |h| −1 ω( λ, x)dx ≤ √ 2 1+ √ + ω( λ, δ) = δ λ λ y−δ     2δ −1  |h|  σ1 (y) A σ1 (y−δ) ea1 (y) , −e + 1+ √ ≤ √ e 1+ N λ λ and according to the corollary to Lemma 5.3.1 ∞

Z

N2

Z

2  √ √ A ω1 ( λ, δ)ω2 ( λ, δ)d (ρ2 (λ) − ρ1 (λ)) ≤ 4δ −2 1 + e2a1 (y) × N ∞

× N2

d (ρ2 (λ) + ρ1 (λ)) 24 ≤ 2 λ δ N

 2
A 1+ exp 2 a1 (y) + b1 (N −1 ) . N

(5.3.9)

1

From formula (5.3.7) and estimates (5.3.8), (5.3.9) with δ = N − 3 we obtain the main inequality Z y
1  h1 − h2 + 1 (q1 (t) − q2 (t)) dt ≤ 24N − 3 (1 + AN −1 )2 exp 2 a1 (y) + b1 (N −1 ) + 2 0  1 1 D(y, N − 3 ) . (5.3.10) + 48

RECONSTRUCTION FROM SPECTRAL FUNCTION

387

1

One can estimate the value D(y, N − 3 ) via sup0≤t≤y |q(t)| using the integral equations  for the kernels Kji (x, t). PROBLEMS 1. Prove the formula 1 h1 − h2 + 2

Z∞

Zx {q1 (t) − q2 (t)}dt =

√ √ ω1 ( µ, x)ω2 ( µ, x)d{ρ1 (µ) − ρ2 (µ)}.

−∞

0

2. Prove that if the boundary-value problems {qj (x)} (j = 1, 2) belong to the set V {α(x)}, then in the domain {x : 2ν > α(x)}, the following inequality holds: ∞  −νx Z  e−νx α2 (x) −2ν(t−x) ≤ e1 (iν, x) − e2 (iν, x) − e 1 − e (q (t) − q (t)) dt . 1 2 2ν ν2 x

Hint. Integrate equation (3.1.7) once. 3. Prove the following generalization of Theorem 5.2.3.: If the scattering data of the two boundary problems {qj (x)} ∈ V {α(x)} (j = 1, 2) for all λ2 ∈ (−∞, N 2 ) differ by no more than δ, i.e. X k

|m2k (2)

−

m2k (1)|

1 + 2π

Z∞ |S1 (λ) − S2 (λ)| dλ < δ, −∞

then |q1 (x) − q2 (x)| ≤ 2δeα1 (x) + 2πhD(x, h) +

16∆(0, N )(1 + 9hα(x)) . 9h2 (1 − 2∆(0, N ))

4. Obtain a similar generalization for formula (5.3.10). 5. Generalize theorems 5.2.2. and 5.3.1. for the case when the scattering data or spectral functions are not equal but are only slightly different on the interval (−∞, N 2 ).

390

391

392

393

CHEL/373.H