Über die Autorin bzw. den Autor

Barry Simon is the IBM Professor of Mathematics and Theoretical Physics at the California Institute of Technology. His books include "Methods of Modern Mathematical Physics and Orthogonal Polynomials on the Unit Circle".

Von der hinteren Coverseite

"This book is an important one. Barry Simon has revolutionized the study of orthogonal polynomials since he entered the field. Many of the fundamental advances he and his students pioneered appear here in book form for the first time. There is no question of his profound scholarship and expertise on this topic."--Doron Lubinsky, Georgia Institute of Technology

"Simon is a leading specialist in orthogonal polynomials and spectral theory, with a very wide mathematical and physical background. This book contains a huge amount of new material found only in research papers. Those interested in orthogonal polynomials will find here many new results and techniques, while specialists in spectral theory will discover deep connections with topics from classical analysis and other areas."--Andrei Martínez-Finkelshtein, University of Almería, Spain

Aus dem Klappentext

"This book is an important one. Barry Simon has revolutionized the study of orthogonal polynomials since he entered the field. Many of the fundamental advances he and his students pioneered appear here in book form for the first time. There is no question of his profound scholarship and expertise on this topic."--Doron Lubinsky, Georgia Institute of Technology

"Simon is a leading specialist in orthogonal polynomials and spectral theory, with a very wide mathematical and physical background. This book contains a huge amount of new material found only in research papers. Those interested in orthogonal polynomials will find here many new results and techniques, while specialists in spectral theory will discover deep connections with topics from classical analysis and other areas."--Andrei Martínez-Finkelshtein, University of Almería, Spain

Auszug. © Genehmigter Nachdruck. Alle Rechte vorbehalten.

Szego's Theorem and Its Descendants

PRINCETON UNIVERSITY PRESS

Contents

Preface.....................................................................................ixChapter 1. Gems of Spectral Theory..........................................................1Chapter 2. Szego's Theorem..................................................................43Chapter 3. The Killip–Simon Theorem: Szego for OPRL...................................143Chapter 4. Sum Rules and Consequences for Matrix Orthogonal Polynomials.....................228Chapter 5. Periodic OPRL....................................................................250Chapter 6. Toda Flows and Symplectic Structures.............................................379Chapter 7. Right Limits.....................................................................418Chapter 8. Szego and Killip–Simon Theorems for Periodic OPRL..........................456Chapter 9. Szego's Theorem for Finite Gap OPRL..............................................477Chapter 10. A.C. Spectrum for Bethe–Cayley Trees......................................591Bibliography................................................................................607Author Index................................................................................641Subject Index...............................................................................647

Chapter One

The central theme of this monograph is the view of a remarkable 1915 theorem of Szego as a result in spectral theory. We use this theme to present major aspects of the modern analytic theory of orthogonal polynomials. In this chapter, we bring together the major results that will flow from this theme.

1.1 WHAT IS SPECTRAL THEORY?

Broadly defined, spectral theory is the study of the relation of things to their spectral characteristics. By "things" here we mean mathematical objects, especially ones that model physical situations. Think of the brain modeled by a density function, or a piece of ocean with possible submarines again modeled by a density function. Other examples are the surface of a drum with some odd shape, a quantum particle interacting with some potential, or a vibrating string with a density function. To pass to more abstract mathematical objects, consider a differentiable manifold with Riemannian metric. To get into number theory, this manifold might have arithmetic significance, say, the upper half-plane with the Poincaré metric quotiented by a group of fractional linear transformations induced by some set of matrices with integral coefficients.

By spectral characteristics, mathematicians and physicists originally meant characteristic frequencies of the object—modes of vibration of the drum or, to state the example that gives the subject its name, the light spectrum produced by a chemical like Helium inside the sun.

Eventually, it was realized that besides the discrete set of frequencies associated with a drum, vibrating string, or compact Riemannian manifold, there were objects with continuous spectrum where the spectral characteristics become scattering or related data. For example, in the case of a brain, the spectral data is the raw output of a computer tomography machine. For quantum scattering on the line, it might be the reflection coefficient.

The process of going from the object to the spectral data or of going from some property of the object to some property of the data is called the direct spectral problem (or direct problem). The process of going from the spectral data to the object or from some aspect of the spectral data to some aspect of the object is the inverse spectral problem (or inverse problem).

The general wisdom is that direct problems are easier than inverse problems, and this is true on two levels: first, on the level of mere existence and/or even specifying the domain of definition; and second, in proving theorems that say if some property holds on one side, then some other property holds on the other.

Almost all these models (tomography is an exception) are described by a differential equation—ordinary or partial—or by a difference equation. In most cases, the object is a selfadjoint operator on some Hilbert space. In that case, the direct problem is usually solved via some variant of the spectral theorem, which says:

Theorem 1.1.1. If A is a selfadjoint operator on a Hilbert space, H, and f [member of] H, there is a measure dµ on R so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.1)

for all t [member of] R.

Remarks. 1. All our Hilbert spaces are complex and <·, ·> is linear in the second factor and antilinear in the first.

2. For a proof, see [14, 361, 369]. Also see Section 1.3 later for the case of bounded A.

3. I have ignored subtle points here when A is an unbounded operator (as happens for differential operators) concerning what it means to be selfadjoint, how e^-itA is defined, and so on. Because we look at difference equations in most of these notes, our A is bounded, and then for n = 0, 1, 2, ..., (1.1.1) is equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.2)

4. We will also consider unitary operators, U, where dµ is now on [partial derivative]D = {z ||z| = 1} and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.3)

for n [member of] Z.

Notice that a spectral measure requires both an operator and a vector, f. Sometimes there is a natural f sometimes not. Sometimes the full spectral measure is overkill—for example, the problem made famous by Mark Kac [212]: "Can you hear the shape of a drum?" asks about whether the eigenvalues of the Laplace–Beltrami operator of a (two-dimensional) compact surface determine the metric up to isometry. The spectral measure typically has point masses at the eigenvalues but also has weights for those masses so has more data than the eigenvalues alone.

It is worth noting that it is arguable whether the shape of a drum problem is a direct or an inverse problem. It asks if the direct map from isometry classes of manifolds to their eigenvalue spectrum is one-one. But on a different level, it asks if an inverse map exists!

By the way, the answer to Kac's question is no (see [181]). For a review of more on this question and its higher-dimensional analogs, see [40, 64, 65, 180, 427].

Here is an example that shows we often do not understand the range of the direct map, and hence also the domain of the inverse map. Let H₀ = -d²/dx² on L²(-8, 8) and consider a function V (x) [member of] L¹_loc(R) so that (H⁰ + 1)^-1 (V + i) ^-1 x (H₀+1)^-1 is compact (e.g., this holds if V (x) [right arrow] 8 as |x| [right arrow] 8 but it also holds for V = W² + W' with W = x²(2 + sin(e^x)) where V is unbounded below). Then

H = H₀ + V (1.1.4)

has spectrum a set of eigenvalues {E_n}⁸_n=1 where E_n [right arrow]. It is well known that this is not sufficient spectral data to determine V.

Here is some additional data that is sufficient. Let H_D be H with a Dirichlet boundary condition at x = 0, that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.5)

where H⁺_D acts on L² (0, 8) and H^-_D acts on L² (-8, 0), and selfadjointness is guaranteed by demanding u(0) = 0 boundary conditions.

Let E^D_n be the eigenvalues of H_D. It is not hard to prove the following:

(i) Eⁿ = E^D_n = E_n+1

(ii) E^D_n = E_n [??] u_n(0) = 0 [??] E^D_n = E^D_n-1

Here u_n is the eigenfunction for H with eigenvalue E_n. Notice that (i) says each (E_n, E_n+1) contains at most one eigenvalue, and if there, it is simple. On the other hand, if E^D_n [member of] {E_j} ⁸_j=1, then it is a doubly degenerate eigenvalue.

If E^D_n [member of] (E_n, E_n+1), as noted E^D_n is simple, so we have a sign s^D_n [member of] {±1}, so E^D_n is an eigenvector of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If E^D_n [member of] {E_n, E_n+1}, s^D_n is undefined. We will see shortly that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a complete set of spectral data and that {V | E_n (V) = E_n(V₀)} is an infinite-dimensional set of potentials. In a situation like this, where some set of the "spectral data" is distinguished but not determining, the set of objects whose spectral data in this subset is the same as for object0 is called the isospectral set of object₀. It is usually a manifold, so we will often call it the isospectral manifold even if we have not proven it is a manifold!

Here is the theorem that describes what I have just indicated:

Theorem 1.1.2 ([165, 166]). If V, W [member of] L¹_loc and E_n(V) = E_n(W), E^D_n (V) = E^D_n (W), s^D_n (V) = s^D_n (W), then V = W (i.e., V [??] {E_n(V), E^D_n (V), s^D_n (V)}⁸_n=1 is one-one). Moreover, if V [member of] L¹_loc and N < 8 are given and E_n, E^D_n, s^D_n are such that

E_n = E_n(V) all n E^D_n = E^D_n (V) all n > N s^D_n = s^D_n (V) all n > N

{E_n, E^D_n} obey (i) and (ii) above, then there is a W with

E_n(W) = E_n E^D_n (W) = E^D_n s^D_n (W) = s^D_n

for all n.

It is an interesting exercise to fix N and picture the topology of the allowed E^D_n, s^D_n. Alas, it is not known precisely what direct data {E^D_n, s^D_n} can occur for a given V. It is definitely not all {E_n, s^D_n} obeying (i), (ii). For example, it cannot happen that

E^D_n = 1/4E_n + 3/4E_n+1 (1.1.6)

for all n.

Open Question 1. What is the range of the map V [??] {E_n(V), E^D_n (V), s^D_n (V)} as V runs through all L¹_loc functions with (H₀ +1)^-1/2 (V + i)^-1 (H₀ + 1)^-1/2 compact, or through all continuous functions obeying V (x) [right arrow] 8 as |x| [right arrow] 8.

Even the most basic isospectral manifolds such as V (x) = x² where E_n(V) = 2n + 1 are not understood.

Open Question 2. Prove that the isospectral manifold of continuous V's with V (x) [right arrow] 8 as x [right arrow] 8 and E_n(V) = 2n + 1 is connected.

I have described this example in detail to emphasize how little we understand even some basic spectral problems.

Having set the stage with a very general overview, we are now going to focus in these notes on two classes of spectral problems: those associated with orthogonal polynomials on the real line (OPRL) and orthogonal polynomials on the unit circle (OPUC). These are the most simple and most basic of spectral setups for several reasons:

(a) As we will see, the construction of the inverse is not only simple and basic, but historically these problems appeared initially as what we will end up thinking of as an inverse problem.

(b) The objects are connected with difference—not differential—operators, so various technicalities that might cause difficulty concerning differentiability, unbounded operators, and so on are absent.

(c) They are, in essence, half-line problems; the parameters in the difference equation are indexed by n = 1, 2, ... or n = 0, 1, 2,....

(c) is a virtue and a flaw. It is a virtue in that, as is typical for half-line problems, one can precisely describe the range of the direct map. It is a flaw in that the methods one develops are often not relevant to go to higher dimensions or, sometimes, even to whole-line problems.

OPRL appear initially in Section 1.2 and OPUC in Section 1.7.

Remarks and Historical Notes. The centrality of spectral theory to modern science can be seen by contemplating the variety of Nobel prizes that are related to the theory—from the 1915 physics prize awarded to the Braggs to the 1979 medicine prize for computer tomography.

1.2 OPRL AS A SOLUTION OF AN INVERSE PROBLEM

Let d? be a measure on R. All our measures will be positive with finite total weight. Normally, we will demand that ? is a probability measure, that is, ?(R) = 1. But for now we only suppose ?(R) < 8. ? is called trivial if L² (R, d?) is finite-dimensional; equivalently, if supp(d?) is a finite set. Otherwise we call ? nontrivial.

If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.1)

for all n, we say d? has finite moments. We will always suppose this. Indeed, we will soon mainly restrict ourselves to the case where ? has bounded support.

If ? is nontrivial with finite moments, every polynomial, P, obeys

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.2)

since the integral can only be zero if ? is supported on the finite set of zeros of P.

Thus, {xⁿ}8_n=0 are independent in L² (R, d?). They may or may not span L². If the support is bounded, they are spanning by the Weierstrass approximation theorem. In the case where the support is unbounded, there is a beautiful theory of when the polynomials span—it is presented in Section 3.8. One of the simplest examples of a case where they are not spanning is exp(-v|x|) dx (see Example 3.8.1 in Sections 3.8 and 3.9 for a discussion).

Thus, we can define monic orthogonal polynomials {P_n(x)}8_n=0 of degree n by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.3)

where p_n is the projection onto the n-dimensional space of polynomials of degree at most n - 1 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.4)

So P_n is determined by

P_n(x) = xⁿ + lower order P_n [perpendicular to] x^j j = 0, ..., n - 1 (1.2.5)

By an obvious induction, we have

(Continues...)

Excerpted from Szego's Theorem and Its Descendantsby Barry Simon Copyright © 2011 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.