Szegos Theorem and its Descendants

Szeg˝ o’s Theorem and Its Descendants: Spectral Theory for L2 Perturbations of Orthogonal Polynomials Barry Simon Cont...

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Szeg˝ o’s Theorem and Its Descendants: Spectral Theory for L2 Perturbations of Orthogonal Polynomials Barry Simon

Contents Chapter 1. Gems of Spectral Theory 1.1. What Is Spectral Theory 1.2. OPRL as a Solution of an Inverse Problem 1.3. Favard’s Theorem, the Spectral Theorem, and the Direct Problem for OPRL 1.4. Gems of Spectral Theory 1.5. Sum Rules and the Plancherel Theorem 1.6. Pólya’s Conjecture and Szeg˝o’s Theorem 1.7. OPUC and Szeg˝o’s Restatement 1.8. Verblunsky’s Form of Szeg˝o’s Theorem 1.9. Back to OPRL: Szeg˝o Mapping and the Shohat–Nevai Theorem 1.10. The Killip–Simon Theorem 1.11. Perturbations of the Periodic Case 1.12. Other Gems in the Spectral Theory of OPUC Chapter 2. Szeg˝o’s Theorem 2.1. Statement and Strategy 2.2. The Szeg˝o Integral as an Entropy 2.3. Carathéodory, Herglotz, and Schur Functions 2.4. Weyl Solutions 2.5. Coefficient Stripping, Geronimus’ and Verblunsky’s Theorems, and Continued Fractions 2.6. The Relative Szeg˝o Function and the Step-by-Step Sum Rule 2.7. The Proof of Szeg˝o’s Theorem 2.8. A Higher-Order Szeg˝o Theorem 2.9. The Szeg˝o Function and Szeg˝o Asymptotics 2.10. Asymptotics for Weyl Solutions 2.11. Additional Aspects of Szeg˝o’s Theorem 2.12. The Variational Approach to Szeg˝o’s Theorem 2.13. Another Approach to Szeg˝o Asymptotics 2.14. Paraorthogonal Polynomials and Their Zeros 2.15. Asymptotics of the CD Kernel: Weak Limits iii

1 1 5 12 21 22 25 28 30 34 42 44 46 49 50 55 59 75 83 90 95 97 103 109 110 116 121 127 133

iv

CONTENTS

2.16. Asymptotics of the CD Kernel: Continuous Weights 2.17. Asymptotics of the CD Kernel: Locally Szeg˝o Weights

138 148

Chapter 3. The Killip–Simon Theorem: Szeg˝o for OPRL 3.1. Statement and Strategy 3.2. Weyl Solutions and Coefficient Stripping 3.3. Meromorphic Herglotz Functions 3.4. Step-by-Step Sum Rules for OPRL 3.5. The P2 Sum Rule and the Killip–Simon Theorem 3.6. An Extended Shohat–Nevai Theorem 3.7. Szeg˝o Asymptotics for OPRL 3.8. The Moment Problem: An Aside 3.9. The Krein Density Theorem and Indeterminate Moment Problems 3.10. The Nevai Class and Nevai Delta Convergence Theorem 3.11. Asymptotics of the CD Kernel: OPRL on [−2, 2] 3.12. Asymptotics of the CD Kernel: Lubinsky’s Second Approach

161 161 162 170 177 184 188 194 205 227 232 239 248

Chapter 4. Sum Rules and Consequences for Matrix Orthogonal Polynomials 255 4.1. Introduction 255 4.2. Basics of MOPRL 256 4.3. Coefficient Stripping 262 4.4. Step-by-Step Sum Rules of MOPRL 268 4.5. A Shohat–Nevai Theorem for MOPRL 273 4.6. A Killip–Simon Theorem for MOPRL 275 Chapter 5. Periodic OPRL 5.1. Overview 5.2. m-Functions and Quadratic Irrationalities 5.3. Real Floquet Theory and Direct Integrals 5.4. The Discriminant and Complex Floquet Theory 5.5. Potential Theory, Equilibrium Measures, the DOS, and the Lyapunov Exponent 5.6. Approximation by Periodic Spectra, I. Finite Gap Sets 5.7. Chebyshev Polynomials 5.8. Approximation by Periodic Spectra, II. General Sets 5.9. Regularity: An Aside 5.10. The CD Kernel for Periodic Jacobi Matrices 5.11. Asymptotics of the CD Kernel: OPRL on General Sets 5.12. Meromorphic Functions on Hyperelliptic Surfaces 5.13. Minimal Herglotz Functions and Isospectral Tori

279 279 282 287 294 316 341 348 356 361 365 373 384 402

CONTENTS

v

Appendix to Section 5.13: A Child’s Garden of Almost Periodic Functions 415 5.14. Periodic OPUC 422 Chapter 6. Toda Flows and Symplectic Structures 6.1. Overview 6.2. Symplectic Dynamics and Completely Integrable Systems 6.3. QR Factorization 6.4. Poisson Brackets of OPs, Eigenvalues, and Weights 6.5. Spectral Solution and Asymptotics of the Toda Flow 6.6. Lax Pairs 6.7. The Symes–Deift–Li–Tomei Integration: Calculation of the Lax Unitaries 6.8. Complete Integrability of Periodic Toda Flow and Isospectral Tori 6.9. Independence of Toda Flows and Trace Gradients 6.10. Flows for OPUC

425 425 428 435 438 446 452 453 457 463 466

Chapter 7. Right Limits 469 7.1. Overview 469 7.2. The Essential Spectrum 470 7.3. The Last–Simon Theorem on A.C. Spectrum 478 7.4. Remling’s Theorem on A.C. Spectrum 484 7.5. Purely Reflectionless Jacobi Matrices on Finite Gap Sets 493 7.6. The Denisov–Rakhmanov–Remling Theorem 495 Chapter 8. Szeg˝o and Killip–Simon Theorems for Periodic OPRL 497 8.1. Overview 497 8.2. The Magic Formula 498 8.3. The Determinant of the Matrix Weight 502 8.4. A Shohat–Nevai Theorem for Periodic Jacobi Matrices 505 8.5. Controlling the ℓ2 Approach to the Isospectral Torus 507 8.6. A Killip–Simon Theorem for Periodic Jacobi Matrices 516 8.7. Sum Rules for Periodic OPUC 518 Chapter 9. Szeg˝o’s Theorem for Finite Gap OPRL 9.1. Overview 9.2. Fractional Linear Transformations 9.3. Möbius Transformations 9.4. Fuchsian Groups 9.5. Covering Maps for Multiconnected Regions 9.6. The Fuchsian Group of a Finite Gap Set 9.7. Blaschke Products and Green’s Functions

521 521 523 543 553 567 575 592

vi

CONTENTS

9.8. Continuity of the Covering Map 609 9.9. Step-by-Step Sum Rules for Finite Gap Jacobi Matrices 616 9.10. The Szeg˝o–Shohat–Nevai Theorem for Finite Gap Jacobi Matrices 618 9.11. Theta Functions and Abel’s Theorem 625 9.12. Jost Functions and the Jost Isomorphism 632 9.13. Szeg˝o Asymptotics 640 Chapter 10.1. 10.2. 10.3. 10.4. 10.5. 10.6.

10. A.C. Spectrum for Bethe–Cayley Trees 649 Overview 649 The Free Hamiltonian and Radially Symmetric Potentials652 Coefficient Stripping for Trees 656 A Step-by-Step Sum Rule for Trees 659 The Global ℓ2 Theorem 660 The Local ℓ2 Theorem 662

Bibliography

667

CHAPTER 1

Gems of Spectral Theory The central theme of this monograph is the view of a remarkable 1915 theorem of Szeg˝o 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 hit on 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 transformation 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. 1

2

1. GEMS OF SPECTRAL THEORY

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 the 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. 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 ϕ ∈ H, there is a measure dµ on R so that Z −itA hϕ, e ϕi = e−ixt dµ(x) (1.1.1)

for all t ∈ R.

Remarks. 1. All our Hilbert spaces are complex and h · , · i is linear in the second factor and antilinear in the first. 2. For a proof, see [14, 353, 360]. Also see Section 1.3 below 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, etc. Because we look at difference equations in most of these lectures, our A is bounded, and then for n = 0, 1, 2, . . . , (1.1.1) is equivalent to Z n hϕ, A ϕi = xn dµ(x) (1.1.2) 4. We will also consider unitary operators, U, where dµ is now on ∂D = {z | |z| = 1} and Z n hϕ, U ϕi = z n dµ(z) (1.1.3)

for n ∈ Z.

1.1. WHAT IS SPECTRAL THEORY

3

Notice that a spectral measure requires both an operator and a vector, ϕ. Sometimes there is a natural ϕ, sometimes not. Sometimes the full spectral measure is overkill—for example, the problem made famous by Mark Kac [207]: “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 [177]). For a review of more on this question and its higher-dimensional analogs, see [39, 60, 61, 176, 417]. Here is an example that shows we often don’t understand the range of the direct map, and so the domain of the inverse map. Let H0 = −d2 /dx2 on L2 (−∞, ∞) and consider a function V (x) ∈ L1loc (R) so that (H0 + 1)−1(V + i)−1 (H0 + 1)−1 is compact (e.g., this holds if V (x) → ∞ as |x| → ∞ but it also holds for V = W 2 +W ′ with W = x2 (2+sin(ex )) where V is unbounded below). Then H = H0 + V

(1.1.4)

has spectrum a set of eigenvalues {En }∞ n=1 where En → ∞. It is well known that this is not sufficient spectral data to determine V. Here is some additional data that is sufficient. Let HD be H with a Dirichlet boundary condition at x = 0, that is, + − HD = HD ⊕ HD

(1.1.5)

− + acts on L2 (0, ∞) and HD acts on L2 (−∞, 0), and selfadwhere HD jointness is guaranteed by demanding u(0) = 0 boundary conditions. Let En0 be the eigenvalues of HD . It is not hard to prove the following: (i) En ≤ EnD ≤ En+1 D (ii) EnD = En ⇔ un (0) = 0 ⇔ EnD = En−1 Here un is the eigenfunction for H with eigenvalue En . Notice that (i) says each (En , En+1 ) contains at most one eigenvalue, and if there, it is simple. On the other hand, if EnD ∈ {Ej }∞ j=1 , then it is a doubly degenerate eigenvalue. If EnD ∈ (En , En+1 ), as noted EnD is simple, so we have a sign σD σnD ∈ {±1}, so EnD is an eigenvector of HDn . If EnD ∈ {En , En+1 },

4


D D ∞ σnD is undefined. We will see shortly that {En }∞ n=1 ∪ {En , σn }n=1 is a complete set of spectral data and that {V | En (V ) = En (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 haven’t proven it is a manifold! Here is the theorem that describes what I’ve just indicated:

Theorem 1.1.2 ([161, 162]). If V, W ∈ L1loc and En (V ) = En (W ), EnD (V ) = EnD (W ), σnD (V ) = σnD (W ), then V = W (i.e., V 7→ 1 {En (V ), EnD (V ), σnD (V )}∞ n=1 is one-one. Moreover, if V ∈ Lloc and ˜ D , σ D are such that N < ∞ are given and Eñ , E n n ñ = En (V ) E EñD = EnD (V )

all n

σ ñD

all n > N

=

σnD (V

)

all n > N

{En , EnD } obey (i) and (ii) above, then there is a W with En (W ) = Eñ

ñD EnD (W ) = E

σnD (W ) = σnD

for all n. It is an interesting exercise to fix N and picture the topology of D ˜D, σ the allowed E n ñ . Alas, it is not known precisely what direct data ˜ D , σ D } can occur for a given V. It is definitely not all {E ñ , σ D } {E n n n obeying (i), (ii). For example, it cannot happen that EnD = 14 En + 34 En+1

(1.1.6)

for all n. Open Question 1. What is the range of the map V 7→ {En (V ), EnD (V ), σnD (V )} as V runs through all L1loc functions with (H0 + 1)−1/2 (V + i)−1 (H0 + 1)−1/2 compact, or through all continuous functions obeying V (x) → ∞ as |x| → ∞. Even the most basic isospectral manifolds such as V (x) = x2 where En (V ) = 2n + 1 are not understood. Open Question 2. Prove that the isospectral manifold of continuous V ’s with V (x) → ∞ as x → ∞ and En (V ) = 2n + 1 is connected. I’ve described this example in detail to emphasize how little we understand even some basic spectral problems.

1.2. OPRL AS A SOLUTION OF AN INVERSE PROBLEM

5

Having set the stage with a very general overview, we are now going to focus in these lectures 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, etc. 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’s a virtue in that, as is typical for halfline problems, one can precisely describe the range of the direct map. It’s 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 dρ be a measure on R. All our measures will be positive with finite total weight. Normally, we will demand that ρ is a probability measure, that is, ρ(R) = 1. But for now we only suppose ρ(R) < ∞. ρ is called trivial if L2 (R, dρ) is finite-dimensional; equivalently, if supp(dρ) is a finite set. Otherwise we call ρ nontrivial. If Z |xn | dρ(x) < ∞

(1.2.1)

for all n, we say dρ has finite moments. We will always suppose this. Indeed, we will soon mainly restrict ourselves to the case where ρ has bounded support. If ρ is nontrivial with finite moments, every polynomial, P , obeys Z 0 < |P (x)|2 dρ(x) < ∞ (1.2.2)

6


since the integral can only be zero if ρ is supported on the finite set of zeros of P . 2 Thus {xn }∞ n=0 are independent in L (R, dρ). They may or may 2 not span L . If the support is bounded, they are spanning by the Weierstrass approximation theorem. One of pthe simplest examples of a case where they are not spanning is exp(− |x| ) dx (see Example 3.8.1 in Sections 3.8 and 3.9 for a discussion). Thus we can define monic orthogonal polynomials {Pn (x)}∞ n=0 of degree n by Pn = πn⊥ [xn ] (1.2.3) where πn is the projection onto the n-dimensional space of polynomials of degree at most n − 1 and πn⊥ = 1 − πn

(1.2.4)

So Pn is determined by Pn (x) = xn + lower order Pn ⊥ xj

j = 0, . . . , n − 1

(1.2.5)

By an obvious induction, we have Proposition 1.2.1. {Pj }nj=0 span Ran(πn+1 ). In particular, Pj /kPj k are an orthonormal basis of this n + 1-dimensional space. So if Q ∈ Ran(πn+1 ), n X Q= hPj , QikPj k−2 Pj (1.2.6) j=0

One gets (1.2.6) by noting Q− rhs of (1.2.6) ⊥ Pk for k = 0, . . . , n since hPj , Pk i = kPj k2 δjk . Here is a key fact for OPRL: Proposition 1.2.2. hPj , xPn i = 0

if j < n − 1

(1.2.7)

Proof. hPj , xPn i = hxPj , Pn i = 0 since xPj has degree j + 1 < n. This leads to the key recursion relation obeyed by OPRL: Theorem 1.2.3. For any nontrivial measure with finite moments, ∞ ∞ there exist {bj }∞ and {aj }∞ so that for n ≥ 0, j=1 in R j=1 in (0, ∞) xPn (x) = Pn+1 (x) + bn+1 Pn (x) + a2n Pn−1 (x) where P−1 (x) ≡ 0 (so we don’t need an=0 ).

(1.2.8)


7

Proof. xPn (x) − Pn+1 (x) is a polynomial of degree n (since the xn+1 terms cancel) and so orthogonal to Pn+1 , that is, hPn+1, xPn i = kPn+1 k2

(1.2.9)

which means the coefficient of Pn+1 in (1.2.6) with Q = xPn is 1. Moreover, the coefficient of Pn−1 is hPn−1 , xPn ikPn−1 k−2 = hPn , xPn−1 ikPn−1k−2 2 kPn k = kPn−1 k

(1.2.10) (1.2.11)

where (1.2.10) follows from the reality of Pj and x, and (1.2.11) uses (1.2.9) for n replaced by n − 1. So we set an =

kPn k kPn+1 k

bn = hPn , xPn ikPn k−2

and (1.2.6) becomes (1.2.8) on account of (1.2.7).

(1.2.12)

The an ’s and bn ’s are called Jacobi parameters. We start labeling with n = 1, but some authors start with n = 0 or even label b from n = 0 but a from n = 1. Also, some reverse the a’s and b’s or use other letters. The formula for (1.2.12) for an implies Theorem 1.2.4. We have that kPn k = an . . . a1 ρ(R)

(1.2.13)

The orthonormal polynomials pn (x) =

Pn (x) kPn k

(1.2.14)

obey xpn (x) = an+1 pn+1 (x) + bn+1 pn (x) + an pn−1 (x) and multiplication by x in the  b1  a1 J= 0 

(1.2.15)

orthonormal set {pj }∞ j=0 has the matrix  a1 0  .. .  b2 a2  (1.2.16) ..  . a2 b3  .. .. .. .. . . . .

8


Remarks. 1. Matrices of the form (1.2.16) are called Jacobi matrices. 2. When supp(dρ) is bounded, {pn }∞ n=0 is a basis, as we have seen. Shortly we will restrict to this case. We now have our direct equation: {an , bn }∞ n=1 defines a second-order difference equation for n = 1, 2, 3, . . . , un+1 = a−1 n ((λ − bn )un − an−1 un−1 )

(1.2.17)

where a0 is picked in a convenient way and λ is a parameter. The solution with u0 = 0 u1 = 1 (1.2.18) is un = pn−1 (λ)

(1.2.19)

In Section 1.4, we will turn to the direct problem of going from {an , bn }∞ n=1 to dρ, but we see at the heart of OPRL is an inverse spectral problem! Central to this language is the idea that going from a difference/differential equation is a direct question. We eventually see (Section 3.2) that the inverse problem has a second method of solution. We note that the Pn (x) for dρ and c0 dρ for any c0 are the same and so also for Jacobi parameters. Thus we will eventually mainly restrict to ρ(R) = 1. Before leaving this introduction, we want to discuss two other ways of understanding OPRL that actually work for positive measures on C, so we pause to define OPs in that case. Let dζ(z) be a positive measure on C so that Z |z|n dζ(z) < ∞ (1.2.20) which is nontrivial (i.e., supp(dζ) is not a finite set of points). Thus we can form monic orthogonal polynomials Ξn (z). Unlike OPRL, Ξn (z) do not obey a three-term recurrence relation because Proposition 1.2.2 uses reality (in general, hΞj , z Ξn i = h¯ z Ξj , Ξn i). Indeed, only OPRL and OPUC (and polynomials for sets affinely related to D and ∂D) are known to obey finite-order recursion relations, and so fit into the scheme of “spectral theory.” 2 We note that {z n }∞ n=0 may not span L (C, dζ) even if supp(dζ) is bounded. For example, if there is an open set U ⊂ C and c so that dζ ≥ cχU d2 z

(1.2.21)

then they are not dense since the closure of the set of polynomials is analytic on U (see the Notes). And, as we will see (Section 2.11,


9

especially Theorem 2.11.5), for measures on ∂D, the issue of density is subtle. But we can define {Ξn (z)}∞ n=0 in any event. Theorem 1.2.5 (Christoffel Variational Principle). Let Mn be the monic polynomials of degree n, that is, Q ∈ Mn means Q(z) = z n + lower order

Then kΞn k2 = min kQk2 Q∈Mn

that is, for all Q ∈ Mn , Z Z 2 |Ξn (z)| dζ(z) ≤ |Q(z)|2 dζ(z)

(1.2.22)

(1.2.23)

with equality if and only if Q = Ξn .

Proof. This follows from the minimization property of orthogonalization, that is, if π is any orthogonal projection in a Hilbert space, k(1 − π)uk2 = min ku − vk2 v∈Ran π

(1.2.24)

It is remarkable how powerful this principle is, given its simplicity. The other general theorem concerns zeros. Theorem 1.2.6. let dζ be a measure obeying (1.2.20) and Mz multiplication by z on polynomials. Let πn be the orthogonal projection in L2 (C, dζ) onto the n-dimensional space of polynomials of degree at most n − 1. Let A = πn Mz πn (1.2.25) on Ran πn . Then (i) The eigenvalues of A are precisely the zeros of Ξn (z). (ii) Each eigenvalue of A has geometric multiplicity 1. (iii) Each eigenvalue z0 of A has algebraic multiplicity equal to the order of z0 as a zero of Ξn (z). (iv) We have that det(z − A) = Ξn (z) (1.2.26) Remark. Recall the geometric multiplicity of z0 is the dimension of {v | (A − z0 )v = 0}. The algebraic multiplicity is the dimension of {v | (A − z0 )ℓ v = 0 for some ℓ}. It is the order of the zero in det(z − A).

Proof. Let v ∈ Ran πn+1 . Then πn v = 0 if and only if v = c Ξn . Thus, if w ∈ Ran πn , w 6= 0, then (A − z0 )w = 0 ⇔ πn (z − z0 )w = 0 ⇔ (z − z0 )w = c Ξn . Moreover, w 6= 0 implies (z − z0 )w 6= 0, so c 6= 0. Ξn (z) = c−1 (z − z0 )w

(1.2.27)

10


implies Ξn (z0 ) = 0, so (i) is half proven. Conversely, if Ξ(z0 ) = 0 (1.2.27) is solved precisely by c Ξn (z) w(z) = (1.2.28) z − z0 which lies in Ran πn . Thus, (i) is proven and so is (ii). The same analysis shows (A − z0 )ℓ w = 0 with (A − z0 )ℓ−1 w 6= 0 if and only if z0 is a zero of Ξn (z) of order at least ℓ, and this proves (iii). (iv) holds since both sides are monic polynomials of degree n with the same zeros counting orders. Corollary 1.2.7 (Fejér’s Theorem). Zeros of Ξn (z) lie in the convex hull of supp(dζ). Proof. If Ξn (z0 ) = 0, there is w ∈ Ran πn , with kwkL2 = 1, so πn (z − z0 )w = 0. Thus, hw, (z − z0 )wi = 0, so Z z0 = hw, zwi = z|w(z)|2 dζ(z) (1.2.29)

Since kwk = 1, |w|2 dζ is a probability measure so the integral lies in the convex hull of supp(w 2dζ) which lies in the convex hull of supp(dζ).

Corollary 1.2.8. Suppose that dρ is a measure on R, with a = min supp(dρ), b = max supp(dρ). Then all the zeros of Pn (x, dρ) lie in [a, b]. Corollary 1.2.9. Let dµ be a measure on ∂D and Φn (z, dµ) the monic orthogonal polynomials. Then the zeros of Φn lie in D. Remark. One can show that if the convex hull of the support of dζ does not lie in a straight line, then zeros lie in the interior of the convex hull of the support of the measure. In particular, in case of Corollary 1.2.9, the zeros lie in D, not merely D. We will prove this explicitly in Theorem 1.8.4. Often, one has an explicit matrix representation of the operator A of (1.2.27), and so an explicit version of (1.2.24). For OPRL, one can take the basis {pj }n−1 j=0 and so get

Theorem 1.2.10. Let Jn.F be the n × n cutoff Jacobi matrix   b1 a1 0   . a1 b2 a2 . .    . . 0 a b  .. ..   2 3 Jn,F =  (1.2.30)  . . . .. .. ..      bn−1 an−1  an−1 bn


11

Then Pn (x) = det(x − Jn,F )

(1.2.31)

Since det(x − A) = xn − Tr(A)xn−1 + O(xn−2) for n × n matrices, we see that X n n Pn (x) = x − bj xn−1 + O(xn−2) (1.2.32) j=1

and, by (1.2.13)/(1.2.14), X n −1 n n−1 bj x + O(xn−2 ) pn (x) = (a1 . . . an ) x −

(1.2.33)

j=1

This provides another way of understanding the recursion (1.2.8). Expand det(x−Jn+1,F ) in minors in the last row. The minor of x−bn+1 is Pn (x) and the minor of −an is an Pn−1 (x).

Remarks and Historical Notes. I would be remiss if I didn’t mention the “classical” OPRL: Jacobi, Laguerre, and Hermite associated, respectively, to the measures (1 + x)α (1 − x)β dx on [−1, 1] with α > −1, β > −1, xα e−x on [0, ∞) with α > −1 and Hermite with 2 e−x dx. Jacobi polynomials with α = β = 0 are Legendre, and with |α| = |β| = 12 are Chebyshev (of four kinds depending on the signs of α and β). Chebyshev with α = β = − 12 and α = β = 12 (of the first and second kind) will occur repeatedly later in these lectures. They obey (up to normalization; Un is normalized but not monic, while Tn is neither the normalized nor monic OP), respectively, Tn (cos θ) = cos(nθ)

(1.2.34)

sin((n + 1)θ) (1.2.35) sin θ These and other specific examples are discussed in detail in Szeg˝o [424] and Ismail [199]. The classical polynomials obey many other relations like Rodriguez formula and second-order (in x) differential equations. This is specific to them; indeed, there is a theorem of Bochner (see [47, 183, 362] and [199, Section 20.1]) that says any set of orthogonal polynomials that obeys a second-order differential equation of the proper form is one of the classical ones! 2 The question of when {xn }∞ n=0 are dense in L (R, dρ) is intimately connected to the issue of determinacy of the moment problem discussed in the Notes to the next section. We will return to this issue in Section 3.9. Un (cos θ) =

12


Analyticity often places restrictions on the density of polynomials. If U ⊂ C is open and dζ ≥ cχU d2 z for some measure on C for which (1.2.20) holds, then by the Cauchy integral formula, for any compact K ⊂ U, we have sup |f (z)| ≤ CK kf kL2 (C,dζ) z∈K

for any function analytic in U and in L2 . It follows that any f in the L2 -closure of the polynomials is analytic on U since the locally uniform limit of analytic functions is analytic. Thus, when (1.2.21) holds, the polynomials do not span L2 . OPRL have their roots in work of Legendre, Gauss, and Jacobi. As a general abstract theory, the key figures were Chebyshev, Markov, Christoffel, and especially, Stieltjes. You can find more history in the books of Szeg˝o [424], Chihara [78], Freud [137], Nevai [314], and Ismail [199]. Closely entwined to the history is the idea of continued fraction expansions of resolvents, an issue we return to in Sections 2.5 and 3.2 and which was pioneered by Jacobi for finite matrices (hence the name Jacobi matrix for (1.2.30)) and Stieltjes. Variational principles like (1.2.22) for OPRL go back to Christoffel. Their use in OPUC with a twist (see Section 2.12 below) is due to Szeg˝o [424]. As a spectral theory tool, they have been especially advocated and exploited by Freud [137] and Nevai [315]. That zeros of OPRL are eigenvalues of truncated Jacobi matrices is well-known in the Schrödinger operator community. I’m unsure who noted it first. The extension to measures on C where there is the complication of nontrivial algebraic multiplicity was arrived at in discussions I had with E. Brian Davies. 1.3. Favard’s Theorem, the Spectral Theorem, and the Direct Problem for OPRL What the orthogonal polynomial community calls Favard’s theorem is the assertion that the map from measures on R (with finite moments) to Jacobi parameters is onto {an , bn }∞ n=1 with an > 0 and bn ∈ R. It is intimately connected to the spectral theorem; indeed, we will prove the spectral theorem for bounded selfadjoint operators in this section (modulo some remarks in the Notes that go from Jacobi matrices to general operators). In the bounded case, we will see the map is also one-one if we restrict to probability measures. Our discussion will be in the three stages: first, finite Jacobi matrices, then bounded, and finally, unbounded (where we will assume, rather than prove, the spectral theorem).

1.3. FAVARD’S THEOREM

13

Consider a trivial probability measure, that is, dρ =

N X

ρj δxj

(1.3.1)

j=1

for and

x1 > x2 > · · · > xN N X

ρj = 1

(1.3.2)

(1.3.3)

j=1

As usual, we can use Gram–Schmidt to define monic polynomials P0 , . . . , PN −1 since our proof of independence of {xj }∞ j=0 in the nonj N −1 trivial case shows that {x }j=0 are independent in this case. We can also use (1.2.3) to define PN (x) as the zero vector in L2 (R, dρ) which, among monic Nth degree polynomials, is unique, namely, N Y PN (x) = (x − xj )

(1.3.4)

j=1

The P ’s obey a recursion relation of the form (1.2.8) for n = 0, 1, 2, . . . , N − 1 and so define b1 , . . . , bN , a1 , . . . , aN −1 and an N × N finite Jacobi matrix. To go backwards, we start with an N × N finite Jacobi matrix, that N −1 is, {bj }N j=1 and {aj }j=1 are given with aj > 0 and bj ∈ R, and we do not (yet) know they come from a measure. We do not have a measure yet, so we cannot define Pj by orthogonality, but we do have recursion coefficients, so we define {Pj }N j=0 inductively by (1.2.8) with P0 (x) ≡ 1, P−1 (x) ≡ 0 (they could also be defined directly by (1.2.31)!), then pj for j = 0, 1, 2, . . . , N − 1 by p0 (x) = 1, and for 1 ≤ j ≤ N − 1, pj (x) =

Pj (x) a1 . . . aj

(1.3.5)

Then pn obey (1.2.15) for n = 0, 1, 2, . . . , N − 2 and

(bN − x)pN −1 (x) + aN −1 pN −2 (x) = −(a1 . . . aN −1 )−1 PN (x)

(1.3.6)

Proposition 1.3.1. Let J ≡ JN ;F be a finite Jacobi matrix given by (1.2.30). (a) Define the vector ~v (x) ∈ CN by vj (x) = pj−1 (x)

j = 1, 2, . . . , N

(1.3.7)

14


Then (J − x)~v (x) = −(a1 . . . aN −1 )−1 δjN PN (x) (b) If w ~ ∈ CN obeys then

[(J − x)w] ~ j=0

j = 1, . . . , N − 1

(1.3.8) (1.3.9)

wj = w1 pj−1(x) (1.3.10) (c) The eigenvalues of J are exactly the set of zeros of PN (x) and each zero has geometric multiplicity 1. (d) The zeros of PN are simple and real. (e) If the zeros of PN are labeled by (1.3.2) and pj−1 (xℓ ) (ϕℓ )j = PN ( j=1 |pj−1(xℓ )|2 )1/2

then the ϕℓ are an orthonormal basis of eigenvectors. (f) If X −1 N 2 2 ρℓ = |(ϕℓ )1 | = |pj−1 (xℓ )|

(1.3.11)

(1.3.12)

j=1

then (1.3.3) holds and {Pj (x)}N j=0 are the OPRL for the measure (1.3.1).

Proof. (a) (1.3.8) is just (1.2.15) for j = 1, . . . , N − 1 and (1.3.6) for j = N. (b) (1.3.10) holds trivially for j = 1 and then inductively by subtracting (1.3.8) from (1.3.10), and noting this implies (wj+1−w1 pj (x)) = (aj )−1 [(x−bj )(wj −w1 pj−1 (x))−aj−1 (wj−1 −w1 pj−2 (x))] (1.3.13) for j = 1, 2, . . . , N − 1 (with a−1 ≡ 0). (c) Any eigenvector obeys (1.3.9) and so must be a multiple of ~v. It obeys [(J − x)~v (x)]N = 0 if and only if PN (x) = 0 by (1.3.8). This argument shows any eigenvector is a multiple of ϕj given by (1.3.9), and so the geometric multiplicity is 1. (d) Define ϕj by (1.3.9). Then hϕk , Jϕℓ i = hJϕk , ϕℓ i implies, using Jϕℓ = xℓ ϕℓ , that (¯ xk − xℓ )hϕk , ϕℓ i = 0 (1.3.14) Taking k = ℓ, we see xk is real since (ϕℓ )1 6= 0 implies hϕℓ , ϕℓ i = 6 0. ′ To see that zeros are simple, suppose PN (xj ) = 0. Let ∂~v (1.3.15) w ~= ∂x x=xj


15

(the components of v are polynomials, hence differentiable). Since PN′ (x1 ) = 0, (1.3.8) implies (J − xj )w = v(xj )

(1.3.16)

That cannot be since it implies

hv(xj ), v(xj )i = hv(xj ), (J − xj )wi = h(J − xj )v(xj ), wi =0

and v1 (xj ) = 1, so hv(xj ), v(xj )i = 6 0. 2 (e) kϕℓ k = 1 is immediate and hϕj , ϕℓ i = 0 for j 6= ℓ by (1.3.14). Since PN (x) has N zeros, the ϕℓ ’s must span the space. (f) Since {ϕℓ }N ℓ=1 are an orthonormal basis, Ukℓ = (ϕℓ )k obeys X U¯kℓ Ukj = δℓj k

that is, U ∗ U = 1, so since it is finite-dimensional, UU ∗ = 1, that is (using (ϕℓ )j real to drop bars), X (ϕℓ )j (ϕℓ )k = δjk (1.3.17) ℓ

This says, by the definitions (1.3.11) and (1.3.12), X ρℓ pj−1 (xℓ )pk−1 (xℓ ) = δjk

(1.3.18)

ℓ

Taking j = k = 1 and using p0 (x) = 1, we see that (1.3.3) holds and −1 (1.3.18) implies that the {pj }N j=0 are orthonormal polynomials for the −1 measure (1.3.1), so {Pj }N j=0 are the monic OPRL. Since PN (xj ) = 0, PN is the monic OPRL for dρ. Remarks. 1. To be self-contained, we have given the standard argument that symmetric matrices have real eigenvalues and have algebraic multiplicities equal to geometric ones. 2. Notice that we have, in essence, just proven the spectral theorem for finite Jacobi matrices. 3. For a more conventional proof that the zeros of OPRL are all real and simple, see Subsection 5 of Section 1.2 of [390]. We have thus proven Theorem 1.3.2 (Favard’s Theorem for Trivial Measures). Every finite N × N Jacobi matrix is the Jacobi matrix of some measure supported on N points.

16


Proof. (f) of the last theorem says the {Pj }N j=0 are the OPRL for dρ defined by (1.3.12) and PN (xj ) = 0. The Jacobi parameters of Pj are the given Jacobi matrix since the polynomials alone obeying (1.2.8) determine a and b inductively by looking at the xN and xN −1 terms on (n) both sides of (1.2.8). For example, if xℓ are the roots of Pn (x), X X n n−1 (n) (n) bn = xℓ − xℓ ℓ=1

ℓ=1

as will occur prominently in Section 8.5.

Theorem 1.3.3. The map from dρ of the form (1.3.1)/ (1.3.3) to −1 N {aj }N j=0 ∪ {bj }j=0 is one-one (and onto by Theorem 1.3.2).

First Proof. Given the Jacobi matrix, JN , of dρ, following the P ′ ′ construction of Theorem 1.3.2, construct a measure dρ′ = N ρ δ j=1 j xj . ′ By construction, xj are the zeros of PN (x; dρ) which are exactly the xj ’s, that is, after renumbering x′j = xj . Moreover, the construction shows the normalized eigenvectors with positive first component are (1.3.11), so since ϕℓ in L2 (R, dρ) or L2 (R, dρ′ ) is the function f (x) = δxℓ x , we have ρℓ = hϕℓ , 1iL2(R,dρ)

= hϕℓ , p0 iL2 (R,dρ) = (ϕℓ )1

= given by (1.3.12) showing ρℓ =

ρ′ℓ .

We want to give a second proof, not because this result is so important or so difficult, but because a slightly more involved proof will yield tools that are useful in the N = ∞ case.

Proposition 1.3.4. (a) Two (probability) measures dρ, dρ′ (supports can be infinite) have the same Jacobi parameters up to n: n {aj }n−1 j=1 ∪ {bj }j=0 if and only if Z Z k x dρ = xk dρ′ (1.3.19)

k = 0, 1, . . . , 2n − 1. (b) Two measures, dρ, dρ′ , each supported at n (possibly different) points are equal if and only if (1.3.19) holds for j = 0, . . . , 2n − 1.

Proof. (a) By (1.2.8), we see that if Jacobi parameters are equal, then Pj (x; dρ) = Pj (x; dρ′ ) (1.3.20)


Multiplying by xℓ , ℓ = 0, . . . , j − 1 and integrating, we see Z j−1 Z ℓ+j ℓ+k x dρ = function of x dρ k=0 Z = xℓ+j dρ′

17

(1.3.21) (1.3.22)

where the function is the same by (1.3.20), and (1.3.19) then follows by induction. As j runs from 0 to n and ℓ from 0 to j − 1, ℓ + j goes from 0 to 2n − 1. Conversely, if (1.3.19) for k = 0, . . . , 2n − 1, the Gram matrices {hxj , xℓ i}0≤j, ℓ≤n−1 are equal which, by the Gram–Schmidt process implies pj (x, dρ) = pj (x, dρ′ ) for 0 ≤ j ≤ n − 1, and so Pj (x, dρ) = Pj (x, dρ′ )

Since n

R

Pn (x) = x − n+ℓ

n−1 X j=0

(1.3.23)

hpj , xn ipj (x)

the moments x dρ, ℓ = 0, . . . , n, then also determine Pn so (1.3.23) also holds for j = n. As noted above, the polynomials determine the a’s and b’s in the recursion relation. (b) As noted in (a), the stated moments determine PN (x) and so ′ N its zeros, and so {xj }N j=1 and {xj }j=1 are identical sets. Then the ρ’s are determined by the equations Z N X ℓ−1 ρj xj = xℓ−1 dρ (1.3.24) j=1

for ℓ = 1, 2, . . . , N since the Vandermonde determinant Y det(xℓ−1 ) = (xi − xj ) j

(1.3.25)

i<j

is nonzero.

Second Proof of Theorem 1.3.3. By (a) of Proposition 1.3.4, the Jacobi parameters determine the first 2n moments and then, knowing the support is n points, the measures by (b) of the proposition. One can combine Theorems 1.3.2 and 1.3.3 and more in Theorem 1.3.5. Fix N. There is a one-one correspondence among each of −1 N (i) Jacobi parameters {aj }N j=1 ∪ {bj }j=1 with bj ∈ R and aj > 0. (ii) Trivial measures of the form (1.3.1) where (1.3.3) holds and ρj > 0.

18


(iii) Unitary equivalence classes of symmetric N × N matrices A with a distinguished cyclic vector, ϕ. Remarks. 1. ϕ is called cyclic if {Aj ϕ}∞ j=0 span the space. For N −1 j N × N matrices, we can instead take {A ϕ}j=0 since if P (A) is the (monic) secular polynomial of A, P (A)Aℓ ϕ = 0 shows inductively that N −1 j {Aj ϕ}∞ j=N are functions of {A ϕ}j=0 . 2. (A, ϕ) and (A′ , ϕ′ ) are unitarily equivalent if and only if there is a unitary U : CN → CN so UAU −1 = A′ and Uϕ = ϕ′ . Proof. (i) ⇔ (ii) is precisely the construction of Section 1.2 combined with Theorems 1.3.2 and 1.3.3. It is easy to see that δ1 = (1, 0, . . . , 0)t is cyclic for a finite Jacobi matrix J. Indeed, if {pℓ }n−1 ℓ=0 are the orthonormal polynomials, then δℓ = pℓ−1 (J)δ1 , so each Jacobi matrix with distinguished δ1 is in an equivalence class. −1 Conversely, if ϕ is cyclic for A, {Aj ϕ}N j=0 must be independent (since they span CN ). Thus, by Gram–Schmidt, we can find −1 polynomials {pj (A)}N j=0 with p0 (A) = 1 so ϕj = pj−1 (A)ϕ, j = 0, . . . , N − 1, is an orthonormal basis. By the Gram–Schmidt construction, hAk ϕ, pj (A)ϕi = 0 if k < j. So by the same argument as in N −1 Section 1.2, there are constant {bj }N j=1 , {aj }j=1 , so Aϕj = aj+1ϕj+1 + bj+1 ϕj + aj ϕj−1

(1.3.26)

for j = 0, . . . , N − 1 where we interpret aN and a0 as 0. Thus hϕj , Aϕk i is a Jacobi matrix! The construction is unitary invariant so the map is from equivalence classes to Jacobi matrices. The two constructions are inverses showing the one-one correspondence. Now we turn to the case of bounded semi-infinite Jacobi matrices. Proposition 1.3.6. A Jacobi matrix (1.2.16) is bounded on ℓ2 if and only if sup |an | + sup |bn | < ∞ (1.3.27) n

n

Proof. bn = hδn , Jδn i while an = hδn+1 , Jδn i so |bn | ≤ kJk and |an | ≤ kJk. Thus, J bounded implies (1.3.27). A diagonal matrix D = {dn δnm } has kDk = supn |dn | , and if A, B are the diagonal matrices with elements a and b, and if Sδn = δn+1 , then J = AS ∗ + B + SA

(1.3.28)

kJk ≤ 2 sup |an | + sup |bn |

(1.3.29)

so n

n


19

We have thus proven sup |an | + sup |bn | ≤ 2kJk ≤ 4 sup |an | + sup |bn | n

n

n

n

(1.3.30)

We can now turn to the main theorem of this section (given our interest in the bounded support regime): Theorem 1.3.7 (Favard’s Theorem for Bounded Jacobi Matrices). ∞ Let {an }∞ n=1 , {bn }n=1 be a set of Jacobi parameters obeying (1.3.27). Then there is a nontrivial measure, dρ, of bounded support so that its Jacobi parameters are the given ones. Proof. Let J be a Jacobi matrix and Jn;F its finite truncations. By Theorem 1.3.2, there are trivial n-point measures, dρn , whose Jacobi n parameters are {aj }n−1 j=0 ∪ {bj }j=0 . By Proposition 1.3.4, Z Z ℓ x dρn = xℓ dρn′ (1.3.31)

for ℓ = 0, 1, . . . , 2 min(n, n′ ) − 1. In particular, for each ℓ, constant for n large, so Z lim xℓ dρn

R

xℓ dρn is

n→∞

exists for each n. By construction, dρn is supported on the eigenvalues of Jn;F and so on [−kJn,F k, kJn;F k], and so on [−kJk, kJk]. Thus, the dρn ’s are supported in a fixed compact set. Since the polynomials are dense in C([−kJk, kJk]), the probability measures, dρn , have a weak limit dρ. This weak limit, by (1.3.31), obeys Z Z ℓ x dρn = xℓ dρ ℓ = 0, . . . , 2n − 1 (1.3.32) By Proposition 1.3.4, the Jacobi parameters of dρ are J.

Remark. Modulo discussion in the Notes, we have just proven the spectral theorem for bounded operators! In the following, we could also discuss cyclic vectors, but we won’t (see the Notes): Theorem 1.3.8. There is a one-one correspondence between bounded Jacobi matrices and nontrivial probability measures of bounded support under the map of measures to Jacobi parameters.

20


Proof. Clearly, if dρ has support [−C, C], then |bn | ≤ |an | ≤

Z

Z

|x| |pn (x)|2 dρ ≤ C |x| |pn (x)| |pn−1(x)| dx ≤ C

so J is bounded. By Favard’s theorem, the map from measures of bounded support to bounded Jacobi parameters is onto. By Proposition 1.3.4, it is one-one. In this monograph, we are mainly interested in the bounded support case, so we will state Favard’s theorem in the unbounded case without giving the proof for now. We will essentially prove it in Section 3.8; see Theorem 3.8.4. Theorem 1.3.9 (Favard’s Theorem). For set of Jacobi paramR any n eters, there is a measure, dρ, on R with |x| dρ(x) < ∞ for all n, which has those Jacobi parameters. The measure may not be unique. This is discussed in Sections 3.8 and 3.9. Remarks and Historical Notes. Favard’s theorem is named after Favard [123] but goes back to Stieltjes [413]. The close connection to the spectral theorem also predates Favard in work of Stone [414] and Wintner [449]; see also Natanson [307], Perron [338], Sherman [373], ´ and the discussion in Marcellán and Alvarez-Nodarse [289]. I am not aware of the approach here appearing elsewhere, but it will not surprise experts and I suspect is known to some. Given any bounded selfadjoint operator, A, on a separable Hilbert space, H, it is not hard to see that one can find {ϕj }N j=1 (N fiℓ m nite or infinite) so that for any ℓ, m, j 6= k, hA ϕj , A ϕk i = 0 and so that {Aℓ ϕj }j,ℓ span H. Thus, Theorem 1.3.7 and Gram–Schmidt 2 imply there is a unitary U from H onto ⊕N j=1 L (R, dµj ) so that −1 (UAU f )m (x) = xfm (x). This is the spectral theorem for bounded operators. The same idea shows that if A has a cyclic vector, ϕ, then applying Gram–Schmidt to {Aj ϕ}∞ j=0 yields an orthonormal basis in which J is a cyclic vector, allowing the two-part equivalence ofTheorem 1.3.8 to extend to the three-part equivalence of Theorem 1.3.5.

1.4. GEMS OF SPECTRAL THEORY

21

1.4. Gems of Spectral Theory In order to explain what I will mean by a gem of spectral theory, I begin by describing a pair of beautiful theorems in the spectral theory of OPRL: Theorem 1.4.1 (Blumenthal–Weyl). Let J be a Jacobi matrix with Jacobi parameters {an , bn }∞ n=1 . If then

an → 1

and

bn → 0

σess (J) = [−2, 2]

(1.4.1) (1.4.2)

Remarks. 1. Recall (see Reed-Simon [356, Section XIII.4]) that σess is defined by σess (J) = σ(J) \ σd (J), where σ(J), the spectrum of J, is {λ | (J − ℓ) does not have aH bounded inverse}, and σd (J) are isolated points λ0 of σ(J), where |z−λ0 |=ε (z − J)−1 dz is finite rank. For J’s with cyclic vector (like Jacobi matrices) and spectral measure dρ, σess (J) is the set of nonisolated points of supp(dρ). 2. See the Notes for a discussion of proof and history. 3. For any a, b ∈ R with a > 0, N(a, b), the Nevai class, is the set of measures where an → a, bn → b. By scaling, σess (J) = [b−2a, b+ 2a] if J ∈ N(a, b). Theorem 1.4.2 (Denisov–Rakhmanov). Let J be a Jacobi matrix with measure dρ and Jacobi parameters {an , bn }∞ n=1 . Suppose (1.4.2) holds and dρ(x) = f (x) dx + dρs (x) (1.4.3) where dρs is singular and (modulo sets of measure 0) Then (1.4.1) holds.

{x | f (x) > 0} = [−2, 2]

Remark. See the Notes for a discussion of proof and history. We will return to this theorem in Section 7.6. These theorems are illuminated by the following: Example 1.4.3. Let an ≡ 12 and bn be the sequence (1, −1, 1, 1, −1, −1, 1, 1, 1, −1, −1, −1, . . . ), that is, 1 k times followed by −1 k times for k = 1, 2, . . . . It is not hard to show σ(J) = σess (J) = [−2, 2], so (1.4.2) is not sufficient for (1.4.1) to hold. Thus, we have a pair of deep theorems that go in opposite directions, but they do not set up equivalences. This leads us to:

22


Definition. By a gem of spectral theory, I mean a theorem that describes a class of spectral data and a class of objects so that an object is in the second class if and only if its spectral data lie in the first class. This idea will be illuminated as we describe gems for OPUC and for OPRL in Sections 1.8 and 1.10 and a non-gem in Section 1.9. In a sense, the overriding purpose of this book is to explore gems of OPRL/OPUC that depend on sum rules with positive coefficients. As we will see, the focus is somewhat narrower than that! And we will discuss some descendants of Szeg˝o’s theorem that are not gems (yet). Remarks and Historical Notes. I find that some listeners object strongly to my use of the term “gem.” I respond that it is a definition and I add that for a mathematician, a definition is not something that can be “wrong.” But if I called them the “Jims of Spectral Theory,” I wouldn’t get the same reaction. And, of course, I used gems because of its connotation. Gems of spectral theory are typically beautiful and hard—but there can be beautiful and hard results that are not necessary and sufficient: Theorem 1.4.2 comes to mind. The Blumenthal–Weyl theorem is named after contributions of Blumenthal [45] and Weyl [445]; Denisov–Rakhmanov after results of Rakhmanov [350, 351] and Denisov [103]; see Sections 9.1 and 9.2 of [391] for further history. Theorem 1.4.1 is a consequence of Weyl’s theorem (see Reed–Simon [356, Sect. XIII.4]) that if C is compact and selfadjoint and A bounded and selfadjoint, then σess (A + C) = σess (A). In Theorem 1.4.1, A = J0 , the Jacobi matrix with an ≡ 1, b≡ 0, and C = J − J0 is compact when (1.4.1) holds. Rakhmanov’s theorem for OPUC is proven in Chapter 9 of [391]. Theorem 1.4.2 is proven in Section 13.4 of that book. As mentioned, we will provide a proof of a more general result in Chapter 7 of the present monograph. 1.5. Sum Rules and the Plancherel Theorem The basic tool we will use is to establish sum rules with positive terms. In this section, we illustrate this with the granddaddy of all spectral sum formulae: that fact that if A = {aij }1≤i,j≤N is a finite matrix and {λj }N j=1 are its eigenvalues, then N X j=1

λj = Tr(A) ≡

N X j=1

ajj

(1.5.1)

1.5. SUM RULES AND THE PLANCHEREL THEOREM

23

The left side is spectral theoretic and the right side involves the coefficients of the object. One standard proof of (1.5.1) is to prove invariance of trace under similarity and the fact that there is a similarity taking A to upper triangular (even Jordan) form. But for us, the “right” proof is to note that the λj are the roots of the secular polynomials, so N Y det(λ1 − A) = (λ − λj )

(1.5.2)

j=1

Since, by expanding the determinant det(λ1 − 1) = λn − Tr(A)λn−1 + · · ·

(1.5.3)

we get (1.5.1). The idea that sum rules occur as Taylor coefficients of suitable analytic functions recurs throughout this book. In the infinite-dimensional case, there are convergence and other issues. Let X be a Banach space. A bounded linear map A : X → X is called finite rank if Ran(A) is finite-dimensional. Every such map has the form N X Ax = ℓj (x)xj (1.5.4) j=1

∗ N ∗ For some {ℓj }N j=1 ⊂ X and {xj }j=1 ⊂ X . It is not hard to show that

Tr(A) =

N X

ℓj (xj )

(1.5.5)

j=1

is independent of the ℓ’s and x’s used in the representation (1.5.4) (essentially by the invariance of trace in the finite-dimensional case). One defines the trace norm of a finite-rank operator by X n X kAk1 = inf kℓj kX ∗ kxj kX A = ℓj ( · )xj (1.5.6) j=1

The nuclear operators, N(X), are the completion of the finite-rank operators in k · k1. It is not hard to see that every such object is associated to an operator and that one can define Tr( · ) on N(X) since |Tr(A)| ≤ kAk1

(1.5.7)

If X is a Hilbert space, then N(X) is called the trace class operators. A celebrated theorem of Lidskii says that

24


Theorem 1.5.1 (Lidskii’s Theorem). If A is a trace class operator on a Hilbert space, H, then σess (A) = {0} and A has nonzero eigenvalues {λj }N j=1 (counting algebraic multiplicity) so that N X

λj = Tr(A)

(1.5.8)

j=1

There are two limitations to note. First, on general Banach spaces, this result is false. Indeed, there is a Banach space, X, with a nuclear operator A so that A2 = 0 (so any eigenvalue is 0) but Tr(A) = 1! (See the Notes.) Second, consider the operator, C, on ℓ2 which is a direct sum C1 ⊕ C2 ⊕ . . . of 2 × 2 matrices αj αj Cj = (1.5.9) −αj −αj

Cj2 = 0, so C has P only eigenvalue zero. Indeed, it is easy to see that σ(C) = {0}. If ∞ j=1 |αj | = ∞, but αj → 0, then C is compact but not trace class. The sum of the eigenvalues is 0. As for the “trace,” the sum of the diagonal matrix elements of C is conditionally convergent to zero, so it looks like a success. But conditionally convergent sums can be rearranged to any value! And rearranged sums are just rearranged bases. The moral is that, due to cancellations, (1.5.8) is subtle as soon as one leaves trace class, and it is unlikely that there is any kind of necessary and sufficient condition directly related to (1.5.8). However, positivity can rescue something. It is not hard to prove Theorem 1.5.2. Let A be a bounded selfadjoint operator on a Hilbert space. Then A2 is trace class if and only if A has a pure point spectrum with eigenvalues {λj (A)}∞ j=1 obeying ∞ X j=1

λj (A)2 < ∞

(1.5.10)

if one writes Tr(A2 ) = ∞ if A2 is not trace class and P In fact, 2 λj (A) = ∞ if A has any non-point spectrum, Theorem 1.5.2 comes from a sum rule X Tr(A2 ) = λj (A)2 (1.5.11) j

There are no cancellations because of positivity. dθ ), one can specialize to operators of the form On ℓ2 (∂D, 2π Z dψ (Af g)(θ) = f (θ − ψ)g(ψ) (1.5.12) 2π

´ ˝ THEOREM 1.6. POLYA’S CONJECTURE AND SZEGO’S

25

where θ − ψ is computed mod 2π. Then λj (Af ) are the Fourier coefficients, Theorem 1.5.2 is the Plancherel theorem, and the sum rule (1.5.11) is Parseval’s equality. As we will see in Section 2.11, Szeg˝o’s theorem can be viewed as a kind of nonlinear Plancherel theorem. Remarks and Historical Notes. The view of Theorem 1.5.2 as a sum rule with positivity, and so a model of Szeg˝o’s theorem as a sum rule has been pushed especially by Killip [217] For a proof of Lidskii’s theorem, see, for example, [380] which obtains it from an equality for trace class operators ∞ Y (1 + zλj (A)) det(1 + zA) =

(1.5.13)

j=1

An analog of (1.5.14) for Hilbert–Schmidt integral operators, namely, −zA

det[(1 + zA)e

]=

∞ Y

[(1 + zλj (A))e−zλj (A) ]

(1.5.14)

j=1

goes back to Carleman [70] in 1921. One can regard him as the father of Theorem 1.5.2. Lidskii’s theorem is named after [275], although the theorem was found somewhat earlier by Grothendieck [182]. Unaware of Grothendieck’s work, Simon [380] rediscovered his approach to the problem. For an introduction to nuclear operators on a general Banach space, see Chapter 10 of Simon [381]. (This book also discusses trace class, Lidskii’s theorem, and proves (1.5.13) and (1.5.14); another reference on those subjects is Gohberg–Krein [166].) In particular, the example mentioned of a nuclear operator with A2 = 0, but Tr(A) = 1 is from Grothendieck [181]. 1.6. P´ olya’s Conjecture and Szeg˝ o’s Theorem Pólya and Szeg˝o have linked names much like Hardy and Littlewood or Laurel and Hardy. This is most of all because of their great twovolume encyclopedia of analysis [345] and because, as part of Szeg˝o’s establishing of a great school of mathematics at Stanford, he brought Pólya to Palo Alto. But they are also linked in the initial history of the main theme of this monograph. As we will see in Section 3.8, Hankel matrices, that is, finite matrices of the form {cj+k }njk=1 are fundamental to the theory of the moment problem on R (since they arise as Gram matrices for {xj }n−1 j=0 ). A

26


Toeplitz matrix, T , is one of the form 1 ≤ j, k ≤ n

tjk = cj−k

(1.6.1)

Just as in the Hankel case, a situation of special interest is when c are the moments of a measure but now on ∂D: Z 2π ck = e−ikθ dµ(θ) (1.6.2) 0

We will, for now, restrict to the case dµs = 0 where w(θ) dµ(θ) = dθ + dµs (1.6.3) 2π that is, to the case Z dθ (1.6.4) ck = e−ikθ w(θ) 2π Define Dn (w) (more generally, Dn (dµ)) to be the determinant of the (n + 1) × (n + 1) Toeplitz matrix c0 c1 . . . cn c−1 c0 . . . cn−1 (1.6.5) Dn (w) = det .. .. .. . . . c ... ... c −n

0

Because of a flurry of activity about moment problems on ∂D unleashed by Carathéodory in 1907 (see the Notes to Section 1.3 of [390]), Toeplitz matrices were all the rage from 1910–1915, and Pólya, a young postdoc, conjectured in [344] that if w > 0 and in L1 , then Z dθ 1/n (1.6.6) lim Dn (w) = exp log(w(θ)) n→∞ 2π In a visit back to his native Budapest, Pólya mentioned this conjecture to Szeg˝o, then an undergraduate, and he proved the theorem below, published in 1915 [418]. At the time, Szeg˝o was nineteen, and when the paper was published, he was serving in the Austrian Army in World War I! Here is the first version of Szeg˝o’s theorem: Theorem 1.6.1 (Szeg˝o’s Theorem). If w(θ) ≥ 0 and Z dθ w(θ) j

(1.7.3)

k=1

and hgj , gk i = 0

Let H be the N × N matrix hjk

  1 = hjk  0

Then the Gram matrices are clearly related by G(f )jk = hfj , fk i

G(g)jk = hgj , gj i

G(g) = H ∗ G(f )H

(1.7.4) (1.7.5)

We thus conclude: Theorem 1.7.1. det(G(f )) =

N Y j=1

kgj k2

(1.7.6)

˝ RESTATEMENT 1.7. OPUC AND SZEGO’S

29

Proof. By (1.7.3), det(H) = det(H ∗ ) = 1 and by (1.7.5), det(G(g)) = |det(H)|2 det(G(f )) = det(G(f ))

Since G(g)jk = kgj k2 δjk , (1.7.6) is immediate.

Given a nontrivial measure, dµ, on ∂D, define the monic OPUC by Z

Φn (z) = z n + lower order z¯j Φn (z) dµ(z) = 0

j = 0, 1, . . . , n − 1

(1.7.7) (1.7.8)

We will use Φn (z; dµ) if we want the dµ dependence to be explicit. Thus, if fj = z j−1 , j = 1, . . . , N, and gj = Φj−1 , j = 1, . . . , N, we see that f and g are related by (1.7.1)/(1.7.2). Recognizing G(f ) as the Toeplitz matrix with Z k j ck−j = hz , z i = e−i(k−j)θ dµ(θ) (1.7.9) we obtain from (1.7.6) that

Corollary 1.7.2. The (N + 1) × (N + 1) Toeplitz determinant, DN (dµ), obeys N Y DN (dµ) = kΦj (z; dµ)k2 (1.7.10) j=0

Szeg˝o also realized a special feature of OPUC that makes Fekete’s 1/N remark that DN +1 /DN has the same limit as DN transparent, namely, Proposition 1.7.3. For each n, kΦn k ≤ kΦn−1 k

(1.7.11)

lim DN (dµ)1/N

(1.7.12)

Thus limn→∞ kΦn k2 exists and equals N →∞

Proof. Since Φn is orthogonal to any polynomial of degree n − 1, it minimizes {kΦn + gk | deg(g) ≤ n − 1}. As a consequence, kΦn k = min{kPn k | Pn (z) = z n + lower order}

Thus, since zΦn−1 is monic and |z| = 1 on supp(dµ), kΦn k ≤ kzΦn−1 k = kΦn−1 k

30


proving (1.7.11). Since kΦn k is decreasing and positive, it has a limit and, of course, (kΦ0 k2 . . . kΦn k2 )1/n then converges to lim kΦn k2 . We thus see an equivalent form of Szeg˝o’s theorem, Theorem 1.6.1: Theorem 1.7.4. (1.6.6) is equivalent to

2 Z

dθ w(θ)

lim Φn z, dθ = exp log(w(θ)) n→∞ 2π 2π

(1.7.13)

Remarks and Historical Notes. Szeg˝o’s great 1920–1921 paper [420] was the first systematic exploration of OPUC, although he had earlier discussed OPs on curves [419]. 1.8. Verblunsky’s Form of Szeg˝ o’s Theorem In this section, we give the final reformulation of Szeg˝o’s theorem as a sum rule and see that it implies a gem of spectral theory. The first element we need is the recursion relation obeyed by the monic OPUC, Φn (z), that will give us the parameters of the direct problem. We first define natural maps δn : L2 (R, dµ) to itself by (δn f )(eiθ ) = einθ f (eiθ )

(1.8.1)

Proposition 1.8.1. (i) δn is an anti-unitary map of L2 to L2 . (ii) If πn is the orthogonal projection onto the span of {z j }n−1 j=0 , then δn maps Ran πn+1 to itself. Indeed, if P ∈ Ran πn+1 , then (δn P )(z) = z n P (1/¯ z)

(1.8.2)

Equivalently, if P (z) =

n X

cj z j

(1.8.3)

j=0

then (δn P )(z) =

n X

c¯n−j z j

(1.8.4)

j=0

(iii) If f ∈ Ran πn+1 and f ⊥ {z, z 2 , . . . , z n }, then f is a multiple of δn (Φn ). Proof. (i) Multiplication by einθ is unitary and f → f¯ is antiunitary. (ii) Immediate from δn (z j ) = z n z j = z n−j . (iii) Since δn is anti-unitary and δn (z j ) = z n−j , δn f ⊥ {1, z, . . . , z n−1 }, so δn f = cΦn , and thus f = δn2 (f ) = c¯δn (Φn ).

˝ THEOREM 1.8. VERBLUNSKY’S FORM OF SZEGO’S

31

We now shift to standard, albeit unfortunate, notation and use Φ∗n for δn (Φn ) and, more generally, P ∗ for δn (P ). It is hoped that in dθ , Φn (z) = z n , Φ∗n = 1, context, the value of n is clear. But for dµ = 2π ∗ ∗ n ∗ n and (Φn ) = z becomes 1 = z . The notation is awful but, as I said, standard. Theorem 1.8.2 (Szeg˝o Recursion Relations). For any nontrivial measure dµ on ∂D, there exists constants αn ∈ C so that Φn+1 = zΦn − α ¯ n Φ∗n

(1.8.5)

Proof. Since Φn and Φn+1 are monic, Φn+1 − zΦn is a polynomial of degree n. Moreover, if j = 1, . . . , n, hz j , Φn+1 − zΦn i = hz j , Φn+1 i − hz j−1 , Φn i =0 so, by (iii) of Proposition 1.8.1, there is αn , so (1.8.5) holds. Applying

∗

on n + 1 degree polynomials, we obtain Φ∗n+1 = Φ∗n − αn zΦn

(1.8.6)

Φ∗n (0) = 1

(1.8.7)

The {αn }∞ n=0 are called Verblunsky coefficients. They are the analog of the Jacobi parameters for OPRL. The reason for the minus sign and complex conjugate will become clear later (see Theorem 2.5.2). Setting z = 0 in (1.8.5) and using the fact that Φn monic implies we have αn = −Φn+1 (0) The following is critical:

(1.8.8)

Theorem 1.8.3. We have that so that

kΦn+1 k2 = (1 − |αn |2 )kΦn k2

(1.8.9)

|αn | < 1

(1.8.10)

and if µ(∂D) = 1, then 2

kΦn k =

n−1 Y j=0

(1 − |αj |2 )

Remark. Of course, more generally, n−1 Y 2 2 kΦn k = (1 − |αj | ) µ(∂D) j=0

(1.8.11)

(1.8.12)

32


Proof. kΦn k2 = kzΦn k2 = kΦn+1 + α ¯ n Φ∗n k2 kΦ∗n k

= kΦn+1 k2 + |αn |2 kΦn k2

(1.8.13)

Φ∗n

since = kΦn k and ⊥ Φn+1 . (1.8.13) implies (1.8.9). kΦn k2 > 0 implies (1.8.10) and (1.8.11) follows by induction. From (1.8.11) and (1.8.5)/(1.8.6), we obtain ϕn+1 = ρ−1 ¯ n ϕ∗n ) n (zϕn − α

ϕ∗n+1

=

∗ ρ−1 n (ϕn

where

− αn zϕn )

ρn = (1 − |αn |2 )1/2 The same calculation that led to (1.8.13) implies

(1.8.14) (1.8.15) (1.8.16)

Theorem 1.8.4. If Φn (z0 ) = 0, then |z0 | < 1. If Φ∗n (z0 ) = 0, then |z0 | > 1. Proof. Since |z0 | < 1 ⇔ |1/z0 | > 1, the first sentence implies the second. If Φn (z0 ) = 0, let P (z) = Φn (z)/(z − z0 ) which is a polynomial of degree n − 1, so orthogonal to Φn . Then kP k2 = kzP k2 = k(z − z0 )P + z0 P k2 = kΦn + z0 P k2

= kΦn k2 + |z0 |2 kP k2

Since kΦn k2 > 0, |z0 | < 1.

(1.8.17)

By Theorem 1.8.3, dµ 7→ {αn (dµ)}∞ n=0 maps the nontrivial measure to D∞ . The following is fundamental to thinking of OPUC as a spectral problem: Theorem 1.8.5 (Verblunsky’s Theorem). The map of dµ 7→ {αn (dµ)}∞ n=0 is a one-one map of nontrivial probability measures onto D∞ . We will prove this in Section 2.5 (see Theorem 2.5.3); see also the Notes to this section. We can now state Verblunsky’s form of Szeg˝o’s theorem; by (1.8.11), the limit on the left of (1.7.13) is just an infinite product: Theorem 1.8.6 (Verblunsky’s Form of Szeg˝o’s Theorem). For any nontrivial probability measure dµ on ∂D with w given by (1.6.3), we have Z ∞ Y dθ 2 (1.8.18) (1 − |αn | ) = exp log(w(θ)) 2π n=0

˝ THEOREM 1.8. VERBLUNSKY’S FORM OF SZEGO’S

33

This is the version we will prove in Chapter 2; see Section 2.7. We note that it has two differences from Szeg˝o’s theorem, even the variant in Theorem 1.7.4. First, we have written it in terms of Verblunsky coefficients, and second, unlike Szeg˝o’s original version, this allows dµs 6= 0. One has the remarkable fact that the left side of (1.8.18) is independent of dµs ! (1.8.18) always holds, although both sides can be zero connected with a “divergent product” on the left and a diverging integral on the right. The two sides are nonzero at the same time, so we get the following gem: Corollary 1.8.7. For nontrivial probability measures dµ on ∂D obeying (1.6.3), Z ∞ X dθ 2 |αn | < ∞ ⇔ log(w(θ)) > −∞ (1.8.19) 2π n=0 Remarks and Historical Notes. The Szeg˝o recursion, (1.8.5), appeared first in 1939 in his famous book on orthogonal polynomials [424]. But at roughly the same time, they appeared in work of Geronimus [152, 153]. The history is murky, but especially as their proofs and presentations are different, it seems like Geronimus’ work was independent but several months later. Interestingly enough, an equivalent form was rediscovered by Levinson [271] about ten years later, and the engineering literature sometimes calls it as the Levinson or Levinson– Szeg˝o algorithm. Five years before Szeg˝o, the αn appeared in work of Verblunsky in two remarkable papers [440, 441] that were mainly ignored for almost seventy years! Verblunsky did not define the αn via a recursion relation, but in [440], he proved there were rational functions ζn (c0 , c1 , . . . , cn−1 ; c¯0 , . . . , c¯n−1 ) ∈ C and Rn (c0 , c1 , . . . , cn−1 ; c¯0 , . . . , c¯n−1 ) ∈ (0, ∞) so that if {cj }n−1 j=0 were moments of some nontrivial measure on ∂D, then the allowed values of cn for nontrivial measures were all the possible values in the open disk of radius Rn in C centered at ζn . He then defined αn−1 by cn = ζn + αn−1 Rn

(1.8.20)

This is discussed in Section 3.1 of [390]. Interestingly enough, the analog of this approach for OPRL was rediscovered by Krein [246], Karlin–Studden [208], and Krein–Nudelman [247], and codified in a book by Dette–Studden [107] who included the analysis of OPUC, thus reinventing [440]!

34


Theorem 1.8.4 goes back to Szeg˝o [420]. The proof we give is due to Landau [257]. [390] has six proofs of the theorem. In [440], Verblunsky also proved Theorem 1.8.5 using his definition of {αn }∞ n=0 . Other proofs of this theorem are presented in [390] and [389]. In particular, we mention the spectral theory proof, the analog of the proof of Favard’s theorem that we gave in Section 1.3. Of course, for that we need an analog of Jacobi matrices. The proper analog, the CMV matrix, will be discussed in Section 2.11. It is due to Cantero, Moral, and Velázquez [66] but essentially was discovered earlier by Amar, Gragg, Reichel, and Watson (see [394]) as a tool in numerical matrix analysis. See Chapter 4 of [390] and [394] for further discussions. Before [390, 391] introduced “Verblunsky coefficient,” the αn ’s had a wide variety of names: reflection coefficient, Schur parameter, Szeg˝o parameter, and Geronimus coefficient. In [441], Verblunsky proved Theorem 1.8.6. In particular, he had the sum rule (1.8.18) and he had a proof that allowed a singular part of the measure. Much of the literature since has attributed this singularpart-allowed result to and work was later and R Krein, whose P Kolmogorov dθ 2 which only proved |αn | = ∞ ⇔ log(w(θ)) 2π = −∞ with a singular part allowed. Others attributed the general result to Geronimus or Szeg˝o—again based on later work. It is also true that KdV sum rules should be viewed as analogs of Verblunsky’s sum rule, but the connection was not realized until many years later. Indeed, the Killip–Simon sum rules discussed in Section 1.10 were discovered in a chain going back to KdV sum rules without knowing of Verblunsky’s work. It was in tracking down the history of (1.8.18) that we uncovered [440, 441]. One of the consequences of Corollary 1.8.7 is the existence ofR mixed spectrum consistent with ℓ2 decay: Given any measure dρs with dρs < 1, there is a measure with a.c. support all of ∂D and that dρs and with P∞ 2 j=0 |αj | < ∞. Not knowing of this, the existence of analogous mixed spectral results for Schrödinger operators was regarded as a significant problem around 2000. 1.9. Back to OPRL: Szeg˝ o Mapping and the Shohat–Nevai Theorem We can translate the gem for OPUC to a result for OPRL using an interesting connection that Szeg˝o found in 1922 [421, 424]. It is connected to the natural conformal bijection of D → C ∪ {∞} \ [−2, 2] by z → E = z + z −1 (1.9.1)

˝ MAPPING AND THE SHOHAT–NEVAI THEOREM 1.9. SZEGO

35

This maps ∂D two-to-one to [−2, 2] by Q

eiθ −→ 2 cos θ

(1.9.2)

(Q♯ f )(eiθ ) = f (Q(eiθ )) = f (2 cos θ)

(1.9.3)

We can use this to map C([−2, 2]), the continuous functions on [−2, 2], to C(∂D): Notice Ran Q♯ is exactly the set of all functions invariant under eiθ → e−iθ . Duality then induces a map Q∗♯ : M+1,1(∂D) → M+,1 ([−2, 2]) between the probability measures by Z Z ∗ f (x)[Q♯ (dµ)](x) = (Q♯ f )(eiθ ) dµ(θ) (1.9.4)

Q∗♯ is onto M+,1 ([−2, 2]), but it is not one to one. For example, if L1 function with f (θ) + f (2π − θ) = 1 and R f is dθany nonnegative dθ dθ f (θ) 2π = 1, then Q♯ (f 2π ) = Q♯ ( 2π ) = π −1 (4 − x2 )−1/2 dx. However, restricted to measures invariant under θ → −θ, Q♯ is one-one, and we denote its restriction to even measures by Sz for Szeg˝o mapping. Thus dρ = Sz(dµ) if and only if dµ(θ) = dµ(−θ) and Z Z x f (θ) dµ(θ) = f arccos dρ(x) (1.9.5) 2

for any f obeying f (−θ) = f (θ). Sz is a bijection between nontrivial even probability measures on ∂D and nontrivial probability measures on [−2, 2]. Because of the impact of symmetry on Szeg˝o recursion, we see z ) ⇔ αn ∈ R for all n dµ even ⇔ Φn (z) = Φn (¯

(1.9.6)

Szeg˝o [421, 424] proved the following:

Theorem 1.9.1. Let dρ = Sz(dµ) for nontrivial probability measures on [−2, 2] and ∂D. Let Pn , pn be the monic and orthonormal OPRL for dρ and Φn , ϕn the monic and orthonormal OPUC for dµ. Then 1 Pn z + = [1 − α2n−1 (dµ)]−1z −n [Φ2n (z) + Φ∗2n (z)] (1.9.7) z kPn k2L2 (dρ) = 2(1 − α2n−1 )−1 kΦ2n k∗L2 (dµ) (1.9.8) 1 pn z + = [2(1 − α2n−1 )]−1/2 z −n (ϕ2n (z) + ϕ∗2n (z)) (1.9.9) z Sketch. (For details, see Theorem 13.1.5 of [391].) The right side Pn of (1.9.7) is a Laurent polynomial of the form j=−n cj z j invariant

36


under z → 1z on account of (1.9.6). Every such Laurent polynomial has the form Qn (z + 1z ) for Qn (·) of degree n. Since Φ2n (0) = −¯ α2n−1 , Φ∗2n (z) = −α2n−1 z 2n + · · · , so Qn is monic. Moreover, by (1.9.5) for ℓ < n, Z Z Qn (x)Qℓ (x) dρ(x) = Φ2n + Φ∗2n (z) z n−ℓ (Φ2ℓ + Φ∗2ℓ ) dµ(z)

(1.9.10)

=0 since Φ2n ⊥ {z, . . . , z 2n−1 } and Φ∗2n ⊥ {z, . . . , z 2n−1 }. Thus, the Qn ’s are the monic OPRL for dρ, that is, we have proven (1.9.7). (1.9.8) follows from (1.9.7) and hΦ2n , Φ∗2n i = hΦ2n , Φ∗2n−1 − α2n−1 zΦ2n−1 i

= −α2n−1 hΦ2n , Φ2n + α ¯ 2n−1 Φ∗2n−1 i

= −α2n−1 kΦ2n k2

(1.9.11)

by using Szeg˝o recursion and orthogonality. (1.9.9) is immediate from (1.9.7) and (1.9.8). There are several other relations we want to note because we will need them in Section 3.11. First, (1.9.9) can be written 1 1 −1/2 −n n pn z + = [2(1 − α2n−1 )] z ϕ2n (z) + z ϕ2n (1.9.12) z z By the same method, one can see 1 1 −1/2 −(n−1) (n−1) pn z + = [2(1 + α2n−1 )] z ϕ2n−1 (z) + z ϕ2n−1 z z (1.9.13) Besides dρ = Sz(dµ), there is a second (non-probability) measure one can associate to dµ, namely, dρ1 (x) ≡ Sz1 (dµ)(x)

= 14 (4 − x2 ) dρ(x)

(1.9.14)

Its orthonormal polynomials are denoted by qn (x). As with the derivation of (1.9.9), one finds −n ϕ2n (z) − zn ϕ2n ( 1z ) 1 −1/2 z 1 q z+ = [2(1 + α2n−1 )] 2 n−1 z z − z −1 (1.9.15)

˝ MAPPING AND THE SHOHAT–NEVAI THEOREM 1.9. SZEGO −1/2

= [2(1 − α2n−1 )]

37

z −(n−1) ϕ2n−1 (z) − z (n−1) ϕ2n−1 ( z1 ) z − z −1 (1.9.16)

This leads to

1 z ϕ2n (z) = (1 − α2n−1 )] pn z + z −1 1 1/2 z − z 1 + [ 2 (1 + α2n−1 )] qn−1 z + 2 z (1.9.17) 1 −(n−1) 1/2 1 z ϕ2n−1 (z) = [ 2 (1 + α2n−1 )] pn z + z −1 1 1/2 z − z 1 + [ 2 (1 − α2n−1 )] qn−1 z + 2 z (1.9.18) −n

[ 12

1/2

−1

When z = eiθ , pn (2 cos θ) and qn−1 (2 cos θ) are real, but z−z2 = i sin θ is pure imaginary, so the absolute value square has no cross term. Thus, we find the formula we will need in Section 3.11 |ϕ2n (eiθ )|2 + |ϕ2n−1 (eiθ )|2 = |pn (2 cos θ)|2 + sin2 θ|qn−1 (2 cos θ)|2 (1.9.19) where we used ([ 12 (1 + α2n−1 )]1/2 )2 + ([ 12 (1 − α2n−1 )]1/2 )2 = 1 to miraculously have α2n−1 drop out! From Theorem 1.9.1, we get the formula relating an , bn and αn : Theorem 1.9.2 (Direct Geronimus Relations). Let dρ = Sz(dµ) for nontrivial probability measures on [−2, 2] and ∂D. Let {an , bn }∞ n=1 be the Jacobi parameters for dρ and {αn }∞ n=0 the Verblunsky coefficients for dµ. Then 2n−2 Y

(i)

(a1 . . . an )2 = 2(1 + α2n−1 )

(ii)

a2n+1

(iii)

bn+1 = (1 − α2n−1 )α2n − (1 + α2n−1 )α2n−2

j=0

= (1 + α2n+1 )(1 −

2 α2n )(1

(1 − αj2 )

(1.9.20)

− α2n−1 )

(1.9.21) (1.9.22)

Remark. (i) holds for n ≥ 1 and (ii)/(iii) for n ≥ 0. For n = 1, (1.9.20) says a21 = 2(1 + α1 )(1 − α02 ), so (1.9.21) holds for n = 1 if we define α−1 = −1 (1.9.23) While α−2 enters in (1.9.22) for n = 0, it is multiplied by (1 +α−1) = 0, so only the “boundary condition” (1.9.23) is needed.

38


Sketch. (For details, see Theorems 13.1.7 and 13.1.12 of [391].) (i) Since 2 1 − α2n−1 = 1 + α2n−1 (1.9.24) 1 − α2n−1 this is a rewriting of (1.9.8) using (1.8.11) and (1.2.13). (ii) This follows from dividing (i) for n+1 by (i) for n using (1.9.24). (iii) This comes from (1.9.7) looking at the O(z n−1 ) terms. By a simple induction from (1.2.8), X n n Pn (x) = x − bj xn−1 + O(xn−2) (1.9.25) j=1

From (1.8.5) and (1.8.6), we get that if

Φn (z) = z n + Cn z n−1 + O(z n−2 )

(1.9.26)

Φ∗n (z) = −αn−1 z n + Dn z n−1 + O(z n−2 )

(1.9.27)

then, by induction,

Cn =

n−1 X

α ¯ j αj−1

(1.9.28)

j=0

(where, as usual, α−1 = −1) and

Dn = −αn−2 − αn−1 Cn−1

These formulae and (1.9.7) imply that n X − bj = C2n−1 − αn−2

(1.9.29)

(1.9.30)

j=1

and this yields (1.9.22).

This lets us “translate” Corollary 1.8.7 to OPRL: Theorem 1.9.3 (Shohat–Nevai Theorem). Let dρ(x) = f (x) dx + dρs (x) be supported on [−2, 2]. Then Z 2 (4 − x2 )−1/2 log f (x) dx > −∞ if and only if

(1.9.31)

−2

lim sup a1 . . . an > 0 If these conditions hold, then lim a1 . . . an

(1.9.32) (1.9.33)


exists in (0, ∞) and

∞ X n=1

and

(an − 1)2 + b2n < ∞

N X (an − 1)

N X

and

n=1

39

(1.9.34)

bn

(1.9.35)

n=1

have limits in (−∞, ∞).

Remarks. 1. We emphasize (1.9.32) is lim sup, that is, it allows lim inf to be 0 so long as some subsequence stays away from 0. 2. This can be rephrased as saying a1 . . . an always has a limit when supp(dρ) ⊂ [−2, 2] since the negation of (1.9.32) is lim a1 . . . an = 0. This is discussed further in Section 3.6. Proof. Let µ be defined by Sz(dµ) = dρ. By (1.9.30), 2

(a1 . . . an ) ≤ 4 Q∞

lim j=0 (1 − αj2 ) P∞ 2 j=0 αj < ∞.

2n−2 Y j=0

(1 − αj2 )

so (1.9.32) implies (the limit always exists) is strictly P 2 positive and thus Conversely, if j αj < ∞, then αj → 0 and so, by (1.9.20), lim a1 . . . an exists in (0, ∞). We have thus proven that ∞ X (1.9.32) ⇒ αj2 < ∞ ⇒ lim a1 . . . an exists in (0, ∞) (1.9.36) j=0

On the other hand, if

dµ = w(θ)

dθ + dµs 2π

(1.9.37)

then, by (1.9.5), w(θ) = 2π|sin θ|f (2 cos θ) (1.9.38) It follows that (changing variables, using x = 2 cos θ ⇒ dx = 2 sin θ dθ or dθ = (4 − x2 )−1/2 dx) Z Z dθ log(w(θ)) > −∞ ⇔ log f (x)(4 − x2 )−1/2 dx > −∞ (1.9.39) 2π

Thus, (1.8.19), (1.9.36), and (1.9.39) imply

lim sup(a1 . . . an ) > 0 ⇔ (1.9.31)

and if this holds, then (1.9.33) has a limit.

40


2 Since bn+1 built out of α2n+j P∞and2an+1 − 1 are P P(j 2= −2,2 −1, 0, 1), we 2 see that if j=0 αj < ∞, then bn < ∞ and (an − 1) < ∞. Since (an + 1) ≥ 1, (an − 1)2 = (a2n − 1)2 /(an + 1)2 ≤ (a2n − 1)2 , so (1.9.34) holds. P∞ 2 2 Finally, when j=0 αj < ∞, an+1 − 1 and bn+1 are the sum of an L1 sequence and a telescoping sequence, so a2n+1 − 1 and bn+1 are summable. Since (a2j − 1) − 2(aj − 1) = (aj − 1)2 is summable, we see that so is an+1 − 1.

We want to emphasize that while Corollary 1.8.7, on which Theorem 1.9.3 is based, is a gem (equivalence of purely spectral condition to purely sufficient condition), Theorem 1.9.3 is not. For it makes the a priori condition that supp(dρ) ⊂ [−2, 2], that is, it is the equivalence of (1.9.31) + supp(dρ) ⊂ [−2, 2] (1.9.40) to

(1.9.32) + supp(dρ) ⊂ [−2, 2]

(1.9.41)

(1.9.40) is purely spectral, but (1.9.41) is not a condition only about the Jacobi parameters. Indeed, supp(dρ) ⊂ [−2, 2] is a very strong restriction if lim sup(a1 . . . an ) > 0. Indeed, it implies strong conditions P PN 2 on the bn ’s ( ∞ n=1 bn < ∞ and n=1 bn conditionally convergent).

Remarks and Historical Notes. The Szeg˝o mapping was introduced by Szeg˝o in [421] and further discussed by him in [424]. Its purpose was to carry over asymptotics of OPUC when the Szeg˝o condition holds to asymptotics of OPRL when the OPRL Szeg˝o condition holds (see Section 3.7). dµ and dρ = Sz(dµ) can be related via their natural transforms Z iθ Z dρ(x) e +z dµ(θ) m(z) = (1.9.42) F (z) = iθ e −z x−z

namely,

F (z) = 2(z − z −1 )m(z + z −1 )

(1.9.43)

This formula is from Geronimus [155]; see also the proof of Theorem 13.1.2 in [391]. The map z → E = z + z −1 may seem miraculous, but it is canonical and uniquely determined. By the Riemann mapping theorem, there is an analytic bijection, g, of D to C ∪ {∞} \ [−2, 2] and it is uniquely determined by g(0) = ∞ and limz→0 zg(z) > 0. This unique map, abstractly guaranteed, is g(z) = z + z −1 . This will become a major theme in Chapter 9.


41

Geronimus [155, 156] found the relations (1.9.21)/(1.9.22). Other proofs can be found in Damanik–Killip [92], Killip–Nenciu [218], and Faybusovich–Gekhtman [124]. The latter two proofs are discussed in Section 13.2 of [391] and in Section 13.3 of the expected second edition of [391] which is posted online at http://www.math.caltech.edu/opuc/newsection13-3.pdf. Szeg˝o found a second natural map on nontrivial symmetric probability measures on ∂D to a large subset of measures on [−2, 2], the map we called Sz1 in (1.9.14). There are, in fact, four natural maps discussed in Section 13.2 of [391] and references therein. We note that all the original papers prior to 2000 use [−1, 1] not [−2, 2], and z → 12 (z + z −1 ). [391] discusses normalized measures (one needs to multiply dρ1 by 2[(1 − |α0 |2 )(1 − α1 )]−1 to normalize). For our purposes in Section 3.11, the unnormalized measure that leads to (1.9.19) is more convenient. Szeg˝o’s book [424] includes (1.9.12)–(1.9.15) (in Section 11.5) and he noted their inverses (in Section 6 of his appendix). The compact consequence in (1.9.19) is from Máté–Nevai–Totik [296]. It is interesting to check these formulae in case dµ = dθ/2π. Then 1 1 √ dx π 4 − x2 1 x = d arccos π 2

Sz(dµ)(x) =

and (Chebyshev polynomials of the first and second kinds) √ pn (2 cos θ) = 2 cos(nθ) √ sin((n + 1)θ) qn (2 cos θ) = 2 sin θ α2n−1 = 0 and, for example, (1.9.18) says √ sin(nθ) 1 √ 1 e−inθ e2niθ = √ 2 cos(nθ) + √ i sin θ 2 sin θ 2 2

(1.9.44) (1.9.45)

(1.9.46) (1.9.47)

(1.9.48)

Theorem 1.9.3 first appeared in Nevai [314] using in part ideas in Shohat [375]. We will eventually see (Theorem 3.6.1) that Theorem 1.9.3 can be extended to situations where there is some point spectrum outside [−2, 2], namely, we will need σess (dµ) = [−2, 2] and X dist(E, σess (dµ))1/2 < ∞ (1.9.49) E∈supp(dµ) E ∈[−2,2] /

42


1.10. The Killip–Simon Theorem As we noted, Theorem 1.9.3 is a spectral result about OPRL related to Szeg˝o’s theorem, but not a gem as we defined it. Here is an OPRL gem that is related to Szeg˝o’s theorem. It will involve the free Jacobi matrix, J0 , whose Jacobi parameters are an ≡ 1 bn ≡ 0 (1.10.1) The OPs for this case are (as is easy to check obey the recursion relations on account of trigonometric addition formula; these are essentially the Chebyshev polynomials of the second kind; see (1.2.35)) Pn (2 cos θ) =

sin(n + 1)θ sin θ

(1.10.2)

The spectral measure is dρ0 (x) =

1 (4 − x2 )1/2 dx 2π

(1.10.3)

so that σ(J0 ) = σess (J0 ) = σac (J0 ) = [−2, 2]

(1.10.4)

Theorem 1.10.1 (Killip–Simon Theorem). Let {an , bn }∞ n=1 be the Jacobi parameters of a Jacobi matrix, J. Then ∞ X (an − 1)2 + b2n < ∞ (1.10.5) n=1

if and only if (a) σess (J) = σess (J0 ) (Blumenthal–Weyl) (b) The eigenvalues En ∈ / σess (J0 ) obey ∞ X dist(En , σess (J0 ))3/2 < ∞ (Lieb–Thirring)

(1.10.6)

(1.10.7)

n=1

(c) The function f of (1.4.3) obeys Z dist(x, R \ σ(J0 ))1/2 log(f (x)) dx > −∞

(Quasi-Szeg˝ o)

σ(J0 )

(1.10.8)

Remarks. 1. (1.10.5) is equivalent to J − J0 being a Hilbert– Schmidt operator (see [166, 372]). 2. (1.10.8) is called “quasi-Szeg˝o” because it looks like the Szeg˝o condition (1.9.30) except − 21 has become 21 , allowing a larger class of f ’s. Similarly, (1.10.7) looks like (1.9.49) except that 12 has become 32 .

1.10. THE KILLIP–SIMON THEOREM

43

The proof of Theorem 1.10.1 will be the main topic of Chapter 3, but to set the stage we want to say something about it. As with Szeg˝o’s theorem, the key is a sum rule. It will involve two somewhat complicated-looking functions, F defined on R\[−2, 2] and G on (0, ∞): F (β + β −1 ) =

1 4

[β 2 − β −2 − log(β 4 )]

β ∈ R \ [−1, 1]

G(a) = a2 − 1 − log(a2 )

(1.10.9)

(1.10.10)

Notice that β 7→ β + β −1 is a bijection of R \ [−1, 1] to R \ [−2, 2] so (1.10.9) defines F . We will eventually show that (Lemma 3.5.3) Z |E| 1 (E 2 − 4)1/2 dE (1.10.11) F (E) = 2 2

which implies and

F (E) > 0 on R \ [−2, 2]

(1.10.12)

F (E) = 23 (|E| − 2)3/2 + O((|E| − 2)5/2 ) We also see that (Lemma 3.5.2) G(a) > 0 on (0, ∞) \ {1} 2

(1.10.13) (1.10.14)

3

G(a) = 2(a − 1) + O((a − 1) )

We also need to define √ Z 2 1 4 − x2 √ Q(ρ) = log 4 − x2 dx 4π −2 2πf (x)

which, given (1.10.3), can be rewritten −1 Z dρ 1 Q(ρ) = − 2 log dρ0 dρ0

(1.10.15)

(1.10.16)

(1.10.17)

whose integral is a relative entropy (see (2.2.1)). As we will show (Theorem 2.2.3), using Jensen’s inequality, Q(ρ) ≥ 0. The sum rule is Theorem 1.10.2. Let dρ be a nontrivial probability measure with associated Jacobi parameters {an , bn }∞ n=1 and σess (dρ) = [−2, 2]. Then ∞ X X Q(ρ) + F (En ) = [ 14 b2n + 12 G(an )] (1.10.18) n=1

This is called the P2 sum rule. Notice all terms on both sides are positive so the sums always make sense, but they may be infinite. Moreover, σess (dρ) = [−2, 2] and LHS of (1.10.18) < ∞ if and only if (a)–(c) of Theorem 1.10.1 holds, on account of (1.10.13) and (1.10.16). On the other hand, using Theorem 1.4.1 and (1.10.15), σess (dρ) =

44


[−2, 2] and RHS of (1.10.18) < ∞ if and only if (1.10.5) holds. Thus, Theorem 1.10.2 implies Theorem 1.10.1. Where will complicated objects like F and G come from? The sum rule of Verblunsky (1.8.18) is a form of Jensen’s equality for analytic functions, hence the logs. In this case, the function is nonvanishing. The sum rule (1.10.18) will come from a Jensen–Poisson equality and involves two Taylor coefficient: the zeroth which has logs and the second without logs. There are terms from the zeros in this case, hence the logs in the sum involving F . These details will unfold in Chapter 3. Remarks and Historical Notes. Theorems 1.10.1 and 1.10.2 are from Killip–Simon [220]. For historical context and the name “P2 ,” see the Notes to Sections 3.1 and 3.4. 1.11. Perturbations of the Periodic Case The material in Chapters 5, 6, and 8 is all connected with analyzing Szeg˝o-like theorems for OPRL (and some related OPUC) where the [−2, 2] of Theorem 1.10.1 is replaced by a union of a finite number of closed bounded intervals, especially the case of perturbations of periodic OPRL. Chapters 5 and 6 discuss periodic OPRL themselves, that is, Jacobi matrices, J0 , where (0)

an+p = a(0) n

(0)

bn+p = b(0) n

(1.11.1)

for some p ≥ 2 and all n = 1, 2, . . . . (In Section 5.14, we also discuss (0) (0) OPUC when αn+p = αn , mainly with p even.) Rather than studying an , bn which approach an ≡ 1, bn ≡ 0 in some sense, we want to discuss (0) (0) approach to J0 . J0 is obviously parametrized by R2p = {(an , bn )pn=1 }. We begin the discussion by describing σ(J0 ), the spectrum of J0 (see Sections 5.2, 5.3, and 5.4): Theorem 1.11.1. σess (J0 ) is the disjoint union of k +1 ≤ p distinct bounded intervals k+1 [ σess (J0 ) = [cj , dj ] (1.11.2) j=1

where

c1 < d1 < c2 < · · · < ck+1 < dk+1

Each of the k gaps (dj , cj+1 ), j = 1, . . . , k, has zero or one point mass. (0)

(0)

Generically, k = p − 1. Indeed, {(an , bn ) | k < p − 1} is a variety of codimension 2 in R2p . If k = p − 1, we say “all gaps are open.”

1.11. PERTURBATIONS OF THE PERIODIC CASE

45

While we will not say a lot about the proof now, we do want to mention one of the key tools. There is a natural polynomial in x, (0) (0) ∆(x; {an , bn }pn=1 ) = ∆(x; J0 ) of exact degree p, so σess (J0 ) = ∆−1 ([−2, 2])

(1.11.3)

We are interested in the analog Theorem 1.10.1 when J0 is a periodic Jacobi matrix. The conjectured analog of the spectral side is obvious: (1.10.6)–(1.10.8) were carefully stated in terms of σess (J0 ) rather than [−2, 2] precisely because they will be one side of the proper periodic theorem. There is an obvious guess for an analog of (1.10.5), namely, ∞ X n=1

2 (0) 2 (an − a(0) n ) + (bn − bn ) < ∞

(1.11.4)

This cannot be right for the following reason. The map (1)

(1)

(1) (1) (1) J1 = {(a(1) n , bn ) | an+p = an , bn+p = bn } → ∆(x, J1 )

(1.11.5)

is a map of R2p to Rp+1 , since ∆ has p + 1 coefficients. As one would expect, generic inverse images of a fixed ∆ are of dimension 2p − (p + 1) = p − 1. In fact, we will show (see Section 5.13) Theorem 1.11.2. For fixed periodic J0 , {J1 | ∆(x, J1 ) = ∆(x, J0 )} is a torus of dimension k where k + 1 = # of components of σess (J0 )

(1.11.6)

This set is called the isospectral torus of J0 which we denote TJ0 . By (1.11.3), if J1 ∈ TJ0 , σess (J1 ) = σess (J0 ), and so J1 also obeys (1.10.6)–(1.10.8), but J1 does not obey (1.11.4). What we need is not ℓ2 approach to a fixed J0 but rather is TJ0 . We define ′ ′ ∞ dm ((an , bn )∞ n=1 , (an , bn )n=1 )

=

∞ X

j=m

e−|j−m| [|aj − a′j | + |bj − b′j |] (1.11.7)

which measures the distances of the tails from each other. We also define dm ((an , bn )∞ n=1 , TJ0 ) =

min

(a′n ,b′n )∈TJ0

dm ((a, b), (a′ , b′ ))

(1.11.8)

It can happen that the minimizing (a′ , b′ ) is m-dependent and that dm ((a, b), TJ0 ) → 0 as m → ∞ without dm ((a, b), J1 ) → 0 for any J1 (although, by compactness of TJ0 , there will be J1 and a subsequence for which dmℓ ((a, b), J1 ) → 0 as ℓ → ∞). Damanik–Killip–Simon [93] have proven:

46


Theorem 1.11.3 (DKS [93]). Let J0 be a fixed periodic Jacobi matrix of period p with all gaps open (i.e., k = p − 1). Let J be another bounded Jacobi matrix with Jacobi parameters (an , bn )∞ n=1 . Then the following are equivalent: (a) (1.10.6), (1.10.7), and (1.10.8) hold. (b) ∞ X dm ((a, b), TJ0 )2 < ∞ (1.11.9) m=1

The proof of this theorem is the main goal of Chapter 8. A key tool will be the study of the matrix ∆(J; J0 ), that is, the matrix obtained by placing J for x in the polynomial ∆(x; J0 ). Since ∆ has degree p, ∆(J) will be a matrix of band width 2p + 1, that is, p diagonals strictly above, p strictly below, and on the main diagonal. Such a matrix can be thought of as “tridiagonal” if we replace a’s and b’s by p × p blocks. We will prove a Killip–Simon theorem for such block Jacobi matrices in Chapter 4, and that will be a main tool in proving Theorem 1.11.3. In the periodic case, σess (J0 ) is a disjoint union, (1.11.2). But not every such union is σess (J0 ) for some periodic J0 . Basically, there is a natural map (harmonic measure), k+1 X k+1 M : {c1 < d1 < c2 < · · · < dk+1} → (θj )j=1 θj > 0; θj = 1 j=1

which is continuous and onto. The allowed σess (J0 ) for periodic J0 ’s with all gaps open is M((c, d)) = ( p1 , . . . , 1p ), and if we drop the demand that all gaps are open, then the range is the set of rational θ’s. For other finite band sets, σess (J0 ) can be that set if we allow certain almost periodic J0 ’s. There is no Killip–Simon-type theorem known in this case, but one-half of a Shohat–Nevai-type theorem is known due to work of Akhiezer, Widom, Aptekarev, and Peherstorfer–Yuditskii. It will be the subject of Chapter 9. Chapter 10 will discuss Szeg˝o-like theorems for perturbations of the graph Laplacian on a Bethe–Cayley tree.

Remarks and Historical Notes. As noted, Theorem 1.11.3 is from Damanik–Killip–Simon [93]. Prior results and historical context are discussed in the Notes to Section 8.1. The history of results mentioned in the last paragraph are in the Notes to Section 9.13. 1.12. Other Gems in the Spectral Theory of OPUC While gems are the leitmotif of this chapter, our choice of topics is motivated by looking at relatives of Szeg˝o’s theorem. We will see

1.12. OTHER GEMS IN THE SPECTRAL THEORY OF OPUC

47

that in this section by mentioning some other gems for OPUC (the Notes discuss OPRL) that will not be discussed further. Here are three theorems in particular: Theorem 1.12.1 (Baxter’s Theorem). Let µ be a probability measure on ∂D of the form (1.6.3) and let {αn }∞ n=1 be its Verblunsky coefficients. Then the following are equivalent: (i) ∞ X |αn | < ∞ (1.12.1) n=0

(ii) dµs = 0,

inf w(θ) > 0 ∞ X |w bn | < ∞

(1.12.2) (1.12.3)

n=−∞

where

w bn =

Z

e−inθ w(θ)

dθ 2π

(1.12.4)

Remark. (1.12.3) implies w is continuous, so the inf in (1.12.2) is a min. Theorem 1.12.2 (Ibragimov’s Form of the Strong Szeg˝o Theorem). Let µ be a probability measure on ∂D of the form (1.6.3) and let {αn }∞ n=1 be its Verblunsky coefficients. Then the following are equivalent: (i) ∞ X n|αn |2 < ∞ (1.12.5) n=0

(ii) dµs = 0, the Szeg˝o condition (1.8.19) holds, and ∞ X b n |2 < ∞ n|L

(1.12.6)

n=1

where

bn = L

Z

e−inθ log(w(θ))

dθ 2π

(1.12.7)

Theorem 1.12.3 (Nevai–Totik Theorem). Let µ be a probability measure on ∂D of the form (1.6.3) and let {αn }∞ n=1 be its Verblunsky coefficients. Let R > 1. Then the following are equivalent: (i) lim sup|αn |1/n ≤ R−1 (ii) µs = 0 and the Szeg˝o function D, defined by (2.9.14), has D −1 (z) analytic in {z | |z| < R}.

48


There are two distinctions between these results and Szeg˝o’s theorem. These only involve µ’s with µs = 0 and with more rapid decay than just ℓ2 , if αn ∼ Cn−s ; Szeg˝o requires s > 12 , but these require s > 1 (and exponential decay in the case of the Nevai–Totik theorem). Remarks and Historical Notes. Baxter’s theorem is from Baxter [32] and is discussed in [390, Ch. 5]. Ibragimov’s form is from Ibragimov [198] and related to Szeg˝o’s work on the second term in Toeplitz determinant asymptotics discussed in the Notes to Section 1.6 where references appear. The Nevai–Totik theorem is from Nevai–Totik [317] and discussed in [390, Ch. 7]. For analogs of Theorems 1.12.1 and 1.12.2 for OPRL, see Ryckman [367, 366]. For an OPRL analog of Theorem 1.12.3, see Damanik– Simon [96].

CHAPTER 2

Szeg˝ o’s Theorem In algebra, when one says a = b, it is a tautology and so uninteresting; while in analysis, when one says a = b, it is two deep inequalities. – attributed to S. Bochner If one only proves a = b by showing a ≤ b and b ≤ a, one has not understood the true reason that a = b. – attributed to E. Noether

In this chapter we will prove Szeg˝o’s Theorem in Verblunsky’s Form (Theorem 1.8.6). Our main thrust will be a proof that extends to the other situations we wish to discuss in later chapters. The Szeg˝o case is simpler than these later ones because the underlying analytic functions have neither zeros nor poles in D, so we will only need that if f is nonvanishing and analyticRin D and log f (z) is in some Hardy class H p dθ (p ≥ 1), then f (0) = exp( log f (eiθ ) 2π ). In later chapters, we have to use Blaschke products to accommodate poles and zeros that can occur. Section 2.1 lays out the strategy of this approach. The last steps establish the sum rule by proving complementary inequalities. One inequality will depend on the realization of integrals involving logs as a relative entropy and semicontinuity properties of entropy—the subject of Section 2.2. Section 2.3 is a mini-course on functions on D and on C+ = {z | Im f > 0} relevant to spectral theory. In Sections 2.4 and 2.5, we turn from generalities back to the specifics of OPUC. By discussing second kind polynomials and Weyl solutions, we can prove the basics, especially coefficient stripping, the relation between dµ and dµ(1) defined by αn (dµ(1) ) = αn+1 (dµ). With those basics, in Section 2.6 we construct the function needed for Step 1 in our strategy, and then we implement this strategy in Section 2.7. The next six sections are extensions and alternate approaches. Section 2.8 discusses higher-order Szeg˝o theorems, Section 2.12 presents Szeg˝o’s variational approach to his theorem, three sections (2.9, 2.10, and 2.13) discuss asymptotics of OPUC and of Weyl solutions, and Section 2.9 has several additional topics. 49

˝ THEOREM 2. SZEGO’S

50

In the last four sections, we study asymptotics of the CD kernel, a subject we return to in Sections 3.11, 3.12, and 5.11. 2.1. Statement and Strategy Given a nontrivial probability measure on ∂D, dθ + dµs (θ) (2.1.1) 2π with dµs singular, recall that we define monic OPUC, Φn (z), and orthonormal ϕn (z) = Φn (z)/kΦn k. Recall that the Verblunsky coefficients {αn (dµ)}∞ n=0 are given by dµ(θ) = w(θ)

αn = −Φn+1 (0)

(2.1.2)

Φ∗n (z) = z n Φn (1/¯ z)

(2.1.3)

The Szeg˝o dual Φ∗n (z) is given by

and the Szeg˝o recursion relations by Φn+1 (z) = zΦn (z) − α ¯ n Φ∗n (z) Φ∗n+1 (z) = Φ∗n (z) − αn zΦn (z) zϕn (z) = ρn ϕn+1 (z) +

α ¯ n ϕ∗n (z)

ϕ∗n (z) = ρn ϕ∗n+1 (z) + αn zϕn (z)

(2.1.4) (2.1.5) (2.1.6) (2.1.7)

where Moreover, if

ρn = (1 − |αn |2 )1/2 n−1 Y

(2.1.8)

ρ−1 j

(2.1.9)

ϕn (z) = κn z n + lower order

(2.1.10)

κn =

j=0

then kΦn k = κ−1 n

We discussed several variants of Szeg˝o’s theorem in the last chapter. In this chapter, our goal is to prove the following (which implies the others and has the gem, Corollary 1.8.7, as a consequence): Theorem 2.1.1 (Verblunsky’s Form of Szeg˝o’s Theorem). For any nontrivial probability measure on ∂D, we have that Z ∞ Y dθ 2 (2.1.11) (1 − |αn | ) = exp log(w(θ)) 2π n=0

2.1. STATEMENT AND STRATEGY

51

R dθ Recall log(w(θ)) 2π can only diverge to −∞, in which case we Q 2 interpret the right side as e−∞ = 0. The product N n=1 (1 − |αn | ) is monotone decreasing in N, so the limit exists although it may be zero. In this section, we describe the overall strategy that we will use. The first problem with (2.1.11) is how one can hope to prove it when R dθ = −∞ where both sides are singular. Our strategy will log(w(θ)) 2π be to find a result that is always finite and always holds. Let dµ1 be the measure defined by dropping α0 and shifting the other α’s down, that is, αj (dµ1 ) = αj+1 (dµ)

(2.1.12)

We call the process “coefficient stripping.” Write dµ1 (θ) = w1 (θ)

dθ + dµs,1 (θ) 2π

(2.1.13)

More generally, let dµN be given by αj (dµN ) = αj+N (dµ)

(2.1.14)

and dθ + dµs,N (2.1.15) 2π Formally, if (2.1.11) holds for dµ and dµ1 and we divide, we get what we will call the step-by-step sum rule Z w(θ) dθ 2 (2.1.16) (1 − |α0 | ) = exp log w1 (θ) 2π dµN = wN (θ)

The key to our proof of (2.1.11) will be to prove that (2.1.16) is always true if suitably interpreted. The phrase “if suitably interpreted” is needed because w(θ) and/or w1 (θ) may vanish on a set of positive measure. What we will prove is dθ that there is a nonnegative function g(θ) so log(g(θ)) ∈ ∩p 0}| = 0

(2.1.34)

where |·| is Lebesgue measure. This is an optimal result in the sense that for any p > 2, there are measures µ which are purely singular but ∞ X n=0

|αn |p < ∞

(2.1.35)

All constructions of such measures have some subtlety but there are many such constructions at this point such as: (i) A method, dubbed Totik’s workshop in [390, Sect. 2.10], due to Totik [431] that shows for any measure, γ, with supp(γ) = D, there is µ mutually equivalent to γ so (2.1.35) holds for all p > 2. (ii) Using Riesz products, Khrushchev [215] constructed singular continuous measures with (2.1.35) for all p > 2; see [390, Sect. 2.11]. (iii) As discussed in [391, Sect. 12.7], if {αj (ω)}∞ j=0 are independent 2 random variables with E(αj (ω)) = E(αj (ω) ) = 0, supω,j |αj (ω)| < 1, supω |αj (ω)| → 0, and for Γ > 0, E(|αj (ω)|2)1/2 = Γj −1/2

for j large (e.g., if βj (ω) are independent, identically distributed random variables, uniformly distributed on {z | |z| = 12 }, one can take αj (ω) = min( 12 , 2Γj −1/2 )βj (ω)), then for a.e. ω, the corresponding measure has no a.c. spectrum. If Γ2 > 1, µ is pure point, and if Γ2 ≤ 1, the spectrum is purely singular continuous of Hausdorff dimension 1 − Γ2 . While the OPUC case is from [391], it is motivated by an OPRL paper of Kiselev, Last, and Simon [222]; see [391] for earlier papers on OPRL with decaying random potentials (iv) It is known that generically slow decay yields purely singular continuous spectrum; see [391, Sect. 12.4]. Explicitly,Pfor any p > 2 ∞ p and C < 1, a dense Gδ in {{αj }∞ j=0 | supj |αj | ≤ C, j=0 |αj | < ∞} in the ℓp metric has an associated measure with purely singular continuous spectrum. Also, for any k < 12 and C > 0, a dense Gδ in k k {{αj }∞ j=0 | kαkC,k = supj (j + 1 + C) |αj | ≤ 1 and j |αj | → 0} in

˝ INTEGRAL AS AN ENTROPY 2.2. THE SZEGO

55

k·kC,k norm has an associated measure with purely singular continuous spectrum. This relies on the Wonderland theorem of Simon [384]. (v) One can construct sparse (i.e., αj mainly zeros with the nonzero p values very far apart) {αj }∞ j=0 in ℓ for all p > 2 so that the associated measures are purely singular continuous; see Golinskii [171] and [391, Sect. 12.5] and see the notes for the motivating Schrödinger operator papers. Lest one thinks decay slower than n−1/2 always means no a.c. spectrum, we note (see [391, Sect. 12.1] and thePreference to Golinskii– Nevai [173] and earlier works there) that if ∞ n=0 |αn+1 − αn | < ∞, then there is pure a.c. spectrum on ∂D \ {1}. 2.2. The Szeg˝ o Integral as an Entropy In this section, we will prove Theorem 2.1.2 as a special case of a more general result concerning relative entropy. This object is defined by Definition. Let µ, ν be two (positive) measures on a compact metric space. Define their relative entropy by ( −∞ if µ is not ν-a.c. R S(µ | ν) = (2.2.1) dµ − log( dν ) dµ if µ is ν-a.c. Notice that if dν is fixed and dµ = g dν, then Z S(g dν | dν) = −g log(g) dν

(2.2.2)

x 7→ −x log x is concave (its second derivative is −1/x) and is sometimes called the entropy function. If dν is a counting measure on a finite set and dµ a probability P measure on the same set with µ({j}) = gj , then RHS of (2.2.2) j −gj log gj , the familiar entropy of statistical mechanics courses.

Example 2.2.1. Let dµ0 = dθ/2π and dµ given by (1.6.3). Then, if w > 0 for a.e. θ, dµ0 is dµ-a.c., and dµ0 /dν = w −1 so log(dµ0 /dν) = − log(w) and Z dθ dθ dθ S w + dµs = log(w(θ)) (2.2.3) 2π 2π 2π If w > 0 for a.e. θ is false, then dµ0 is not dµ-a.c., and both sides of (2.2.3) are infinite and (2.2.3) still holds. We thus see that the Szeg˝o integral is a relative entropy. That will also be the case for other objects in sum rules, for example, two times the negative of the function Q in (1.10.16) (see (1.10.17)).


56

The key to controlling S is Proposition 2.2.2 (Linear Variational Principle for the Entropy). Let E(X) be the family of strictly positive continuous functions on X. Then S(µ | ν) = inf S(f ; µ, ν) (2.2.4) f ∈E(X)

where S(f ; µ, ν) =

Z

f (x) dν(x) −

Z

(1 + log(f (x)) dµ(x)

(2.2.5)

Sketch. (For details, see Lemma 2.3.3 of [390].) Define for b > 0, x > 0, Qb (x) = xb−1 − 1 − log(x) (2.2.6) ′ −1 −1 ′′ −2 Then Qb (x) = b − x and Qb (x) = x . Thus, Qb is convex in x and its derivative vanishes at x = b where Qb (x) = − log b. Since a smooth convex function with a zero derivative at some point takes its minimum at the point where the derivative vanishes, we have Qb (x) ≥ − log(b)

(2.2.7)

Suppose dµ is dν-a.c. Let g = dµ/dν and A = {x | g(x) 6= 0}. Then dν = χX\A dν + g −1 dµ (2.2.8) and, for f ∈ E, Z Z S(f ; µ, ν) = f (x) dν(x) + Qg(x) (f (x)) dµ(x) (2.2.9) X\A A Z ≥ − log(g(x)) dµ(x) (2.2.10) = S(µ | ν)

(2.2.11)

where (2.2.10) follows from (2.2.7). If g is continuous and strictly positive, choose f = g. Then S(g; g dν, ν) = S(µ | ν)

(2.2.12)

which proves (2.2.4) in case dµ = g dν with g continuous and nonvanishing. The proof can be completed using two approximation arguments. One approximates any g by strictly positive continuous g’s to prove (2.2.4) in the general case where µ is ν-a.c. The other uses very large g’s approximately supported on a set, A, where ν(A) = 0, µ(A) > 0 to show the RHS of (2.2.4) is −∞ if µ is not ν-a.c. As an immediate corollary, we have

˝ INTEGRAL AS AN ENTROPY 2.2. THE SZEGO

57

Theorem 2.2.3. S(µ | ν) is jointly concave and jointly weakly upper semicontinuous in µ, ν. Moreover, if µ(X) = ν(X) = 1

(2.2.13)

S(µ | ν) ≤ 0

(2.2.14)

then Remarks. 1. Joint concavity means for 0 ≤ θ ≤ 1,

S(θµ1 + (1 − θ)µ0 | θν1 + (1 − θ)ν0 ) ≥ θS(µ1 | ν1 ) + (1 − θ)S(µ0 | ν0 ) (2.2.15) w w 2. Upper semicontinuity means µn −→ µ, νn −→ ν implies lim sup S(µn , νn ) ≤ S(µ, ν)

(2.2.16)

Proof. S(f ; µ, ν) is linear and weakly continuous jointly in µ, ν for any f ∈ E(X). Thus, by (2.2.4), S(µ | ν) is concave and upper semicontinuous. Noticing that if (2.2.13) holds, then S(f ≡ 1; µ, ν) = 0, we obtain (2.2.14) from (2.2.4). Corollary 2.2.4 (≡ Theorem 2.1.2). If N is given by (2.1.26), then (2.1.28) holds. Proof. Follows from (2.2.16) since dθ N(dµ) = −S 2π

by (2.2.3).

µ

(2.2.17)

Example 2.2.5. Here are some examples that show S is only upper semicontinuous and not continuous. Let N −1 1 X dθ dµ∞ = dµN = δ2πj /N 2π N j=0 w

Then dµN −→ dµ∞ but

S(dµ∞ | dµN ) = −∞

S(dµ∞ | dµ∞ ) = 0

(2.2.16) holds, but clearly, there is no equality. Another example where measures are mutually a.c. is dθ 1 dθ dµ∞ = dµN = 1 + cos(Nθ) 2π 2 2π w

Then dµN −→ dµ∞ , and by scaling,

S(dµ∞ | dµN ) = S(dµ∞ | dµ1 ) < 0 = S(dµ∞ | dµ∞ )


58

Finally, we note the more usual proof of (2.2.14). It depends on Theorem 2.2.6 (Jensen’s Inequality). If F is convex on Rn , then for any probability measure dµ on Rn , Z Z F ~x dµ(~x) ≤ F (~x) dµ(~x) (2.2.18) Remark. As our proof shows, this result holds if F is defined on a convex set, A, in Rn so long as dµ is supported there. Proof. Convexity implies for each j, (Dj+ F )(x0 ) = limy↓0 [F (x0 + yδj ) − F (x0 )]/y exists for each x0 ∈ Rn , and for all x, Pick x0 =

R

F (x) − F (x0 ) ≥ (x − x0 ) · (D + F )(x0 )

~x dµ(x) and integrate (2.2.19) dµ0 to get (2.2.18).

(2.2.19)

Alternate Proof of (2.2.14). Since − log(·) is convex on (0, ∞), Jensen’s inequality implies that if dµ = g dν and A = {x | g(x) 6= 0}, then Z S(µ | ν) = log(g −1) dµ A Z −1 ≤ log g dµ A

= log(ν(A)) ≤ 0

Remarks and Historical Notes. Entropy was discovered in thermodynamics and understood in statistical mechanics. That entropy P has the form of − pj log pj is a discovery of Boltzmann. Variational principles go back to Gibbs. His variational principle in this context says: Z Z g S(µ | ν) = inf log e dν − g dµ (2.2.20) g∈C(X)

It is not hard to prove his relation from (2.2.4); see Section 10.6. For discussion of entropy in statistical mechanics, see Israel [200], Ruelle [364, 365], or Simon [383]. For a mathematical discussion of entropy, see Carl–Stephani [69], Ellis [117], Gray [178], Ohya–Petz [321], or Parry [325]. While he didn’t know it was entropy he was using, Verblunsky [441] proves the Gibbs variational principle for the Szeg˝o integral, namely, Z R g e dµ dθ R dθ = exp log(w(θ)) inf 2π exp( g 2π )

´ 2.3. CARATHEODORY, HERGLOTZ, AND SCHUR FUNCTIONS

59

and used it to prove a semicontinuity result. The use of entropy in proving sum rules was then rediscovered by Killip–Simon [220]. 2.3. Carath´ eodory, Herglotz, and Schur Functions One of the surprises (but which I’ve already strongly hinted at) is that complex analysis is a central tool of the spectral analysis of orthogonal polynomials. We will eventually see that techniques from Riemann surface theory, namely, Abelian integrals (see Sections 5.12, 5.13, and 9.11) and covering spaces (see Sections 9.2–9.5) will enter. In this section, we discuss more conventional boundary value theory. Definition. A Carathéodory function is an analytic function, F (z), on D so Re F (z) > 0 F (0) = 1 (2.3.1) A Herglotz function is an analytic function, G(z), on so that on C+ ,

C+ = {z | Im z > 0}

Im G(z) > 0 A Schur function is an analytic function, f , on D so that |f (z)| ≤ 1

(2.3.2) (2.3.3) (2.3.4)

Remarks. 1. Herglotz functions are also called Pick functions or Nevanlinna functions. 2. By the maximum principle, either (2.3.4) can be strengthened to |f (z)| < 1 that is, f : D → D, or else f is constant f (z) = w0 ∈ ∂D

(2.3.5)

Example 2.3.1. The following shows the close connection between Herglotz functions and OPRL. Let dρ be a measure on R with Z (1 + |x|)−1 dρ(x) < ∞ (2.3.6)

Let

dρ(x) x−z Z dρ(x) Im m(z) = Im z |x − z|2 m(z) =

Then

so m is Herglotz.

Z

(2.3.7) (2.3.8)


60

R Suppose now that dρ has compact support and dρ(x) = 1. Then writing (x − z)−1 = −z −1 − x(x − z)−1 z −1 (2.3.9) we see that

m(z) = −z −1 + O(z −2 )

(2.3.10)

This motivates a definition: Definition. A discrete m-function is a Herglotz function, m(z), so that for some bounded interval I ⊂ R, we have that m(z) has an analytic continuation from C+ to C \ I with z ∈ R \ I ⇒ Im m(z) = 0

(2.3.11)

and (2.3.10) holds. It is easy to see that, given the analyticity assumption, (2.3.11) is equivalent to (2.3.12) m(¯ z ) = m(z) We will shortly prove (see Theorem 2.3.6) that every discrete mfunction has the form (2.3.7) for a probability measure dρ on I. For now, we note Proposition 2.3.2. Suppose m(z) has the form (2.3.7) where supp(dρ) ⊂ [−R, R] for some R. Let cn be the moments of dρ: Z cn = xn dρ(x)

(2.3.13)

(2.3.14)

Then for |z| > R, we have an absolutely convergent series m(z) = −

∞ X

cn z −(n+1)

(2.3.15)

n=0

Proof. Immediate from the geometric series expansion, uniformly and absolutely convergent on|z| > R + ε for each ε > 0, (x − z)

−1

=−

∞ X

xn z −(n+1)

(2.3.16)

n=0


61

If R is the minimum value for which (2.3.13) holds, it is easy to see the Taylor series at infinity (2.3.15) diverges if |z| < R. We will eventually find Padé approximants (see the remark after Proposition 3.2.8) that converge on all of C \ I. Indeed, the numerator and denominator will be orthogonal polynomials! Equivalently, we will find continued fraction expansions in terms of the Jacobi parameters. We will see all this in Section 3.2 and its OPUC analog in Section 2.5. To figure out the OPUC analog of (2.3.7), we need the complex Poisson representation: Proposition 2.3.3. Let f be analytic in a neighborhood of D. Then for z ∈ D, Z iθ e +z iθ dθ f (z) = i Im f (0) + Re f (e ) (2.3.17) eiθ − z 2π Proof. f has a Taylor series converging for |z| < 1 + ε f (z) =

∞ X

an z n

(2.3.18)

n=0

so iθ

Re f (e ) = Re a0 +

1 2

∞ X

(an einθ + a ¯n e−inθ )

n=1

Thus

Z

−inθ

e

dθ Re f (e ) = 2π iθ

( Re a0 1 a 2 n

if n = 0 if n > 0

On the other hand, for |w| < 1, X ∞ ∞ X 1+w n = (1 + w) w =1+2 wn 1−w n=0 n=1

(2.3.19)

(2.3.20)

so that for |z| < 1,

∞ X eiθ + z 1 + ze−iθ = =1+2 z n e−inθ iθ −iθ e −z 1 − ze n=1

(2.3.21)

Therefore, by (2.3.19), ∞ X n 1 RHS of (2.3.17) = i Im a0 + Re a0 + 2 z ( 2 an ) n=1

= f (z)

62


It is useful to note that iθ e +z 1 − |z|2 Re iθ = iθ >0 e −z |e − z|2 In particular, iθ 1 − r2 e + reiϕ Re iθ = e − reiϕ 1 + r 2 − 2r cos(θ − ϕ) the celebrated Poisson kernel. Proposition 2.3.3 motivates:

(2.3.22)

(2.3.23)

Definition. The Carathéodory function of a probability measure dµ on ∂D is given by Z iθ e +z dµ(θ) (2.3.24) F (z) = eiθ − z This R is a Carathéodory function since (2.3.22) implies Re F (z) > 0 and dµ(θ) = 1 implies F (0) = 1. Our three classes of functions are clearly related. The map 1+z z→i (2.3.25) 1−z

maps D bijectively and biholomorphically onto C+ . If G is a function on C+ with values in C, then G is Herglotz if and only if

−i[G(i[ 1+z ]) − Re G(i)] 1−z F (z) = (2.3.26) Im G(i) is a Carathéodory function. And the association 1 + zf (z) F (z) = (2.3.27) 1 − zf (z) sets up a one-one correspondence between Carathéodory functions and Schur functions. To see this, one needs the Schwarz lemma: Proposition 2.3.4 (Schwarz Lemma). If f is a Schur function with f (0) = 0, then f (z)/z is also a Schur function. Proof. Let g(z) = f (z)/z. Then g is analytic and for 0 < r < 1, 1 max |g(z)| = max |g(z)| ≤ r −1 max |f (z)| ≤ |z|≤r |z|=r |z|=r r Taking r ↑ 1, we see max |g(z)| ≤ 1 (2.3.28) |z| 0, for 0 < r < 1,

dθ (2.3.31) dµr (θ) = Re F (reiθ ) 2π R defines a measure, with dµr (θ) = 1 since F (0) = 1. Moreover, since F (rz) is analytic in a neighborhood of D, (2.3.17) implies Z iθ e +z F (rz) = dµr (θ) (2.3.32) eiθ − z which, by (2.3.21), implies Z einθ dµr (θ) = r n cn

(n > 0)

It Rfollows that dµr (θ) is a family of measures where limr→1 einθ dµr (θ) = cn for n > 0, by reality, the limit is R and iθ c¯−n for n < 0. Thus, by the fact that | f (e ) dµr (θ)| ≤ kf k∞ and the density of Laurent polynomials in C(∂D), dµr has a weak limit dµ. Taking r → 1 in (2.3.32), we obtain Z iθ e +z F (z) = dµ(θ) (2.3.33) eiθ − z

By (2.3.26), this translates to a Herglotz representation for Herglotz functions (see the Notes). One could use that to analyze discrete mfunctions, but we will instead use a direct argument that mimics the above proof.


64

Theorem 2.3.6 (Herglotz Representation for Discrete m-functions). A function m(z) on C+ is a discrete m-function if and only if m has the form (2.3.7) for some probability measure dρ supported on a bounded interval in R. Proof. Suppose (2.3.13) holds and pick δ > 0 and M > R + δ + 1. Let Γ1 be the contour going clockwise around the rectangle centered at 0 with width 2(R + 1) and height 2δ and Γ2 be the circle of radius M centered at zero going counterclockwise. If y ∈ R and R + δ + 1 < |y| < M, we have Z 1 m(z) m(y) = dz (2.3.34) 2πi Γ1 ∪Γ2 z − y The contribution of Γ2 is dominated in absolute value by 1 1 (2πM) sup |f (z)| 2π M − |y| |z|=M

which goes to zero as M → ∞. Thus, for |y| > R + δ + 1, Z 1 m(z) dz m(y) = 2πi Γ1 z − y

(2.3.35)

A similar analysis shows for y ′ 6= δ > 0, ( Z ∞ 1 m(x + iδ) m(iy ′ ) y ′ > δ dx = 2πi −∞ x + iδ − iy ′ 0 y′ < δ

Taking this for y ′ > δ and y ′′ = δ − (y ′ − δ) and subtracting, we get Z ∞ dx(y ′ − δ) m(x + iδ) 2 = πim(iy ′ ) ′ − δ)2 x + (y −∞

Taking the imaginary part, multiply by y ′, and taking y ′ → ∞ (using Im m(x + iy) = O(1/x2 )), we get Z ∞ 1 Im m(x + iδ) dx = 1 (2.3.36) −∞ π Now let

dρδ (x) = χ[−R,R] (x) and see (2.3.36) implies

Z

1 Im m(x + iδ) dx π

dρδ (x) → 1

and that (2.3.35) implies for y real with |y| > R + 1, Z dρδ (x) → m(y) x−y

(2.3.37) (2.3.38)

(2.3.39)


65

Since {(x − y)−1 | |y| > R + 1} is total in C([−R, R]), we have that dρδ has a limit dρ and (2.3.7) holds first for z ∈ R \ [−R − 1, R + 1] and then by analytic continuation for z ∈ C+ . Our proofs showed Theorem 2.3.7. If F is a Carathéodory function with associated measure dµ in (2.3.26), then w-lim Re F (reiθ ) r↑1

dθ = dµ(θ) 2π

(2.3.40)

If m is a discrete m-function and dρ the associated measure in (2.3.7), then 1 w-lim Im m(x + iδ) dx = dρ(x) (2.3.41) δ↓0 π Define H p (D) by Definition. Let 0 < p < ∞. An analytic function, f , on D is said to lie in H p (D) if and only if 1/p Z iθ p dθ kf kp ≡ sup |f (re )| 0, can be written uniquely as a product of an inner and an outer function, that is, if iθ {zj }N j=1 are its zeros in D \ {0} (N finite or infinite), then for some e , ℓ ≥ 0, and nonnegative singular measure, dν, on ∂D, Y Z iθ N dθ e +z iθ iθ ℓ f (z) = e z bzj (z) exp log|f (e )| − dν(θ) eiθ − z 2π j=1

(2.3.71)

If f is analytic in a neighborhood of D, then dν = 0 and N < ∞. In that case when ℓ = 0, (2.3.71) at z = 0 is a celebrated formula of Jensen. Thus, (2.3.71) is sometimes called the Poisson–Jensen formula. Its importance is due to Nevanlinna. The next fact we need concerns the issue of whether boundary values determine an H p function. The key fact is: Theorem 2.3.20. Let f ∈ H p (D), 0 < p < ∞, and let f (eiθ ) be dθ its boundary values. Then {eiθ | f (eiθ ) = 0} has 2π measure zero. If p iθ iθ iθ f, g ∈ H and f (e ) = g(e ) for e ∈ Σ with |Σ| > 0, then f = g. Sketch. Using Jensen’s formula, one proves that dθ log− |f (eiθ )| 2π < ∞ (where, for x > 0, log− (x) = max(0, − log(x)) and this implies |{eiθ | f (eiθ ) = 0}| = 0. The second statement follows from the first since f − g ∈ H p . R

Theorem 2.3.21. (a) If f and g are distinct Carathéodory functions, then Σ = {eiθ | f (eiθ ) = g(eiθ )} has |Σ| = 0. Similarly, |{eiθ | f (eiθ ) = c}| = 0 for each c ∈ C (except for the case c = 1, f ≡ 1). (b) If m, n are distinct discrete m-functions, |{x ∈ R | m(x + i0) = n(x + i0)}| = 0

(2.3.72)

Similarly, |{x ∈ R | m(x + i0) = c}| = 0 for any c ∈ C. Proof. (a) e−f , c−g ∈ H ∞ (D), so this follows from the previous theorem. (b) eim , ein are bounded and, by mapping C+ to D, we can apply the previous theorem. We have just seen that if Σ ⊂ R has positive measure, then the map m → m(· + i0) ↾ Σ is one-one. Later (see Theorem 7.4.7), we will need to know this map has a continuous inverse. Since Im m(x + i0) ≥ 0, for t ≥ 0, |eitm(x+i0) | ≤ 1, so we will topologize the functions by saying


71

mn → m weakly on Σ if and only if for each g ∈ L2 (Σ, dx) and t positive and rational, we have Z Z itmn (x+i0) g(x)e dx → g(x)eitm(x+i0) dx (2.3.73) Σ

Σ

Theorem 2.3.22. Let e ⊂ R be compact and let Σ ⊂ e be a Borel set of strictly positive Lebesgue measure. Let M(e) be the set of functions, m(z), which are discrete m-functions of the form (2.3.7) where supp(dρ) ⊂ e. Topologize M(e) with the topology of uniform convergence on compact subsets of C+ . Topologize functions on Σ with positive imaginary part by (2.3.73). Then RΣ : m 7→ m(· + i0) ↾ Σ is a continuous map with a compact range and a continuous inverse on this range.

Proof. M(e) is compact and we have proven above that RΣ is one-one, so continuity automatically implies a compact range and a continuous inverse. In H 2 (D), uniform convergence on compacts is equivalent to weak convergence of boundary values in L2 (∂D) (since {einθ } are total and given by Taylor coefficients). By mapping C+ to D and noting that eitm is bounded, we get functions in H ∞ , so in H 2 . There is one final topic concerning Schur functions that we want to discuss: the Schur algorithm and Schur parameters. Given a Schur function, f , either f (z) ≡ eiθ = γ0 or else f (0) ≡ γ0 ∈ D

(2.3.74)

In the latter case, we can look at Mγ0 (f (z)) which is a Schur function vanishing at 0, so, by the Schwarz lemma, f1 (z) = is also a Schur function and f (z) =

1 f (z) − γ0 z 1 − γ¯0 f (z)

(2.3.75)

γ0 + zf1 (z) 1 + z¯ γ0 f1 (z)

(2.3.76)

We can iterate this process. If f1 (z) ≡ eiθ = γ1 , we stop. Otherwise we set f1 (0) = γ1 and define f2 via the analog of (2.3.75). In this way we associate with a Schur function a sequence of numbers {γj }N j=0 with either N < ∞ and then |γj | < 1 j = 0, . . . , N − 1

|γN | = 1

(2.3.77)

72


or else N = ∞ and then for all j, |γj | < 1

j = 0, 1, . . .

(2.3.78)

We also get a sequence {fj }∞ j=0 (with f = f0 ) of Schur functions called the Schur iterates. If we define S to be the set of potential Schur parameters, that is, a finite sequence obeying (2.3.77) or an infinite sequence obeying (2.3.78). S has a natural topology of convergence of each γj (with the (n) (∞) rule that if the limit γ (∞) has N < ∞, we only require γj → γj for j ≤ N) in which S is compact. The main theorem about the Schur algorithm is: Theorem 2.3.23. The map from Schur functions to S is a bijection and for a sequence, {gn }∞ n=1 , of Schur functions and another Schur function, g∞ , convergence of γ(gn ) to γ(g∞ ) is equivalent to convergence of gn (z) to g(z) uniformly on compact subsets of D. Moreover, γ(f ) has N < ∞ if and only if f is a phase factor times a finite Blaschke product. We want to sketch the proof of this theorem. Proposition 2.3.24. A Schur function f has N < ∞ if and only if it is Blaschke product of order N times a phase factor. Proof. We will use induction in N. N = 0 is obvious. f has N = n > 0 if and only if there is γ0 ∈ D and f1 with N = n−1 so (2.3.76) holds. If f1 is a Blaschke product of order n − 1, then zf1 has order n and so is analytic in a neighborhood of D with winding number n as a map of ∂D to ∂D. Since eiθ → (γ0 + eiθ )/(1 + γ¯0 eiθ ) is a bijection of ∂D with positive derivative it preserves winding number, so f also has winding number n, and so, by the argument principle, it has n zeros. By Proposition 2.3.15, it is a Blaschke product of order n. Conversely, if f is a Blaschke product of order n, the above winding number argument shows zf1 has n zeros, and so f1 has n − 1 zeros. So, by Proposition 2.3.11 again, f1 is a Blaschke product of order n − 1. Thus, f is a Blaschke product of order n (times a phase factor) if and only if f1 is a Blaschke product of order n − 1. This plus induction completes the proof. Lemma 2.3.25. For any finite sequence (γ0 , . . . , γn−1) ∈ D , there is a function f whose Schur parameters are γ0 , γ1, . . . , γn−1 , 0, 0, . . . , 0, . . . . n


73

Proof. If n = 0, take f (z) = 0 which has γ0 = 0, f1 (z) = 0, and so Schur parameters (0, 0, . . . ). Now given any finite sequence, we can suppose inductively we have f1 with Schur parameters (γ1 , γ2, . . . , γn−1 , 0, . . . ) and define f by (2.3.76). Lemma 2.3.26. Let f and g be two Schur functions with Schur parameters γn (f ) and γn (g). Then f (j) (0) = g (j)(0)

j = 0, 1, . . . , n

(2.3.79)

if and only if γj (f ) = γj (g)

j = 0, 1, . . . , n

(2.3.80)

Proof. Since w → M−γ0 (w) is a smooth bijection near w = 0 and = 1 − |γ0 |2 , we have for w small,

′ M−γ (0) 0

M−γ0 (w) = (1 − |γ0 |2 )w + O(w 2)

(2.3.81)

Thus, if γ0 (f ) = γ0 (g), f (z) − g(z) = (1 − |γ0 |2 )z(f1 (z) − g1 (z)) + O(z 2 (f1 (z) − g1 (z)) (2.3.82) which proves that if γ0 (f ) = γ0 (g), then f (z) − g(z) = O(z n ) ⇔ f1 (z0 ) − g1 (z) = O(z n−1 ) By induction, one obtains the result.

Theorem 2.3.27. If f and g are Schur functions and (2.3.81) holds, then |f (z) − g(z)| ≤ 2|z|n+1 (2.3.83) for all z ∈ D.

Proof. Let h(z) = 12 (f (z) − g(z)) which is a Schur function that, by Lemma 2.3.26, obeys h(z) = O(z n+1 ). By repeated use of the Schwarz lemma, h(z)/z n+1 is a Schur function so |h(z)| ≤ |z|n+1 which is (2.3.83). Proof of Theorem 2.3.23. If γj (f ) = γj (g) for all j and γj (f ) ∈ D∞ , then |f (z) − g(z)| ≤ 2|z|n for all n, so taking n → ∞, f = g on D. If γN (f ) ∈ ∂D for some N, f and g can both be obtained via a finite Schur algorithm, and so are equal. Thus f → γ(f ) is one-one. If γ ∈ D∞ , let f [N ] be the Schur function with Schur parameters (γ0 , . . . , γN , 0, 0, . . . ) guaranteed by Lemma 2.3.25. Thus, by Theorem 2.3.27, if M > N, |f [N ] (z) − f [M ] (z)| ≤ 2|z|N

(2.3.84)


74

so f [N ] (z) is Cauchy and converges uniformly on compact subsets of D to f (z) obeying |f (z) − f [N ] (z)| ≤ 2|z|N (2.3.85) By Lemma 2.3.26, γj (f ) = γj (f [N ] ) = γj for j < N. Thus, we have shown f → γ· (f ) is onto.

Remarks and Historical Notes. Most of the material is standard analysis textbook fare; see Duren [113] or Rudin [363]. The exception is the material on the Schur algorithm due to Schur [371]. The argument we use for that follows Schur except that he has a weaker bound than (2.3.83). That bound is from [114]. Riesz’s theorem (Proposition 2.3.8) fails for p = 1 or p = ∞. However, it is a theorem of Kolmogorov (see [113]) that if (2.3.44) holds for p = 1, then f ∈ ∩p pn−1 (on account of (2.5.19)), we see 1 1 < pn+1 pn pn pn−1 and thus, by (2.5.22), qn+1 qn−1 q q n n < − − (2.5.28) pn+1 pn pn−1 pn which, with (2.5.25), proves (2.5.21). Finally, by ξn ≥ 1, we have pn ≥ pn−1 ≥ pn−2 and so pn ≥ 2pn−2 . Thus, for n ≥ 0, p2n+1 ≥ p2n ≥ 2n

Thus, by (2.5.21)/(2.5.22), q 1 1 n x − ≤ ≤ 2n−1 pn pn pn+1 2 which implies (2.5.23).

(2.5.29)

There are two extensions of this that are often also called continued fractions. First, fn is often replaced by fn (y) = ξn +

ζn y

(2.5.30)

These are sometimes called “extended continued fractions.” Second, ξn (and ζn ) may becomes functions of an external variable, x or z. This


88

happens most especially when one or both are linear functions of x or z. In the context of (2.5.30), we find (2.5.19) becomes pn = ξn pn−1 + ζn pn−1

(2.5.31)

If ζn = −a2n−1 and ξn = x − bn , (2.5.31) has the form of the OPRL recursion relation (1.2.8), which suggests that continued fractions have a close connection to OPRL. We will see this in Section 3.2—indeed, m(z) has a continued fraction expansion, (3.2.31), whose approximants, (3.2.43), have the OPRL Pn in the denominator. Because 1 − |α|2 α+w =α+ (2.5.32) 1 − αw ¯ −¯ α + w1 the Schur algorithm has the flavor of a continued fraction also (see the Notes). In any event, it suggests that we use a matrix approach to iterate the Schur algorithm and, in essence, the resulting matrix is the inverse of (2.5.1). We have the following: Theorem 2.5.5. Given {αj }∞ j=0 , for j = 0, 1, 2, . . . , there exist polynomials Aj and Bj of degree at most j so that n−1 ∗ Y zBn−1 (z) −A∗n−1 (z) 2 −1/2 Tn (z) = (1 − |αj | ) (2.5.33) −zAn−1 (z) Bn−1 (z) j=0

Moreover,

1 zf (z) In particular,

. =

Bn−1 (z) A∗n−1 (z) ∗ zAn−1 (z) zBn−1 (z) f [n] (z) =

1 zfn (z)

An (z) Bn (z)

(2.5.34)

(2.5.35)

Remarks. 1. As in the proof of Theorem 2.3.23, f [n] (z) is the Schur function with Schur parameters (γ0 , γ1 , . . . , γn , 0, 0 . . . ). 2. An and Bn are called the Wall polynomials. 3. An , Bn are not monic; indeed, they may not be of degree n. They are normalized by Bn (0) = 1, so Bn∗ (z) is monic. 4. The ∗ in (2.5.33) is the one suitable for degree n − 1 polynomials. Proof. Since A(α, z) is affine in z, Tn (z) has components which are polynomials of degree at most n. For the ∗ relations note that if J = J −1 = ( 01 10 ), then JA(α, z)J −1 = z A(α, 1/¯ z), so JTn (z)J −1 = z n Tn (1/¯ z)

(2.5.36)

2.5. COEFFICIENT STRIPPING

89

which leads to the ∗ -relation. (2.5.34) follows from (2.5.1) and the lemma below. (2.5.35) comes from setting fn+1 = 0 in the relation (2.5.34) with n replaced by n + 1. Lemma 2.5.6. Let ad − bc 6= 0 and . a b v= w c d

then

. w=

d −b v −c a

Proof. Follows immediately from a −b a b = (ad − bc)1 −c d c d

(2.5.37)

(2.5.38)

(2.5.39)

The recursion Tn (z) = A(αn−1 , z)Tn−1 (z) immediately implies recursion relations for the Wall polynomials ∗ An (z) = An−1 (z) + γn zBn−1 (z)

(2.5.40)

Bn (z) = Bn−1 (z) + γn zA∗n−1 (z)

(2.5.41)

A∗n (z) Bn∗ (z)

+ γ¯n Bn−1 (z)

(2.5.42)

+ γ¯n An−1 (z)

(2.5.43)

= =

zA∗n−1 (z) ∗ zBn−1 (z)

A0 (z) = γ0

(2.5.44)

B0 (z) = 1

(2.5.45)

Moreover, (2.4.26) and (2.5.33) lead to the Pintér–Nevai formulae ∗ Φn (z) = zBn−1 (z) − A∗n−1 (z)

Ψn (z) =

∗ zBn−1 (z)

+

A∗n−1 (z)

(2.5.46) (2.5.47)

Remarks and Historical Notes. Coefficient stripping formulae for OPUC go back to Peherstorfer [331]. The name is from [390]. Geronimus’ theorem was found by him in 1944 [154]. There are four proofs of it in [390] and one in [389]. See the notes in [390], especially Section 3.2, for a discussion of the history of these proofs. Verblunsky’s theorem was the main result in his paper [440], although his definition of αn was different (see the Notes to Section 1.8). [390] has four proofs of it. In the later literature, it is often called “Favard’s Theorem for the Unit Circle.” Continued fractions for reals were formally defined by Wallis in the 1600s with important contributions by Euler, Lagrange, and Galois.

90


Knopp [226] and Ryll-Nardzewski [368] found a Lebesgue a.c. measure under which x → y1 (x) is ergodic and so he determined the a.e. typical distributions of the ξ’s. That a rational, x, has a finite string of ξn ’s is a consequence of the Euclidean algorithm. Write x = p/q and then p = ξ 0 q + r1 q1 = ξ1 r1 + r2 r1 = ξ 2 r2 + r3 .. . until rk = 0. Theorem 2.5.4 is largely classical. Khinchin’s book [213] is the standard reference. When the ζ’s become functions, we have the “analytic theory of continued fractions” whose founders are Jacobi, Chebyshev, Markov, and Stieltjes. Wall’s book [443] is the classic exposition. The name “Wall polynomials” is due to Khrushchev [214], although the polynomials appear already in Schur’s paper [371]. The Pintér– Nevai formula appeared in their paper [339]. As discussed by Geronimus [154] and Khrushchev [214], there is a continued fraction expansion associated to f for which the f [N ] are only half the approximants. 2.6. The Relative Szeg˝ o Function and the Step-by-Step Sum Rule Our goal in this section is to prove (2.1.16). Suppose we can find a nonvanishing analytic function (δD)(z) which is in H p (D) so that |(δD)(eiθ )|2 =

w(θ) w1 (θ)

(2.6.1)

Then Jensen’s formula, essentially the fact that log|δD(z)|2 is harmonic, will lead to Z 1 w(θ) 2 |δD(0)| = exp log dθ (2.6.2) 2π w1 (θ) This is a simplification. The existence of singular inner functions shows that f ∈ H p (D) and nonvanishing on D, even H ∞ (D) is not enough for Z dθ iθ |f (0)| = exp log|f (e )| (2.6.3) 2π Rather, we need a condition that implies f is outer, for example, that log f is in some H p (D), p ≥ 1. In fact, log δD(z) will be in ∩p 0 since: Lemma 2.6.1. If F (eiθ ) ≡ limr↑1 F (reiθ ) exists (and is finite) and Re(F (eiθ )) > 0, then f (eiθ ) = limr↑1 f (reiθ ) exists and |f (eiθ )| < 1

(2.6.9)

In particular, for a.e. θ with w(θ) > 0, lim r↑1

1 − r 2 |f (reiθ )|2 =1 1 − |f (reiθ )|2

(2.6.10)


92

Proof. Re F ≥ 0 implies 1 + F is bounded away from zero so (2.3.29) implies f has a limit. If w(θ) = Re F (eiθ ) > 0, then 1 − |f (eiθ )|2 = 1 −

|F − 1| 4 Re F (eiθ ) = >0 |F + 1| |1 + F (eiθ )|2

(2.6.11)

proving (2.6.9). Since F (reiθ ) has a limit for a.e. θ and lim|f (eiθ )|2 < 1 implies (2.6.10), we have the final assertion. This allows us to state the main properties of δ0 D(z). Theorem 2.6.2. δ0 D(z) is analytic in D and nonvanishing. Moreover, \ log δ0 D(z) ∈ H p (D) (2.6.12) p 0, then

Z

lim |δ0 D(reiθ )|2 = r↑1

iθ

2

log|δ0 D(e )| dθ

w(θ) w1 (θ)

(2.6.13)

(2.6.14)

and that up to sets of dθ-measure 0, {θ | w(θ) > 0} = {θ | w1 (θ) > 0}

so if this set is all of ∂D (up to sets of measure 0), then Z w(θ) 1 2 1 − |α0 | = exp log dθ 2π w1 (θ)

(2.6.15)

(2.6.16)

Remarks. 1. (2.6.16) is the step-by-step sum rule. 2. (δ0 D)(z) has a pole at a point z0 in ∂D if z0 is an isolated pure point of dµ, so δ0 D may not itself lie in H 1(D). But since any Carathéodory function lies in H p (D) for all p < 1 (see the Notes to Section 2.3) and (1 − zf )−1 = 21 (1 + F ), we see (1 − zf )−1 is in all H p (D), p < 1. Since |δ0 D(z)| ≤ 4(1 − |α0 |2 )−1/2 |(1 − zf )−1 |, we see δ0 D ∈ ∩0 0}. Then we have that Z w(θ) dθ −iθ e log = α0 − α1 + α ¯ 0 α1 (2.6.20) w1 (θ) 2π Proof. By (2.3.52), if h ∈ H 1 (D), Z Z Z dθ −iθ iθ dθ −iθ iθ dθ 1 1 e Re h(e ) = 2 e h(e ) + 2 eiθ h(eiθ ) 2π 2π 2π 1 ′ = 2 h (0)

(2.6.21)

Taking h(z) = 2 log(δ0 D)(z) and using (2.6.14), we see that (2.6.20) is equivalent to 1 2

By (2.6.7), 1 2

h′ (0) = α0 − α1 + α ¯ 0 α1

(2.6.22) (2.6.23)

h(z) = log(1 − zf1 ) − log(1 − zf ) + log(1 − α ¯ 0 f ) − log(ρ0 ) (2.6.24)

94


and we need the O(z) terms. Since f (0) = α0 , f1 (0) = α1 , and log(1 − zg) = −zg + O(z 2 ), the first two terms have O(z) terms −α1 and α0 . The last term is z-independent. That leaves the third term. We note first that α0 + zf1 (z) = α0 + (1 − |α0 |2 )α1 z + O(z 2 ) (2.6.25) f (z) = 1+α ¯ 0 zf1 (z) so 1−α ¯ 0 f = (1 − |α0 |2 ) + α ¯ 0 α1 (1 − |α0 |2 )z + O(z 2 ) so

= (1 − |α0 |2 )(1 + α ¯ 0 α1 z + O(z 2 ))

log(1 − α ¯ 0 f ) = log(1 − |α0 |2 ) + α ¯ 0 α1 z + O(z 2 ) proving (2.6.23).

(2.6.26) (2.6.27)

Notes and Historical Remarks. The relative Szeg˝o function was defined by Simon in [390, Sect. 2.9]. If it were not for the Killip–Simon analog for OPRL [220], he might not have found it. There the m-function was a natural object whose boundary values are log(Im m/ Im m1 ) (see (3.4.9)). For some OPUC issues, the analog of m(z) is F (z) but not here. Simon noted the basic step-by-step sum rule, (2.6.16) and also the higher-order (2.6.20). The symbol δ0 D is used because, making the α-dependence explicit, it is natural to define δn D inductively by ∞ ∞ (δn D)(z, {αj }∞ j=0 ) = (δn−1 D)(z, {αj }j=0 ) + (δ0 D)(z, {αj+n }j=0 ) (2.6.28) connected to stripping off n α’s. There are two alternate ways of writing (δ0 D)(z). One, due to Killip–Nenciu (unpublished), is noted in [390, Prop. 2.9.4]: ρ0 1 − zf1 (δ0 D)(z) = (2.6.29) 1+α ¯ 0 zf1 1 − zf The other is in terms of

M(z) = z(1 + α0 )(1 + F (z)) + (1 + α ¯ 0 )(1 − F (z))

introduced in [391, Sect. 11.7] for

(δ0 D)(z) = (2zρ0 )−1 M(z)

(2.6.30)

We note—as will be important later (see Section 5.14)—that if dµ has a gap in its essential spectrum, δ0 D(z) has an analytic continuation to [C \ ∂D]∪ gap with poles in the gap precisely at pure points of dµ and zeros at pure points of dµ1 . This is also a property of m for OPRL. These properties hold since, by (2.3.57), z0 is a pure point of dµ if and

˝ THEOREM 2.7. THE PROOF OF SZEGO’S

95

only if F has a pole at z0 , and by (2.3.27)/(2.3.29), this happens if and only if 1 − z0 f (z0 ) = 0. 2.7. The Proof of Szeg˝ o’s Theorem We have all the pieces lined up and can follow the strategy of Section 2.1 to prove Theorem 2.1.1. We begin by iterating (2.6.16). Theorem 2.7.1 (Iterated Step-by-Step Sum Rule). Let dµ obey (2.1.1) and let dµN be given by (2.1.14) and obey (2.1.15). Then up to sets of measure zero, {θ | w(θ) 6= 0} = {θ | wN (θ) 6= 0}

(2.7.1)

and if this set is all of ∂D, then \ w(θ) dθ p log ∈ L ∂D, wN (θ) 2π p −∞, by (2.7.2), log(wN (θ)) 2π > −∞ and Z Z Z w(θ) dθ dθ dθ log = log(w(θ)) − log(wN (θ)) (2.7.13) wN (θ) 2π 2π 2π

˝ THEOREM 2.8. A HIGHER-ORDER SZEGO

97

so (2.7.3) becomes N −1 Z Z dθ dθ Y 2 (1 − |αj | ) = exp log(w(θ)) exp log(wN (θ)) 2π j=0 2π

(2.7.14)

By (2.2.14) and (2.2.3), Z dθ exp log(wN (θ)) ≤1 2π

(2.7.14) and (2.7.15) imply (2.7.12).

(2.7.15)

Proof of Theorem 2.1.1. Taking N → ∞ in (2.7.12), we get Z ∞ Y dθ 2 (1 − |αj | ) ≥ exp log(w(θ)) (2.7.16) 2π j=0

This and (2.7.9) imply (2.1.11).

Remarks and Historical Notes. This is the proof of Section 2.3 of [390] but is close to Verblunsky’s proof [441] and motivated by Killip–Simon [220]. 2.8. A Higher-Order Szeg˝ o Theorem In this section, we want to use the tools of this chapter to prove the following gem: Theorem 2.8.1. Let dµ obey (2.1.1). Then Z dθ (1 − cos(θ)) log(w(θ)) > −∞ 2π if and only if ∞ X (i) |αn |4 < ∞

(2.8.1)

(2.8.2)

n=0

(ii)

∞ X

|αn+1 − αn |2 < ∞

∞ X

|αn |2 < ∞

n=0

(2.8.3)

Remarks. 1. Since w ∈ L1 , the integral in (2.8.1) is either absolutely convergent or diverges to −∞. 2. If αn = (n + 1)−β , then

n=0

(2.8.4)

98


if and only if β > 12 . (2.8.2)/(2.8.3) hold if and only if β > 41 , so β ∈ ( 14 , 12 ] provides examples where (2.8.1) holds, but the integral is −∞ if the 1 − cos(θ) is dropped. Thus, log(w(θ)) has a nonintegral divergence at 0 = 0, but (1−cos(θ)) log(w(θ)) is still integrable; indeed, it should be w(θ) ∼ exp(−cθ−γ ) for 1 < γ < 3. The analogous weight for OPRL is studied in [242]. 3. It is known (see the Notes) that if ∞ X n=0

|αn+1 − αn | < ∞

(2.8.5)

then w(θ) is continuous and positive on (0, 2π) (but can vanish at θ = 0 rather rapidly). αn = (n + 3)−1/3 + (−1)n (n + 3)−2/3 is an example where (2.8.2)/(2.8.3) hold, but neither (2.8.4) nor (2.8.5) holds. As with Szeg˝o’s theorem, the key is a step-by-step sum rule: Theorem 2.8.2. Let dµ obey (2.1.1) and suppose w(θ) > 0 for a.e. θ. Then Z w(θ) dθ (1 − cos(θ)) log = log(1 − |α0 |2 ) − Re(α0 − α1 + α ¯ 0 α1 ) w1 (θ) 2π (2.8.6) Proof. Immediate from (2.6.16), (2.6.20), and 1 − cos(θ) = 1 − 21 (eiθ + e−iθ )

(2.8.7)

In applying the strategy of the last section, a key role is played by “positivity” which in the context there meant log(1 − |α0 |2 ) ≤ 0. If α0 = 0, then RHS of (2.8.6) = − Re(α1 ), which can have either sign, so there isn’t any strict positivity or negativity. However, the right side can be rewritten as something negative plus something that telescopes—and that we will see is enough. We will need the function g(α) = − log(1 − |α|2) − |α|2 Lemma 2.8.3. Let A < 1. Then, if |α| ≤ A, we have 2 1 |α|4 |α|4 ≥ g(α) ≥ 1 − A2 2 2

(2.8.8)

(2.8.9)


99

Proof. Let G(y) = − log(1 − y) − y

so

(2.8.10)

1 1 −1 G′′ (y) = (2.8.11) 1−y (1 − y)2 In particular, G(0) = G′ (0) = 0, so by Taylor’s theorem with remainder, if 0 ≤ y ≤ A2 , then G′ (y) =

G(y) =

y2 1 y 2 ′′ G (z(y)) = 2 2 (1 − z(y))2

(2.8.12)

for some z(y) ∈ [0, A2 ]. Thus, since (1 − z)−2 is decreasing on [0, A2 ], 2 y2 1 y2 ≥ G(y) ≥ 2 1 − A2 2

which implies (2.8.9).

By simple algebra, RHS of (2.8.6) = −g(α0 )− 12 |α0 −α1 |2 − 12 (|α0 +1|2 −|α1 +1|2) (2.8.13) This gives Theorem 2.8.4 (Iterated Step-by-Step Sum Rule). Let dµ obey (2.1.1) and let dµN be given by (2.1.14) and obey (2.1.15). Then up to sets of measure zero, {θ | w(θ) 6= 0} = {θ | wN (θ) 6= 0}

(2.8.14)

and if this set is all of ∂D, then \ w(θ) dθ p log ∈ L ∂D, wN (θ) 2π p −∞, and any N, e

1 (1−|1+α0 |2 ) 2

N −1 Y j=0

2

1

(1 − |αj |2 )e|αj | e− 2 |αj −αj+1 | 1

2

2

≥ e 2 (1−|1+αN | ) exp(−L(wN )) exp(L(w))

(2.8.24)


where L(w) =

Z

(1 − cos(θ)) log(w(θ))

dθ 2π

101

(2.8.25)

Proof. As in the proof of Proposition 2.7.4, immediate from (2.6.29) since we can separate the log(w(θ)) and log(wN (θ)) integrals. Theorem 2.8.8. For any nontrivial measure, (2.8.18) holds. Proof. If S(d˜ µ0 | dµ) = −∞, the right side of (2.8.18) is zero, and then by (2.8.19), the left side is zero and equality holds. Thus we can suppose S(d˜ µ0 | dµ) > −∞ (2.8.26) Then by (2.8.24) and (2.8.20) (which implies monotonicity of the left side), we have that where

LHS of (2.8.18) ≥ ea+b exp(L(w))

(2.8.27)

a = lim inf 12 (1 − |1 + αN |2 )

(2.8.28)

b = lim inf(−L(wN ))

(2.8.29)

Clearly, a ≥ 12 (1 − 4) = − 32 and by (2.8.22) and S ≤ 0, b ≥ −C, so (2.8.24) implies LHS of (2.8.18) ≥ e−3/2 e−C RHS of (2.8.18)

(2.8.30)

Since we are supposing (2.8.26) holds, LHS of (2.8.18) > 0, so ∞ ∞ X X g(αj ) + |αj − αj+1|2 < ∞ (2.8.31) j=0

j=0

and thus, by (2.8.9),

∞ X j=0

This implies

|αj |4 < ∞

(2.8.32)

lim|αj | = 0

(2.8.33)

lim sup L(wn ) ≤ L(w∞ ≡ 1) = 0

(2.8.34)

and so, a = 0. w Moreover, by (2.8.33), dµn −→ dµ0 , so by (2.8.22) and the semicontinuity of the entropy, so b ≥ 0. Thus, by (2.8.27) and (2.8.19),

RHS of (2.8.18) ≥ LHS of (2.8.18) ≥ eb RHS of (2.8.18)

(2.8.35)


102

which, with b ≥ 0, implies b = 0, and (2.8.18) holds.

Remark. The extra steps were needed to show a = 0 and b = 0 and so get the sum rule, but for the corollary, Theorem 2.8.1, (2.8.30) suffices and we can give a shorter argument. Proof of Theorem 2.8.1. (2.8.1) holds if and only if RHS of (2.8.18) > 0. (2.8.2)/(2.8.18) hold if and only if LHS of (2.8.18) > 0. Thus, Theorem 2.8.8 implies Theorem 2.8.1. Remarks and Historical Notes. Theorem 2.8.1 and the sum rule (2.8.18) first appeared in Section 2.8 of [390]. Rather than using a relative Szeg˝o function and a step-by-step sum rule, the proof there exploits Szeg˝o’s theorem. Our proof here is patterned after Simon– Zlatoˇs [402] who proved a more complicated result (see below). My motivation in seeking those results was that the OPRL analog of Szeg˝o’s theorem was the C0 sum rule of Section 3.6. I felt there had to be an OPUC analog of the P2 sum rule of Killip–Simon [220] with positivity. Even before the higher-order sum rule discussed here, there were higher-order sum rules without full positivity for OPRL. These are discussed in the Notes to Section 3.6. In [390], Simon conjectured a generalization of Theorem 2.8.1, namely, Conjecture 2.8.9. Fix θ1 , . . . , θk distinct in [0, 2π) and m1 , . . . , mk strictly positive integers. Then Z Y k dθ [1 − cos(θ − θj )]mj log w(θ) > −∞ (2.8.36) 2π j=1 if and only if ∞ Y X k −iθj mj [δ − e ] α n=0

j=1

where ℓ = 1 + maxj=1,...,k mj

n

2 + |αn |2ℓ < ∞

(2.8.37)

For n = 1, this can be obtained by rotation covariance from Theorem 2.8.1. For n = 2, it was proven by Simon–Zlatoˇs [402]. It is open for general n, but there are partial results in Golinskii–Zlatoˇs [174]. The argument used to get (2.8.35) is borrowed from Simon–Zlatoˇs [401] where (δα)n = αn+1 (2.8.38)

˝ FUNCTION AND SZEGO ˝ ASYMPTOTICS 2.9. THE SZEGO

103

As mentioned, if (2.8.5) holds, w(θ) is strictly positive and continuous on (0, 2π), a result of Peherstorfer–Steinbauer [334]. For a history of related results and a proof, see Section 12.1 of [391]. It is a conjecture of Last [264] that if ∞ X |αn+1 − αn |2 < ∞ (2.8.39) n=0

then w(θ) > 0 for a.e. θ. For the OPRL analog of this result, see TK. 2.9. The Szeg˝ o Function and Szeg˝ o Asymptotics

In his great 1920 paper [420], Szeg˝o realized that his earlier result on Toeplitz determinants allowed very strong asymptotic results on the OPUC, ϕ∗n (z), in D (and then two years after [421], he realized he could use this to obtain asymptotics for OPRL; see Section 3.7). While an aside from our main thrust, we would be remiss to ignore this beautiful and simple result. n Consider a sequence {xn }∞ n=1 and three senses in which xn ∼ β for some β ∈ C \ {0}: (i) Root asymptotics: x1/n →β (2.9.1) n If xn is complex, we have an issue of phase, but can fix it by looking only at |xn |1/n . (ii) Ratio asymptotics: xn+1 →β (2.9.2) xn (iii) Power asymptotics (also called Szeg˝ o asymptotics) for some c ∈ C \ {0}: xn →c (2.9.3) βn It is easy to see that Proposition 2.9.1. Power asymptotics ⇒ ratio asymptotics ⇒ root asymptotics. Before turning to the subtle Szeg˝o asymptotics for Φ∗n (z), we want to discuss an elementary result on ratio asymptotics: Theorem 2.9.2. Let Φn (z) be the monic OPUC associated to a nontrivial probability measure on ∂D with Verblunsky coefficients {αn }∞ n=0 . Then lim

n→∞

Φ∗n+1 (z) = 1 uniformly on D ⇔ lim αn = 0 n→∞ Φ∗n (z)

(2.9.4)

x-ref?


104

Moreover, if either side of (2.9.4) holds, then uniformly on compact subsets of D: Φn (z) lim =0 (2.9.5) n→∞ Φ∗ n (z) Remarks. 1. If limn→∞ αn = 0, ρn → 1 so ϕ∗n+1 (z)/ϕ∗n (z) also goes ∗ ∗ to 1. On the other hand, ϕ∗n+1 (0)/ϕ∗n (0) = ρ−1 n so ϕn+1 (0)/ϕn (0) → 1 implies αn → 0. 2. It is not true that Φ∗n+1 (z)/Φ∗n (z) → 1 uniformly on compact subsets of D implies αn → 0; see the Notes. 3. The proof shows that if (2.9.4) holds for single z in ∂D, then αn → 0. Proof. By (1.8.5), ∗ Φn+1 (z) Φn (z) Φ∗ (z) − 1 = |αn | |z| Φ∗ (z) n n

(2.9.6)

By the lemma below, the right side is bounded for z ∈ D by |αn | and equal to |αn | if |z| = 1. Thus (2.9.4) holds. Next suppose |αj | → 0 and fix z0 ∈ D. Pick nj → ∞ so that |Φnj +1 (z0 )| |Φn (z0 )| → lim sup ∗ ≡ q(z0 ) ∗ |Φnj +1 (z0 )| |Φn (z0 )|

(2.9.7)

By (1.8.5), ∗ Φnj (z0 ) Φ∗nj (z0 ) Φnj (z0 ) Φnj +1 (z0 ) Φ∗ (z0 ) ≤ |z0 | Φ∗ (z0 ) Φ∗ (z0 ) +|αn+j+1| Φ∗ (z0 ) (2.9.8) nj +1

nj +1

nj

nj +1

Using (2.9.4) and taking j → ∞, we get q(z0 ) ≤ |z0 |q(z0 )

so q(z0 ) = 0. Thus, Φn (z)/Φ∗n (z) → 0 pointwise. So, by the lemma and Vitali’s theorem, the convergence is uniform on compact sets. Lemma 2.9.3. For n ≥ 1,

Φn (z) z ∈ ∂D ⇒ ∗ = 1 Φ (z) n Φn (z) z ∈ D ⇒ ∗ < 1 Φ (z) n Φn (z) z ∈ C \ D ⇒ ∗ > 1 Φn (z)

(2.9.9) (2.9.10) (2.9.11)

Remark. Indeed, up to phase, Φn (z)/Φ∗n (z) is the Blaschke product of the zeros of Φn (z) which lie in D.

˝ FUNCTION AND SZEGO ˝ ASYMPTOTICS 2.9. THE SZEGO

105

Proof. (2.9.9) is immediate from the definition of Φ∗n , (2.9.10) then follows from analyticity (using Theorem 1.8.4) and the maximum principle, and (2.9.11) follows from ∗ Φn (1/¯ z ) Φn (z) = (2.9.12) Φn (1/¯ z ) Φ∗ (z) n

We now turn to Szeg˝o asymptotics. A key role will be played by: Definition. Let dµ be a nontrivial probability measure on ∂D of the form (2.1.1) If the Szeg˝o condition holds, that is, Z dθ > −∞ (2.9.13) log(w(θ)) 2π

then the Szeg˝o function, D(z), is defined by Z iθ e +z dθ D(z) = exp log(w(θ)) eiθ − z 4π Note that Szeg˝o’s theorem says that ∞ Y D(0) = ρn = lim (ϕ∗n (0))−1 n=0

n→∞

(2.9.14)

(2.9.15)

Note also the 1/4π, not 1/2π, in (2.9.14). It is responsible for: Proposition 2.9.4. Whenever the Szeg˝ o condition holds, D(z) ∈ H (D) and is nonvanishing on D. Indeed, Z dθ ≤1 (2.9.16) sup |D(reiθ )|2 2π 0≤r 1, we have that z −n ϕn (z) → D(1/¯ z)

(iii)

Z

|ϕn (eiθ )|2 dµs → 0

−1

(2.9.28)

as n → ∞

(iv) Define Dac (eiθ )−1 in L2 (∂D, dµ) by ( D(eiθ )−1 a.e. θ w.r.t. iθ −1 Dac (e ) = 0 a.e. dµs (θ)

dθ 2π

(2.9.29)

(2.9.30)

Then (v)

ϕ∗n (eiθ ) → Dac (eiθ )−1 ϕ∗n (eiθ )D(eiθ )

(vi)

→1

in L2 (∂D, dµ)

(2.9.31)

dθ in L ∂D, 2π

(2.9.32)

|ϕn (eiθ )2 | dµ(θ) →

2

dθ 2π

(2.9.33)

weakly as measures in ∂D. (vii) We have uniformly on compact subsets of D that ϕn (z) → 0

(2.9.34)

Remarks. 1. We will prove later (see Theorem 2.13.5) that if the Szeg˝o condition fails, then |ϕ∗n (z)|−1 → 0

(2.9.35)

uniformly on compact subsets of D. 2. We have not been explicit about the results for ϕn , for example, −1 −inθ e ϕn (eiθ ) → Dac (eiθ ) . 3. (2.9.33) holds in much greater generality than the Szeg˝o condition. For example, Rakhmanov [350, 351] has proven (2.9.33) so long as w(θ) > 0 and Khrushchev [216] has found necessary and sufficient conditions for (2.9.33), namely, (2.9.33) holds if and only if for

108


all ℓ, αn αn+ℓ → 0. These issues are discussed in Chapter 9, explicitly Sections 9.3 and 9.7, of [391]. 4. (2.9.28) is called Szeg˝ o asymptotics. 5. It is a result of Nevai–Totik [317] that if |αn | → 0 exponentially, then D(z)−1 has an analytic continuation beyond D and (2.9.27) holds; see Chapter 7 of [390]. Q Proof. (i), (iii), (v). Since ∞ j=0 ρj converges, RHS of (2.9.1) → 0 ∗ iθ iθ 2 so ϕn (e )D(e ) → 1 in H (∂D) (proving (v)) and (2.9.29) holds. For any f ∈ H 2 , one can extend (2.3.51) to get Z 1 − r2 dϕ iθ f (re ) = f (eiϕ ) (2.9.36) 2 1 + r − 2r cos(θ − ϕ) 2π

Thus H 2 -norm convergence implies uniform convergence on compact subsets of D. (ii) Immediate from (2.9.27) and (1.8.2). (iv) Since w(θ) = |D(eiθ )|2 , Z Z Z dθ ∗ iθ iθ −1 2 ∗ iθ 2 |ϕn (e )−Dac (e ) | dµ = |ϕn (e )| dµs + |D(eiθ )ϕ∗n (eiθ )−1| 2π goes to zero by (2.9.29) and (2.9.32). w

dθ (vi) |ϕn (eiθ )|2 dµs −→ 0 by (2.9.29) and |ϕn (eiθ )|2 w(θ) 2π = dθ iθ iθ 2 dθ |D(e )ϕn (e )| 2π → 2π by (2.9.32). (vii) Immediate from (2.9.27) and (2.9.28) (which holds since the Szeg˝o condition implies |αn | → 0).

Remarks and Historical Notes. Root asymptotics are discussed further in Theorem 2.15.3. The key calculation (2.9.24) and its consequences in Theorem 2.9.6 are in Szge˝o’s great paper [420]. Khrushchev [216] (see also Section 9.5 of [391]) has a general study of what kind of ratio asymptotics can occur for OPUC. In particular, these references discuss examples of Máté–Nevai [295] where one has ratio asymptotics uniformly on compact subsets of D for which αn 9 0. For OPRL, the analog is studied by Simon [388]. In this section, we discussed pointwise limits in D but only L2 limits on ∂D. That is because one cannot prove pointwise limit theorems on ∂D if one assumes only the Szeg˝o condition. Under stronger hypotheses, one can prove pointwise theorems; for example, see Szeg˝o’s book [424, Chapter XII], Freud’s book [137, Sections V.4–5], and Section 2.5 of the planned second edition of [390].

2.10. ASYMPTOTICS FOR WEYL SOLUTIONS

109

2.10. Asymptotics for Weyl Solutions Recall that in Section 2.4 we defined the Weyl solution z ∈ D by (2.4.25)/(2.4.27) and proved that (see (2.4.38)) z −n gn∗ (z) → 0

gn (z) ∗ (z) gn

for

(2.10.1)

As an aside on the aside that was the last section, we will prove here that Theorem 2.10.1. If the Szeg˝ o condition holds, then uniformly on compact subsets of D, z −n gn (z) → 2D(z)

(2.10.2)

z −n gn (z) → 0

(2.10.3)

If the Szeg˝o condition fails, then uniformly on compacts.

Our proof will require the following result from Section 2.13 (see Theorem 2.13.5): Proposition 2.10.2. If the Szeg˝ o condition fails, ϕ∗n (z)−1 → 0

(2.10.4)

uniformly on compact subsets of D.

Proof of Theorem 2.10.1. We apply (2.4.43) with a0 = b0 = −1, so an = −ϕn (z), bn = −ϕ∗n (z), and a0 g0∗(z) − b0 g0 (z) = −(−1 + F ) + (1 + F ) = 2

and we have

ϕ∗n (z)(z −n gn (z)) − ϕn (z)(z −n gn∗ (z)) = 2

(2.10.5)

Suppose the Szeg˝o condition fails. By (2.4.47) and (2.9.10), so (2.10.5) implies which implies

|ϕn (z)gn∗ (z)| ≤ |z| |ϕ∗n (z)gn (z0 )|

|ϕ∗n (z)z −n gn (z)| ≤ 2 + |z| |ϕ∗n (z)z −n gn (z)|

|z −n gn (z)| ≤ 2(1 − |z|)−1 |ϕ∗n (z)|−1 (2.10.6) so if the Szeg˝o condition fails, (2.10.4) implies (2.10.3). Now suppose the Szeg˝o condition holds. By (2.10.1) and (2.9.34), lim ϕn (z)z −n gn∗ (z) = 0

n→∞

(2.10.7)


110

and (2.10.5) implies lim z −n gn (z) = 2 lim ϕ∗n (z)−1 = 2D(z) n→∞

by (2.9.27).

Remarks and Historical Notes. Theorem 2.10.1 is due to Peherstorfer [329] and is related to OPRL results of Damanik–Simon [95] on the equivalence of Jost and Szeg˝o asymptotics in that case. One can argue in this case that the “Jost function” is (2D(z))−1 or (2D(z))−1 (1 + F ). [329] also has some results on asymptotics on ∂D when w(θ) has regularity properties. 2.11. Additional Aspects of Szeg˝ o’s Theorem In this section, we discuss several additional issues connected with Szeg˝o’s theorem. Szeg˝ o’s Theorem as a Nonlinear Plancherel Theorem. As mentioned in Section 1.5, there is a “small coupling” limit of Szeg˝o’s theorem in which it becomes the Plancherel theorem so that Szeg˝o’s theorem is a kind of nonlinear Plancherel theorem. dθ Suppose that f ∈ L∞ (∂D, 2π ) is real-valued and obeys Z dθ =0 (2.11.1) f (θ) 2π

Then for |λ| < kf k−1 ∞,

wλ (θ) = 1 + λf (θ)

(2.11.2)

is a weight for a probability measure dµλ = wλ (θ) Clearly,

Z

dθ 2π

(2.11.3)

Z Z dθ dθ 1 dθ = λf (θ) − 2 λ2 f (θ)2 + O(λ3 ) log wλ 2π 2π 2π = − 12 λ2 kf k22 + O(λ3 ) (2.11.4)

by (2.11.1). On the other hand: Proposition 2.11.1. αn−1 (dµλ) = λ

Z

e−inθ f (θ)

dθ + O(λ2) 2π

(2.11.5)

˝ THEOREM 2.11. ADDITIONAL ASPECTS OF SZEGO’S

111

Proof. We begin by proving that Φn (z) = z n + O(λ)

(2.11.6)

For {Φj }n−1 j=0 is an orthogonal basis for the polynomials of degree n − 1 so n−1 X n Φn (z) = z − hΦj , z n iΦj kΦj k−2 (2.11.7) j=0

(2.11.6) certainly holds for n = 0, so inductively we have kΦj k2 = 1 + O(λ) for j = 0, . . . , n − 1. Moreover, since Z z ℓ dµλ = O(λ) for ℓ 6= 0 (2.11.8) n we have for j < n that hΦ R j , z i = O(λ), proving (2.11.6) from (2.11.7). Szeg˝o recursion and Φn+1 dµ = h1, Φn+1 i = 0 implies (for any dµ) Z Z iθ iθ e Φn (e ) dµ(θ) = α ¯ n Φ∗n (eiθ ) dµ(θ) (2.11.9)

Now

Z

Φ∗n dµ = h1, Φ∗n i = hz n , Φn i = kΦn k2 = 1 + O(λ)

(2.11.10)

by (2.11.6). In (2.11.6), Rthe O(λ) term is a polynomial whose coefficients are O(λ), so since zz m dµ = O(λ) by (2.11.8), Z LHS of (2.11.9) = ei(n+1)θ dµ(θ) + O(λ2 ) by (2.11.6). Thus (2.11.9) implies (2.11.5).

Thus n n Y X (1 − |αj |2 ) = 1 − λ2 |fˆj | + O(λ3 ) j=0

where

(2.11.11)

j=0

Z

dθ (2.11.12) 2π Therefore, formally (i.e., ignoring the passage n → ∞ which can be subtle), Szeg˝o’s theorem says fˆj =

1−λ

2

∞ X j=0

e−ijθ f (θ)

|fˆj |2 + O(λ3) = exp(1 − 12 λ2 kf k22 + O(λ3 ))

(2.11.13)


112

Thus the small coupling limit of Szeg˝o’s theorem is the Plancherel theorem, and one can think of Szeg˝o’s theorem as a nonlinear Plancherel theorem. Szeg˝ o’s Theorem and the Density of the Polynomials. In dθ L (∂D, 2π ), the closure of the polynomials is H 2 (D), that is, analytic functions on D. The polynomials are not dense in L2 . One can ask for which measures the polynomials are dense in L2 (∂D, dµ) and we will answer that here. The initial steps have nothing to do with Szeg˝o’s theorem: 2

Proposition 2.11.2. Let P be the projection onto the closure of the polynomials in L2 (∂D, dµ) for dµ a nontrivial probability measure on ∂D. Then ∞ Y (1 − |αj |2 )1/2 k(1 − P )z k = lim kΦn k = −1

n→∞

(2.11.14)

j=0

Proof. The second equality is (1.8.11), so we focus on the first. Let P{f1 ,...,fn } be the projection onto the span of f1 , . . . , fn . By Proposition 1.8.1(iii), Φ∗n is the projection of 1 onto the span of {z, . . . , z n }, kΦ∗n k = k(1 − P{z,...,z n} )1k

= k[z −1 (1 − P{z,...,z n} )z]z −1 k

= k(1 − P{1,...,z n−1 } )z −1 k

(2.11.15) (2.11.16)

since multiplication by z −1 is a unitary that maps P{z,...,z n} to P{1,...,z n−1 } . Since kΦn k = kΦ∗n k, (2.11.14) follows from (2.11.16) by taking n → ∞. Lemma 2.11.3. If z −1 is in the closure of the span of the polynomials in L2 (∂D, dµ), so is z −ℓ for all ℓ. Proof. We use induction in ℓ. So suppose for polynomials Qn and Rn , Qn → z −ℓ and Rn → z −1 in L2 (∂D, dµ). Then (k · k = L2 -norm and k · k∞ the L∞ (∂D) norm) kz −ℓ−1 − Qn Rm k = kz −ℓ−1 − z −1 Qn + z −1 Qn − Qn Rm k ≤ kz −ℓ − Qn k + kQn k∞ kz −1 − Rm k

(2.11.17)

since kz −1 k∞ = 1. Given ε, pick n so kz −ℓ −Qn k < 2ε . Having chosen n, −ℓ−1 pick m so kz −1 −Rm k < 2ε kQn k−1 has a polynomial within ∞ . Then z 2 −ℓ−1 2 ε in L -norm and z is in the L -closure of the polynomials.


113

Theorem 2.11.4. Let dµ be a nontrivial probability measure on ∂D with Verblunsky coefficients {αj }∞ j=0 . The polynomials are dense in 2 L (∂D, dµ) if and only if ∞ Y j=0

(1 − |αj |2 ) = 0

(2.11.18)

Proof. If (2.11.18) fails, Proposition 2.11.2 shows that z −1 is not in the closure and so the closure is not all of L2 . Conversely, if (2.11.18) holds, by Proposition and Lemma 2.11.3, all Laurent polynomiP2.11.2 2 als (i.e., finite sums kj=−k c z j ) are in the closure of the polynomials. 1 j By Weierstrass’ theorem, the Laurent polynomials are dense in the continuous functions in k · k∞ , so in L2 , and so the polynomials are dense. Because of Szeg˝o’s theorem, we have Theorem 2.11.5 (Kolmogorov’s Density Theorem). Let dµ be a probability measure on ∂D of the form (2.1.1). Then the polynomials are dense in L2 (∂D, dµ) if and only if Z dθ = −∞ (2.11.19) log(w(θ)) 2π

As a final remark about the density result, if the Szeg˝o condition holds so the span of {z n }∞ n=1 is not dense, one can ask for explicit 2 functions in L (∂D, dµ) in the orthogonal complement. One can take g(θ) = e−iθ D(eiθ )−1 χS (θ)

(2.11.20)

dθ where S is a set of full 2π -measure whose complement supports dµs . For then Z Z dθ inθ e g(θ) dµ = ei(n+1)θ D −1 |D|2 2π Z dθ = ei(n+1)θ D(eiθ ) 2π =0 dθ since D ∈ H 2 (∂D, 2π ) and z n+1 D(z) vanishes at z = 0.

Szeg˝ o’s Theorem and CMV Matrices. One of the important aspects of Jacobi matrices is that they all act on the same space, ℓ2 ({1, 2, . . . }) so operator comparison and cancellation methods are available. Here we want to present a similar matrix representation for OPUC and show how it allows an expression of the Szeg˝o function

114


as a Fredholm determinant. CMV matrices will appear again later in Sections 6.10 and 8.7. Definition. The CMV basis, {χj }∞ j=0 , is the orthonormal basis 2 for L (∂D, dµ) obtained by applying Gram–Schmidt to the sequence 1, z, z −1 , z 2 , z −2 , . . . . The alternate CMV matrix basis, {xj }∞ j=0 , is ob−1 −2 tained from 1, z , z, z , . . . . Remark. We saw above that the {ϕj }∞ j=0 may or may not be a basis because the polynomials might or might not be dense. Since ∞ Laurent polynomials are always dense, the {χj }∞ j=0 and {xj }j=0 are always bases. It is not hard to see that for n = 0, 1, 2, . . . , χ2n (z) = z −n ϕ∗2n (z)

(2.11.21)

χ2n+1 (z) = z −n ϕ2n+1 (z)

(2.11.22)

x2n (z) = z −n ϕ2n (z)

(2.11.23)

x2n+1 (z) = z

−n−1

ϕ∗2n+1 (z)

(2.11.24)

xj (z) = χj (1/¯ z)

(2.11.25)

The CMV matrix, C, is just multiplication by z in the {χj }∞ j=0 basis, ˜ is multiplication by z in the {xj }∞ and the alternate CMV matrix, C, j=0 basis. Thus, Ckℓ = hχk , zχℓ i C˜kℓ = hxk , zxℓ i (2.11.26) By (2.11.25) and unitarity of C,

C˜kℓ = Cℓk

that is, C˜ = C t . C is a five-diagonal matrix with the form  α ¯0 α ¯ 1 ρ0 ρ1 ρ0 0 0 α1 α0 −ρ1 α0 0 0  ρ0 −¯  0 α ¯ ρ −¯ α α α ¯ ρ ρ  2 1 2 1 3 2 3 ρ2 C= ρ2 ρ1 −ρ2 α1 −¯ α3 α2 −ρ3 α2  0  0 0 0 α ¯ 4 ρ3 −¯ α4 α3 ... ... ... ... ...

(2.11.27)

... ... ... ... ... ...

      

(2.11.28)

with a 2 × 3 block at the top and then 2 × 4 blocks clustered about the diagonal. The easiest way to see this is to use: Proposition 2.11.6. Define Lkℓ = hχk , zχℓ i

Mkℓ = hxk , χℓ i

(2.11.29)


115

Then C = LM

(2.11.30)

L = Θ0 ⊕ Θ 2 + Θ4 ⊕ . . .

(2.11.31)

and

M = 11×1 ⊕ Θ1 ⊕ Θ3 ⊕ . . .

where 11×1 is the 1 × 1 identity and Θj is the 2 × 2 matrix α ¯j ρj Θj = Θ(αj ) = ρj −αj

(2.11.32)

(2.11.33)

Sketch of Proof. (2.11.30) follows from the fact that {xj }∞ j=0 is a basis. (2.11.31)/(2.11.32) are a restatement of the Szeg˝o recursion for the ϕ’s: zϕn (z) = ρn ϕn+1 (z) + α ¯ n ϕ∗n (z) ϕ∗n (z) = ρn ϕ∗n+1 (z) + αn zϕn (z)

(2.11.34) (2.11.35)

C0 is the CMV matrix associated to dθ/2π. In terms of the trace ideals, Ip ([166, 381]), one can show ∞ X j=0

|αj |p < ∞ ⇔ C − C0 ∈ Ip

(2.11.36)

for 1 ≤ p < ∞. By the general theory of trace ideals, if A ∈ I1 , one can define det(1 + A), and if A ∈ I2 , det2 (1 + A), which is formally det(1 + A)e−Tr(A) , and actually det((1 + A)e−A ) (since A ∈ I2 ⇒ (1 + A)e−A − 1 ∈ I1 ). One has P Theorem 2.11.7. If ∞ j=0 |αj | < ∞, then If

P∞

j=0 |αj |

where

2

D(0)D(z)−1 = det((1 − zC)(1 − zC0 )−1 )

< ∞, then

D(0)D(z)−1 = det2 ((1 − zC)(1 − zC0 )−1 )ezw1 w1 = α0 −

∞ X

αn α ¯ n−1

(2.11.37)

n=1

Remarks and Historical Notes. T. Tao and C. Thiele have emphasized the view of Szeg˝o’s theorem as a kind of nonlinear Plancherel theorem.


116

x-ref?

The density of polynomials results (i.e., Theorem 2.11.5) are due to Kolmogorov [234]. It was Krein [243] who realized the connection to OPUC. We return to the Kolmogorov density theorem in Section 3.9. CMV matrices are named after [66], although the history is complicated due to early work in the numerical linear algebra community; see the discussion in [394]. For details of the proof of Proposition 2.11.6, see [390, Section 4.2]. (2.11.36) is a result of Golinskii–Simon that appeared in Section 4.3 of [390]. Theorem 2.11.7 is due to Simon in that section where a proof is given. Note that (4.2.57) of [390] is wrong; see the erratum at http://www.math.caltech.edu/opuc.html. 2.12. The Variational Approach to Szeg˝ o’s Theorem While we are emphasizing step-by-step sum rule approaches to Szeg˝o’s theorem, we should present Szeg˝o’s variational proof from his great 1920 paper [420]. On one technical point—which, as I’ll explain, Szeg˝o didn’t address in 1920—we will provide an elegant resolution of Helson–Lowdenslager [192]. We begin with Proposition 2.12.1. Let dµ be a nontrivial probability measure on ∂D of the form (2.1.1) with Verblunsky coefficients {αn }∞ n=0 . Then ∞ Y (1 − |αn |2 ) = lim kΦ∗n k2 (2.12.1) n=0

n→∞

= inf{kP k2 | P a polynomial, P (0) = 1}

(2.12.2)

and if dµs = 0, this is = inf{kf k2 | f ∈ H ∞ (D), f (0) = 1} Remark. Here kgk is given by Z 2 kgk = |g(eiθ )|2 dµ(θ)

(2.12.3)

(2.12.4)

and in (2.12.3), we use the dθ-a.e. boundary values, which is why we suppose dµs = 0. Proof. (2.12.1) is just (1.8.11) and kΦn k = kΦ∗n k. Since Φ∗n is the projection of 1 onto the orthogonal complement of {z, . . . , z n }, kΦ∗n k2 ≤ k1 + a1 z + · · · + an z n k2

for any a1 , . . . , an ∈ C, so

(2.12.5)

kΦ∗n k = inf{kP k2 | P is a polynomial of degree at most n, P (0) = 1} (2.12.6)

˝ THEOREM 2.12. THE VARIATIONAL APPROACH TO SZEGO’S

117

which implies (2.12.2) by taking n → ∞. Since each polynomial, P, is in H ∞ , RHS of (2.12.3) ≤ RHS of (2.12.2)

On the other hand, if f ∈ H ∞ , f (reiθ ) → f (eiθ ) pointwise for dθ-a.e. θ (by (2.3.48)), and so a.e. dµ since we are expressing dµs = 0. Since f is bounded, we have convergence in L2 (∂D, dµ). For r < 1, the Taylor polynomials for g(z) = f (rz) converge uniformly to f (reiθ ), and so for any f ∈ H ∞ , we can find polynomials with Pn with Pn (0) = f (0) and Pn → f in L2 (∂D, dµ). Thus, RHS of (2.12.3) ≥ RHS of (2.12.2)

One side of Szeg˝o’s theorem follows from Jensen’s inequality that Z Z h(x) h(x) dγ(x) (2.12.7) e dγ(x) ≥ exp

for any probability measure dγ.

Theorem 2.12.2. For any polynomial P and any nontrivial probability measure dµ on ∂D of the form (2.1.1), we have Z Z dθ iθ 2 2 |P (e )| dµ ≥ |P (0)| exp log(w(θ)) (2.12.8) 2π In particular,

∞ Y

(1 − |αn | ) ≥ exp

n=0

Z

2

Z

dθ log(w(θ)) 2π

(2.12.9)

Proof. We have Z dθ iθ 2 |P (e )| dµ(θ) ≥ |P (eiθ )|2 w(θ) 2π Z dθ = exp(2 log|P (eiθ )| + log(w(θ))) 2π Z Z dθ dθ iθ ≥ exp 2 log|P (e )| exp log(w(θ)) 2π 2π

by Jensen’s inequality. The lemma below completes the proof of (2.12.8). (2.12.9) then follows from (2.12.2). Lemma 2.12.3. For any polynomial P, Z dθ log|P (eiθ )| ≥ log|P (0)| 2π

(2.12.10)


118

Proof. Suppose first that P is nonvanishing on D. Then log(P (z)) is analytic in D, continuous on D, and so log|P (z)| = Re log(P (z)) is harmonic. So (2.12.10) holds with equality. The same is true if P has zeros on ∂D by a limiting argument. If P (0) = 0, (2.12.10) is trivial. Let {zj }ℓj=0 are all the zeros of P in D and no zj is zero. Define Q(z) =

ℓ Y 1 − z¯j z P (z) z − z j j=1

(2.12.11)

Then |Q(eiθ )| = |P (eiθ )| and Q is nonvanishing on D. So by the special case at the start of the theorem, Z Z dθ dθ iθ = log|Q(eiθ )| log|P (e )| 2π 2π = log|Q(0)| ℓ Y −1 = log |zj | P (0) j=1

≥ log|P (0)|

since |zj |−1 ≥ 1.

Theorem 2.12.4. Suppose dµs = 0. Then Z dθ log(w(θ)) RHS of (2.12.3) ≤ exp 2π

(2.12.12)

Proof. For ε > 0, define Z eiθ + z dθ fε (z) = exp 1 − iθ log(w(θ) + ε) e −z 4π −1 = Lε gε (z)

(2.12.13)

with Lε = exp and

Z

dθ log(w(θ) + ε) 4π

Z

(2.12.14) (2.12.15)

eiθ + z dθ gε (z) = exp log(w(θ) + ε) (2.12.16) eiθ − z 4π Clearly, gε (0) = Lε , so fε (0) = 1. Moreover, log(w(θ) + ε) ≥ log(ε), so |gε (z)| ≥ ε1/2 , and so, |fε (z)| ≤ ε−1/2 Lε < ∞

˝ THEOREM 2.12. THE VARIATIONAL APPROACH TO SZEGO’S

and f ∈ H ∞ . Thus, by (2.12.3) and dµs = 0, Z dθ RHS of (2.12.3) ≤ |fε (eiθ )|2 w(θ) 2π

119

(2.12.17)

But, by (2.3.62),

|fε (eiθ )|2 = L2ε (w(θ) + ε)−1 so

Z

so

|fε (eiθ )|2 w(θ)

dθ ≤ L2ε 2π

(2.12.18) (2.12.19)

RHS of (2.12.3) ≤ lim L2ε = RHS of (2.12.12) ε↓0

proving (2.12.12).

This completes the discussion of what Szeg˝o had in 1920. We want to end by saying something about allowing dµs 6= 0 in this proof, an issue first addressed in print by Szeg˝o in 1958 [179], although by other means (close to our entropy arguments earlier in this chapter) Verblunsky [441] allowed dµs 6= 0 in 1934. The idea used by Szeg˝o is to find suitable polynomials to “mask” dµs . Szeg˝o did this by hand, others (e.g., Garnett [142]) use peak functions, and Section 2.5 of [390] has a construction using boundary values of Carathéodory functions of singular measures. Instead, we want to present here a simple and elegant argument due to Helson–Lowdenslager [192]. Given dµ, a positive measure on D, define Sdµ ⊂ L2 (∂D, dµ) to be the closure of polynomials, P , with P (z) = 0. Let Hdµ be the ⊥ orthogonal projection of the function 1 to Sdµ , so Z 2 iθ 2 kHdµ kL2 (dµ) = min |P (e )| dµ(θ) P (0) = 1 (2.12.20) polynomials

Moreover, H is the unique function which is a norm limit of polynomials P with P (0) = 1 and so that Z Hdµ (θ)e−ikθ dµ(θ) = 0 k = 1, 2, . . . (2.12.21)

Proposition 2.12.5. Suppose that dµ has the form (2.1.1). We have that |Hdµ |2 dµs (θ) = 0 (2.12.22) and

Z

Hdµ (θ)e−ikθ w(θ)

dθ =0 2π

(2.12.23)


120

Proof. {P | polynomial, P (0) = 0} is an ideal in the set of polynomials. Thus, for any k > 0, H(1 + αeikθ ) ∈ 1 + Sdµ , so if for α ∈ C we define Z Fk (α) = |Hdµ (θ)(1 + αeikθ )|2 dµ (2.12.24) then

Fk (α) ≥ Fk (0)

(2.12.25)

Fk (α) = Fk (0) + Re(αck ) + dk |α|2

(2.12.26)

Expanding

we see (2.12.25) implies ck = 0, that is, Z |Hdµ (θ)|2 eikθ dµ(θ) = 0

(2.12.27)

for all k > 0. But taking complex conjugates, we conclude the measure |H|2 dµ has all k 6= 0 Fourier coefficients zero, from which we conclude that dθ |H|2 dµ = c (2.12.28) 2π This immediately implies (2.12.22), and (2.12.22) plus (2.12.21) implies (2.12.23). This allows us to prove Theorem 2.12.6. Let dµ have the form (2.1.1). Then Z Z dθ iθ 2 inf |P (e )| dµ(θ) P (0) = 1 = inf |P (eiθ )|2 w(θ) 2π for the inf over all polynomials.

P (0) = 1 (2.12.29)

Proof. Since Hdµ obeys (2.12.23) and, by kf kL2 (w dθ ) ≤ kf kL2 (dµ)

(2.12.30)

2π

dθ we see that Hdµ is an L2 (w 2π ) limit of polynomials with P (0) = 1. We have Hw dθ = Hdµ (2.12.31) 2π

Thus, by (2.12.22), RHS of (2.12.29) = =

Z Z

|Hdµ |2 w(θ)

dθ 2π

|Hdµ |2 dµ

= LHS of (2.12.29)

˝ ASYMPTOTICS 2.13. ANOTHER APPROACH TO SZEGO

121

Thus, Szeg˝o’s theorem for dµs = 0 implies the theorem for dµs 6= 0, and we have a new proof of Theorem 2.12.7. If µ has the form (2.1.1), then Z dθ 2 lim kΦn k = exp log(w(θ)) n→∞ 2π

(2.12.32)

Remarks and Historical Notes. As noted, the basic ideas when dµs = 0 are in Szeg˝o’s 1920 paper [420]. The beautiful argument in Proposition 2.12.5 which relies on the fact that {P | P (0) = 0} is an ideal is due to Helson–Lowdenslager [192]. 2.13. Another Approach to Szeg˝ o Asymptotics In this section, we want to discuss another approach to Szeg˝o asymptotics. Central is a formula of considerable interest as a tool in OPUC: Theorem 2.13.1 (CD Formula). Let {ϕn }∞ n=0 be the normalized OPUC for a nontrivial probability measure, dµ. Then for any z, ζ with z ζ¯ 6= 1, n X ¯ −1 ϕj (ζ) ϕj (z) = [ ϕ∗n+1 (ζ) ϕ∗n+1 (z) − ϕn+1 (ζ) ϕn+1 (z)](1 − ζz) j=0

(2.13.1)

Remarks. 1. The quantity in (2.13.1) is called the CD kernel (or Christoffel–Darboux kernel) and denotes Kn (ζ, z). We will study it further in Sections 2.14–2.17, 3.12, and 5.11. 2. This is called the Christoffel–Darboux formula because they proved an analog for OPRL (see Subsection 1.2.9 of [390] and [398]). It is due to Szeg˝o [424]. 3. This result is an analog of the Wronskian relation for solutions of −u′′ + V u = zu, −w ′′ + V w = ζw with u(0) = w(0) = 0. Then, Z a ¯ (z − ζ) w(x) u(x) dx = w ′ (a) u(a) − u′ (0) w(0) 0

and the first proof is similar.

First Proof. Taking the conjugate of (1.8.15) for ζ and multiplying by (1.8.15) for z and subtracting the same for (1.8.14), we get ¯ ϕn (ζ) ϕn (z) ϕ∗n+1 (ζ) ϕ∗n+1 (z) − ϕn+1 (ζ) ϕn+1 (z) = ϕ∗n (ζ) ϕ∗n (z) − ζz (2.13.2)


122

2 since the cross terms cancel and ρ−2 n (1 − |αn | ) = 1. Thus,

¯ ϕn (ζ) ϕn (z) + [ ϕ∗ (ζ) ϕ∗ (z) − ϕn (ζ) ϕn (z)] LHS of (2.13.2) = (1 − ζz) n n (2.13.3) which leads to (2.13.1) if we iterate.

Second Proof. Fix first ζ ∈ ∂D. Then ϕ∗n+1 (ζ) ϕ∗n+1 (z) − ϕn+1 (ζ) ϕn+1 (z) vanishes if ζ = z since |ϕ∗n+1 (ζ)| = |ϕn+1 (ζ)|. Thus, for some polynomial h of degree n, ϕ∗n+1 (ζ) ϕ∗n+1 (z) − ϕn+1 (ζ) ϕn+1 (z) = (ζ − z)h(z)

(2.13.4)

Since ϕn+1 , ϕ∗n+1 ⊥ {z j }nj=1 , we see hz j , (ζ − z)hi = 0

j = 1, . . . , n

(2.13.5)

or ¯ j−1 , hi = |ζ|2hz j , hi = hz j , hi ζhz

(2.13.6)

so, by induction (with C = h1, hi), ¯ jC hz j , hi = (ζ)

j = 0, 1, . . . , n

(2.13.7)

and thus for any polynomial Q of degree at most n, hQ, hi = C Q(ζ)

(2.13.8)

Since {ϕj }nj=0 is an orthonormal basis, h(z) =

n X j=0

we find h(z) = C

hϕj , hiϕj (z)

n X

ϕj (ζ) ϕj (z)

(2.13.9)

(2.13.10)

j=0

Equating powers of z n+1 in (2.13.4), ∗ C ϕn (ζ) = ρ−1 n [ ϕn+1 (ζ) + αn ϕn+1 (ζ) ] = ζ¯ ϕn (ζ)

(2.13.11) (2.13.12)

by the inverse recursion relation (2.4.6). Thus, C = ζ¯ proving (2.13.1) ¯ the formula for ζ ∈ ∂D. By analyticity (both sides are analytic in ζ), holds for all ζ. Corollary 2.13.2. For |z| < 1,

|ϕ∗n (z)| ≥ (1 − |z|2 )1/2

(2.13.13)

˝ ASYMPTOTICS 2.13. ANOTHER APPROACH TO SZEGO

123

Proof. Taking ζ = z in (2.13.1), |ϕ∗n (z)|2

2

2

= |ϕn (z)| + (1 − |z| )

Since ϕ0 (z) = 1, (2.13.13) is immediate.

n−1 X j=0

|ϕj (z)|2

(2.13.14)

Corollary 2.13.3. Fix z0 ∈ D. Then either sup |ϕ∗n (z0 )| < ∞

(2.13.15)

lim |ϕ∗n (z0 )| = ∞

(2.13.16)

n

or else n→∞

(2.13.15) holds if and only if ∞ X j=0

|ϕj (z0 )|2 < ∞

(2.13.17)

Proof. By (2.13.14), 2

(1 − |z0 | )

n−1 X j=0

2

|ϕj (z0 )| ≤

|ϕ∗n (z0 )|2

≤

n X j=0

|ϕj (z0 )|2

(2.13.18)

This shows (2.13.15) ⇔ (2.13.17), and that if (2.13.17) fails, then (2.13.16) is true. Proposition 2.13.4. Let {fn (z)}∞ n=1 be a family of nonvanishing analytic functions on a connected open subset, Ω, of C. Suppose that for each compact K ⊂ Ω, CK = inf |fn (z)| > 0

(2.13.19)

lim sup |fn (z)| = ∞

(2.13.20)

z∈K,n

Then either (a) For every z ∈ Ω,

N →∞ n≤N

with convergence uniform on compact K ⊂ Ω, or (b) For every compact K ⊂ Ω, DK = sup |fn (z)| < ∞

(2.13.21)

z∈K,n

Proof. Let gn (z) = fn (z)−1 so the gn are uniformly bounded on compact subsets of Ω. Suppose (2.13.21) fails for some K. Then we can find zj ∈ K and nj so |fnj (zj )| ≥ j

(2.13.22)


124

By passing to a subsequence, we can suppose zj → z∞ ∈ K and gnj has a limit g∞ . By (2.13.22), g∞ (z∞ ) = 0 so, by Hurwitz’s theorem and the fact that each gn is nonvanishing, g∞ ≡ 0, that is, (2.13.20) holds uniformly on compacts. Theorem 2.13.5. Let dµ be a nontrivial probability measure on ∂D of the form (2.1.1). Then (i) If Z dθ = −∞ (2.13.23) log(w(θ)) 2π then for each R < 1, inf |ϕ∗n (z)| → ∞

n,|z|≤R

(ii) If

Z

(2.13.24)

dθ > −∞ 2π

(2.13.25)

sup |ϕ∗n (z)| < ∞

(2.13.26)

log(w(θ))

then for each R < 1, n,|z| 0, we see, if eiθ bn−1 (eiθ ) = eiγ(θ)

(2.14.27)

then γ ′ (θ) > 0 (2.14.28) which implies that zeros are simple. (iii) γ(θ) defined by (2.14.27) is strictly monotone, so if β = e−iγ0 , then between two solutions of γ(θ) = γ0 (mod 2π), every value of γ(θ) (mod 2π) is taken once. (iv) By Szeg˝o recursion, zΦn−1 − α ¯ n−1 Φ∗n−1 Φn = Φ∗n Φ∗n−1 − αn−1 zΦn−1 zbn−1 − α ¯ n−1 = η(zbn−1 , α ¯ n−1 ) = 1 − αn zbn−1

bn =

(2.14.29)

In the interval (z1 , z2 ), zbn−1 goes through a 2π change of phase. eiθ → η(eiθ , α ¯ n−1) is a monotone bijection of ∂D to ∂D, so bn also goes through a 2π change of phase. Thus, zbn goes through more than a 2π change of phase, and so zbn = β¯˜ must have a solution. (v) Suppose that Φn (z; β) has zeros at e±iϕ and µ is supported in −iϕ iϕ (e , e ). Let Φn (z; β)z P (z) = (z − eiϕ )(z − e−iϕ ) This is a polynomial because of the assumed zeros, and since P z −1 has degree n − 2, P is a linear combination of z, z 2 , . . . , z n−1 . This is orthogonal to Φ∗n−1 and to zΦn−1 , and so to Φn (z; β). Thus, Z z |Φn (z; β)|2 dµ = 0 (2.14.30) iϕ (z − e )(z − e−iϕ ) But

(eiθ − eiϕ )(eiθ − e−iϕ ) = (eiθ + e−iθ ) − (eiϕ + e−iϕ ) eiθ

132

so


1 z = >0 iϕ −iϕ (z − e )(z − e ) z=eiθ cos θ − cos ϕ

for |θ| < ϕ, that is, on supp(dµ). Thus (2.14.30) cannot hold. By rotation covariance, any pair of zeros in a gap can be rotated to this case. We can make the consequences of (iv) explicit: Corollary 2.14.5. Let (z0 , . . . , zn−1 ) and (w0 , . . . , wn ) be the ze˜ respectively, counted counterclockwise. ros of Φn (z; β) and Φn+1 (z; β), Then one of the following happens: (a) Φn and Φn+1 have a single zero in common which, by cyclic relabeling, we can suppose is z0 = w0 . In that case, each of the n intervals (z0 , z1 ), (z1 , z2 ), . . . , (zn−1 , zn ), (zn , z0 ) has exactly one w. (b) Φn and Φn+1 have no zeros in common, in which case among the n intervals, (z0 , z1 ), . . . , (zn , z0 ), one has exactly two w’s and each of the others has exactly one w. Proof. Follows from the fact that each of the n intervals (z0 , z1 ), . . . , (zn , z0 ) must contain at least one w. There is only one other w left. Remarks and Historical Notes. For properties (iii)–(v) for OPRL, see Section 1.2 of [390]. The gap property (property (iii)) comes as follows: If x0 , x1 are two zeros of Pn in (a, b) which is disjoint from supp(dµ), then Pn /(x − x0 )(x1 − x) is of degree n − 2, so orthogonal to Pn , so Z |Pn |2 (x − x0 )−1 (x1 − x)−1 dµ(x) = 0

But (x − x0 )−1 (x1 − x)−1 is positive on supp(dµ). This classical argument motivated the final proof in the section. For purposes of Gaussian quadrature on ∂D, POPUC were introduced by Jones, Nj˚ astad, and Thron [205]. Their zeros and other properties have been studied by Golinskii [170], Cantero–Moral–Velázquez [65], Wong [450], and Simon [396]. Our discussion here using CD kernels is influenced by Wong [450]. Most of Theorem 2.14.4 is from [65, 170] with parts from [396]. The use of bn and of the recursion (2.14.29) is due to Khrushchev [214].

2.15. ASYMPTOTICS OF THE CD KERNEL: WEAK LIMITS

133

2.15. Asymptotics of the CD Kernel: Weak Limits This is the first of three sections on the asymptotics of the CD kernel for OPUC, Kn (w, z), especially when |w| = |z| = 1 and w = z or |w − z| is small. In this section, we will say something about limits 1 of n+1 Kn (eiθ , eiθ ) dµ(θ) as a measure. We start by relating it to limits of the zero counting measure for paraorthogonal polynomials. Given a measure dµ on ∂D, we let dνn be the zero counting measure for Φn , that is, νn is a pure point measure with νn ({w}) = n−1 × (multiplicity of w as a zero of Φn )

(2.15.1)

(β)

Similarly, for any β ∈ ∂D, we let νn be the zero counting measure for the POPUC Φn (z; β) (all multiplicities are one). Finally, we define 1 KN (eiθ , eiθ ) dµ(θ) (2.15.2) N +1 R (β) which is a probability measure on ∂D, since |ϕj |2 dµ = 1. νn is a probability measure on ∂D and νn on D. Here is a result that says they have the same weak limits: dµ(N ) (θ) =

Theorem 2.15.1. For any ℓ = 1, 2, . . . and any β, Z Z z ℓ dµ(N ) − z ℓ dν (β) ≤ 2ℓ N +1 N +1 Z Z z ℓ dνN +1 − z ℓ dν (β) ≤ 2ℓ N +1 N +1

(2.15.3) (2.15.4) (β )

w

j In particular, for a subsequence, N(1) < N(2) < . . . , dνN (j)+1 −→ dν∞

w

if and only if dµ(N (j)) −→ dν∞ (for one, and then for all choices of {βj }), and in that case, for any ℓ = 1, 2, . . . , Z Z ℓ lim z dνN (j)+1 = z ℓ dν∞ (z) (2.15.5) j→∞

w

Conversely, if (2.15.5) holds for some dν∞ on ∂D, then dµ(N (j)) −→ dν∞ . Proof. ϕ0 , . . . , ϕN are a basis for Ran(πN +1 ), so with A = πN +1 Mz πN +1 , Z

N

ℓ

z dνN +1

1 1 X = Tr(Aℓ ) = hϕj , (Aj )ℓ ϕj i N +1 N + 1 j=0

(2.15.6)


134

and similarly, Z

N

z

ℓ

(β) dνN +1

1 X (N +1) ℓ = hϕj , (Uβ ) ϕj i N + 1 j=0

(2.15.7)

By definition of KN , Z

N

z ℓ dµ(N ) =

1 X hϕj , z ℓ ϕj i N + 1 j=0

(2.15.8)

(N +1)

If j ≤ N − ℓ, (Aj )ℓ ϕj = (Uβ )ℓ ϕj = z ℓ ϕj , so the terms in the sum cancel for such j’s. Since |hϕj , z ℓ ϕj i| ≤ 1 and similarly for A and (N +1) Uβ for any j, the remaining terms contribute at most 2ℓ/(N + 1) to the difference of the sums. This proves (2.15.3) and (2.15.4) R R (β) For dµ(N ) and dνN +1 , we have measures on ∂D so z −ℓ dη = z ℓ dη. Polynomials in z and z −1 are dense in the continuous functions on ∂D, R so weak convergence is equivalent to convergence of z ℓ dη (for all (βj ) ℓ ≥ 0) which happens for one of dµ(N (j)) and dνN (j)+1 if and only if it happens for both (by (2.15.3)). And convergence then implies (2.15.5). For the converse, note that (2.15.5) implies convergence of the moments (β) of dνN (j)+1 by (2.15.4). This is especially useful since there is a class of measures dµ for (β) which w-lim dνn can be seen to be dθ/2π. Proposition 2.15.2. Consider the conditions (a)

lim (ρ0 . . . ρn−1 )1/n = 1

n→∞

(2.15.9)

n−1

(b)

1X lim |αj |2 = 0 n→∞ n j=0

(2.15.10)

n−1

(c)

1X |αj | = 0 lim n→∞ n j=0

(2.15.11)

Then (a) ⇒ (b) ⇔ (c). If sup |αn | = R < 1 n

then (b) ⇒ (a) also.

(2.15.12)


135

Proof. (b) ⇔ (c). Since |αj | < 1, we have that |αj |2 < |αj |. This and the Schwarz inequality imply 2 X n−1 n−1 n−1 1X 1X 1 |αj | ≤ |αj |2 ≤ |αj | (2.15.13) n j=0 n j=0 n j=0 (a) ⇒ (b). We have that 2

2

− log|ρj | = |αj | + so

∞ X 1 k=2

k

|αj |2k ≥ |αj |2

(2.15.14)

n−1

1X |αj |2 ≤ − log[(ρ0 . . . ρn−1 )2/n ] n j=0

(2.15.15)

Thus (a) ⇒ lim(− log(ρ0 . . . ρn−1 )2/n ) = 0 ⇒ (b). (b) ⇒ (a) if (2.15.12) holds. If (2.15.12) holds, then for some K (can be taken −R−1 log(1 − R)), so

− log|ρj |2 ≤ K|αj |2

n−1 KX |αj |2 ≥ − log[(ρ0 . . . ρn−1 )2/n ] n j=0

so (b) plus the fact that ρj < 1 implies (a).

(2.15.16)

Definition. Let µ be a measure on ∂D. If lim (ρ0 . . . ρn−1 )1/n = 1

n→∞

we say µ is regular. Regularity has two important consequences: Theorem 2.15.3. Let µ be a measure on ∂D which is regular. Then for any z ∈ C \ D, we have lim |Φn (z)|1/n = lim |ϕn (z)|1/n = |z|

n→∞

n→∞

(2.15.17)

Remark. The proof shows the convergence is uniform on compact subsets of C \ D. Proof. Since (ρ1 . . . ρn )1/n → 1, we need only prove the result for Φn . Suppose |z| > 1. By Szeg˝o recursion and |Φn (z)| ≥ |Φ∗n (z)| if |z| > 1 (see (2.9.11)), we have (|z| − |αn |)|Φn (z)| ≤ |Φn+1 (z)| ≤ (|z| + |αn |)|Φn (z)|

(2.15.18)


136

Since |z| > 1 holds, there is a K(|z|) so that for all n, Moreover, if |z| > 1,

1 − |αn | |z|−1 ≥ exp(−K|αn |)

(2.15.19)

1 + |αn | |z|−1 ≤ exp(|αn |)

(2.15.20)

Thus, (2.15.18) plus induction implies X X n−1 n−1 |Φn (z)| exp −K |αj | ≤ ≤ exp |αj | |z|n j=0 j=0

(2.15.21)

(2.15.11) thus implies (2.15.17) for Φn . This proves (2.15.17) for |z| > 1 and the limit is uniform in θ, for z = reiθ with r > 1 fixed. By the maximum principle, for any r > 1, |Φn (eiθ )| ≤ sup |Φn (reiϕ )|

(2.15.22)

ϕ

This plus the uniformity implies for any r > 1, lim sup sup |Φn (eiθ )|1/n ≤ r θ

Since r is arbitrary, the lim sup is at most 1. Since the ρ’s for the second kind polynomials are the same, we have lim sup|ψn (eiθ )|1/n ≤ 1

(2.15.23)

|ϕn (eiθ )| |ψn (eiθ )| ≥ 1 This plus (2.15.22) implies

(2.15.24)

lim inf|ϕn (eiθ )|1/n ≥ 1

(2.15.25)

But by (2.4.57),

and so (2.15.17) for |z| = 1.

Theorem 2.15.4. Let µ be a measure on ∂D which is regular. Then dθ w-lim dµ(n) = (2.15.26) n→∞ 2π and for any {βj } ∈ ∂D,

dθ (2.15.27) n→∞ 2π Proof. By Theorem 2.15.1, it suffices to prove for ℓ ≥ 1, Z z ℓ dνn (z) → 0 (2.15.28) R since dθ/2π is the unique measure on ∂D with eiℓθ dη(θ) = 0 for ℓ > 0. w-lim dνn(βn ) =


137

Let dν∞ be an arbitrary weak limit point of dνn . For |z| > 1, log|z − w| is continuous for w ∈ D, so Z Z log|z − w| dνn(w) → log|z − w| dν∞(w) (2.15.29) Since

Z 1 log|Φn (z)| = log|z − w| dνn(w) n (2.15.17) implies for |z| > 1, Z w log 1 − dν∞ (w) = 0 z

(2.15.30)

(2.15.31)

In the region |z| > 1, uniformly in |w| ≤ 1, log|1 − wz | is the real part of an analytic function, so Z w log 1 − dν∞ (w) = 0 (2.15.32) z since we first see it is an imaginary constant and then, by taking |z| → ∞, we see the constant is zero. Now X j ∞ w 1 w log 1 − = (2.15.33) z j z j=1

uniformly in |w| ≤ 1 and |z| ≥ 2, so interchanging the sum and integral, we see Z w j dν∞ (w) = 0 (2.15.34)

for j ≥ 1, proving (2.15.28).

We have thus proven that if dµ is regular, then 1 dθ w dθ iθ iθ Kn (e , e ) w(θ) + dµs −→ n+1 2π 2π

(2.15.35) w

1 When the Szeg˝o condition holds, (2.9.30) says n+1 Kn dµs −→ 0, and one might hope that this is true more generally (indeed, see Theorem 2.17.7), which leads us to a natural guess that under suitable hypotheses, pointwise in θ, 1 Kn (eiθ , eiθ )w(θ) → 1 (2.15.36) n+1 It is precisely this surmise that we explore in the next two sections. Of course, it cannot hold at points with w(θ) = 0. Note, however, if dµs = 0, (2.15.35) implies that if the left side of (2.15.36) converges uniformly, the limit must be 1.

138


Remarks and Historical Notes. Theorem 2.15.1 is from Simon [400]. Regularity will be discussed more extensively in Section 5.9, mainly in the context of OPRL. In particular, its history is discussed in the Notes to that section. That regularity implies zeros are distributed according to an “equilibrium” measure (which is dθ/2π for ∂D) is a major theme of that section. The proof of (2.15.28) is essentially potential theoretic—this is discussed in Section 5.5. 2.16. Asymptotics of the CD Kernel: Continuous Weights In this section, we will study the asymptotics of the CD kernel for continuous nonvanishing weights and apply this to obtain a refined estimate on the zeros of POPUC. We will call a function, f , on ∂D “continuous” on an interval I = [α, β] (i.e., α, β ∈ ∂D and I is the set of points between α and β going counterclockwise from α to β) if, as a function on ∂D, it is continuous at each z ∈ [α, β]. This is stronger than saying the restriction of f to I is continuous on I; in particular, it says something if α = β and I is a single point. Here is the main theorem of this section: Theorem 2.16.1 (Levin–Lubinsky [270]). Let dµ be a regular probability measure on ∂D of the form dµ = w(θ)

dθ + dµs 2π

(2.16.1)

Suppose, for any interval I = [α, β] ⊂ ∂D, (a) supp(dµs ) ∩ I = ∅ (b) w is “continuous” on I and nonvanishing there. Then (1) (Diagonal Asymptotics) For any A < ∞, uniformly in z∞ ∈ I, and sequences zn ∈ ∂D with n|zn − z∞ | ≤ A for all n, 1 Kn (zn , zn ) → w(z∞ )−1 n+1

(2.16.2)

(2) (Lubinsky Universality) For any A < ∞, uniformly in z∞ ∈ I, and a, b ∈ R with |a|, |b| ≤ A, we have i

e 2 (a−b) sin 12 (a − b) Kn (z∞ eia/n , z∞ eib/n ) → 1 Kn (z∞ , z∞ ) (a − b) 2

(2.16.3)

More generally, the limit of Kn (zn , wn )/Kn (z∞ , z∞ ) is the right side of (2.16.3) so long as zn , wn ∈ ∂D, |zn − z∞ | < A/n, |wn − z∞ | < A/n, and (zn /wn )n → ei(a−b) .

2.16. ASYMPTOTICS OF THE CD KERNEL: CONTINUOUS WEIGHTS 139

As a most important application of Lubinsky universality, we will analyze the fine structure of the spacing of the zeros of the POPUC (see Theorem 2.16.10). The most important tools in the proof will be a variational formula for Kn (z, z) and Lubinsky’s inequality which relates off-diagonal asymptotics to diagonal asymptotics. Here is the variational formalism. Define the Christoffel function by Z iθ 2 λn (z0 ; dµ) = inf |Pn (e )| dµ(θ) deg Pn ≤ n; Pn (z0 ) = 1 Pn (2.16.4) If z = 0, λn (0) = kΦ∗n k2 and Szeg˝o’s theorem gives the asymptotics of λn . Here is the connection to the CD kernel: Proposition 2.16.2. The minimizer of (2.16.4) is given by Kn (z0 , z) Kn (z0 , z0 )

(2.16.5)

λn (z0 ) = Kn (z0 , z0 )−1

(2.16.6)

Pn (z; z0 ) = and

Proof. Expand any trial polynomial n X Pn (z) = aj ϕj (z)

(2.16.7)

j=0

Then the normalization condition says n X aj ϕj (z0 ) = 1

(2.16.8)

j=0

while

2

kPn k =

n X j=0

|aj |2

(2.16.8) and the Schwarz inequality says X n 2 |aj | Kn (z0 , z0 ) ≥ 1

(2.16.9)

(2.16.10)

j=0

so

λn (z0 ) ≥ Kn (z0 , z0 )−1 On the other hand, the choice aj =

ϕj (z0 ) Kn (z0 , z0 )

(2.16.11)

(2.16.12)


140

that is, Pn given by the right side of (2.16.5) has kPn k2 =

Kn (z0 , z0 ) = Kn (z0 , z0 )−1 2 Kn (z0 , z0 )

Thus, (2.16.6) is the minimum and (2.16.5) the minimizer.

(2.16.13)

Remark. λn (0) = kΦ∗n k2 and (2.16.6) is just the CD formula at ζ = z = 0. For comparison purposes, it will be useful to consider all positive but not necessarily normalized measures. If µ ˜ = µ/µ(∂D), then the monic Φn are the same, that is, Φn (z; dµ) = Φn (z; d˜ µ)

(2.16.14)

so we define αn (dµ) = αn (d˜ µ)

ρn (dµ) = ρn (d˜ µ)

(2.16.15)

Thus, kΦn (dµ)k = ρ0 . . . ρn−1 µ(∂D)1/2

(2.16.16)

lim kΦn (dµ)k1/n = 1 ⇔ (ρ0 . . . ρn )1/n → 1 Thus, if we define regularity as

(2.16.17)

so

1/n

lim kΦn (dµ)kL2 (dµ) = 1

(2.16.18)

then µ regular ⇔ µ ˜ regular. It is also easy to see that Theorem 2.16.1 for probability measures ˜ implies the result for any positive µ by comparing w to w˜ and K to K. From the definition (2.16.4) and (2.16.6), we immediately have (note that µ ≤ µ∗ only makes sense because we allow nonnormalized measures) Corollary 2.16.3. For any two measures on ∂D, for all n, z ∈ C, µ ≤ µ∗ ⇒ λn (z, µ) ≤ λn (z, µ∗ ) ⇔

Kn∗ (z, z)

≤ Kn (z, z)

(2.16.19)

(2.16.20)

We will prove Theorem 2.16.1 by a comparison technique. We thus need one example where we can prove the theorem by calculation. The example will be dµ0 = dθ/2π! Theorem 2.16.4. Fix any A < ∞. Let dθ dµ0 = 2π (0)

(so w ≡ 1) and let Kn be its CD kernel. Then


(i) 1 K (0) (zn , zn ) → 1 (2.16.21) n+1 n uniformly for all zn ∈ ∂D for which zn → z∞ ∈ ∂D and n|zn − z∞ | ≤ A

(2.16.22)

for all A. (ii) Uniformly for z∞ ∈ ∂D and a, b real with |a|, |b| ≤ A, we have (0)

Kn (z∞ eia/n , z∞ eib/n ) (0)

Kn (z∞ , z∞ )

i

e 2 (a−b) sin 12 (a − b) → 1 (a − b) 2

(2.16.23)

Remark. If a = b, sin( 12 (a − b))/ 12 (a − b) is interpreted as 1. Proof. (i) Neither zn → z∞ nor (2.16.22) is needed (!) since Kn(0) (eiθ , eiθ ) = n + 1

(2.16.24)

for all eiθ ∈ ∂D since |ϕn (eiθ )| = 1. (ii) If a = b, this is immediate by (2.16.24). If a 6= b, since iθ inθ ϕ(0) n (e ) = e

we have, by summing a geometric series (or by using the CD formula), Kn(0) (eiθ+ia/n , eiθ+ib/n ) = Since

1 − ei(a−b)(n+1)/n 1 − ei(a−b)/n

n(1 − ei(a−b)/n ) → −i(a − b)

and ieiu/2 2 sin u2 = (eiu − 1), we get (2.16.23).

(2.16.25)

That the measure is regular will provide a key estimate on the minimizer (2.16.5): Lemma 2.16.5. Let µ be a regular measure on ∂D. Then for any ε > 0, there is a δ and C so that the minimizer, Pn (z, z0 ), of (2.16.5) obeys |Pn (z, z0 )| ≤ Ceεn (2.16.26) for all z, z0 with

|z|, |z0 | ∈ (1 − δ, 1 + δ)

(2.16.27)

Proof. Let δ = eε/4 − 1. By regularity and Theorem 2.15.3, for all m and |z| = 1 + δ, |ϕm (z)| ≤ C1 e3εm/8

(2.16.28)


142

for some C1 . By the maximum modulus, this holds for z, z0 obeying (2.16.27). Thus, since Kn (z0 , z0 ) ≥ 1, |Pn (z, z0 )| ≤ nC12 e3εn/4

so (2.16.26) holds for suitable C.

(2.16.29)

Here is a main tool for shifting from a nice case like µ0 to a less nice case. We state it in greater generality than used to prove Theorem 2.16.1 because of the needs of the next section. Theorem 2.16.6 (Nevai Comparison Theorem). Let µ, µ♯ be two regular measures on ∂D of the form dθ dθ dµ = w + dµs dµ♯ = w ♯ + dµ♯s (2.16.30) 2π 2π Suppose z0 = eiθ0 ∈ D obeys (1) dµs = dµ♯s for z ∈ (z0 e−iδ , z0 eiδ ) for some δ > 0. (2) For all ε sufficiently small, there is aε > 1 so for |θ − θ0 | < ε, we have ♯ a−1 (2.16.31) ε w(θ) ≤ w (θ) ≤ aε w(θ) and lim aε = 1 (2.16.32) ε→0

(3) For some zn ∈ D, zn → z0 , and every ℓ(n) with n2 < ℓ(n) < 2n, 1 lim Kn (zℓ(n) , zℓ(n) ) = B 6= 0 (2.16.33) n→∞ n + 1 Then 1 lim Kn♯ (zn , zn ) = B (2.16.34) n→∞ n + 1 Moreover, this is uniform in zn in the sense that if (2.16.33) holds (with the same B) for all zn → z0 , there are, for any ε, a δ and N0 so that if n > N0 and |zn − z0 | < δ, then B − 1 K ♯ (zn , zn ) < ε (2.16.35) n n+1

There is also uniformity in z0 : If w and w ♯ are continuous and nonvanishing on an interval in ∂D and we have dµs = dµ♯s in a neighborhood of I and (2.16.31) is replaced by w(θ) w ♯ (θ) w(θ) ≤ ♯ ≤ aε w(θ0 ) w (θ0 ) w(θ0 ) for |θ − θ0 | < ε (aε the same for all θ0 ), and (2.16.33) holds uniformly in z0 ∈ I where B(z0 ) is z0 -dependent, then (2.16.34) holds with B in 0) (2.16.34) replaced by B(z0 ) ww(z ♯ (z ) . 0 a−1 ε


Proof. We will leave the two uniformity statements to the reader and focus on the case of a single z0 and single sequence zn → z0 . We will construct Nevai trial functions to put into the variational principle (2.16.4) for λ♯n (z0 ), the Christoffel functions for µ♯ . Fix ε > 0 and write n = n(ε) + m(ε) where |nε − m(ε)| < 1 and n(ε) = n − m(ε). Let z + zn q (n) (z) = (2.16.36) 2zn which obeys η (n) (n) (n) iθ sup |q (e )| = cos q (zn ) = 1; sup |q (z)| = 1; 0 and C < ∞. (2) follows from the second relation in (2.16.37) and (3) from the third relation in (2.16.37) and (2.16.26) (where the “ε” of (2.16.26) is picked so e“ε” < cos(η)−ε/(1−ε) ). Use Qn (z) as a trial function for λ♯n (zn ), breaking up the integral into z = eiθ with |θ − θ0 | ≤ η and > η, writing the contributions as λ♯n;≤η and λ♯n;≥η . By (2.16.40), 2 −2K(ε,η)n λ♯n;≥η ≤ Cη,ε e

(2.16.41)

For λn;≤η , we use the fact that, by dµs = dµ♯s and by (2.16.31), dµ♯ ↾ (θ0 + η, θ0 − η) ≤ aη dµ ↾ (θ0 + η, θ0 − η)

(2.16.42)

if η < δ, so

λ♯n;≤η ≤ a2η λn(ε) (zn )

(2.16.43)

since the contribution of |θ − θ0 | > η to λn(ε) is positive. Note first that limn→∞ ne−2Kn = 0. Thus, by (2.16.33) and lim(n + 1)/(n(ε) + 1) = (1 − ε)−1 , lim sup(n + 1)λ♯n (zn ) ≤ a2η (1 − ε)−1 B

(2.16.44)

144


Since ε and η are arbitrary, we can take them to zero and use (2.16.32) to see lim sup(n + 1)λ♯n (zn ) ≤ B (2.16.45)

In the other direction, we switch the roles of µ and µ♯ . We define m(ε) so |nε − m(ε)| < 1 but now define n(ε) = n + m(ε)

(2.16.46)

q (n) (z) is still defined as before, but now the Nevai trial function is Q♯n(ε) (z) = Pn♯ (z, zn )q (n) (z)m(ε)

(2.16.47)

Use in Q♯n(ε) (z) as a trial function of λn(ε) (zn ), breaking the integral into two pieces, λn;≤η and λn;≥η , for |θ − θ0 | ≤ η and |θ − θ0 | > η. By (2.16.40), 2 −2K(η,ε)n λn;≥η ≤ Cη,ε e which λn;≤η ≤ a2η λ♯n (zn )

Multiply by n and use ne−2Kn → 0, plus lim nλn(ε) (zn ) = n B lim n(ε) = (1 + ε)−1 B to see (1 + ε)−1B ≤ a2η lim inf nλ♯n (zn ) n→∞

(2.16.48)

Again, we take η ↓ 0 and then ε to 0 and so, with (2.16.45), we obtain (2.16.34). Proof of Theorem 2.16.1, part (1). Let us denote µ by µ♯ dθ and then take µ = w(z∞ ) 2π . All the hypotheses of the Nevai com1 parison theorem hold with B = w(z∞ )−1 since n+1 Kn (z, z) = w(z∞ )−1 for any z! Thus, by that theorem, (2.16.2) holds. The reader should check the uniformity statements. For the second part, the key is Theorem 2.16.7 (Lubinsky’s Inequality). Let µ ≤ µ∗ . Then, for any z, w ∈ C, we have |Kn (z, w) − Kn∗ (z, w)|2 ≤ Kn (w, w)[Kn (z, z) − Kn∗ (z, z)]

(2.16.49)

For this, we need a critical property of the CD kernel: Theorem 2.16.8 (CD Reproducing Property). For any polynomial Qn of degree at most n and all w ∈ C, Z Kn (ζ, w)Qn(ζ) dµ(ζ) = Qn (w) (2.16.50)


In particular, for any z, w ∈ C, Z Kn (z, ζ)Kn (ζ, w) dµ(ζ))Kn(z, w)

(2.16.51)

Remark. One way to understand this is that Kn is the integral kernel of πn+1 , the projection onto polynomials of degree at most n 2 (2.16.51) is then an expression that πn+1 = πn+1 . Pn Proof. If Qn (ζ) = j=0 aj ϕj (ζ), then (2.16.50) is just Z Kn (ζ, w)ϕj (ζ) dµ(ζ) = ϕj (w) (2.16.52)

which is immediate. Since Kn (z, ζ) is a polynomial of degree n in ζ, (2.16.50) implies (2.16.51). Proof of Theorem 2.16.7. Since Kn (z, w)−Kn∗ (z, w) is a polynomial of degree n in w, we have Z Kn (ζ, w)[Kn (z, ζ) − Kn∗ (z, ζ)] dµ(ζ) = Kn (z, w) − Kn∗ (z, w)

(2.16.53)

Thus, by the Schwarz inequality, where

LHS of (2.16.49) ≤ 1 · 2 1 = =

Z

Z

|Kn (ζ, w)|2 dµ(ζ)

(2.16.54)

(2.16.55)

Kn (w, ζ)Kn(ζ, w) dµ(ζ)

= Kn (z, w)

(2.16.56)

by (2.16.52), while Z 2 = |Kn (z, ζ) − Kn∗ (z, ζ)|2 dµ(ζ) Z Z Z 2 ∗ = |Kn (ζ, z)| − 2 Re Kn (ζ, z)Kn (z, ζ) dµ(ζ) + |Kn∗ (ζ, z)|2 dµ(ζ) ≤ Kn (z, z) − 2Kn∗ (z, z) + Kn∗ (z, z)

= Kn (z, z) −

Kn∗ z, z)

(2.16.57)

(2.16.58)

The first Kn (z, z) in (2.16.57) comes from the same calculation that went from (2.16.55) to (2.16.56), while the last term comes from first using dµ ≤ dµ∗ and then doing the same calculation for K ∗ , µ∗ . The middle term in (2.16.57) is just (2.16.50) for Qn (ζ) = Kn∗ (z, ζ).


146

Lemma 2.16.9. Let µ, ν be two positive measures on ∂D. Suppose µ is regular. Then µ ∨ ν, their sup, is also regular. Remark. For any two measures, µ, ν, one shows there is a smallest η larger than µ and ν. This is denoted µ ∨ ν. It is discussed in [109, 202]. Proof. Since ρj (µ ∨ ν) ≤ 1, we have

1/n

lim sup kΦn ( · , d(µ ∨ ν))kL2 (µ∨ν) ≤ 1 On the other hand, since Φn ( · , dµ) is a minimizer and µ ≤ µ ∨ ν, kΦn ( · , dµ)kL2(µ) ≤ kΦn ( · , d(µ ∨ ν))kL2 (µ)

so, by regularity,

= kΦn ( · , d(µ ∨ ν))kL2 (µ∨ν) 1/n

lim inf kΦn ( · , d(µ ∨ ν))kL2 (µ∨ν) ≥ 1

Completion of the Proof of Theorem 2.16.1. Let µ∗ = µ ∨ (w(z∞ ) dµ0). Then w, w ∗ and w0 ≡ w(z∞ ) (the constant weight) are continuous and agree at z∞ . Thus, by part one, uniformly in |a|, |b| < A, 1 K ♯ (eic/n z∞ , eic/n z∞ ) → w(z∞ )−1 (2.16.59) n+1 n for c = 0, a, or b, and for Kn♯ associated to any of µ∗ , µ, or w(z∞ ) dµ0 (where we used the lemma to assure µ∗ is regular). Apply (2.16.49) where Kn is associated to w(z∞ ) dµ0 and Kn∗ is associated to µ∗ . Here we divide by Kn (z∞ , z∞ ). By part (1) of the theorem,

so

Kn (z∞ eib/n , z∞ eib/n ) Kn∗ (z∞ eib/n , z∞ eib/n ) − →0 Kn (z∞ , z∞ ) Kn (z∞ , z∞ )

Kn∗ (z∞ eia/n , z∞ eia/n ) → RHS of (2.16.3) (2.16.60) Kn (z∞ , z∞ ) where we used (2.16.25). Now we use part (1) of the theorem again to replace Kn (z∞ , z∞ ) in (2.16.60) by Kn∗ (z∞ , z∞ ). Next, we use Lubinsky’s inequality for µ ≤ µ∗ and, in the same same way, transfer the limit to ratios of Kn for µ. The more general assertion at the end of Theorem 2.16.1 follows by going through the proof and seeing it gives the stronger result. Finally, we want to turn to the zeros of POPUC.


Definition. Given any measure dµ on ∂D and β, z∞ ∈ ∂D, √ is defined for j = 0, ±1, ± · · · , ±[ n ] to be “successive” (0),(β) zeros Φ(z; β) where zn is the first one going counterclockwise from (1),(β) (−1),(β) z∞ (or z∞ itself), zn the next, etc., and zn going clockwise. If, for each j ∈ Z and β ∈ ∂D, (j+1),(β) (j),(β) (z∞ ) − zn (z∞ ) zn lim →1 (2.16.61) e2πi/n (j),(β) zn (z∞ )

we say there is clock behavior at z∞ . If the limit in (2.16.61) is uniform in z∞ ∈ I ⊂ ∂D, we say there is uniform clock behavior on I. Remark. The names comes from the fact that the zeros are spaced like numerals on a clock. Theorem 2.16.10 (Freud–Levin Theorem). If the hypotheses of Theorem 2.16.1 hold, we have uniform clock behavior on I for each fixed β. Proof. Fix z∞ ∈ I. Let βn be defined so Φn (z∞ ; βn ) = 0. By (2.14.20), the other zeros of Φn ( · ; βn ) are the zeros of Kn (z, βn ). By the uniformity of convergence of (2.16.3), Kn changes sign at points asymptotic to e2πi/n z∞ , and so Φn has a zero within a slightly larger interval. By Theorem 2.14.4(iii), Φn (z, β) has a zero within 2πi (1+o(1)) n (0),(β) 2πi of z∞ . Thus, |zn (z∞ ) − z∞ | ≤ n (1 + o(1)). By (2.16.3), there are no zeros in (0),(β) −iπ/n (0),(β) iπ/n (zn (z∞ )e , zn (z∞ )e ) and the next zero has argument 2πi (1 + o(1)) greater. Repeating this, we get clock behavior. n 1 Remarks and Historical Notes. That n+1 Kn (z∞ , z∞ ) → w(z∞ )−1 for smooth w’s (or its equivalent for Christoffel weights) goes back to the first half of the twentieth century. Nevai [315] describes the history, Freud’s key role, and applications. Its great generality is a result of Máté–Nevai–Totik [296], discussed in the next section. In the context of OPRL on [−1, 1], the ability to wiggle z∞ and the importance of doing so was noted by Lubinsky [282]. In the same paper, he noted what we call Lubinsky’s inequality and used it to prove Lubinsky universality. For smooth w’s, this universality result goes back at least to Freud [137] and was studied in the context of random matrices using Riemann–Hilbert techniques (see [98, 250]) but nothing like Lubinsky’s generality. The extension to OPUC is in Levin–Lubinsky [270].


148

The idea of localizing trial functions for one problem using [ 12 (z + z0 )] (or its equivalent for OPRL) goes back to Nevai [314]. We name the Nevai trial functions and Nevai comparison theorem after this work. At the very end of his book, Freud [137] noted that universality (or its OPRL analog) implied clock spacing for the zeros. It was Levin–Lubinsky [269] who applied this idea and Lubinsky’s very general universality result to get clock behavior in general. Since it was Levin who rediscovered Freud’s result that universality implies clock behavior, we call Theorem 2.16.10 the Freud–Levin theorem. Earlier Last–Simon [267], using very different methods, had the best clock behavior results for OPRL, and Simon [393] had some clock behavior results for (P)OPUC, but the two Levin–Lubinsky papers [269, 270] have the strongest results on clock behavior. Lubinsky [281] has a second interesting approach to universality; see the Notes to Section 3.12. We note the clock behavior here is only local. At opposite ends of the circle, the zeros have about n/2 zeros in between and the errors can add up so that there is no result on, say, asymptotically opposite zeros for even n. nε

2.17. Asymptotics of the CD Kernel: Locally Szeg˝ o Weights In this final section on asymptotics of the CD kernel for OPUC supported on all of ∂D, we consider the case of noncontinuous weights. To even state the main theorem, we need to recall some basic harmonic analysis. dθ Definition. Let f ∈ L1 ([a, b], dx) (or L1 (∂D, 2π )). A point x in [a, b] (or ∂D) is called a Lebesgue point of f if and only if Z x+ε −1 lim (2ε) |f (y) − f (x)| dy = 0 (2.17.1) ε↓0

x−ε

In particular, at a Lebesgue point, the maximal function, Mf , obeys Z x+a −1 (Mf )(x) ≡ sup (2a) |f (y)| dy < ∞ (2.17.2) a>0

x−a

Three fundamental results we will need (and discuss in the Notes) are: Theorem 2.17.1. For f ∈ L1 (dx), a.e. x in [a, b] (or ∂D) is a Lebesgue point. We also need an analog of (2.17.1)/(2.17.2) for singular measures:

˝ WEIGHTS 2.17. CD KERNEL: LOCALLY SZEGO

149

Theorem 2.17.2. Let dµs be a singular measure on [a, b] (or ∂D). Then for Lebesgue almost every x, we have lim (2ε)−1 µs (x − ε, x + ε) = 0

ε→0

(2.17.3)

Remark. It is also known for µs -a.e. x that the limit is infinite. If dµ has the form (2.1.1), we say eiθ ∈ ∂D is a Lebesgue point for dµ if (2.17.3) holds for x = eiθ and µs , the singular part of dµ, and eiθ is a Lebesgue point for the weight w. Theorem 2.17.3 (Fatou’s Theorem). Let f ∈ H 1 (D) with boundary values denoted by f (eiθ ). If eiθ0 is a Lebesgue point of f ↾ ∂D, then nontangential boundary values are given by f (eiθ0 ), that is, for any ε > 0, π 1 −iθ iθ 0 lim sup |f (z)−f (e )| |z| > 1− , |arg(1−ze )| ≤ (1−ε) =0 n→∞ n 2 (2.17.4) Definition. We say dµ obeying (2.1.1) is locally Szeg˝ o on I = [α, β] ⊂ ∂D if and only if Z dθ log(w(eiθ )) > −∞ (2.17.5) 2π I If w obeys a local Szeg˝o condition, we can find w˜ obeying a global ˜ be the Szeg˝o function for w. Szeg˝o condition with w˜ ↾ I = w. Let D ˜ If ♯ int ♯ ˜ w is a second extension and z0 ∈ I , then D − D is analytic near z0 , ˜ if and only if it is for D ♯ . Thus, being a so z0 is a Lebesgue point for D Lebesgue point is independent of global Szeg˝o extension. We will say z0 is a Lebesgue point of the local Szeg˝o function in that case. Note. Being a Lebesgue point for log(w) is not sufficient to be a Lebesgue point of D. One needs to also be a Lebesgue point for the conjugate function of log(w). Here are the main results of this section: Theorem 2.17.4 (MNT Theorem [296]). Let µ be a regular measure on ∂D which is locally Szeg˝ o on I. Let eiθ0 ∈ I be a point where w(θ0 ) 6= 0 and be Lebesgue point for both µ and for the local Szeg˝ o function. Let zn ∈ ∂D be a sequence obeying sup n|zn − eiθ0 | ≡ A < ∞

(2.17.6)

n

Then

1 Kn (zn , zn ) = w(eiθ0 )−1 (2.17.7) n+1 Moreover, for each A, the limit is uniform in zn obeying (2.17.6). lim

n→∞


150

Theorem 2.17.5 (Findley’s Theorem [128]). Under the hypotheses of Theorem 2.17.4, we have that for any A < ∞, uniformly in |a|, |b| < A, i e 2 (a−b) sin 12 (a − b) Kn (eiθ0 eia/n , eiθ0 eib/n ) → (2.17.8) 1 Kn (eiθ0 , eiθ0 ) (a − b) 2 More generally, the limit relation holds for Kn (zn , wn )/Kn (eiθ0 , eiθ0 ) if zn , wn ∈ ∂D, |zn − eiθ0 | < A/n, |wn − eiθ0 | < A/n, and (zn /wn )n → ei(a−b) . We sill see later that Findley’s theorem implies a local clock behavior for the zeros of POPUC. Two other other theorems we will prove do not even require a local Szeg˝o condition. We will use the first in the proof of Theorem 2.17.4: Theorem 2.17.6 (Máté–Nevai Upper Bound [294]). For any measure dµ on ∂D of the form (2.1.1) and any Lebesgue point, z0 , of dµ, lim sup (n + 1)λn (zn ) ≤ w(z0 )

(2.17.9)

n→∞

for any sequence zn ∈ ∂D with

sup n|zn − z0 | < ∞

(2.17.10)

n

Remark. This includes points where w(z0 ) = 0. Theorem 2.17.7 (Simon [400]). If I = (α, β) is an open interval in ∂D, if µ is regular, and w(z) > 0 for a.e. z ∈ (α, β), then Z 1 dθ iθ iθ (i) (2.17.11) n + 1 Kn (e , e )w(θ) − 1 2π → 0 I Z 1 (ii) Kn (eiθ , eiθ ) dµs(θ) → 0 (2.17.12) n + 1 I

We now turn to the proof of these four theorems, starting with the third and fourth:

Lemma 2.17.8. Let λ be a finite positive measure on R. For x∞ ∈ R, define for t > 0 1 L(t) = λ([x∞ − t, z∞ + t]) (2.17.13) 2t Let h(s) be a continuous, even L1 (R, dx) function on R with Suppose

0 ≤ s ≤ t ⇒ h(s) ≥ h(t) ≥ 0

(2.17.14)

lim L(t) = 0

(2.17.15)

t↓0


and xn → x∞ with

A = sup n|xn − x∞ | < ∞

Then

lim

n→∞

Z

nh(n(x − xn )) dλ(x) = 0

151

(2.17.16) (2.17.17)

Remarks. 1. Continuity of h is not needed. 2. (2.17.18) below is often called the Layer Cake Principle; see Lieb–Loss [276]. Proof. Let dν = −dh as a Stieltjes measure on (0, ∞), so h(s) = ν([s, ∞)), or equivalently, Z ∞ h(s) = χ[−t,t] (s) dν(t) (2.17.18) 0

Thus, Z Z t t dν(t) nh(n(x − xn )) dλ(x) = nλ xn − , xn + n n Z t+A t+A ≤ nλ x∞ − , x∞ + dν(t) n n Z t+A = L (t + A) dν(t) (2.17.19) n R R R Since dν(t) = h(0) and 2t dν = h(s) ds, we see (t + A) dν(t) is a finite measure. By hypothesis, kLk∞ < ∞ and limn→∞ L( t+A )=0 n for all t, so by the dominated convergence theorem, (2.17.19) goes to 0. Proof of Theorem 2.17.6. Let n 1 X ij(θ−ϕ) iθ iϕ e Qn (e , e ) = n + 1 j=0

(2.17.20)

precisely the λn (eiϕ ) minimizer for dθ/2π, that is, (0) iθ iϕ iθ iϕ Kn (e , e )/Kn (e , e ). Of course, one can sum the geometric series (essentially a special case of the CD formula!) 1 ei(n+1)(θ−ϕ) − 1 n + 1 ei(θ−ϕ) − 1 (θ − ϕ)) 1 ein(θ−ϕ)/2 sin( n+1 2 = 1 n+1 sin( 2 (θ − ϕ))

Qn (eiθ , eiϕ ) =

which is “essentially” the classical Dirichlet kernel, and Fn (eiθ , eiϕ ) ≡ (n + 1)|Qn (eiθ , eiϕ )|2

(2.17.21) (2.17.22)


152

=

(θ − ϕ)) 1 sin2 ( n+1 2 2 1 n + 1 sin ( 2 (θ − ϕ))

(2.17.23)

which is exactly the classical Fejér kernel. It has the following properties: Z dθ (a) [Fn (eiθ , eiϕ )] =1 (2.17.24) 2π (b) sup |Fn (eiθ , eiϕ )| = n + 1 (2.17.25) θ,ϕ

(c) where

|Fn (eiθ , eiϕ )| ≤ (n + 1)G((n + 1)(θ − ϕ))

(2.17.26)

π2 π2 (2.17.27) G(x) = min 2 , x 4 (a) is immediate from (2.17.20) and the orthogonality of eijθ . (b) follows from |Q| ≤ 1 and Q(eiθ , eiθ ) = 1. To get (c), we note that (sin x)/x is monotone decreasing for x in [0, π/2], so for |θ − ϕ| < π, 2 |θ − ϕ| 1 sin (θ − ϕ) ≥ (2.17.28) 2 π 2

Thus with

˜ |Fn (eiθ , eiϕ )| ≤ (n + 1)G((n + 1)(θ − ϕ))

(2.17.29)

π 2 sin2 ( x2 ) ˜ G(x) = (2.17.30) x2 ˜ ˜ G(x) ≤ π 2 /x2 since sin2 (x/2) ≤ 1 and G(x) ≤ π 2 /4 since sin2 (x/2) ≤ x2 /4. Since Qn is a valid trial function in (2.16.4), Z (n + 1)λn (zn ) ≤ (n + 1)|Qn (z, zn )|2 dµ(z) Z = Fn (z, zn ) dµ(z) (2.17.31) Z ≤ w(z0 ) + (n + 1)G((n + 1)(θ − ϕn )) dλ(θ) (2.17.32)

where zn = eiϕn , z0 = eiϕ0 dθ + dµs (θ) 2π Here (2.17.32) comes from (2.17.24) and (2.17.26). dλ(θ) = |w(θ) − w(ϕ0 )|

(2.17.33)


153

Lemma 2.17.8 is applicable since z0 , being a Lebesgue point of dµ, implies the L(t) associated to λ obeys (2.17.13). Proof of Theorem 2.17.7. By compactness of the measures, η, on ∂D, with η(∂D) ≤ 1, pick a subsequence n(j) so dθ 1 Kn(j) (eiθ , eiθ )w(θ) → dη1 (θ) n(j) + 1 2π

(2.17.34)

1 Kn(j) (eiθ , eiθ ) dµs (θ) → dη2 (θ) n(j) + 1

(2.17.35)

and

By Theorem 2.15.4 and the regularity assumption, dη1 + dη2 =

dθ 2π

(2.17.36)

On the other hand, by Theorem 2.17.6 and (2.16.6), lim inf

1 Kn (eiθ , eiθ )w(θ) ≥ 1 n+1

(2.17.37)

for a.e. θ ∈ I (this uses w(θ) > 0 a.e. on I). Thus, by Fatou’s lemma, for any positive continuous function, f , supported in I, Z Z 1 dθ Kn (eiθ , eiθ )w(θ)f (θ) f (θ) dη1 (θ) = lim n→∞ I n + 1 2π Z 1 dθ Kn (eiθ , eiθ )w(θ) f (θ) ≥ lim inf n+1 2π ZI dθ ≥ f (θ) (2.17.38) 2π I by (2.17.37). This means dη1 ↾ I ≥

dθ 2π

(2.17.39)

so, by (2.17.36), dθ dη2 ↾ I = 0 (2.17.40) 2π By compactness of the set of measures, (2.17.12) holds and Z 1 dθ |I| Kn (eiθ , eiθ )w(θ) = (2.17.41) 2π 2π I n+1 dη1 ↾ I =

This and (2.17.37) implies (2.17.26).


154

We turn to the proof of Theorem 2.17.4. The Máté–Nevai upper bound provides half the result, so we only need a lower bound. We will suppose for now that a global Szeg˝o condition holds. Throughout, zn obeys (2.17.6). Since we will be using analytic continuation of D(z) and Kn (z, zn ) from ∂D to D, the following lemma will be useful: Lemma 2.17.9. Let Qn be a polynomial of degree at most n with no zeros in D. Let z0 ∈ ∂D and 0 < s < 1. Then n 1+s |Qn (sz0 )| ≥ |Q(z0 )| > e−n(1−s) |Q(z0 )| (2.17.42) 2 Proof. For 0 < t < 1, t t2 1 1 t − log 1 − = + +··· < t + + ... = t 2 2 4 2 4

Let t = (1 − s) so 1 −

which implies

t 2

= 12 (1 + s) and see 1+s − log e−(1−s) (2.17.45) 2 showing the second inequality in (2.17.42). rotation covariance, we can suppose z0 = 1. Any such Qn (z) = QBy n c j=1 (z − zj ) with zj ∈ / D, so it suffices to prove the case n = 1 and c = 1, that is, that for any 0 < s < 1 and z1 ∈ / D, we have 1+s |1 − sz1 | ≥ |1 − z1 | (2.17.46) 2 To prove this, fix s ∈ (0, 1) and let g(w) =

|1 − sw| |1 − w|

(2.17.47)

g is harmonic on (C \ D) ∪ {∞}, so its minimum on C \ D is taken on ∂D. If x = Re w and w ∈ ∂D,

(1 − sx)2 + s2 (1 − x2 ) (1 − x)2 + (1 − x2 ) (1 + s2 ) − 2sx = ≡ h(x) (2.17.48) 2 − 2x Since h′ (x) = 2(1 − s2 )/(2 − 2x)2 > 0, we see on [−1, 1], h(x) takes its minimum at x = −1, that is, g(w) is minimized at w = −1 where g(−1) = 1+s . 2 g(w)2 =


155

We will be looking at points with s ∼ 1 − ε/n, so define for ε > 0, −1 ε xn (ε) = 1 + zn (2.17.49) n Without loss, we suppose z0 = 1. Here is the key inequality that will prove the result: Proposition 2.17.10. Let Pn (z) be the minimizer for λn (zn ), that is, Pn (z) =

Kn (zn , z) Kn (zn , zn )

(2.17.50)

Suppose we prove that X ∞ Z iθ iθ j −ijθ dθ lim lim sup Pn (e )D(e )xn (ε) e =0 ε↓0 2π n→∞ j=n+1

(2.17.51)

Then lim inf(n + 1)λn (zn ) ≥ w(1)

(2.17.52)

Remarks. 1. The intuition about why (2.17.51) is reasonable comes from the following. Since Pn is concentrated near eiθ = 1 and D is “reasonable” near 1, we would expect to be able to replace D(eiθ ) by D(1). But then, since deg Pn ≤ n and j ≥ n + 1, the integral is zero. 2. By Proposition 2.14.3 and Theorem 2.14.4, Pn has all its zeros on ∂D, hence none in D, so it obeys Lemma 2.17.9. Proof. Since Pn (z)D(z) lies in H 2 , its Taylor coefficients are given by integrals over ∂D, so ∞ Z X dθ Pn (xn (ε))D(xn (ε)) = Pn (eiθ )D(eiθ )xn (ε)j e−ijθ (2.17.53) 2π j=0 so if iθ

Hn (e , ε) =

n X

xn (ε)j e−ijθ

(2.17.54)

j=0

and if En (ε) = Pn (xn (ε))Dn (xn (ε))− then (2.17.51) implies

Z

Pn (eiθ )D(eiθ )Hn (eiθ , ε)

lim lim sup |En (ε)| = 0 ε↓0

n→∞

dθ (2.17.55) 2π (2.17.56)


156

Since {e−ijθ }nj=0 are Z

dθ -orthonormal 2π

and |xn (ε)| ≤ 1,

n

dθ X |Hn (e , ε)| = |xn (ε)|2j ≤ (n + 1) 2π j=0 iθ

2

(2.17.57)

Thus, by the Schwarz inequality and |D(eiθ )|2 = w(θ), Z 2 Z Pn (eiθ )D(eiθ )Hn (eiθ , ε) dθ ≤ (n + 1) |Pn (eiθ )|2 w(θ) dθ 2π 2π Z ≤ (n + 1) |Pn (eiθ )|2 dµ = (n + 1)λn (zn )

(2.17.58)

Thus, by (2.17.56), lim inf(n + 1)λn (zn ) ≥ lim lim inf [|Pn (xn (ε))|2 |D(xn (ε))|2 ] (2.17.59) ε↓0

n→∞

Since Pn (zn ) = 1, Lemma 2.17.9 implies that −1 ε |Pn (xn (ε))| ≥ exp −n 1 − 1 + n −1 ε = exp −ε 1 + n ≥ exp(−ε)

(2.17.60)

Since 1 is a Lebesgue point of D, Fatou’s lemma implies that D(xn (ε)) → D(1). Moreover, |D(1)|2 = w(1). Thus, (2.17.59) becomes lim inf(n + 1)λn (zn ) ≥ lim[e−2ε w(1)] = w(1) ε↓0

proving (2.17.52).

(2.17.61)

Thus, we need to prove (2.17.51). Once we have Theorem 2.17.5, we will know |Pn (z)| = |Kn (z, zn )|/Kn (zn , zn ) is asymptotically less than 1 for z ∈ ∂D with |z − 1| ≤ B/n. At this point, we only need a weaker bound. Proposition 2.17.11. Let Pn (z) be the minimizer (2.17.50). Then for any B finite, e lim sup |Pn (eiθ )| ≤ √ (2.17.62) n→∞ |θ| 0} and let m(z) v0 = −1

(3.2.10)

Then

vn = Tn (z)v0 (3.2.11) 2 2 obeys v ∈ ℓ (i.e., n=0 kvn k < ∞), and for any w0 ∈ C with 2 Tn (z)w0 ∈ ℓ , we have that w0 is a multiple of v0 . 2

P∞

To prove this, we introduce the analog of ψn : Definition. The second kind polynomials, qn (z), are defined for n = 0, 1, 2, . . . by Z pn (x) − pn (y) qn (x) = dµ(y) (3.2.12) x−y and for n = −1, q−1 (x) = −1 (3.2.13)

164

3. THE KILLIP–SIMON THEOREM

Theorem 3.2.2. The vector qn (z) wn = (3.2.14) an qn−1 (z) solves wn = Tn (z)w0 (3.2.15) t where w0 is (0, −1) Moreover, for n ≥ 1, qn (x) is a polynomial of degree n − 1. Indeed, n−1 qn (x) = a−1 1 pn−1 (x; {aℓ+1 , bℓ+1 }ℓ=1 )

(3.2.16)

the OPRL for the once stripped measure with Jacobi parameters {aℓ+1 , bℓ+1 }∞ ℓ=1 . Proof. The recursion relation (1.2.15) obeyed by pn implies Z xpn (x) − ypn (y) an+1 qn+1 (x) + bn+1 qn (x) + an qñ−1 (x) = dµ(y) x−y Z x−y = xqn (x) + pn (y) dµ x−y = xqn (x) + δn (3.2.17) where q˜−1 = 0 and otherwise q˜j = qj . Since a0 q−1 = −1, we see for n ≥ 0, an+1 qn+1 (x) + (bn − x)qn (x) + an qn−1 (x) = 0 which, given (3.2.5)/(3.2.6), implies that (3.2.15) holds. Since a1 q1 +(b1 −x)q0 +a0 q−1 = 0, we see, using q0 = 0, a0 q−1 = −1, that q1 = 1/a1 , so qn obeys qn+1 q1 n = Tn ({aj+1, bj+1 }j=1 ; z) (3.2.18) an+1 a1 with initial conditions (q1 , q0 ) = 1/a1 (1, 0), which immediately implies (3.2.16). This in turn implies qn is a polynomial of degree n − 1. Thus

pn (z) −qn (z) Tn (z) = (3.2.19) an pn−1 (z) −an qn−1(z) The analog of Proposition 2.4.6 is more powerful since there is no z n factor: Proposition 3.2.3. If rn and sn solve tn+1 t1 = Tn tn t0 then rn+1 sn − rn sn+1 = r1 s0 − r0 s1

(3.2.20) (3.2.21)

3.2. WEYL SOLUTIONS AND COEFFICIENT STRIPPING

165

In particular, an (qn (x)pn−1 (x) − qn−1 (x)pn (x)) = 1

(3.2.22)

sn+1 Proof. By (3.2.20), with Rn = ( rn+1 rn sn ), we have Rn = Tn R0 , so det(Rn ) = det(Tn ) det(R0 ). By (3.2.9), (3.2.21) holds.

Proof of Theorem 3.2.1. Clearly, vn = (gn , an gn−1 ) where gn (z) = m(z)pn (z) + qn (z) (3.2.23) Z Z Z dµ(x) dµ(x) pn (x) = pn (z) − pn (z) + dµ(x) x−z x−z x−z Z pn (x) = dµ = hpn , (· − z)−1 i (3.2.24) x−z

Since {pn }∞ n=0 is an orthonormal basis, Z ∞ X dµ(x) 2 |gn (z)| = ≤ |Im z|−2 2 |x − z| n=0

(3.2.25)

so v ∈ ℓ2 since {an }∞ n=1 is bounded. On the other hand, if wn ≡ Tn (z)w0 = (hn , an hn−1 ), then hn gn−1 − gn hn−1 = h0 g−1 − h−1 g0

(3.2.26)

by (3.2.21). If wn ∈ ℓ2 , hn ∈ ℓ2 , so hn gn−1 − gn hn−1 ∈ ℓ1 . Thus, LHS of (3.2.26) is in ℓ1 . Since the right side is constant, the constant must be zero, which implies (h0 , h−1 ) is a multiple of (g0 , g−1 ). Remarks. 1. As we will see, gn actually decays exponentially; see Proposition 3.2.6. This plus (3.2.26) shows that any other solution must grow exponentially. 2. We note that Z Z 1 1 Im m(z) dµ(x) = Im dµ(x) = 2 |x − z| Im z x−z Im z

so (3.2.25) can be rewritten as ∞ X n=0

|m(z)pn (z) + qn (z)|2 =

Im m(z) Im z

(3.2.27)

3. We will call gn (z) the Weyl solution, although we note that [95, 391] define the Jost solution by wn (z) = −gn−1 (z + z1 ). We will discuss the reasons for the differing conventions in the Notes to Section 3.7. Given Theorem 3.2.1, coefficient stripping is immediate:

166


Theorem 3.2.4 (Coefficient Stripping for OPRL, aka Stieltjes Expansion). Let m(z) be the m-function of a Jacobi matrix with Jacobi parameters {an , bn }∞ n=1 and let m1 be the m-function of the once stripped Jacobi matrix, that is, the one with parameters {an+1 , bn+1 }∞ n=1 . Then 1 m(z) = (3.2.28) b1 − z − a21 m1 (z)

Remarks. 1. To make the analog to the Schur algorithm precise, note that if m(z) is any discrete m-function, −m(z)−1 is also Herglotz and analytic on R \ I with (2.3.11). But (2.3.10) fails since −m(z)−1 ∼ z. But it can be seen that for some a > 0 and b and discrete m-function m, ˜ −m(z)−1 = z − b + a2 m(z) ˜ This is the analog of the Schur algorithm. This theorem says that a, b are the first two Jacobi parameters and m ˜ = m1 . 2. For a more streamlined proof, see Section 10.3. Proof. By the uniqueness in Theorem 3.2.1, m(z) m1 (z) z − b1 −1 =c −1 −1 a21 0

(3.2.29)

for some c. Since

n Tn−1 ({aj+1, bj+1 }n−1 j=1 ; z)A(a1 , b1 ; z) = Tn ({aj , bj }j=1 ; z)

(3.2.30)

2 implies that Tn−1 ({aj+1 , bj+1 }n−1 j=1 ; z) [LHS of (3.2.29)] lies in ℓ . This means [(z − b1 )m(z) + 1] −m1 (z) = [a21 m(z)] which is equivalent to (3.2.28).

In Theorem 3.7.6, we will extend (3.2.28) and relate gk /gk−1 to the m-function of a stripped J. As for the OPUC case, we thus have a continued fraction expansion for m(z), 1 m(z) = (3.2.31) a21 b1 − z − a22 b2 − z − b3 − z − · · · (3.2.12) provides a second way to go from dρ to {an , bn }∞ n=1 . This completes the results we need going forward, but we would like to make some additional remarks.


167

First, we want to note a connection to the spectral theorist’s Green’s function, and second, deduce exponential decay from that. Define for z ∈ C+ , Gkℓ (z) = hδk , (J − z)−1 δℓ i (3.2.32) Proposition 3.2.5. We have that for z ∈ C+ , Gn1 (z) = gn−1 (z)

(3.2.33)

where g is given by (3.2.23). Remark. More generally, one can show that Gkℓ (z) = Gℓk (z), and for k ≤ ℓ, Gkℓ = pk−1 (z)gℓ−1 (z) (3.2.34) Proof. We have proven that g·−1 (z) is the unique ℓ2 solution of [(J − z)(u)]n = 0 for n ≥ 2. But clearly, u = (J − z)−1 δ1 obeys the same equation, so (3.2.33) holds up to a single overall constant (a priori constant in n but not necessarily in z). But G11 (z) = m(z) = g0 (z) showing the constant is one. Proposition 3.2.6. For any Q > 2 supn |an |, we have −|k−ℓ| |Im z| |Gkℓ (z)| ≤ CQ,z 1 + Q

(3.2.35)

In particular, for each z ∈ C+ , gn (z) decreases exponentially in z. We will prove this using the method of Combes–Thomas [88] which depends on: Lemma 3.2.7. Let A be a (possibly unbounded) selfadjoint operator and J a bounded operator. Suppose z0 ∈ / σ(J) and that J(s) = eisA Je−isA

(3.2.36)

originally defined for s ∈ R has an analytic continuation to SK0 ≡ {s | |Im s| < K0 } with z0 ∈ / σ(J(s)) for all s ∈ SK0 . Let ϕ ∈ D(eK0 |A| ). Then (J − z0 )−1 ϕ ∈ D(eKA ) for all K ⊂ (−K0 , K0 ). Proof. It is a simple general fact that follows from the spectral theorem that η ∈ D(eKA ) for all K ⊂ (−K0 , K0 ) if and only if eisA η, defined initially for s ∈ R has a (Hilbert-space) analytic continuation to SK0 . For s real, eisA (J − z0 )−1 ϕ = (J(s) − z0 )−1 eisA ϕ

(3.2.37)

168


Under the hypotheses of the theorem, the right side of (3.2.37) has an analytic continuation and thus, so does the left. Proof of Proposition 3.2.6. We will prove the result when CQ,z is also ℓ-dependent. That it can be chosen ℓ-independent follows from detailed estimates implicit in (3.2.37). Of course, our application is to a fixed ℓ, namely, ℓ = 0. Let A be multiplication by n on ℓ2 , that is, Aδn = nδn . Then for s real, eisA Je−isA is the tridiagonal matrix with bn on the main diagonal and e±is an off-diagonal. This has an analytic continuation to all of C. Moreover, for K real, kJ(±iK) − Jk ≤ 2 sup |an | |eK − 1|

(3.2.38)

n

Since k(J − z)−1 k = |Im z|−1 and

(J(±iK) − z) = (J − z)−1 (1 + (J − z)−1 (J(±iK) − J))

(3.2.39)

we see z0 ∈ / σ(J(±iK)) so long as

|eK − 1|Q|Im z|−1 ≤ 1

(3.2.40)

or e|K| ≤ eK0 ≡ 1 +

Im z Q

SincePδℓ ∈ D(eK0 |A| ), the lemma implies (J − z)−1 δℓ ∈ D(e|K|A), that is, n |(e|K| )n Gnℓ |2 < ∞, which implies (3.2.36). Finally, we want to prove that on C+ , m(z) = lim − n→∞

qn (z) pn (z)

(3.2.41)

something closely related to the fact that qn + mpn ∈ ℓ2 . It can be derived from qn + mpn ∈ ℓ2 but only by getting lower bounds on pn which is tricky (but see the Notes). Instead, we will proceed with a result of independent interest: Proposition 3.2.8. Let JF ;n be the truncated n × n Jacobi matrix of (1.2.30). Let −1 Gn;F kℓ (z) = hδk , (Jn;F − z) δℓ i

(3.2.42)

Then Gn;F 11 (z) = −

qn (z) pn (z)

(3.2.43)


169

Proof. By (1.2.31), pn (z) = (a1 . . . an )−1 det(z − Jn;F ). By (3.2.16), qn (z) is (a1 . . . an )−1 times the 11 minor of z − Jn;F . Taking −1 into account that Gn;F 11 (z) = −hδ1 , (z − Jn;F ) δ1 i, we see that (3.2.43) is just Cramer’s rule. Remark. By (2.3.15) and Proposition 1.3.4, m(z)+qn (z)pn (z)−1 = O(z −2n−1 ) at infinity which, given the degrees of p and q, implies that (−qn /pn ) are Padé approximants about infinity. This convergence of Padé approximants in all of C+ is a (special case of a) result of Stieltjes [413]. Theorem 3.2.9. For k, ℓ fixed and z ∈ C+ , Gn;F k,ℓ (z) → Gk,ℓ (z)

(3.2.44)

In particular, (3.2.41) holds. Proof. View Jn;F as acting on ℓ2 by embedding it in a matrix with all zeros. Clearly, for any ϕ ∈ ℓ2 , Jn;F ϕ → Jϕ. Thus for z ∈ C+ , k[(Jn;F − z)−1 − (J − z)−1 ]ϕk ≤ k(Jn;F − z)−1 (J − Jn;F )(J − z)−1 ϕk

≤ |Im z|−1 k(J − Jn;F )(J − z)−1 ϕk → 0

Taking ϕ = δℓ yields (3.2.44).

We remark that one can also prove coefficient stripping from (3.2.44) without using Theorem 3.2.1, and it is often done that way. Remarks and Historical Notes. While we present this as the OPRL analog of OPUC results, the history is the opposite! Jacobi [201] essentially wrote down the finite matrix terminating continued fraction expansion (3.2.31). Stieltjes [413] wrote the infinite N case. Wall [443] calls them J-fractions. Similarly, Weyl solutions were first discussed (as Jost solutions) for OPRL; see, for example, Case [72]. We note that if supp(dρ) = [−2, 2] so there is a dµ on ∂D via the Szeg˝o mapping theorem, then there are formulae relating the Weyl solutions for dρ to those of dµ. While it may predate that, the use of the transfer matrix (3.2.1) with the extra an /an−1 in the second factor that leads to det(T ) = 1 is borrowed from Damanik–Killip–Simon [93]. For other proofs of (3.2.35), see the use of CD kernels and/or potential theory in Stahl– Totik [408] and Simon [395]. These references (see also [94]) also provide direct proofs that pn (z) is bounded below as n → ∞ for any z ∈ C+ and so direct proofs of (3.2.41) from (3.2.27).

170


3.3. Meromorphic Herglotz Functions In the proof of Szeg˝o’s theorem, a key role was played by the fact that a nonvanishing analytic function on D (in this case, (δ0 D)(z)), f (z), with log f ∈ H 1 (D) has a Poisson–Jensen representation Z iθ e +z dθ iθ f (z) = exp log(f (e )) (3.3.1) eiθ − z 2π

if f (0) > 0. For OPRL, the analog of δ0 D will be the m-function moved to D by the map (1.9.1), that is, M(z) = −m(z + z −1 )

(3.3.2)

(we will see the reason for the minus sign shortly). This function has zeros and poles in D so it cannot be represented in the form (3.3.1)! There is a standard method for controlling zeros of H p functions, namely, via Blaschke products which we discussed in Section 2.3. As we will see, one needs a variant on the products. To frame the change, we remark that one proves the convergence part of Proposition 2.3.16 by noting breiθ (z) = br (e−iθ z) (3.3.3) and br (z) − 1 = − so that sup |z|≤R

X j

(1 − r)(z + 1) 1 − rz

|bzj (z) − 1| ≤

1+R X (1 − |zj |) 1−R j

(3.3.4) (3.3.5)

The absolute convergence of Blaschke products will sometimes be relevant so the p = 1 case of the following is important (as will p = 3): Proposition 3.3.1. Let {Ej }∞ j=1 ⊂ R \ [−2, 2] and define βj in R \ [−1, 1] by Ej = βj + βj−1 (3.3.6) Then for any p > 0, ∞ ∞ X X (|Ej | − 2)p/2 < ∞ ⇔ (1 − |βj |−1 )p < ∞ j=1

(3.3.7)

j=1

Proof. Follows immediately from |Ej | − 2 = |βj |(|βj |−1 − 1)2

(3.3.8)

3.3. MEROMORPHIC HERGLOTZ FUNCTIONS

171

The convergence result P PJ in Proposition 2.3.16 is thus an analog of We j |aj | < ∞ ⇒ j=1 aj convergent for numerical sums. j+1 will instead need the analog of (−1) aj > 0, |aj | ≥ |aj+1 |, and P |aj | → 0 ⇒ Jj=1 aj convergent, the result for alternating sums. ∞ Theorem 3.3.2. Let {zj }∞ j=1 and {pj }j=1 be subsets of (−1, 1) so that

(a) (b)

|zj | → 1 as j → ∞ ∞ X |zj − pj | < ∞

(3.3.9) (3.3.10)

j=1

Then, with b given by (2.3.67), N Y bzj (z) → B∞ (z) b (z) p j j=1

as N → ∞ uniformly on compact subsets of C \ S where −1 ∞ S = {pj }∞ j=1 ∪ {zj }j=1 ∪ {±1}

Moreover, on ∂D \ {±1},

|z| = 1 ⇒ |B∞ (z)| = 1

(3.3.11)

(3.3.12) (3.3.13)

If zj > pj , let Ij = (pj , zj ) and σj = +1, and if zj < pj , let Ij = (zj , pj ) and set σj = −1. Define, for x ∈ (−1, 1), X N(x) = σj χIj (x) (3.3.14) j

If

then in C+ ∩ D,

N∞ ≡ kNk∞ < ∞

(3.3.15)

|arg B∞ (z)| < πN∞

(3.3.16)

Remarks. 1. (3.3.9)/(3.3.10), of course, imply |pj | → 1 also. 2. The convergence as a function with values in C ∪ {∞} is uniform away from ±1. 3. By (a), (b), any x ∈ (−1, 1) lies in at most finitely many Ij and the sum in (3.3.14) is uniformly convergent on each (−1 + ε, 1 − ε). N∞ is an integer (if finite). 4. In (3.3.16), we mean the continuous branch of arg B∞ with limε↓0 arg B∞ (x + iε) = 0 at points in (0, δ) for small δ, where B∞ (x) > 0.

172


5. In the case where the z’s and p’s interlace so N∞ = 1, it can happen that the set of values of arg B∞ (z) is either (0, π) or (−π, 0). Before proving this theorem, we want to note that (3.3.15) implies (3.3.10) for suitable orderings of the z’s and p’s. Lemma 3.3.3. Suppose that 0 ≤ z1 ≤ z2 ≤ . . .

0 ≤ p1 ≤ p2 ≤ . . . Then

∞ X j=1

|zj − pj | =

Z

0

(3.3.17) (3.3.18)

1

|N(x)| dx

(3.3.19)

so N∞ < ∞ implies (3.3.10). Proof. We claim that if Ij ∩ Ik 6= 0, then σj = σk . Suppose that j < k and σj = 1, σk = −1 (the other cases are similar). Then zj < pj while pk < zk so Ij ∩ Ik 6= ∅ implies pk < pj , contrary to (3.3.18). Thus, |N(x)| = #{j | x ∈ Ij } (3.3.20) since there are no χIj − χIk cancellations. This in turn implies (3.3.19). Remarks. 1. The above proof also shows that if x ∈ {zj }∞ j=1 ∪ {pj }∞ , then j=1 N(x) = #(j | zj < x) − #(j | pj < x)

(3.3.21)

2. One can handle the situation where we consider (−1, 1) instead of (0, 1) with ±1 limit points of the z’s and p’s by labeling {zj }∞ j=−∞ and {pj }∞ with j=−∞ z−2 ≤ z−1 < 0 ≤ z0 ≤ z1 < . . .

(3.3.22)

One still has (3.3.19). We begin the proof of Theorem 3.3.2 with two lemmas: Lemma 3.3.4. Let Q and K be two compact sets in C with Q a real interval and K ∩ [Q ∪ Q−1 ] = ∅ (3.3.23) For z ∈ K, x ∈ Q, define

˜b(z, x) = z − x 1 − xz

(3.3.24)

3.3. MEROMORPHIC HERGLOTZ FUNCTIONS

173

Then there is a constant C depending only on Q, K so for all x, w ∈ Q, ˜ 1 − b(z, x) ≤ C|x − w| (3.3.25) ˜b(z, w)

Remarks. 1. Of course, if z ∈ D, ˜b(z, x) = sgn(−x)b(z, x). b is normalized by b(0, x) > 0 (for x 6= 0 and b′ (z, 0)|z=0 > 0) which is convenient for products of b’s not to oscillate, but for us here, smoothness of b in x is more important. The x, w cancellations control oscillations. 2. By (3.3.23), ˜b is analytic and nonvanishing in z, x for z ∈ K, x ∈ Q. 3. In our applications, we will take Q = [−1, 1] and K ⊂ C+ or K ⊂ C− or else Q = [a, 1] or [−1, a] and K = {z | |z| ≤ a − ε}. Proof. Clearly, (3.3.25) follows from inf |˜b(z, w)| > 0 z∈K x∈Q

(3.3.26)

and

|˜b(z, w) − ˜b(z, x)| ≤ C1 |x − w| In turn, since Q is connected, (3.3.27) is implied by ∂ ˜b(z, w) < ∞ sup z∈K ∂w

(3.3.27)

(3.3.28)

w∈Q

The required (3.3.26)/(3.3.28) are immediate by compactness, given analyticity and nonvanishing of ˜b on K × Q.

Lemma 3.3.5. Fix z ∈ C+ ∩ D. Define arg(˜b(z, x)) for x ∈ (−1, 1) by requiring continuity and arg(˜b(z, x = 0)) = arg(z) ∈ (0, π). Then (i) arg(˜b(z, x)) ∈ (0, π) (ii) (iii) (iv) ∂ ∂x

∂ arg(˜b(z, x)) ∂x ∂ Im z arg(˜b(z, x)) < 2 ∂x |x − z|2 Z 1 ∂ arg(˜b(z, x)) dx = π −1 ∂x

0
0 (3.3.37) Im(z) Example. Since z 7→ z + z −1 = E maps Im z > 0 to Im E < 0, the function M of (3.3.2) is an MH function. Theorem 3.3.6. Let f be an MH function. Then all its zeros, M {zj }M j=−N , and poles {pj }j=N , lie on R∩D and interlace. Their Blaschke product, K Y bzj (z) (3.3.38) b (z) j=−K pj converges as K → ∞ to a function, B∞ , which obeys (i) B∞ is analytic on C \ ({zj−1 } ∪ {pj } ∪ {±1}). (ii) |B∞ (eiθ )| = 1 for eiθ ∈ ∂D \ {±1}. (iii) |arg B∞ | ≤ 2π iθ For a.e. θ, limr↑1 f (re ) exists and is nonzero with Z log|f (eiθ )| p dθ < ∞ 2π for all p ∈ [1, ∞). Moreover, f has a representation Z iθ e +z dθ iθ f (z) = σB∞ (z) exp log|f (e )| eiθ − z 2π with σ = ±1. Explicitly, f (0) 6= 0, ∞ ⇒ σ = sgn(f (0)) f (0) = 0 ⇒ σ = +1

(3.3.39)

(3.3.40)

(3.3.41)

(3.3.42a) (3.3.42b)

176


f (0) = ∞ ⇒ σ = −1

(3.3.42c)

Remarks. 1. One can improve (3.3.39) to ≤ π. 2. In line with the discussion after Theorem 2.3.19, we will call (3.3.41) the Poisson–Jensen formula for MH functions. Proof. In the neighborhood of any finite-order zero or pole of a meromorphic function, f , takes values with all possible arguments, so (3.3.40) implies that all the zeros and poles lie on (−1, 1). As one goes around a circle centered on (−1, 1) which intersects R at points in (−1, 1) which are neither zeros nor poles, arg f can change by at most π in each half-plane, so at most 2π over all. Thus, by the argument principle, each such circle has |# of poles inside − # of zeros inside| ≤ 1 This counts multiplicity. So zeros and poles are simple and must interlace. Thus, the intervals (zj , pj ) are disjoint and M X

j=−N

|zj − pj | < 2

(3.3.43)

(typically, N = M = ∞). Clearly, |zj | → 1 and N∞ = 1. Theorem 3.3.2 is thus applicable and implies (i)–(iii). Define f (z) (3.3.44) g(z) = B∞ (z) g is nonvanishing, so log(g(z)) defined with Im[log(g(0))] = 0 or π is analytic in D. By (3.3.39) and arg|f (z)| ≤ π on D ∩ C+ , we see |Im log(g(z))| ≤ 3π

(3.3.45)

so by M. Riesz’s theorem (Proposition 2.3.8), log(g(z)) ∈ ∩p 0 for z0 6= 0, we see σ=

f (0) |f (0)|

(3.3.46)

3.4. STEP-BY-STEP SUM RULES FOR OPRL

177

if f (0) 6= 0, ∞, which is (3.3.42a). If f (0) = 0, B∞ (z) has a factor z, f so σ = sgn f ′ (0) > 0 since ∂∂ Im ≥ 0 and simple zeros implies f ′ (x) > 0 Im z on (−1, 1). This proves (3.3.42b). This also implies residues of poles are negative, which implies (3.3.42c). Remarks and Historical Notes. Theorem 3.3.6 was first proven by Simon [387] as a tool for proving OPRL sum rules. Our proof follows his, given Theorem 3.3.2. He proved that theorem only for alternating zj and pj . Following a suggestion of Killip, the presentation in [391] P (see Proposition 13.8.2 and Theorem 13.8.3) emphasized that only j |zj − pj | < ∞ was needed for convergence. The extension of Theorem 3.3.2 essentially to the form we have it was needed by Damanik–Killip–Simon [93] to get sum rules for matrixvalued OPRL (see Section 4.4). Our proof here, by using Lemmas 3.3.4 and 3.3.5, is somewhat simpler than theirs. It is worth emphasizing that there is some magic going on here and explaining where the magic comes from. In the usual analysis of Nevanlinna functions, f (or if we allow poles, functions of bounded characteristic), one assume some weak bounds on |f (reiθ )| as r ↑ 1. These bounds imply information on the number of zeros (by Jensen’s inequality—essentially one goes from bounds on |f (reiθ )| to some control of arg(f (reiθ ))) and this allows construction and control of a Blaschke product, B. One proves that f /B has the same kind of growth property. Here, we make no a priori assumptions on |f (z)| but instead on arg f , which it turns out implies bounds on |f (z)|. The magic in both the usual analysis for Nevanlinna functions and the one here is, in essence, M. Riesz’s duality. The difference is that in the usual case, one goes from Re log|f | to Im log|f | and here we go in the opposite direction. There is also a difference in how the Blaschke products are controlled. 3.4. Step-by-Step Sum Rules for OPRL At this point, we are ready to turn the crank. Find the right function, write down a Poisson–Jensen formula for it, and obtain step-bystep sum rules as Taylor coefficients in the Poisson–Jensen formula. The magic, of course, is in picking the right function—it will be the m-function! The analog of (2.6.17) is: Theorem 3.4.1 (Nonlocal Step-by-Step Sum Rule). Let J be an infinite Jacobi matrix with σess (J) ⊂ [−2, 2] and J1 the once-stripped N± (1)± N1,± matrix. Let {En± }n=1 (resp. {En }n=1 ) be the eigenvalues of J (resp.

178


± J1 ) with ±En± > 2 and |En+1 | < |En± |. Let M (resp. M1 ) be given by (3.3.2) for the m-function of J (resp. J1 ). Then M is an MH function. Its poles in D are at

{β −1 | β + β −1 = En± , |β| > 1} ≡ P

(3.4.1)

{β −1 | β + β −1 = En(1)± , |β| > 1} ≡ Z

(3.4.2)

{θ | Im M(eiθ ) 6= 0} = {θ | Im M1 (eiθ ) 6= 0}

(3.4.3)

and its zeros in D are at z = 0 and

Moreover, M(reiθ ), M1 (reiθ ) have limits as r ↑ 1 for a.e. θ, and up to sets of measure zero, and

Im M(eiθ ) log Im M1 (eiθ )

dθ ∈ L ∂D, 2π p 0 if Im z 6= 0, M is an MH function. Poles of m are at {En± } so M has poles precisely in P . By (3.2.28), m has zeros precisely at points where m1 has poles, and so M has zeros precisely on Z ∪ {0} (M(z) = 0 since m(E) → 0 as |E| → ∞). Since M is an MH function, Theorem 3.3.6 implies that M(reiθ ) dθ has a limit M(eiθ ) as r ↑ 1, and since log|M| ∈ ∩p m and only depend on matrix elements of J, bn , an , with |n − 21 (k + ℓ)| ≤ m + 1. It follows that Tn ( 21 J) − 0 ⊕ Tn ( 12 J1 ) has zero matrix element except in a block of size at most (n + 2) × (n + 2). It is thus trace class. As we have seen (see Theorem 3.2.9), M(z) = lim hδ1 , (E(z) − Jm;F )−1 δ1 i m→∞

detm−1 (E(z) − (J1 )m−1;F ) m→∞ detm (E(z) − Jm;F ) detm (E(z) − 0 ⊕ (J1 )m−1;F ) = lim E(z)−1 m→∞ detm (E(z) − Jm;F ) detm (1 − [0 ⊕ (J1 )m−1;F ]/E(z)) = lim E(z)−1 m→∞ detm (1 − Jm;F /E(z)) = lim

(3.4.21) (3.4.22) (3.4.23)

where E(z) = z + z −1 . (3.4.21) is Cramer’s rule, (3.4.22) uses detm (E(z) − 0 ⊕ B) = E(z) detm−1 (E(z) − B) for any (m − 1) × (m − 1) matrix B. Once we have two detm ’s, we can use detm (E(z) − B) = E(z)m detm (1 − B/E(z)). By (3.4.18), if B has eigenvalues h1 , . . . , hm and |z| is small, X m B hj log detm 1 − = log 1 − E(z) E(z) j=1 =

∞ X 2 [mTn (0) − Tr(Tn ( 21 B))]z n (3.4.24) n n=1

Thus, (3.4.23) implies ∞ X 2 Tr[Tn ( 12 J) − Tn (0 ⊕ 12 J1 )] log(M(z)E(z)) = n n=1

(3.4.25)

where we took limm→∞ by first noting, since Tn ( 21 J) and Tn (0 ⊕ 12 J1 ) agree except on an (n + 2) × (n + 2) block, that the coefficients for fixed n are m-independent for m large, and then noting that the convergence in (3.4.21) is uniform in z for |z| small. Next we note that E(z) = (1 + z 2 )/z and that by Lemma 3.4.3, ∞ X 2 2 log(1 + z ) = − Tn (0)z n n n=1 Thus

M(z) log z

= log(M(z)E(z)) − log(1 + z 2 )

182


=

∞ X 2 {Tn (0) + Tr(Tn ( 12 J) − Tn (0 ⊕ 21 J1 ))}z n n n=1

∞ X 2 = Tr(Tn ( 12 J) − 0 ⊕ Tn ( 12 J1 ))z n n n=1

since Tn (0 ⊕ 21 J1 ) = Tn (0) ⊕ Tn ( 12 J1 ).

Let us write out (3.4.20) explicitly for n = 1, 2. T1 (x) = x, so Tr(T1 ( 12 J) − 0 ⊕ T1 ( 12 J1 )) =

1 2

b1

Next, T2 (x) = 2x2 − 1. Thus, since

Tr(1 − 0 ⊕ 1) = 1

and

Tr(( 12 J)2 − 0 ⊕ ( 12 J1 )2 ) = 14 (b21 + 2a21 )

(for the sum of the squares of all matrix elements is involved), we see that Tr(T2 ( 21 J) − 0 ⊕ T2 ( 12 J1 )) = 12 b21 + (a21 − 1) Thus (3.4.20) says M(z) log = b1 z + ( 12 b21 + a21 − 1)z 2 + O(z 3 ) z

(3.4.26)

Theorem 3.4.6 (Step-by-Step Case Sum Rules). We have (C0 ) Define Z 2π 1 Im M1 (eiθ ) Z(J | J1 ) = log dθ (3.4.27) 4π 0 Im M(eiθ ) Then

− log(a1 ) = Z(J | J1 ) + (Cn ) For n ≥ 1, we have

X j,±

[log(|pj |) − log(|zj |)]

2 [Tr(Tn ( 12 J) − 0 ⊕ Tn ( 12 J1 ))] = Sn + Eñ n

(3.4.28)

(3.4.29)

where

Z 2π 1 Im M1 (eiθ ) Sn = − log cos nθ dθ 2π 0 Im M(eiθ ) X (zjn − pnj ) − (zj−n − p−n j ) ˜ En = n j,±

(3.4.30) (3.4.31)

3.4. STEP-BY-STEP SUM RULES FOR OPRL

183

Proof. Given (3.4.10), (3.4.13), and (3.4.20), these are just the Taylor coefficients of the log’s of the two sides of (3.4.5) (we have defined C0 as the negative of the zeroth coefficient). Here we need to note that since Im M1 (e−iθ ) Im M1 (eiθ ) = Im M(eiθ ) Im M(e−iθ ) we can replace e−inθ by 12 (e−inθ + einθ ) = cos nθ. We want to note explicitly the C1 formulae using (3.4.26): b1 = S1 + E˜1

(3.4.32)

and the combination C0 + 12 C2 called the P2 sum rule: Corollary 3.4.7 (Step-by-Step P2 Sum Rule). Define Z 2π 1 Im M1 (eiθ ) Q(J | J1 ) = log sin2 θ dθ 4π 0 Im M(eiθ ) F (E) = 14 [β 2 − β −2 − log β 4 ]

where |β| > 1 and E = β + β −1 , Then 1 4

G(a) = a2 − 1 − log(a2 )

b21 + 21 G(a1 ) = Q(J | J1 ) +

X j,±

[F (Ej± (J)) − F (Ej± (J1 ))]

(3.4.33) (3.4.34) (3.4.35) (3.4.36)

The P in P2 is for “positive” and comes from the fact that the left side of (3.4.36) is positive and the right side is a difference of a positive term for J and a positive term for J1 . We will discuss this further in the next section. Remarks and Historical Notes. The nonlocal step-by-step sum rule is from Simon [387], but he was motivated by the earlier stepby-step sum rule in Killip–Simon [220] and followup in Simon–Zlatoˇs [401]. The Case sum rules are named after Case [72, 73]. He did not have them in step-by-step form nor was he careful about conditions for them to hold, but he had the idea of looking at Taylor coefficients of a Poisson–Jensen representation of the Jost function, which is an iterated M-function; see (3.7.27). He did not have a general formula for the functions of the Jacobi parameters, but knew they were polynomials in the a’s and b’s and found the first few. The formula in terms of Chebyshev polynomials is due to Killip–Simon [220]. The positivity of P2 is a discovery of Killip–Simon [220].

184


3.5. The P2 Sum Rule and the Killip–Simon Theorem Our goal in this section is to prove (3.1.5) and use that to prove Theorem 3.1.1. As a preliminary, we need to study the functions Q, F, G of (3.1.5). Lemma 3.5.1. Let ρ have the form (1.4.3) and let M be its Mfunction, given by (2.3.2). Let Q(ρ) be given by (1.10.16). Then Z 2π 1 sin θ Q(ρ) = log sin2 θ dθ (3.5.1) 4π 0 Im M(eiθ ) = − 12 S(µ0 | µ) (3.5.2) where dµ = Sz−1 (dρ) is given by (1.9.5) and 1 dµ0 = sin2 θ dθ π In particular, Q(ρ) ≥ 0 and w dρn −→ dρ ⇒ lim inf Q(ρn ) ≥ Q(ρ)

(3.5.3) (3.5.4) (3.5.5)

Remark. If dρ0 is given by (1.10.3), then dµ0 = Sz(dρ0 ) and (3.5.2) is (1.10.17). Proof. By (3.3.2), for θ ∈ [0, π], we have that Im M(eiθ ) = Im m(2 cos θ) = πf (2 cos θ)

(3.5.6)

by (2.3.56). If weR first use θ → −θ symmetry to write the integral in π (3.5.1) as (2π)−1 0 and then make the change of variables x = 2 cos θ √ and use 4 − x2 = 2 sin θ, we obtain (3.5.1). (3.5.2) is just the definition of entropy. (3.5.4) is then just (2.2.14) and (3.5.5) is Theorem 2.2.3. Lemma 3.5.2. Let G be given by (1.10.10). Then G(a) > 0 and near a = 1,

on (0, ∞) \ {1}

G(a) = 2(a − 1)2 + O((a − 1)3 )

(3.5.7) (3.5.8)

Proof. We compute

G′ (a) = 2(a − a−1 )

G′′ (a) = 2 + 2a−2

(3.5.9)

so G(1) = G′ (1) = 0, G′′ (a) ≥ 0, and G′′ (1) = 4. Since G is strictly convex, its minimum is at a = 1, proving (3.5.7). (3.5.8) is just Taylor’s theorem at a = 1.

3.5. P2 SUM RULE

185

Lemma 3.5.3. Let F be given by (1.10.9). Then Z |E| 1 (E 2 − 4)1/2 dE F (E) = 2

(3.5.10)

2

We have that

F (E) > 0 and for |E| near 2 and in (2, ∞),

on R \ [−2, 2]

(3.5.11)

F (E) = 23 (|E| − 2)3/2 + O((|E| − 2)5/2 )

(3.5.12)

Proof. Differentiating (1.10.9) with respect to β yields (1 − β −2 )F ′ (β + β −1) = 21 (β + β −3 − 2β −1 ) so F ′ (β + β −1 ) =

1 2

(β − β −1 )2 = 12 (β − β −1 ) (β − β −1 )

(3.5.13)

If E = β + β −1 , then (E 2 − 4)1/2 = |β| − |β|−1, so (3.5.13) says that if E > 2, then F ′ (E) = 12 (E 2 − 4)1/2 (3.5.14) From (1.10.9), limE↓2 F (E) = 0, so (3.5.14) implies (3.5.10). This in turn implies (3.5.11) and (3.5.12) if we note that with y = E − 2, (E 2 − 4)1/2 = (y(y + 4))1/2 = 2y 1/2 + O(y 3/2)

(3.5.15)

Proposition 3.5.4 (P2 Sum Rule for Finite Rank J −J0 ). Suppose that for some N, an = 1, bn = 0 for all n ≥ N. Then the number of En outside [−2, 2] is finite and (3.1.5) holds with each term finite. Proof. Define dρm to be the m-times stripped measure, that is, the measure with Jacobi parameters {an+m , bn+m }∞ n=1 and Jm its Jacobi matrix. By iterating (3.4.36), we find Q(J | Jm ) < ∞ and Q(J | Jm ) +

X j,±

[F (Ej± (J))

−

F (Ej± (Jm ))]

=

m X

[ 14 b2n + 12 G(an )]

n=1

(3.5.16) By hypothesis, J −J0 is finite rank. Thus, by the min-max principle (see Subsection 1.4.9 of P [390]), each of (−∞, −2) and (2, ∞) has at most N eigenvalues, so F (Ej± (J)) is finite. (3.5.16) for m = N is (3.1.5) since then Q(J | Jm ) = Q(ρ) (by Lemma 3.5.1) and there are no Ej± (JN ).

186


Theorem 3.5.5 (P2 Sum Rule). Let J be a Jacobi matrix with σess (J) = [−2, 2]. Then, with ρ its spectral measure, Q(ρ) +

X

F (E) =

∞ X

[ 14 b2n + 12 G(an )]

(3.5.17)

n=1

E ∈σ / ess (J)

Each term is positive, including +∞, and (3.5.17) holds in the sense that either both sides are infinite or both are finite and equality holds. Proof. Define J (m) by (m) ak

(m)

bk

( ak k ≤ m − 1 = 1 k≥m ( bk k ≤ m = 0 k ≥ m+1

(3.5.18) (3.5.19)

s

Then J (n) −→ J. It follows by the min-max principle that for ℓ fixed, lim sup ∓Eℓ± (J (n) ) ≤ ∓Eℓ (J)

(3.5.20)

n→∞

Since F ≥ 0, it follows that for any L L X

F (Eℓ± (J))

ℓ=1,±

= lim inf n→∞

≤ lim inf n→∞

L X

ℓ=1,±1 ∞ X

F (Eℓ± (J (n) )) F (Eℓ± (J (n) ))

(3.5.21)

ℓ=1,±1

Since the right side of (3.5.21) is L-independent, we can take L → ∞. Moreover, sinceR the spectral measure ρ(n) for J (n) convergesR weakly to that for J (i.e., xℓ dρ(n) = hδ1 , (J (n) )ℓ δ1 i → hδ1 , (J)ℓ δ1 i = xℓ dρ), (3.5.5) says Q(ρ) ≤ lim inf Q(ρ(n) ) By Proposition 3.5.4, ∞ ∞ X X (n) ± (n) ± Q(ρ) + F (Eℓ (J)) ≤ lim inf Q(ρ ) + F (Eℓ (J )) ℓ=1,±

≤ lim inf =

∞ X j=1

X n

b2j

j=1

b2j + G(aj )

+

ℓ=1,± n X

G(aj )

j=1

(3.5.22)

3.5. P2 SUM RULE

187

by the positivity of b2j and G(aj ). Thus, we need only prove that ∞ X

b2j

j=1

+ G(aj ) ≤ Q(ρ) +

∞ X

F (Eℓ± (J))

(3.5.23)

ℓ,±

If the right side of (3.5.23) is ∞, there is nothing to prove, so suppose it is finite. Then Q(ρ) < ∞ and Q(J | J1 ) finite (which is always true) proves Q(ρ1 ) < ∞ and P

Q(J | J1 ) = Q(ρ) − Q(ρ1 )

Similarly, ℓ,± F (Eℓ± (J)) < ∞ and interlacing proves that the sum for the right of (3.4.36) is a difference of separate J and J1 sums. Thus X X Q(J) + F (Eℓ± (J)) = 41 b21 + 12 G(a1 ) + Q(J1 ) + F (E ± (J1 )) ℓ,±1

ℓ,±1

(3.5.24)

Iterating this n times and noting Q ≥ 0, F ≥ 0, we get Q(J) +

X

F (Eℓ± (J))

ℓ,±

≥

n X

( 41 b2j + 12 G(aj ))

j=1

Taking n → ∞ yields (3.5.23),

Proof of Theorem 3.1.1. If (3.1.1) holds, then J − J0 is compact, so (3.1.2) holds by Weyl’s theorem on invariance of the essential spectrum under compact perturbations (see Subsection 1.4.15 of [390]). By (3.1.5), (3.5.8), and (3.5.23), Q(ρ) < ∞ ⇒ (3.1.4) and

X j,±

F (Ej± (J)) < ∞ ⇒ (3.1.3)

by (3.5.12). Conversely, if (i)–(iii) hold. By (i), we have the sum rule (3.5.17). By (ii), (iii), and (3.5.12), LHS of (3.5.17) is finite, so ∞ X n=1

[ 14 b2n + 21 G(an )] < ∞

Thus G(an ) → 0, an → 1, and

P∞

2 2 n=1 bn + (an − 1)

(3.5.25) < ∞ by (3.5.8).

Remarks and Historical Notes. See the Notes to Section 3.1 for the history.

188


Just as for OPUC where, once one goes to slower than ℓ2 decay, there could be no a.c. spectrum; one can use any of the methods described in the Notes to Section 2.1. In particular, one has the following theorem, which we will need later (see Section 10.2): Theorem 3.5.6. There exist Jacobi parameters so that each of the matrices Jm with parameters {an+m , bn+m }∞ n=1 has only dense pure point spectrum in [−2, 2] (and, in particular, no a.c. spectrum) and so that an ≡ 1 and |bn | ≤ Cn−1/2 (3.5.26) For example, one can do this with decaying random potentials; see [222]. 3.6. An Extended Shohat–Nevai Theorem While it is missing positivity, the C0 sum rule is useful and can be used to prove the following: Theorem 3.6.1 (Extended Shohat–Nevai Theorem). Let dρ(x) = f (x) dx + dρs (x) N

± with σess (J) = [−2, 2]. Suppose that {En± }n=1 are the pure points of dρ in ±(2, ∞) and that X (|En± | − 2)1/2 < ∞ (3.6.1)

n,±

Then

if and only if

Z

2

−2

(4 − x2 )−1/2 log f (x) dx > −∞

lim sup a1 . . . an > 0 If these conditions hold, then

(3.6.3)

lim a1 . . . an exists in (0, ∞) and lim

n→∞

Moreover,

∞ X n=1

n=1

(3.6.4)

(an − 1)2 + b2n < ∞

N X (an − 1)

and

(3.6.2)

N X

bn

(3.6.5)

(3.6.6)

n=1

have limits in (−∞, ∞). If (3.6.3) fails, the limit in (3.6.4) exists and is 0.

3.6. AN EXTENDED SHOHAT–NEVAI THEOREM

189

As a first preliminary, we need Lemma 3.6.2. Define 1 Z(ρ) = 4π Then Z(ρ) = where dµ = Sz−1 (dρ).

Z

− 21

2π

0

sin θ log dθ Im M(eiθ )

dθ S dµ − 2π

1 2

log 2

Proof. Suppose dµ has the form (1.9.37). Then Z 2π 1 1 dθ 1 dµ = log dθ −2 S 2π 4π 0 w(θ)

(3.6.7)

(3.6.8)

(3.6.9)

On the other hand, by (1.9.38) and (3.5.6)

1 sin θ = 2 w(θ) 2 sin θ Im M(eiθ )

and so, (3.6.8) is implied by Z 1 log(2 sin2 θ) dθ = − 21 log 2 4π

(3.6.10)

(3.6.11)

Let f (z) = log(|1 − z 2 |) which is harmonic in D and continuous in the closure. Thus, Z 1 log(|1 − e2iθ |) dθ 0 = f (0) = 2π Z 1 = log(2|sin θ|) dθ 2π Z 1 = log(4 sin2 θ) dθ 4π which is (3.6.11). Next we need approximation results for the eigenvalue sums: Lemma 3.6.3. Define E0 (J) =

X

log|βj± (J)|

(3.6.12)

j,±

which may be +∞. Define J (n) by (3.5.18)/ (3.5.19). Then, (a) For any n, E0 (J (n) ) ≤ E0 (J) + 2 sup |bm | + 4 sup {|am |, 1} m

m

(3.6.13)

190


(b) If an → 1, bn → 0, and E0 (J) < ∞, then

lim E0 (J (n) ) = E0 (J)

(3.6.14)

n→∞

Proof. Let J˜(n) be J (n) with an replaced by 0. Then J˜(n) is a direct sum of Jn;F and J0 . Since Jn;F is a restriction of J, we have ±E ± (J˜(n) ) = ±E ± (Jn;F ) ≤ ±E ± (J) (3.6.15) j

j

j

by the min-max principle (see Subsection 1.4.9 of [390]). Since 0 an 1 1 1 −1 1 = 2 an − an 0 1 1 −1 1

(3.6.16)

J (n) − J˜(n) is the sum of a positive rank one and negative rank one perturbations. For Ej− , the positive term can only move eigenvalues up, while the negative one interlaces. We have the opposite for Ej+ . Thus ± ±Ej+1 (J (n) ) ≤ ±Ej± (J) (3.6.17) We get (3.6.13) from this if we note ±E1 (J (n) ) ≤ kJ (n) k ≤ sup|bm | + 2 sup{|am |, 1} m

(3.6.18)

m

If an → 1 and bn → 0, then kJ (n) − Jk → 0, which implies that for each fixed ℓ, Eℓ± (J (n) ) → Eℓ± (J). If E0 (J) < ∞, we can use dominated convergence to get (3.6.14). Remark. Since Eℓ± (J (n) ) → Eℓ± (J) if kJ (n) − Jk → 0, (3.6.14) holds even if E0 (J) is infinite. Lemma 3.6.4. Define Jm to be the m-th stripped Jacobi matrix, that is, the one with Jacobi parameters {an+m , bn+m }∞ n=1 . Then, with E0 given by (3.6.12), E0 (Jm ) ≤ E0 (J) (3.6.19) and if an → 1, bn → 0, and E0 (J) < ∞, then lim E0 (Jm ) = 0

m→∞

(3.6.20)

Proof. Jm is the restriction of J to ℓ2 ({δℓ }∞ ℓ=m+1 ), so by the minmax principle, ±Ej± (Jm ) ≤ ±Ej± (J) (3.6.21) so (3.6.19) follows by monotonicity of log|β(E)| in E. If an → 1, bn → 0, then kJn k → 2, so since Jn − J0 is compact, Ej± (Jm ) → ±2 for each j. By (3.6.21) and dominated convergence of E0 (J) < ∞, we have (3.6.20).


191

We are now ready to prove the relevant sum rule as two halves: Proposition 3.6.5. If E0 (J) < ∞, then X n Z(ρ) ≤ lim inf − log(aj ) + E0 (J) + 2 sup |bm | + 4 sup {|an |, 1} n→∞

m

j=1

m

(3.6.22)

and if an → 1, bn → 0, then

X n log(aj ) + E0 (J) Z(ρ) ≤ lim inf − n→∞

(3.6.23)

j=1

Proof. Let J (n) be given by (3.5.18)/(3.5.19) and let ρ(n) be the corresponding measure. By (3.4.30), iterated n + 1 times (so (J (n) )n = J0 ), X n−1 (n) (n) log(aj ) (3.6.24) Z(ρ ) ≤ E(J ) + − j=1

We get (3.6.22) by using (3.6.13) and taking n → ∞ along a sequence P that takes − n−1 j=1 log(aj ) to its lim inf. By (3.6.8), Z is lower semicontinuous, so we get (3.6.22). For (3.6.23), we use (3.6.14) instead of (3.6.13).

Proposition 3.6.6. If σess (J) = [−2, 2], Z(ρ) < ∞, and E0 (J) < ∞, then X n lim sup − log(aj ) ≤ Z(ρ) − E0 (J) (3.6.25) n→∞

j=1

P Proof.± Since Z(ρ) < ∞ ⇒ Q(ρ) < ∞ and E0 < ∞ ⇒ j,±1 F (Ej (J)) < ∞, Theorem 3.1.1 implies ∞ X n=1

so, in particular,

(an − 1)2 + b2n < ∞

(3.6.26)

an → 1 bn → 0 (3.6.27) By (3.4.30), since Z(ρ | ρ1 ) < ∞ and the E0 (J1 ) ≤ E0 (J) (see (3.6.19)), − log(a1 ) = Z(ρ) − Z(ρ1 ) − E0 (J) + E0 (J1 )

so iterating, n X − log(aj ) = Z(ρ) − Z(ρn ) − E0 (J) + E0 (Jn ) j=1

(3.6.28)

(3.6.29)

192

3. THE KILLIP–SIMON THEOREM w

By (3.6.27) and (3.6.20), E0 (Jn ) → 0. Moreover, since ρn −→ ρJ0 , the measure for J0 and Z(ρJ0 ) = 0, we have lim inf Z(ρn ) ≥ Z(ρJ0 ) = 0

(3.6.30)

so (3.6.29) implies (3.6.25).

Proof of Theorem 3.6.1. If (3.6.3) holds, then X n log(aj ) < ∞ lim inf −

(3.6.31)

1

Since (3.6.1) ⇒ E0 (J) < ∞, Z(ρ) < ∞ by (3.6.22). But then, as in the last proof, we obtain (3.6.26) and so (3.6.27), and thus (3.6.23) holds. On the other hand, if Z(ρ) < ∞ and (3.6.11) holds, then by (3.6.25), X n lim sup − log(aj ) < ∞ (3.6.32) 1

A fortiori, (3.6.31) holds, so (3.6.23) holds. Thus, X X n n lim sup − log(aj ) ≤ Z(ρ) − E0 (J) ≤ lim inf − log(aj ) j=1

j=1

It follows that the limit exists and X n lim − log(aj ) = Z(ρ) − E0 (J)

(3.6.33)

j=1

This proves (3.6.4) and (3.6.5) follows from Theorem 3.1.1. (3.6.33) for Jn and (3.6.20) let us strengthen (3.6.30) to lim Z(ρn ) = 0

n→∞

(3.6.34)

P Finally, we turn to the conditional convergence of n1 bj . By iterating the step-by-step C1 Case sum rule, (3.4.32), we get n X 1

where

bj = T (J) − T (Jn ) + E1 (J) − E1 (Jn )

1 T (J) = − 2π and

Z

sin θ log cos θ dθ Im M(eiθ )

E1 (J) =

X j,±1

[βj± − (βj± )−1 ]

(3.6.35)

(3.6.36) (3.6.37)


193

Because Z(ρ) < ∞ and E1 (J) is convergent, we can separate P out the terms in the step-by-step sum rule. Clearly, to prove lim n1 bj exists, it is sufficient to prove that lim T (Jn ) = 0

(3.6.38)

lim E1 (Jn ) = 0

(3.6.39)

n→∞ n→∞

The second result has a proof identical to (3.6.20). For the first, we define Z 2π sin θ 1 ± T (J) = log (1 ± cos θ) dθ (3.6.40) 2π 0 Im M(eiθ ) = 2Z(J) ± T (J) (3.6.41) As in the proof of (3.6.8), one sees that dθ ± T (J) = −S (1 ± cos θ) dµ + c 2π

(3.6.42)

for a constant c, so T ± is lower semicontinuous. Since dµJn → dµJ0 and T ± (J0 ) = 0, we see that lim inf T ± (J) ≥ 0

(3.6.43)

Remark. (3.6.33) is the C0 sum rule. (3.6.34), (3.6.41) and (3.6.43) imply (3.6.38).

For one application, we need the following which we state without proof since the application is peripheral (but see the Notes): Theorem 3.6.7 (Hundertmark–Simon [196]). For any J with σess (J) = [−2, 2], X

((Ej± )2

j,±

1/2

− 4)

≤

∞ X n=1

(|bn | + 4|an − 1|))

(3.6.44)

This implies that Theorem 3.6.8. If ∞ X n=1

|bn | + |an − 1| < ∞

(3.6.45)

then the Szeg˝o condition Z(J) < ∞ holds. Remark. (3.6.45) is equivalent to saying J − J0 is trace class.

194


P Q Proof. Clearly, ∞ |an − 1| < ∞ implies nj=1 aj has a nonzero n=1 P ± limit. By (3.6.44), (|Ej | − 2)1/2 < ∞, so Theorem 3.6.1 implies Z(ρJ ) < ∞. Remarks and Historical Notes. That Z(ρ) and E0 (J) < ∞ implies Q that Jj=1 aj has a limit is a result of Peherstorfer–Yuditskii [336]. This result was rediscovered and the converse proven by Killip–Simon [220] as part of their analysis of sum rules. Our proof here follows [220] with some important refinements of Simon–Zlatoˇs [401]. Theorem 3.6.7 is due to Hundertmark–Simon [196] (see Section 13.8 of [391] for a historical discussion and an exposition of their proof). Theorem 3.6.8 is from Killip–Simon [220]. It settled a conjecture of Nevai [316]. There have been a variety of papers that attempt to find higherorder sum rules and associated gems: [106, 252, 253, 254, 262, 308, 453]. 3.7. Szeg˝ o Asymptotics for OPRL Szeg˝o [420] related his OPUC theorem to asymptotics of OPUC (and later to asymptotics of OPRL, as we will discuss in the Notes, but we want to go beyond that here). So it is natural to ask about the relation of OPRL asymptotics to the ideas of this chapter—and that is what we will do in this section. To see what we are seeking, consider the case J0 (i.e., an ≡ 1, bn ≡ 0) where we have seen the OPRL are given by Chebyshev polynomials of the second kind, (1.10.2), which can be conveniently rewritten using the map z + z −1 from D to C \ [−2, 2]: 1 z n+1 − z −(n+1) = (3.7.1) pn,0 z + z z − z −1

so for all z ∈ D \ {0}, since z −1 dominates z, 1 n z pn,0 z + →1 z

(3.7.2)

This leads to Definition. We say orthogonal polynomials, {pn (x)}∞ n=0 , have Szeg˝o asymptotics at z0 ∈ D \ {0} if and only if 1 n lim z0 pn z0 + (3.7.3) n→∞ z0

˝ ASYMPTOTICS FOR OPRL 3.7. SZEGO

195

√ exists and is nonzero. We define the limit to be ( 2D(z))−1 (the reason for this choice of symbol and normalization will be explained in the Notes). Our main theorem, proven below, is Theorem 3.7.1 (Damanik–Simon [95]). Suppose the Jacobi parameters, {an , bn }∞ n=1 , of a measure, dρ, obey n Y (a) lim aj exists in (0, ∞) (3.7.4) n→∞

(b)

lim

n→∞ ∞ X

(c)

j=1

j=1 n X

bj

exists in R

(3.7.5)

j=1

(aj − 1)2 + b2j < ∞

(3.7.6)

Then the limit in (3.7.3) exists for all z in D \ {0} and is nonzero (precisely) for all z so that z + z −1 ∈ / σ(J). The convergence is uniform on compact subsets of D. Conversely, if there is ε > 0 so that (3.7.3) holds uniformly for |z| = r for all r ∈ (0, ε), then (a)–(c) hold. Remarks. 1. Since (a)–(c) imply σess (J) = [−2, 2], {z | z + z −1 ∈ σ(J)} is a discrete set in D ∩ R. 2. One can also ask about suitable L2 convergence on ∂D. This is true and proven in [95] (but not in the form (3.7.3); see the Notes), but the proof is much more involved than the proof in this section. The following predated [95]: Corollary 3.7.2 (Peherstorfer–Yuditskii [336]). Let dρ(x) = f (x) dx + dρs (x) N

± with σess (J) = [−2, 2]. Suppose that {En± }n=1 are the pure points of dρ in ±(2, ∞) and that X (|En | − 2)1/2 < ∞ (3.7.7)

n,±

and

Z

2

−2

(4 − x2 )−1/2 log f (x) dx > −∞

(3.7.8)

Then (3.7.3) holds for all z ∈ D \ {0} uniformly on compacts and the limit is nonzero (precisely) for z so that z + z −1 ∈ / σ(J). Proof. Immediate from Theorems 3.6.1 and 3.7.1.

196


In Section 2.10, we saw that for OPUC, asymptotics of Weyl solutions and polynomials are related and that is a theme we will use here but in the opposite direction from the OPUC case (where we went from asymptotics of polynomials to Weyl solutions). Since Tn (z) is entire and m(z) is analytic on C \ σ(J), the Weyl solution, gn (z), is defined via (3.2.23) on all of C \ σ(J). Definition. We say the Weyl solutions, gn (z), have Jost asymptotics at z0 ∈ D \ [{0} ∪ {z | z + z −1 ∈ σ(J)}] if 1 −(n+1) lim −z0 gn z0 + (3.7.9) n→∞ z0 and is nonzero. We define the limit to be 1/u(z0 ). u is called the Jost function.

Example 3.7.3. For J0 (i.e., an ≡ 1, bn ≡ 0), we have 1 z n+1 − z −n−1 1 1 pn z + = qn z + = pn−1 z + (3.7.10) z z − z −1 z z

and (recall M(z) = z and M(z) = −m(z + 1z )) 1 = −z (3.7.11) m z+ z so by algebra, 1 = −z n+1 (3.7.12) gn z + z which explains the reason for the minus sign and n + 1 in (3.7.9). In [95] and [391], the term “Weyl solution” is used for 1 wn (z) = −gn−1 z + (3.7.13) z The Jost solution is defined by

1 un (z) = −u(z)gn−1 z + z

(3.7.14)

so un (z) ∼ z n . We will also prove the following below:

Theorem 3.7.4. The conditions (a)–(c) of Theorem 3.7.1 imply that for all z ∈ D\[{0}∪{z | z+z −1 ∈ σ(J)}], one has Jost asymptotics uniformly on compact subsets of this set. Conversely, if one has Jost asymptotics uniformly on |z| = ε for all sufficiently small ε, then (a)– (c) hold.


197

We first show Jost asymptotics is equivalent to Szeg˝o asymptotics because of the following lemma: Lemma 3.7.5. Suppose an → 1

bn → 0

(3.7.15)

Then the limit y∞ of (3.7.3) exists if and only if the limit x∞ of (3.7.9) exists and 1 x∞ y∞ = (3.7.16) 1 − z2 Remark. We use the language of two-sided Jacobi matrices of Sections 5.2 and 5.4. Proof. Let Gnm (z) = hδn , (J−(z+z −1 ))−1 δm i

−1 −1 G(0) )) δm i nm (z) = hδn , (J0 −(z+z (3.7.17) where J0 is the two-sided Jacobi matrix with an ≡ 1, bn ≡ 0. By (3.7.15), if (u(k) )n = uk+n , Ju(k) → J0 u for any u, so for any z ∈ D, (0) lim Gnn (z) = G00 (z) = −(z − z −1 )−1 (3.7.18) n→∞

(0)±

where we use (5.4.59) and un

= z ±n . By (5.4.41),

Gnn (z) = pn−1 (z + z −1 )gn (z + z −1 )

(3.7.19)

(since the Wronskian of p·−1 and g· is 1). Thus, if yn is the quantity in (3.7.3) and xn in (3.7.9), (3.7.18) says that z zyn−1 xn → (3.7.20) 1 − z2 proving that xn → x∞ if and only if yn−1 → y∞ and that (3.7.16) holds. Proof of Theorem 3.7.1 Given Theorem 3.7.4. (a)–(c) imply (3.7.15) by (3.7.6), so Jost asymptotics implies Szeg˝o asymptotics. Conversely, since near z = 0 (z + 12 = ∞), X n 1 1 n 2 z pn z + = 1+z bj + O(z ) (3.7.21) z a1 . . . an j=1 so Szeg˝o asymptotics implies (a), (b), and so (3.7.15), and thus, by the lemma, Jost asymptotics. Remark. By (3.7.16), we have u(z) =

(1 − z 2 ) √ (D(z) 2 )

(3.7.22)

198


To get from asymptotics of z n gn to information on the Jacobi parameters, we need a relation between {gn }∞ n=0 and {M(z, Jn )}. Theorem 3.7.6. We have for k ≥ 1, m(x, Jk ) = −

gk (x) ak gk−1 (x)

(3.7.23)

and m(x, J) = g0 (x)

(3.7.24)

Proof. We have that for k ≥ 1,

n+k Tn+k ({aj , bj }j=1 , x) = Tn ({aj+k , bj+k }nj=1, x)Tk ({aj , bj }kj=1 , x) (3.7.25) m(z) Thus, applying this to −1 , we find gk m(z, Jk ) =c (3.7.26) ak gk−1 −1

since there is a unique ℓ2 solution. This implies (3.7.23). (3.7.24) is just the initial condition for g. Corollary 3.7.7. We have that Y n n 1 M(z) Y M(z, Jk ) −(n+1) −z gn z + = aj z z z j=1 k=1

(3.7.27)

Proof. This follows from (3.2.27) and (3.2.28). The minus comes from n factors in (3.7.23) and (n + 1) factors in M(z) = −m(z + z −1 ). Proof of the half of Theorem 3.7.4 that Jost asymptotics ⇒ (a)–(c). By (3.7.27), −z −(n+1) gn (z + z1 ) has a removable singularity at z = 0 and defines a function ηn (z) analytic in R \ {z | z + z −1 ∈ σ(J)} (zeros of M(z) include poles of M1 , etc.). Thus convergence of ηn uniformly on |z| = r implies convergence of the Taylor coefficients of ηn . In particular, ηn (0) = (a1 . . . an ) (3.7.28) has a finite limit. This limit is nonzero since ηn is nonvanishing on {z | |z| ≤ ε} and the limit is not identically zero. By (3.7.27) and (3.4.26), log(ηn (z)) = βn + γn z + ϕn z 2 + O(z 2 )

(3.7.29)


199

where βn =

n−1 X

log(aj )

ϕn =

j=1

n X

bj

γn =

j=1

n X j=1

(a2j − 1 + 12 b2j ) (3.7.30)

That γn has a finite limit is thus immediate. As usual, we combine n X ϕn − 2βn = [G(aj ) + 12 b2j ] (3.7.31) j=1

so positivity and conditional convergence imply convergence. For the other direction, we need a general result about asymptotics of difference equations which we state without proof (but see the Notes).

Theorem 3.7.8 (Discrete Hartman–Wintner Theorem). Let B0 be a d×d diagonal matrix whose diagonal elements, λ1 , . . . , λd , obey |λi | = 6 |λj | for all i 6= j. Let {(δB)(n)}∞ obey n=1 (i) (δB)(n)kk → 0 (3.7.32) as n → ∞ for k = 1, . . . , d. (ii) X |(δB)(n)kj |2 < ∞ (3.7.33) n

for all k 6= j. (iii) Let

B(n) = B0 + δB(n) For all n, we suppose that

(3.7.34)

det B(n) 6= 0

(3.7.35)

Then for each j, there exists u(j), so n Y (λj + δB(ℓ)jj )−1 [B(n) . . . B(1)u(j)] → δj

(3.7.36)

ℓ=1

the vector with (δj )k = δjk .

End of the proof of Theorem 3.7.4. Suppose z ∈ D and define the 2 × 2 matrices −1 1 1 1 −1 B(n) = A(an , bn ; z + z ) −1 z −1 z z where A is given by (3.2.2). z 0 1 B0 ≡ = −1 0 z z −1

(a)–(c) hold. Let 1 z

(3.7.37)

The conjugating matrix is chosen because −1 1 z + z −1 −1 1 1 (3.7.38) z 1 0 z −1 z

200


is diagonal. By (c), if δB is given by (3.3.46), then X kδB(n)k22 < ∞ n

implying (3.7.33). (3.7.32) holds since (c) implies an → 1, bn → 0. Since |z| < 1, |z| = 6 |z −1 P|.n P (a)Q and (b) imply ℓ=1 (δB(ℓ))jk has a limit and nℓ=1 |δB(ℓ)jk |2 < ∞, so nj=1 [1 + z −1 (δB(ℓ))11 ] has a limit. By (3.7.36), there exist some initial conditions so that z −n Tn ({aj , bj }nj=1, z + z −1 )˜ u0 (1)

has a nonzero limit. This gives an ℓ2 solution, and so the ℓ2 solution has Jost asymptotics. The following is known to hold in the generality of Theorem 3.7.1 (see the Notes), but the proof is easier with the stronger hypotheses discussed in Theorem 3.6.1. Theorem 3.7.9. Let dρ = f (x) dx + dρs (x) where (3.7.7) and (3.7.8) hold. Then u has nontangential boundary values a.e. on ∂D and 2 Z u(eiθ(x) )ei(n+1)θ(x) ] pn (x) − Im[¯ (3.7.39) f (x) dx → 0 sin(θ(x)) and

Z

|pn (x)|2 dµs (x) → 0

(3.7.40)

Here θ(x) ∈ [0, π] is given, as usual, by x = 2 cos(θ(x))

(3.7.41)

and one has [sin(θ(x))]−1 u(eiθ(x) ) ∈ L2 (R, f (x) dx).

Proof. Let us begin by defining u on D by Z iθ Y e +z sin θ dθ u(z) = bp± (z) exp log j eiθ − z Im M(eiθ ) 4π j,±

(3.7.42)

where, as usual (see (3.5.6)), Im M(eiθ(x) ) = πf (x)

(3.7.43)

and p± j are the points in D with ± −1 Ej± = (p± j ) + (pj )

(3.7.44)


201

By (3.7.7), the Blaschke product in (3.7.42) converges, and by (3.7.8), the second factor exists. As in the proof for p(z) for OPUC (see Proposition 2.9.4), this second factor, E(z), has (1−z 2 )E(z)−1 in H 2 (D), so u(z) has boundary values obeying sin(θ(x)) f (x) = (3.7.45) π|u(eiθ(x) )|2 We will prove that u(z), defined by (3.7.42), obeys (3.7.39) and then use that to prove it is also an inverse of the limit in (3.7.9), so it agrees with our previous definition. To facilitate calculation, define u¯(eiθ ) 2i sin θ − iθ k (e ) = k + (eiθ ) k + (eiθ ) =

(3.7.46) (3.7.47)

u(m) (z) = Bm (z)E(z) Y Bm (z) = bp±j (z)

(3.7.48) (3.7.49)

1≤j≤m,±

u(m) (3.7.50) u We let k · kf and h , if be the L2 (R, f (x) dx) norm and inner product, and k · ks, the L2 (R, dµs ) inner product. Clearly, (3.7.39)/(3.7.40) is equivalent to − km = k−

kpn − z n+1 k + − z n+1 k − k2f + kpn k2s → 0

(3.7.51)

where z(x) = eiθ(x) . This will follow from kpn k2f + kpn k2s = 1

(3.7.52)

kk ± k2f =

hz n+1 k + , z

1 2 −(n+1) −

k if → 0

lim hz −n−1 k − , pn if =

n→∞

(3.7.53) 1 2

(3.7.54) (3.7.55)

(3.7.52) is the normalization condition. To get (3.7.53), use (3.7.45) and (3.7.46) to see Z 2 |u(eiθ(x) )|2 sin(θ(x)) + 2 kk kf = d(2 cos θ) (3.7.56) 2 iθ 2 −2 4 sin (θ(x)) π|u(e )| Z π 1 dθ 1 = = (3.7.57) 2 0 2 π

202


To prove (3.7.54), note that a calculation similar to one just done shows 2 Z 2π iθ dθ n+1 + −(n+1) − −(2n+2)iθ u(e ) hz k , z k i= e (3.7.58) iθ |u(e )| 2π 0

goes to zero since [u/¯ u]2 ∈ L2 (∂D, dθ/2π). Finally, note

− − |hz −n−1 (k − − km ), pn if | ≤ kk − − km kf

= kk − (B∞ − Bm )kf →0

(3.7.59)

as m → ∞, by the dominated convergence theorem. Thus, uniformly in n, − hz −n−1 km , pif → hz −n−1 k − , pif (3.7.60) so, to prove (3.7.55), it suffices to prove − lim lim hz −n−1 km , pif = 12 (3.7.61) m→∞ n→∞

We compute (using |u(m) | = |u| on ∂D) Z 2 (m) n+1 sin θ u z − pn (2 cos θ) 2 sin θ dθ hkm , pn i = π|u(m) |2 −2 2i sin θ Z 2π −1 −z 1 dz n+1 z = z pn z + 2 z 2πizu(m) (z) 0 Z 2π 1 1 dz 2 n = (1 − z )z pn z + 2 0 z 2πizu(m) (z)

(3.7.62)

m (zu(m) )−1 has poles at z = 0 and at {p± j }j=1 . Using the fact that zn pn (z + 1z ) z=0 = (a1 . . . an )−1 , we see that − hkm , pn i =

where ε(m) = n

1 B∞ (0) (a1 . . . an )−1 u(0)−1 + ε(m) n 2 Bm (0)

1 X 1 2 ± n−1 [1 − (p± pn (Ej± ) (m) ′ ± j ) ](pj ) 2 j=1,...,m;± (u ) (pj )

This is a finite sum. Since X n

|pn (Ej± )|2

we see

sup n,j=1,...,m,±

ρs (Ej± ) ≤ 1

|pn (Ej± )| < ∞

(3.7.63)

(3.7.64)

(3.7.65) (3.7.66)


203 (m)

n−1 so, since supj=1,...,m,± |p± → 0 as n → ∞, we have εn j | − lim hkm , pn i =

n→∞

1 B∞ (0) u(0)−1 lim (a1 . . . an )−1 n→∞ 2 Bm (0)

→ 0. Thus, (3.7.67)

One can rewrite (3.6.37) as lim (a1 . . . an )−1 = u(0)

n→∞

(3.7.68)

Since Bm (0) → B∞ (0) as m → ∞, (3.7.67) implies (3.7.61). This completes the proof of (3.7.51) and so of (3.7.39) and (3.7.40) for u defined by (3.7.42). All that remains is to prove that this u is the same as the previously defined Jost function. One can rewrite (3.7.39) as saying (with z = eiθ(x) ) that u(z) u(z) 2n+2 1 n − + z →0 (3.7.69) z pn z + z 1 − z2 1 − z2 in k · kf . Since

and (3.7.45) says

u(z) = E(z)B∞ (z) 2 sin2 θ dθ |E(z)|2 π 1 − z 2 2 dθ = E(z) 2π

(3.7.70)

f (x) dx =

(3.7.71)

we see that (3.7.69) is the same as 1 E(z) 2 −1 (1 − z )E(z) pn 1 + B∞ (z) → 0 (3.7.72) − B∞ (z) − z 2n+2 z E(z) in L2 (∂D, dθ/2π) norm. Since E(z)/E(z)B∞ (z) ∈ L∞ (∂D) ⊂ L2 (∂D, dθ/2π), we see that the last term goes to zero weakly in L2 (∂D, dθ/2π). Since (1 − z 2 )E(z)−1 and B∞ (z) are in H 2 (∂D), we see that weakly in H 2 (∂D), 1 2 −1 (1 − z )E(z) zn pn z + − B∞ (z) (3.7.73) z goes to zero, so the function goes to zero uniformly on compact subsets of D. Thus, we have Szeg˝o asymptotics with √ −1 u(z) 2 D(z) = (3.7.74) 1 − z2

204


By (3.7.22) and Lemma 3.7.5, we have Jost asymptotics with the Jost function given by the u of (3.7.42). Remark. As a bonus, we get the explicit formula (3.7.42) for the Jost function. Remarks and Historical Notes. The main theorems, Theorems 3.7.1 and 3.7.4, of this section are from Damanik–Simon [95]. There is earlier work. Szeg˝o asymptotics for OPRL with supp(dρ) = [−2, 2] (i.e., no bound states) is a result of Szeg˝o [421]. He used the fact, (1.9.9), that 1 n z pn z + = [2(1 − α2n−1 )]−1/2 [ϕ2n (z) + ϕ∗2n (z)] z Since α2n−1 → 0, ϕ2n (z) → 0, and ϕ∗2n (z) → D(z)−1 (see Theorem 2.9.6), we see in this case that 1 1 n z pn z + → √ z ( 2 D(z)) hence the definition of “D” in cases when there are bound states and D cannot be defined via the Szeg˝o map. Nevai [314] extended this to allow finitely many {Ej± }. Corollary 3.7.2 was then found by Peherstorfer– Yuditskii [336]. [95] had a different proof of Lemma 3.7.5; the proof here is from Christiansen–Simon–Zinchenko [84]. Jost asymptotics, Jost solution, and Jost function all come from an analogy to work of Jost [206] who studied solutions of the Schrödinger equation, −u′′ (x) + V u(x) = k 2 u(x), with asymptotics u(x) ∼ eikx . This Jost solution had a value at x = 0, called the Jost function. The use of Jost functions in OPRL was pioneered by Case [72, 73, 150]. If one has Jost asymptotics and the Jost function is a Nevanlinna function, then the hypotheses of Theorem 3.6.1 are valid, and conversely. That the Jost function is a Nevanlinna function with trivial singular inner factor is a result explicit in Killip–Simon [220] and implicit in Peherstorfer–Yuditskii [335]. Case labels his Jacobi parameters starting at n = 0, and as a result, he has various factors of z −1 in Jost function formulae. It is to avoid such factors that Killip–Simon used the now common convention to start labelling at n = 1. It is also why the solution in [95, 391] start at n = 1, not n = 0. Theorem 3.7.8 is due to Coffman [87] and is an analog of a continuum result (ODE) of Hartman–Wintner [187]. A pedagogic presentation of this theorem and additional history

3.8. THE MOMENT PROBLEM: AN ASIDE

205

will appear in the second edition of [390] and is available at http://www.math.caltech.edu/opuc/newsection13-3.pdf. [95] has two other proofs of the direction (a)–(c) ⇒ Jost asymptotics. One uses Fredholm determinant formulae for Jost functions and the other a renormalized inner-outer factorization. Theorem 3.7.9 and the proof we give is due to Peherstorfer– Yuditskii [335]. The same theorem for the more general context of Theorem 3.7.1 is in Damanik–Simon [95], but the proof is different. One cannot define u by (3.7.42) because the Blaschke product and Szeg˝o integrals may diverge. Instead, one needs to use “renormalized” Blaschke products and Poisson representations. Then the trick of re−1 placing B∞ by Bm with only finitely many poles in Bm is not available and a different method is needed. Notice that our proof of Theorem 3.7.9 also provides an independent proof of Szeg˝o asymptotics for pn on D when (3.7.2) and (3.7.8) hold. It also only needs lim sup(a1 . . . an )−1 ≥ u(0) which is the easier half of the proof of (3.7.66) (i.e., of (3.6.37)). It then implies the full (3.7.66). Notice that (3.7.39) expands pn in terms of e±inθ(x) , not un (eiθ(x) ) and its conjugate. The product of un and u0 is not necessarily L2 , but since e±inθ(x) are in L∞ , their products with u0 are in L2 . 3.8. The Moment Problem: An Aside In the next section, we will discuss an application of Szeg˝o’s theorem for OPUC to the moment problem on the real line. This section is background but also illustrates the use of OPRL and, in particular, transfer matrices to study the moment problem. The moment problem in its primeval form is: Moment Problem: First Form. Given a sequence {cn }∞ n=0 of real numbers, when does there exist a nontrivial measure, dµ, on R with Z xn dµ(x) = cn (3.8.1)

When a solution exists, is it unique? If it is not unique, what is the structure of the set of solutions? Of course, for (3.8.1) to make sense, one needs Z |x|n dµ < ∞ (3.8.2)

206


By structure of the set of solutions, we mean is it closed in the weak topology? (This is not obvious since xn is not bounded.) Is it of finite or infinite dimension? Among the solutions, are there any which are pure point or singular continuous or purely absolutely continuous? If there exists a unique solution, we call the moment problem determinate, and if there are multiple solutions, indeterminate. Since we can replace cn by cn /c0 , we can and will always suppose that c0 = 1. Often the cn are given by (3.8.1), so existence is trivial. The moment problem then becomes: Moment Problem: Second Form. Suppose cn is a sequence given by (3.8.1) for some nontrivial probability measure, dµ0, on R obeying (3.8.2). Is dµ0 the unique measure obeying (3.8.1) for the given cn , or are there others? If there are others, what is the structure of the solutions? Example 3.8.1. Fix 0 < α real and let cn be given by Z −1 cn = Nα xn exp(−|x|α ) dx (3.8.3) R where Nα = exp(−|x|α ) dx is a normalization constant. Below (see later in this section and then in the next) we will show that this problem is determinate if α ≥ 1 and indeterminate if 0 < α < 1. There is an obvious necessary condition on the cn ’s for there to be any nontrivial measure. Proposition 3.8.2. If a solution of the moment problem exists, then for each n = 1, 2, . . . , the Hankel determinants Hm ({cn }∞ n=0 ) = det((cj+k−2 )1≤j,k≤m )

(3.8.4)

m Proof. Let {αj }m j=1 lie in C . Then 2 Z m−1 m X X α ¯ j αk cj+k−2 = αj xj dµ

(3.8.5)

are strictly positive.

j,k=1

j=0

so Hm is positive as the determinant of a strictly positive matrix.

We will see below (see Theorem 3.8.4) that, conversely, if Hm > 0 for all m, then the moment problem is soluble. For now, we note that it is easy to see that if each Hm is positive, there exists a unique nondegenerate inner product on polynomials with h1, xm i = cm

(3.8.6)


207

This inner product defines OPs both monic and normalized and Jacobi ∞ parameters {an , bn }∞ n=1 ∈ ((0, ∞) × R) . Thus, we have:

Moment Problem: Third Form. Given a set of Jacobi parameters, ∞ {an , bn }∞ n=1 ∈ ((0, ∞) × R) , when does there exist a measure, dµ, whose Jacobi parameters are {an , bn }∞ n=1 ? If one exists, is it unique? If it is not unique, what is the structure of the set of solutions? Existence is essentially Favard’s theorem discussed in Section 1.3. Jacobi parameters determine moments, so an inner product on polynomials and (3.8.4) holds. Thus, Problems 1 and 3 are equivalent. We will see (see Theorem 3.8.4) that in this form, the moment problem always has solutions, that is, any set of Jacobi parameters can occur. Proposition 3.8.3. Fix k ≥ 1. Let {cn }2k n=0 be a set of moments with (3.8.4) strictly positive for m = 1, . . . , k + 1. Then the set of measures in R obeying (3.8.1) for n = 0, . . . , 2k − 1 and Z x2k dµ ≤ c2k (3.8.7) is a nonempty set, compact of weak-∗ convergence (i.e., R in the topology R dµℓ → dµ if and only if f (x) dµℓ → f (x) dµ for all bounded continuous functions on R). Proof. The {cn }2k n=0 define an inner product on polynomials of degree up to k, so orthonormal polynomials {pj }kj=0, and so Jacobi parameters {an , bn }kn=1 . Choose any value for bk+1 and so get a (k + 1)×(k +1) finite Jacobi matrix, Jk+1;F . Let dµ be the spectral measure for this matrix and vector δ1 . Then dµ obeys (3.8.1) for n = 0, . . . , 2k, so there is a solution proving the set is nonempty; indeed, we can suppose equality in (3.8.7). Using the fact that the probability measures on [−R, R] are compact, it is easy to see that the set of probability measures on R obeying Z dµ(x) ≤ c2k R−2k (3.8.8) |x|≥R

is compact. Here we use k ≥ 1. (3.8.7) implies (3.8.8). Thus, we need only prove that the set, S, of µ’s obeying (3.8.1) for m ≤ 2k − 1 and (3.8.7) is weakly closed. Let  n  |x| ≤ R x n (3.8.9) fn;R (x) = R x≥R  (−R)n x ≤ −R

208


and suppose dµℓ ∈ S converges weakly to dµ. Then Z f2k;R dµℓ ≤ c2k

so

Z

f2k;R dµ ≤ c2k

(3.8.10) (3.8.11)

and (3.8.7) holds by the monotone convergence theorem. By dominated convergence, (3.8.7) implies that for any m = 1, . . . , 2k − 1, Z Z lim

R→∞

fm;R dµ =

Moreover, for any finite ℓ, Z Z m |fm;R − x | dµℓ ≤ 2

|x|≥R

xm dµ

(3.8.12)

|x|m dµℓ

2k−m x ≤2 |x|m dµ R |x|≥R Z

≤ 2R−(2k−m) c2k

(3.8.13)

so converges for each ℓ uniformly in ℓ. This plus (3.8.12) plus R (3.8.12) xm dµ = cm implies dµ obeys (3.8.1) for n = 0, . . . , 2k − 1. We thus have existence:

Theorem 3.8.4. A set, {cn }∞ n=0 , of real numbers with c0 = 1 has solutions of the moment problem if and only if each Hm ({cn }∞ n=0 ) (given by (3.8.4)) is strictly positive. Any set of Jacobi parameters ∞ {an , bn }∞ is the Jacobi parameter of some measure. n=1 ∈ ((0, ∞) × R) Remark. The second sentence is essentially Favard’s theorem in the general case; see Theorem 1.3.9. Proof. Let Sk be the set of measures given by Proposition 3.8.3. Since Sk is compact and nonempty, and Sk+1 ⊂ Sk , we see ∩k Sk is nonempty. This plus Proposition 3.8.2 proves the first sentence in this proposition. As noted, the first and third forms of the moment problem are equivalent, thus proving the second sentence. To go further and analyze uniqueness, we need to briefly study unbounded selfadjoint operators. A densely defined operator, A, on a Hilbert space, H, has a domain D(A) ⊂ H, a dense subspace, and is a linear map of D(A) into H. Associated to A is its graph, Γ(A) ⊂ H×H, defined by Γ(A) = {(ϕ, Aϕ) | ϕ ∈ D(A)} (3.8.14)


209

Γ(A) is always a subspace of H × H. A is called closed if and only if Γ(A) is closed. B is an extension of A if and only if Γ(A) ⊂ Γ(B), that is, D(A) ⊂ D(B) and B ↾ D(A) = A. Given an operator, A, we define D(A∗ ) to be those ϕ ∈ H for which there is an η ∈ H with hη, γi = hϕ, Aγi

(3.8.15)

hA∗ ϕ, γi = hϕ, Aγi

(3.8.16)

for all γ ∈ D(A). η is uniquely determined if it exists since D(A) is dense. We then set η = A∗ ϕ, so for all γ ∈ D(A), η ∈ D(A∗ ). A∗ is called the adjoint of A. A∗ is defined to be the maximal operator so that (3.8.16) holds. If D(A∗ ) is dense, then it is easy to see that A∗ is a closed operator. Note that there is a relation between extension and adjoint: An operator is called

A ⊂ B ⇒ B ∗ ⊂ A∗

Hermitian ⇔ A ⊂ A∗ Selfadjoint ⇔ A = A∗

Essentially selfadjoint ⇔ A ⊂ A∗ = (A∗ )∗

Notice that if A is Hermitian, then A∗ is densely defined and we can define (A∗ )∗ . Proposition 3.8.5. Let A be a Hermitian operator and let z = x + iy ∈ C \ R. Then (i) For all ϕ ∈ D(A), k(A − z)ϕk2 = k(A − x)ϕk2 + y 2 kϕk2

(ii) A is closed ⇔ Ran(A − z) is closed. (iii) A∗∗ is the smallest closed extension of A, so we write A¯ = A∗∗ ¯ (iv) A∗ = A∗∗∗ . Moreover, if A is Hermitian, so is A. (v) Ran(A − z) = Ran(A¯ − z) (vi) ¯ + ker(A∗ − z) + ker(A∗ − z¯) D(A∗ ) = D(A) (vii) A is essentially selfadjoint if and only if ker(A∗ − z) = ker(A∗ − z¯) = {0}

(3.8.17)

(3.8.18)

(3.8.19) (3.8.20) (3.8.21)

210


Remark. (3.8.20) holds in the sense of algebraic direct sum, that is, any ψ ∈ D(A∗ ) is uniquely the sum of three vectors, one in each space. Proof. (i) (3.8.17) follows from noting that the cross term h(A − x)ϕ, iyϕi + hiyϕ, (A − x)ϕi = 0

(3.8.22)

by Hermiticity. (ii) By (3.8.17),

(ϕ, Aϕ) 7→ (A − z)ϕ (3.8.23) is a metric space equivalence of Γ(A) and Ran(A − z), so one space is complete if and only if the other is. (iii) Let J : H → H by Jhϕ, ψi = hψ, −ϕi. Then Γ(A∗ ) = J[Γ(A)⊥ ] = [JΓ(A)]⊥

(3.8.24)

Since J 2 = −1, we see Γ(A∗∗ ) = [−Γ(A)]⊥⊥ = Γ(A). Thus, A∗∗ is closed and is the smallest closed extension. (iv) A∗ is closed by (3.8.24), so (3.8.18) implies A∗ = A∗∗∗ . Thus, A ⊂ A∗ implies A∗∗ ⊂ A∗ = (A∗∗ )∗ . (v) As noted in the proof of (ii), (3.8.23) is a metric space equivalence, so it takes closures to closures. ¯ ϕ+ ∈ ker(A∗ − z), ϕ− ∈ ker(A∗ − z¯), and (vi) If ψ ∈ D(A), ϕ+ + ϕ− + ψ = 0

Then applying (A∗ − z) and then (A∗ − z¯), we see ¯ ϕ− = i(2 Im z)−1 Aψ ¯ ϕ+ = −i(2 Im z)−1 Aψ

(3.8.25)

(3.8.26) (3.8.27)

so ϕ+ = −ϕ− , which implies ϕ+ = ϕ− = 0, and then ψ = 0. This proves uniqueness. If η ∈ D(A∗ ), since Ran(A¯ − z) + Ran(A¯ − z)⊥ = H ⊥ ¯ ¯ ϕ− ∈ ker(A∗ −¯ and Ran(A−z) = ker(A∗ −¯ z ), we can find ψ ∈ D(A), z) so that (A∗ − z)η = (A¯ − z)ψ + (A∗ − z)ϕ− Thus, ϕ+ = η − ψ − ϕ− ∈ ker(A∗ − z). ¯ = D(A∗ ) if and only if (3.8.21) holds. (vii) By (3.8.20), D(A)

Given any sequence {un }∞ n=1 , define J u, a new sequence, by (J u)n = an un+1 + bn un + an−1 un−1

(3.8.28)


211

where a0 = 0. Define an operator, A, by D(A) = {u | un = 0 for large n}

Au = J u

(3.8.29)

Then A : D(A) → D(A) ⊂ ℓ2 is a densely defined operator. Theorem 3.8.6. (i) We have that for any u ∈ D(A) and any sequence v that (both sums are finite) ∞ ∞ X X v¯n (Au)n = (J v)n un (3.8.30) n=1

n=1

(ii) We have that and

D(A∗ ) = {u ∈ ℓ2 | J (u) ∈ ℓ2 }

(3.8.31)

A∗ u = J (u)

(3.8.32)

∗

(iii) If u, v ∈ D(A ), then

hu, A∗vi − hA∗ u, vi = − lim W (¯ u, v)(n)

(3.8.33)

Wn (f, g) = an (fn+1 gn − fn gn+1 ) (iv) If u, v ∈ D(A ) and

(3.8.34)

n→∞

where ∗

hu, A∗ vi − hA∗ u, vi = 6 0

then both

(3.8.35)

¯ u, v ∈ D(A∗ ) \ D(A)

Remark. (iii) includes the assertion that the limit exists. Proof. (i) is a simple summation by parts. (ii) If u ∈ ℓ2 and J (u) ∈ ℓ2 , then (3.8.30) proves u ∈ D(A∗ ) and A∗ u = J (u). Conversely, if u ∈ D(A∗ ) and η ∈ A∗ , then by (3.8.30), η − A∗ u is a sequence with ∞ X (η − J (u))n wn = 0 n=1

for all w ∈ D(A). Picking wn = δkn shows η = J (u), proving (3.8.32) and so J (u) ∈ ℓ2 . (iii) By a direct calculation, N X n=1

[¯ un J (v)n − J (u)n vn ] = W (¯ u, v)N

from which (3.8.33) is immediate.

(3.8.36)

212


¯ then (iv) If u ∈ D(A),

¯ vi = hu, A∗vi hA∗ u, vi = hAu,

¯ so (3.8.35) fails; similarly, if v ∈ D(A).

(3.8.37)

For each z ∈ C, we define two sequences, π(z), ξ(z), by π(z)n = pn−1 (z) ξ(z)n = qn−1 (z)

(3.8.38)

Of course, W (π, ξ) is constant and, by (3.2.22), W (π, ξ)n = −1

(3.8.39)

Lemma 3.8.7. If dµ solves the moment problem and Z dµ(x) mµ (z) = x−z

(3.8.40)

then ξ(z) + m(z)π(z) ∈ ℓ2 for any z ∈ C \ R. Proof. By (3.2.24),

ξn (z) + m(z)πn (z) = hpn−1 , (· − z)−1 i So, by Bessel’s inequality, Z X 2 |ξn (z) + m(z)πn (z)| ≤ n

=

dµ(x) |x − z|2

Im mµ (z) Im z

(3.8.41)

(3.8.42) (3.8.43)

Note that if {pn−1 }∞ n=1 is an orthonormal basis, we have that equality holds in (3.8.42)/(3.8.43). Here is one of the main results on the moment problem: Theorem 3.8.8. The following are equivalent: (i) For one z0 ∈ C \ R, π(z0 ) ∈ ℓ2 . (ii) For one z0 ∈ C \ R, ξ(z0 ) ∈ ℓ2 . (iii) A is not essentially selfadjoint. (iv) For all z0 ∈ C \ R, π(z0 ) ∈ ℓ2 and ξ(z0 ) ∈ ℓ2 . (v) The moment problem is indeterminate. Remark. We will eventually show (see Theorem 3.8.15) that (iv) can be replaced by all of C.


213

Proof. We will show that (i) ⇔ (ii) ⇔ (iii) so (iii) ⇔ (iv) will be automatic. We will then prove (v) ⇒ (i). We will postpone the proof that (iii) ⇒ (v). (i) ⇔ (ii). By Theorem 3.8.4, the moment problem has solutions. So for some mµ (z) 6= 0, ξ(z0 ) + mµ (z0 )π(z0 ) ∈ ℓ2 . This implies π(z0 ) ∈ ℓ2 ⇔ ξ(x0 ) ∈ ℓ2 . (i) ⇔ (iii). There is a unique sequence solving J u = z0 u

(3.8.44)

and un=1 = 1 and no solution with un=1 = 0. This is given by u = π. Thus, by Theorem 3.8.6(ii), ker(A∗ − z0 ) 6= {0} ⇔ π(z0 ) ∈ ℓ2

(3.8.45)

Since π(¯ z0 ) = π(z0 ), we see ker(A∗ − z0 ) 6= {0} ⇔ ker(A∗ − z¯0 ) 6= {0} By Proposition 3.8.5(vii), A essentially selfadjoint ⇔ π(z0 ) ∈ / ℓ2

(3.8.46)

proving (i) ⇔ (iii). (iii) ⇔ (iv). Obviously, (iv) ⇒ (i) ⇒ (iii). But since (iii) ⇒ (i) for any z0 , it implies it for all z0 . Not (i) ⇒ not (v). Since π(z0 ) ∈ / ℓ2 , there is at most one m(z0 ) with ξ(z0 )+m(z0 )π(z0 ) ∈ ℓ2 . So for any two µ’s solving the moment problem and all z0 ∈ C \ R, mµ1 (z0 ) = mµ2 (z0 ), so µ1 = µ2 , that is, we have not (v). The following depends only on (v) ⇒ (i): Corollary 3.8.9. If ∞ X n=1

a−1 n = ∞

(3.8.47)

then the moment problem is determinate. In particular, if a moment problem is indeterminate, then lim an = ∞

n→∞

(3.8.48)

214


Proof. If π(z0 ) ∈ ℓ2 , then so is ξ(z0 ), and thus, a−1 n = (qn (z0 )pn−1 (z0 ) − qn−1 (z0 )pn−1 (z))

(3.8.49)

1

(by (3.2.22)) lies in ℓ . Therefore, (3.8.47) implies not (i) implies not (v). Lemma 3.8.10. For any {aj }nj=1 ∈ Rn , we have n X

(a1 . . . aj )

−1/j

j=1

≤ 2e

n X

a−1 j

(3.8.50)

j=1

Proof. We have 1 + x ≤ ex so (1 + n1 )n ≤ e and thus, inductively, nn ≤ en n!

It follows that

(3.8.51)

−1 −1 1/j (a1 . . . aj )−1/j = [a−1 (j!)−1/j 1 (2a2 ) . . . (jaj )]

≤ ej

−2

j X

ka−1 k

(3.8.52)

k=1

by the arithmetic-geometric mean inequality. Thus, n n n X X X k −1/j −1 (a1 . . . aj ) ≤e ak j2 j=1 k=1 n X

≤ 2e since

j=k

a−1 k

(3.8.53)

k=1

n ∞ X X k 1 ≤ 2k =2 2 j j(j + 1) j=k j=k

Corollary 3.8.11 (Carleman’s criterion). If ∞ X −1/2n c2n =∞

(3.8.54)

(3.8.55)

n=1

then the moment problem is determinate.

Proof. Since pn (x) = (a1 . . . an )−1 xn + lower order, h(a1 . . . an )−1 xn , pn i = 1

(3.8.56)

and thus, by the Schwarz inequality, −1/2n

c2n

≤ (a1 . . . an )−1/n

(3.8.57)


By (3.8.50), we see (3.8.55) implies (3.8.47).

215

Example 3.8.1, revisited. If α ≥ 1, Z Z n α x exp(−|x| ) ≤ 2 + xn exp(−|x|1 ) = 2 + 2n! ≤ 4nn

and

(3.8.58)

1 (3.8.59) 8n Thus, (3.8.55) holds, and so the moment problem is determinate. To get the last step in the proof of Theorem 3.8.8, we need to analyze selfadjoint extensions of A when A¯ 6= A∗ , that is, operators B with A¯ ⊂ B = B ∗ . Since A¯ ⊂ B implies B ∗ ⊂ A∗ , we have A¯ $ B = B ∗ $ A∗ (3.8.60) −1/2n

c2n

≥

where B 6= A¯ and B 6= A∗ comes from A¯ 6= A∗ 6= A∗∗ . In our case where ¯ has dimension 2, we must thus have dim(D(B)/D(A)) = D(A∗ )/D(A) 1 which simplifies the analysis. ¯ has dimension 2. Then Theorem 3.8.12. Suppose D(A∗ )/D(A) ∗ (i) D(B) = D(A) + [ϕ] with ϕ ∈ D(A ) \ D(A) is the domain of a selfadjoint extension (i.e., A∗ ↾ D(B) is selfadjoint) if and only if hϕ, A∗ ϕi ∈ R

(3.8.61)

ϕ + tψ 1 + |t|

(3.8.62)

(ii) Suppose ϕ, ψ = D(A∗ ) with hϕ, A∗ ψi, hψ, A∗ ϕi, hϕ, A∗ ϕi, hψ, A∗ ψi all real. Let t ∈ R ∪ {∞} and let ϕt =

(where ϕ∞ is interpreted as ψ). Then ¯ + [ϕt ] D(Bt ) = D(A) Bt = A∗ ↾ D(Bt )

(3.8.63)

describes all the selfadjoint extensions of A. Proof. (i) By (3.8.60), D(B)/D(A) is of dimension 1, so for every selfadjoint extension, B, D(B) always has the claimed form. Since ϕ ∈ D(B), hϕ, A∗ϕi = hϕ, Bϕi (3.8.64) is real. Conversely, if (3.8.61) holds and η ∈ D(A), then hϕ + η, A∗ (ϕ + η)i = hϕ, A∗ ϕi + hη, Aηi + hϕ, Aηi + hAη, ϕi (3.8.65)

216


is real, so A∗ ↾ D(A) + [ϕ] has real expectation values. By polarization, it is Hermitian. Since A¯ $ B ⊂ B ∗ $ A∗ , we see that D(B ∗ ) must be D(B) since every subspace between D(B) and D(A∗ ) is either D(B) or D(A∗ ). Thus, B = B ∗ . ¯ (ii) We have, for all η ∈ D(A), Imhϕ + αψ + η, A∗ (ϕ + αψ + η)i = (Im α)[hϕ, A∗ψi − hψ, A∗ ϕi]

¯ with A∗ (βϕ + αψ + η) = Since there is α, β ∈ C and an η ∈ D(A) ∗ i(βϕ + αψ + η), we conclude that hϕ, A ψi − hψ, A∗ ϕi = 6 0. It follows that (3.8.61) holds for ϕ + αψ if and only if α ∈ R. Given (i), this proves (ii). Below (see Theorem 3.8.15), we will prove that if A is not selfadjoint for the concrete Jacobi matrix, then not only is π(z0 ), ξ(z0 ) ∈ ℓ2 for z0 ∈ C \ R but also for z0 ∈ R. We use that for now for z0 = 0. We have J (π(0)) = 0 J (ξ(0)) = δ · 1 (3.8.66)

so if A is the operator of J restricted to finite sequences, by Theorem 3.8.6(ii), we have hξ(0), J (π(0))i = hπ(0), J (π(0))i = hξ(0), J (ξ(0))i = 0 hπ(0), J (ξ(0))i = 1

(3.8.67) (3.8.68)

By Theorem 3.8.6(iv), we have π(0), ξ(0) ∈ D(A∗ ) \ D(A) and, by Theorem 3.8.12, there is a one-parameter family, {Bt }t∈R∪{∞} , of selfadjoint extensions with ¯ + [ξ(0) + tπ(0)] D(Bt ) = D(A)

(3.8.69)

Proposition 3.8.13. Suppose A¯ is not essentially selfadjoint. (i) For each z0 ∈ C \ R, we have π(z0 ), ξ(z0) ∈ D(A∗ ) \ D(A). For each t, there is an at (z0 ) ∈ C so that ξ(z0 ) + at (z0 )π(z0 ) ∈ D(Bt )

(3.8.70)

and for every such z0 , all at (z0 ) are distinct as t varies. (ii) hδ1 , (Bt − z0 )−1 δ1 i = at (z0 ) (3.8.71) In particular, if A¯ is not selfadjoint, there are multiple solutions to the moment problem. Remark. The spectral measures for Bt which solve the moment problem are called the von Neumann solutions of the moment problem.


217

Proof. As noted in the proof of Theorem 3.8.8, ker(A∗ − z0 ) = [π(z0 )]

(3.8.72)

Moreover, by (3.8.66) (extended from z0 = 0), (A∗ − z0 )ξ(z0 ) = δ · 1

(3.8.73)

Thus, every solution of (A∗ − z0 )η = δ0 has the form η = ξ(z0 ) + cπ(z0 )

(3.8.74)

(Bt − z0 )−1 δ1 = ξ(z0 ) + at (z0 )π(z0 )

(3.8.75)

So for some at (z0 ) ∈ C, Let ηt be the right side of (3.8.75). By (3.8.33), hπ(¯ z0 ), A∗ ηt i − hA∗ π(¯ z0 ), ηt i = 1

(3.8.76)

we conclude that ηt ∈ D(A∗ ) \ D(A), so

¯ + [ηt ] D(Bt ) = D(A)

(3.8.77)

which implies that the ηt are distinct for distinct t. Finally, by (3.8.75), hδ1 , (Bt − z0 )−1 δ1 i = at (z0 ) proving (3.8.71).

(3.8.78)

Next, we turn to the claim that in the indeterminate case, π(z0 ), ξ(z0 ) ∈ ℓ2 also for z0 ∈ R. We depend on a useful general perturbation theorem. ˜ ∞ Theorem 3.8.14. Suppose {Aj }∞ j=1 and {Aj }j=1 are two sequences of bounded operators with bounded inverses, and define Tn = An . . . A1 Tñ = Añ . . . A˜1 Bk = T −1 (A˜k − Ak )Tk−1

(3.8.79) (3.8.80) (3.8.81)

k

where T0 = T˜0 = 1. Then (1) We have for each n, kTn−1Tñ k

≤ exp

X n j=1

kBj k

(3.8.82)

218


(ii) If ∞ X n=1

then

kBn k < ∞

(3.8.83)

lim Tn−1 Tñ

(3.8.84)

n→∞

exists and is given by lim

n→∞

Tn−1 Tñ

=1+

∞ X

−1 ˜ Bj Tj−1 Tj−1

(3.8.85)

j=1

(iii) If ∞ X

kTn k2 < ∞

(3.8.86)

∞ X

kTñ k2 < ∞

(3.8.87)

n=1

and (3.8.83) holds, then

n=1

Remark. By (3.8.81) and (3.8.85), we get lim Tn−1 Tñ = 1 +

n→∞

∞ X j=1

Tj−1 (A˜j − Aj )T˜j−1

(3.8.88)

Proof. Noticing that Tk−1 Ak Tk−1 = 1

(3.8.89)

Tk−1 A˜k Tk−1 = 1 + Bk

(3.8.90)

we have Therefore, −1 ˜ Tn−1 Tñ = (Tn−1 Añ Tn−1 )(Tn−1 An−1 Tn−2 ) . . . (T1−1 A˜1 T0 )

= (1 + Bn ) . . . (1 + B1 )

(3.8.91)

(i) Thus, kTn−1 Tñ k

X n n Y ≤ (1 + kBj k) ≤ exp kBj k j=1

(3.8.92)

j=1

(ii) By (3.8.91), we have Tn−1 Tñ

=1+

n X j=1

Bj (1 + Bj−1 ) . . . (1 + B1 )

(3.8.93)


=1+

n X

−1 ˜ Bj Tj−1 Tj−1

219

(3.8.94)

j=1

By (3.8.82),

−1 ˜ kBj Tj−1 Tj−1 k

X ∞ ≤ kBj k exp kBk k

(3.8.95)

k=1

so the sum is absolutely convergent, implying that the limit exists and is given by (3.8.85). (iii) By (3.8.82), X ∞ ˜ kTn k ≤ kTn k exp kBj k (3.8.96) j=1

so (3.8.86) implies (3.8.87).

To apply this to moment problems, Tn , An , . . . will be 2 × 2 transfer matrices, but we will want to modify from the definition in Section 3.2. There we added an an to the lower component of vectors to get a transfer matrix of determinant one. With an ’s bounded from above, this is normally harmless, but here our an ’s are unbounded so we will modify. Given Jacobi parameters {an , bn }∞ n=1 , we define (with a0 ≡ 1) z−bn −an−1 an an An (z) = (3.8.97) 1 0 so pn (z) pn−2 (z) = An (z) (3.8.98) pn−1 (z) pn−2 (z) and pn (z) −qn (z) Tn (z) = (3.8.99) pn−1 (z) −qn−1 (z) to be compared with (3.2.19). Now det(Tn ) 6= 1 but rather det(Tn ) = a−1 n

and, thus

−qn−1 (z) qn (z) Tn (z) = an −pn−1 (z) pn (z) Our perturbation will be to change z to w, so w−z 0 An (w) − An (z) = an 0 0 −1

(3.8.100)

(3.8.101)

(3.8.102)

and

Bn ≡ Tn (z)−1 (An (w) − An (z))Tn−1 (z)

(3.8.103)

220


The an in (3.8.101) and the a−1 n in (3.8.102) cancel! Thus, with Nn (z) = (|pn (z)|2 + |pn−1 (z)|2 + |qn (z)|2 + |qn−1 (z)|2 )1/2

(3.8.104)

we obtain kBn k ≤ |w − z| Nn (z)Nn−1 (z)

(3.8.105)

and by the Schwarz inequality ∞ X n=1

2

Nn (z) < ∞ ⇒

∞ X n=1

kBn k < ∞

(3.8.106)

Thus, we can apply Theorem 3.8.14 and find Theorem 3.8.15. If π(z), ξ(z) are both in ℓ2 for a single z, then π(w), ξ(w) are in ℓ2 for any w ∈ C and lim Tn (z)−1 Tn (w)

(3.8.107)

n→∞

exists. One defines four functions, A(z), B(z), C(z), and D(z), by −B(z) −A(z) −1 lim Tn (z)Tn (w = 0) = (3.8.108) D(z) C(z) n→∞ and the Nevanlinna matrix by N(z) =

A(z) C(z) B(z) D(z)

(3.8.109)

By (3.8.88), (3.8.99), (3.8.101), and (3.8.102), we get Proposition 3.8.16. The Nevanlinna matrix is given by A(z) = z

∞ X

qn (0)qn (z)

(3.8.110)

n=0

B(z) = −1 + z C(z) = 1 + z

∞ X

qn (0)pn (z)

(3.8.111)

n=0 ∞ X

(3.8.112)

pn (0)pn (z)

(3.8.113)

pn (0)qn (z)

n=0

D(z) = z

∞ X n=0

These functions are entire functions obeying |A(z)| ≤ Cε exp(ε(z))

(3.8.114)


221

and similarly for B, C, D. Near z = 0, B(z) = −1 + O(z)

(3.8.115)

D(z) = D0 z + O(z 2 )

(3.8.116)

D0 > 0

(3.8.117)

where Proof. The formulae follow from the earlier equations. (3.8.115) is immediate, as is (3.8.116) where D0 =

∞ X

pn (0)2 > 0

(3.8.118)

n=0

To get (3.8.114), we note that

Bk (z) = zbk (3.8.119) P∞ with bk a constant matrix with k=1kbk k < ∞. Thus, n N Y X k(1+BN ) . . . (1+Bk )k ≤ (1+|z| kbj k) exp |z| kbj k (3.8.120) j=1

j=n+1

from which (3.8.114) follows.

We can express the resolvent of the selfadjoint extensions, Bt , in terms of the Nevanlinna matrix: Theorem 3.8.17. Consider an indeterminate moment problem. For t ∈ R ∪ {∞} and z ∈ C \ R, the resolvent of the selfadjoint extensions, Bt , are given by where for z, w ∈ C,

(δ1 , (Bt − z)−1 δ1 ) ≡ F (z, t)

(3.8.121)

C(z)w + A(z) D(z)w + B(z)

(3.8.122)

F (z, w) ≡ − by

2 Proof. Given a sequence, {sn }∞ n=1 , we let Rn (s) ∈ C be defined

Rn (s) = (sn+1 , sn )

(3.8.123)

and we define wn : C2 × C2 → C by wn ((α, β), (γ, δ)) = an (αδ − βγ)

(3.8.124)

Wn (f, g) = wn (Rn (f ), Rn (g))

(3.8.125)

so that

222


Constancy of the Wronskian for solutions of the same difference equation shows that for any z ∈ C and u, v ∈ C2 , wn (Tn (z)u, Tn (z)v) = w0 (u, v)

(3.8.126)

By (3.8.33), if f, g ∈ D(Bt ), then lim wn (Rn (f ), Rn (g)) = 0

n→∞

since hf, Bt gi = hBt f, gi by Hermiticity of Bt . Since t Rn (ξ(0) + tπ(0)) = Tn (0) 1 at (z0 ) Rn (ξ(z0 ) + at (z0 )π(z0 )) = Tn (z0 ) 1 we see, by (3.8.127), that t at (z0 ) lim wn Tn (0) , Tn (z0 ) =0 n→∞ 1 1 So, by (3.8.126), t at (z0 ) −1 lim w0 , Tn (0) Tn (z0 ) =0 n→∞ 1 1 By the existence of the limit, for some constant c, at (z0 ) t −1 = cTn (z0 ) Tn (0) 1 1 Given (3.8.108), this implies (3.8.121)/(3.8.122).

(3.8.127)

(3.8.128) (3.8.129)

(3.8.130)

(3.8.131)

(3.8.132)

Lemma 3.8.18. For z ∈ C+ , {F (z, t) | t ∈ R ∪ {∞}} is a circle in the upper complex plane. F (z, · ) maps C+ to the interior of the disk bounded by the circle. Proof. By (3.8.121), F maps R ∪ {∞} to C+ and not to ∞, so the image is a circle in C. Suppose [F (z, · )]−1(∞) lies in C− . Then F (z, · ) maps C− to the outside of the circle, and so C+ to the inside. Since, for z ∈ C+ , it can never lie in R, by continuity, [F (z, · )]−1 (∞) is either always in C− (or always in C+ ), so it suffices to show this for z = iε, that is, that Im(−B(iε)/D(iε)) < 0 for ε small and positive. This follows from (3.8.120)/(3.8.121). Next, we relate solutions of the moment problem to asymptotics of the Stieltjes transform.


223

Proposition 3.8.19. Let µ be a probability measure on R solving (3.8.1) and let Z dµ(x) Gµ (z) = (3.8.133) x−z Let N X RN (µ; iy) = Gµ (iy) + (−i)n+1 y −n−1cn (3.8.134) n=0

Then

|RN (µ; iy)| ≤

(

cN +1 y −N −2 1 (c + cN +2 )y −N −2 2 N

N odd N even

(3.8.135)

Conversely, if G(z) is a Herglotz function, so RN , given by (3.8.134), is O(y −N −2) for each N, then G is given by (3.8.133) for some measure µ solving (3.8.1). Proof. If (3.8.133) holds and µ obeys (3.8.1), write −1 N X x −1 n n −n−1 −N −1 N +1 −N −2 (x − iy) = x (−i) y + (−i) x y 1− iy n=0 (3.8.136) to see that RN , given by (3.8.134), is given by −1 Z x N +1 −N −2 N +1 RN (µ; iy) = (−i) y x 1− dµ(x) (3.8.137) iy Since |1− iyx | ≥ 1 for x, y real, the N odd case of (3.8.135) is immediate. For N even, use the fact that for such N, |x|N +1 ≤ 12 xN + xN +2

(3.8.138)

lim |y|−1|G(iy)| = 0

(3.8.139)

For the converse, start with the Herglotz representation, (2.3.87). Since (3.8.134)/(3.8.135) imply y→∞

we see that A = 0. They also imply that yG(iy) → ic0

from which one first sees (with ρ replaced by µ) Z dµ(x) = c0 since

Im yG(iy) =

Z

y2 dµ(x) x2 + y 2

(3.8.140)

(3.8.141)

(3.8.142)

224


and we can use the monotone convergence theorem, and then that there is a cancellation of real parts that implies (3.8.133). From (3.8.134)/(3.8.135), one sees inductively, using (3.8.136), that Z (iy)2 x2n−1 dµ(x) + iγc2n−1 → c2n (3.8.143) x − iy which implies, taking real and imaginary parts, that Z y 2x2n c2n = lim dµ(x) (3.8.144) y→∞ x2 + y 2 Z 2 2n−1 y x c2n−1 = lim dµ(x) (3.8.145) y→∞ x2 + y 2 R 2n Monotone convergence and the first of these implies x dµ = c2n R 2n−1 and then dominated convergence and (3.8.145) implies x dµ = c2n−1 . Corollary 3.8.20. For z ∈ C+ , let

D(z) = {F (z, w) | Im w > 0}

(3.8.146)

be the disk of Lemma 3.8.18. If G has the form (3.8.134) where µ solves (3.8.1), then G(z) ∈ D(z) (3.8.147) for all z ∈ C+ . Conversely, if G is an analytic function on C+ obeying (3.8.150), then G has the form (3.8.133) for some µ obeying (3.8.1). Proof. By Proposition 3.8.19, Gµ (iy) has an asymptotic series G(iy) ∼ −

∞ X

(−i)n+1 y −n−1cn

(3.8.148)

n=0

uniformly in the von Neumann solutions. Since these solutions fill out the circle at the boundary of D(z), the estimates hold in all on D(z), so G solves the moment problem by Proposition 3.8.19. Conversely, by (3.8.43), if µ solves the moment problem, where

Gµ (z) ∈ ∆(z)

(3.8.149)

Im w 2 ∆(z0 ) = w kξ(z0 ) + wπ(z0 )k ≤ Im z0 This set is given by a quadratic inequality in Re w, Im w whose quartic term is |w|2kπ(z0 )k2 . Such a set always describes a disk or the empty set. Since equality holds in (3.8.43) for von Neumann solutions, ∂∆(z) = ∂D(z), so ∆(z) = D(z) and (3.8.149) is (3.8.147).


225

Here is the main result on the description of the solutions of the moment problem in the indeterminate case: Theorem 3.8.21 (Nevanlinna’s Parametrization). Let {cn }∞ n=1 be the moments of an indeterminate problem, and let A, B, C, D be the elements of the Nevanlinna matrix, and F given by (3.8.122). There is a one-one correspondence between H, the set of all analytic functions, ϕ, of C+ to C+ so that µϕ is given by Z dµϕ (x) = F (z, ϕ(z)) (3.8.150) x−z

The von Neumann solutions correspond to ϕ(z) ≡ t and all other solutions have Ran(ϕ) ⊂ C+ .

Proof. Any function of the form G(z) ≡ F (z, ϕ(z)) has G obeying (3.8.147) by Lemma 3.8.18. Conversely, if G obeys (3.8.150), then, because F (z, · ) is a bijection of C taking C+ to D(z), there is a unique ϕ obeying G(z) = F (z, ϕ(z)) with ϕ analytic or infinite. By the open mapping theorem, either ϕ(z) = t ∈ R ∪ {∞} or Ran(ϕ) ∈ C+ . Given Corollary 3.8.20, this proves the theorem. This allows further analysis of solutions, of which the following is typical: Theorem 3.8.22. (i) The von Neumann solutions of an indeterminate moment problem are discrete pure point measures. (ii) If ϕ is a rational Herglotz function, dµϕ is pure point. (iii) The positions of the pure points and weights of the von Neumann solutions are real analytic in t. The positions are nonconstant. (iv) There are always purely a.c. and purely s.c. solutions of an indeterminate problem. Proof. (i), (ii) In these cases, Gµ has an analytic continuation to an entire meromorphic function. (iii) This follows from analyticity of A, B, C, D and the form of F (z, t). (iv) If dµt is the von Neumann solution associated to Bt and dν(t) is a probability measure, then Z dην (x) = dµt (x) dν(t) (3.8.151) is a solution of the moment problem. By (iii), dην is a.c. (resp. s.c.) if dν is a.c. (resp. s.c.).

226


Remarks and Historical Notes. The critical paper on the moment problem is by Stieltjes [413]. Earlier, Chebyshev had asked about uniqueness for Gaussian measures. The approach via selfadjoint operators was pioneered by Stone [414] and the transfer matrix connection was exploited especially by Simon [386], which we follow in much of this section. For other presentations, see Akhiezer [13] and Shohat– Tamarkin [376]. The name von Neumann solutions comes from Simon [386], after von Neumann’s theory of selfadjoint extensions. Such solutions are called N-extremal in Akhiezer [13]. The Nevanlinna parametrization is from [319]. A further result (see [13, 386]) is that the polynomials are dense in L2 (R, dµ) if and only if dµ is a von Neumann solution and their closure has finite codimension if and only if the Nevanlinna function, ϕ, is rational. All these solutions are extreme points in the convex set of solutions of the moment problem, proving that the extreme points are dense. Carleman’s criterion (Corollary 3.8.11) is due to Carleman [71]. The awkward terminology (at least in English) “determinate” and “indeterminate” comes from the French. While Stieltjes was Dutch, his paper [413] is in French. There are actually two moment problems discussed in the next section: what we have called “the moment problem” (i.e., solution of (3.8.1) with the measure allowed to be supported anywhere on R) is more properly the Hamburger moment problem. The Stieltjes moment problem is the problem one gets by restricting to measures supported on [0, ∞). There is a simple relation between the two problems. Let dρ0 be a probability measure on [0, ∞) with moments cn . Define d˜ ρ0 on R by d˜ ρ0 (x) = 21 [χ[0,∞) (x) dρ(x2 ) + χ(−∞,0] (x) dρ(x2 )]

(3.8.152)

( 0 xn d˜ ρ0 (x) = cn/2

(3.8.153)

and let Γn =

Z

n odd n even

(3.8.152) sets up a one-one correspondence between all solutions of the Stieltjes moment problem with moments cn and all solutions of the Hamburger moment problem with moments Γn symmetric under x → −x. It is a basic fact that any indeterminate Hamburger moment problem with vanishing odd moments has multiple solutions that are invariant under x → −x, namely, the von Neumann solutions with t = 0 and t = ∞. This implies immediately that Theorem 3.8.23. Let (dρ0 , cn ) be a measure and set of moments on [0, ∞). Let (d˜ ρ0 , Γn ) be given by (3.8.152)/ (3.8.153). Then the Stieltjes

3.9. THE KREIN DENSITY THEOREM

227

moment problem for {cn } is determinate (resp. indeterminate) if and only if the Hamburger moment problem for {Γn } is determinate (resp. indeterminate). Theorem 3.8.23 goes back at least to Chihara [79] and appears also in Berg [41] and Simon [386]. 3.9. The Krein Density Theorem and Indeterminate Moment Problems If one sought a connection between measures on ∂D and measures on R, one might not think first of x = z + z −1 , which is quadratic, but rather 1−z x=i (3.9.1) 1+z which is a fractional linear mapping of D to C+ and its inverse i−x (3.9.2) i+x For the version of Szeg˝o’s theorem that gives asymptotics of the leading term in OPs, this is not useful—it relates polynomials in z to i−x polynomials in i+x —or what is the same time polynomials in (i + x)−1 since i−x 2i = −1 + (3.9.3) i+x i+x Krein [244] realized it could be used to transfer the density theorem (Theorem 2.11.5), which gives criteria for when {einθ }∞ n=0 span 2 L (∂D, dµ), to a continuum analog: z=

Theorem 3.9.1 (Krein’s Density Theorem [244]). Let dρ = F (x) dx + dρs

(3.9.4)

be a finite measure on R. Then the span of {eiαx }α≥0 is dense in L2 (R, dρ(x)) if and only if Z ∞ log F (x) dx = −∞ (3.9.5) 2 −∞ 1 + x Remark. As usual, F ∈ L1 implies that the integral with log+ F is finite, so the integral is either convergent or it diverges to −∞. As a first preliminary for the proof, we need Lemma 3.9.2. For any finite measure dρ on R, the span of {(i + iαx ∞ x)−n }∞ }α=0 . n=0 is the same as the span of {e

228


Proof. Suppose that f is orthogonal to (i+x)−n for n = 0, 1, 2, . . . . Then since 1 − ix = −i(i + x), we see that if Z F (w) = f (x)(1 − wx)−1 dρ(x) (3.9.6) n

which is analytic in C+ , then ddwFn (i) = 0 for all n. So F = 0 and thus (taking derivatives of F ), we have that f is orthogonal to (1 − wx)−n for all w ∈ C+ and n = 0, 1, 2, . . . . Since for α ≥ 0, −n iαx 1− → eiαx (3.9.7) n )−n | ≤ 1, we have convergence in L2 so f pointwise in x with |(1 − iαx n is orthogonal to {eiαx }α≥0 . Conversely, if f is orthogonal to {eiαx }α≥0 , we have f orthogonal to (1 − iβx)−1 for all β > 0 since Z ∞ eiβαx e−α dx = (1 − iβx)−1 (3.9.8) 0

and the integral converges weakly in L2 (R, dρ). But then, by analyticity of F (given by (3.9.6), F is zero on C+ so its derivatives at i are all zero and f is orthogonal to {(i + x)−n }∞ n=0 .

As a second preliminary, we introduce an analog of the Szeg˝o map, Sz, of Section 1.9. Notice that the boundary value of (3.9.1) on ∂D is θ iθ (3.9.9) x(e ) = tan 2

Thus we define the Krein map Kr : M+,1 (∂D) → M+,1(R ∪ {∞}) by dρ = Kr(dµ) if Z Z g(θ) dµ(θ) = g(2 arctan(x)) dρ(x) (3.9.10) Kr is a one-one correspondence between {µ ∈ M+,1(∂D) | µ({−1}) = 0} and measures dρ in M+,1 (R). Notice also that if

dθ + dµs 2π and dρ is given by (3.9.4), then because (3.9.9) says dµ(θ) = w(θ)

we have that

dx 1 + x2 = dθ 2 θ 2 θ w(θ) = π sec F tan 2 2

(3.9.11)

(3.9.12) (3.9.13)


229

or

w(2 arctan(x)) (3.9.14) 1 + x2 Proof of Theorem 3.9.1. By Lemma 3.9.2 and (3.9.3), iαx are {e }α≥0 is dense in L2 (R, dρ(x)) if and only if polynomials in i−x i+x dense in L2 (R, dρ(x). Pick dµ on ∂D so dρ = Kr(dµ) and let F (x) = π −1

by

V : L2 (∂D, dµ) → L2 (R, dµ)

(3.9.15)

(Vf )(x) = f (2 arctan(x)) (3.9.16) By (3.9.10), V is unitary, and if Mg is multiplicative by g, we have (z = eiθ ) VMz V −1 = M(i−x)/(i+x) (3.9.17) iαx 2 It follows that {e }α≥0 is dense in L (R, dρ) if and only if polynomials in z are dense in L2 (∂D, dµ). By Theorem 2.11.5, this is true if and only if Z 2π dθ log(w(θ)) = −∞ (3.9.18) 2π 0 where w is given by (3.9.11). Since Z 2π dθ log sec θ 2π < ∞ 2 0 (for there is only a single logarithmic singularity at θ = π), (3.9.13) says that (3.9.18) is equivalent to Z 2π θ dθ log F tan = −∞ (3.9.19) 2 2π 0 2dx Changing variables to x = tan( θ2 ) so dθ = 1+x 2 , we see that (3.9.19) is equivalent to Z 1 ∞ dx log(F (x)) = −∞ π −∞ 1 + x2 which is (3.9.5).

Corollary 3.9.3 (Krein). Let dρ have the form (3.9.5) and suppose Z ∞ log F (x) dx > −∞ (3.9.20) 2 −∞ 1 + x and that Z ∞ |x|n dρ(x) < ∞ (3.9.21) −∞

230


for all n so the polynomials lie in L2 (R, dρ). Then the polynomials are not dense in L2 (R, dρ). Remarks. 1. We will see many examples soon where (3.9.20) holds. 2. It is known (see Theorem 3.8.22) that there are discrete measures (F ≡ 0 so (3.9.20) fails) with the polynomials not dense so the converse of Corollary 3.9.3 does not hold. Proof. If (3.9.20) holds, then the span of {eiαx }α≥0 is not dense by Krein’s density theorem. Find a nonzero f ∈ L2 with Z f (x) eiαx dρ(x) = 0 (3.9.22)

for all α ≥ 0. By (3.9.21), for any Rf in L2 , the integral in (3.9.22) is C ∞ with derivatives given by (i)n f (x)xn eiαx dρ(x). In particular, taking derivatives at α > 0 and taking α ↓ 0, Z f (x) xn dρ(x) = 0 (3.9.23) that is, f is orthogonal to the polynomials.

This has applications to the theory of moments. Corollary 3.9.4 (Krein). If dρ0 has the form (3.9.5), (3.9.20), and (3.9.21), then the moment problem is indeterminate. Proof. By Corollary 3.9.3, the polynomials are not dense in L (R, dρ0 ). By Theorem 3.8.8, if the moment problem is determinate, then the unique solution of the moment problem has the polynomials dense. Thus, if ρ0 exists, the problem is indeterminate. 2

Example 3.8.1, revisited. If α < 1, Z α log(e−|x| ) dx > −∞ 1 + x2 so, by Krein’s result, the problem is indeterminate. Thus, we see dρα is determinate (resp. indeterminate) if α ≥ 1 (resp. 0 ≤ α < 1). Example 3.9.5. Stieltjes [413] showed that the log normal measure 2

π −1/2 χ0,∞) (x) e−(log x) dx is indeterminate. One can see this from Krein’s criteria since Z (log x)2 dx < ∞ (3.9.24) 1 + x2


231

(in fact, Stieltjes showed the Stieltjes moment problem is indeterminate—this follows from a translation of Krein’s criterion that we discussed in the Notes to the last section). In this case, the moments can be written down explicitly cn = exp( 41 (n + 1)2 ) and one can even write down explicit measures with the same moments: For θ ∈ [−1, 1], 2

dρ0 (x) = π −1/2 χ(0,∞) (x)[1 + θ sin(2π log x)] e−(log x) dx also solves the moment problem. This moment problem is further discussed in Christiansen [81] and references therein. Example 3.9.6. Hamburger [185] showed that the Stieltjes moment problem for √ π x χ[0,∞)(x) exp − dx [log(x)]2 + π 2 is indeterminate. This follows from the Krein criterion for that case (see the Notes). Remarks and Historical Notes. The Krein density theorem (Theorem 3.9.1) appeared in Krein [244] with a proof essentially identical to the one here. He refers to Kolmogorov [234] for the density theorem on the disk with no mention of the connection of the entropy integral to Szeg˝o, although earlier in 1945, he wrote a paper [243] on extensions of Kolmogorov’s density theorem that discusses Szeg˝o’s work. Interestingly enough, probably a sign of World War II solidarity, both 1945 papers were in English! Lp versions of the Krein density theorem are due to Akhiezer [11]. Corollary 3.9.4 seems to have appeared first in Akhiezer’s book [13] on the moment problem (see p. 87) and is attributed to Krein (without any reference). The proof he gives first shows Corollary 3.9.3—we follow his arguments for both corollaries. Theorem 3.8.23 allows Corollary 3.9.4 to be translated to: Corollary 3.9.7. If dρ0 has the form (3.9.4) and (3.9.21) holds, and if dρ0 is supported on [0, ∞) and Z ∞ log(F (x)) dx √ > −∞ (3.9.25) 1+x x 0 then the Stieltjes problem is indeterminate.

232


This shows the Hamburger example of Example 3.9.6 is borderline for indeterminacy. The orthogonal polynomials associated to various explicit indeterminate problems are included in what is called the Askey scheme. Among them are the Stieltjes–Wigert polynomials associated to the measure of Example 3.9.5 [81] and the q-Laguerre and 1/q-Hermite (see, e.g., Christiansen [80]). While there is no strict converse to Corollary 3.9.3, there is a weak variant of the converse: If the polynomials are not dense, there is always a measure with the same moments for which (3.9.20) holds. Indeed, there is—among all measures solving the moment problem—a unique one that maximizes the integral in (3.9.20); see Berg [41] or Gabardo [140]. As mentioned after Theorem 2.11.5, when the Szeg˝o condition holds, one can use the Szeg˝o function, D, to find an explicit function orthogonal to all polynomials. One can also do this directly in the case of R providing a “direct” proof of Corollary 3.9.3. In fact, by using an analog of D 2 , one gets G analytic in the R nupper plane whose boundary values obey |G(x + i0)| = F (x) and x G(x + i0) dx = 0. Then dρ − Re(G(x + i0) dx gives an explicit second measure with the same moments (since F − Re G ≥ 0, it is a positive measure). This is discussed in Simon [386]. 3.10. The Nevai Class and Nevai Delta Convergence Theorem Recall a measure on R is said to lie in the Nevai class for [−2, 2] if and only if its Jacobi parameters obey an → 1

bn → 0

(3.10.1)

In preparation for carrying over the limit theorems for CD kernels of Sections 2.15–2.17 from ∂D to [−2, 2], we focus here on two theorems of Nevai [314] and a consequence. Here are the three results: Theorem 3.10.1. Let pn (x, dρ) be the normalized OPRL for a measure in the Nevai class. Then for x ∈ [−1, 1], we have lim |pn (x, dρ)|2 Kn (x, x)−1 = 0

n→∞

(3.10.2)

Theorem 3.10.2 (Nevai’s Delta Convergence Theorem). Let Qn (x, x0 ) be the minimizer in the Christoffel function Z 2 λn (x0 , dρ) = min |Xn (x, x0 )| dρ(x) deg X ≤ n; Xn (x0 ) = 1

(3.10.3)

3.10. THE NEVAI CLASS AND DELTA CONVERGENCE THEOREM

233

for a measure dρ in the Nevai class. Then for all x0 ∈ [−1, 1], the probability measure dξn (x) ≡ λn (x0 )−1 |Qn (x, x0 )|2 dρ(x)

(3.10.4)

converges weakly to a point mass at x0 .

Theorem 3.10.3. Let dµ and g dµ be two measures in the Nevai class where g is such that there are polynomials R0 , R1 so that R02 g and R12 g −1 are bounded continuous functions in some bounded open interval containing supp(dρ). Then for any x0 ∈ (−2, 1), lim

n→∞

λn (x0 , g dρ) = g(x0 ) λn (x0 , dρ)

(3.10.5)

Remarks. 1. All limits are uniform on [−2, 2] as our proofs show. 2. These results also hold at point masses in supp(dµ). 3. The R0 , R1 condition says g and g −1 have finitely many zeros and the vanishing is of finite order in that |g(x)| ≥ C|x − x0 |ℓ for some integer ℓ and x near x0 . 4. We will need Theorem 3.10.3 for the case g(x) = 14 (4 − x)2 in connection with Theorem 3.11.9. We will show Theorem 3.10.1 is equivalent to Theorem 3.10.2 and the two together imply Theorem 3.10.3, and then we will turn to the more subtle proof of Theorem 3.10.1. The Christoffel–Darboux kernel (aka CD kernel) is defined by n X Kn (x, y) = pn (x)pn (y) (3.10.6) j=0

for x, y ∈ R.

Theorem 3.10.4 (CD Formula). For all x 6= y,

an+1 (pn+1 (x)pn (y) − pn+1 (y)pn (x)) x−y

(3.10.7)

Ln (x, y) = an+1 (pn+1 (x)pn (y) − pn+1 (y)pn (x))

(3.10.8)

xpn (x) = an+1 pn+1 (x) + bn+1 pn (x) + an pn (x)

(3.10.9)

Kn (x, y) = Proof. Let Take

multiply by pn (y) and subtract the expression obtained by interchanging x and y. Then (x − y)pn (x)pn (y) = Ln (x, y) − Ln−1 (x, y)

(3.10.10)

234


This plus induction (starting with p−1 (x) = 0 and K−1 = 0) yields (3.10.7). As in the proof of Proposition 2.16.2, one immediately obtains that λn (x0 ) = Kn (x, x0 )−1

(3.10.11)

and that the minimizer is Kn (x, x0 ) Kn (x0 , x0 ) As in the proof of Theorem 2.16.8, Z Kn (x, y)Kn (y, z) dρ(y) = Kn (x, z) Qn (x, x0 ) =

which, in particular, implies that Z λn (x0 )−1 [Qn (x, x0 )]2 dρ(x) = 1

(3.10.12)

(3.10.13)

(3.10.14)

First we will show that Theorem 3.10.2 implies Theorem 3.10.1 and is equivalent if inf an > 0 (3.10.15) n

Proposition 3.10.5. If (3.10.2) holds for x = x0 , then the measure dξn converges weakly to δx0 , a point mass at x0 . Conversely, if w (3.10.15) holds and if dξn −→ δx0 , then (3.10.2) holds. Proof. We begin with three preliminaries. Since the dξn are probability measures with supports inside a fixed compact set, it is easy to see that Z w dξn −→ δx0 ⇔ (x − x0 )2 dξn (x) → 0 (3.10.16)

Secondly, by the CD formula (3.10.7) and orthogonality of {pk } in L2 (R, dρ), Z a2 [pn (x)2 + pn+1 (x)2 ] (3.10.17) (x − x0 )2 dξn = n+1 Kn (x0 , x0 ) Finally, we claim

|pn (x0 )|2 Kn (x0 , x0 )−1 → 0 ⇒ |pn+1 (x0 )|2 Kn (x0 , x0 )−1 → 0 (3.10.18) For

pn+1 (x0 )2 + Kn (x0 , x0 ) = Kn+1 (x0 , x0 ) so |pn+1 (x0 )2 | Kn (x0 , x0 ) →0⇒ →1 Kn+1 (x0 , x0 ) Kn+1 (x0 , x0 )

(3.10.19)


⇒

|pn+1 (x0 )|2 →0 Kn (x0 , x0 )

235

(by (3.10.19) again)

Now we turn to the theorem. If (3.10.2)R holds, then since supn |an+1 | < ∞, (3.10.18) and (3.10.17) imply (x − x0 )2 dξn → 0, which implies the weak convergence. Conversely, the weak convergence plus (3.10.16), (3.10.17), and (3.10.15) imply (3.10.2). Proof of Theorem 3.10.3 given Theorems 3.10.1 and 3.10.2. Since dµ = g −1 (g dµ), there is a symmetry in hypothesis and it suffices to prove that λn (x0 , g dρ) ≤ g(x0 ) (3.10.20) lim sup λn (x0 , dρ) Let ℓ be the degree of R0 . By (3.10.2), lim

n→∞

λn (x0 , dρ) =1 λn−ℓ (x0 , dρ)

(3.10.21)

λn (x0 , g dρ) ≤ g(x0 ) λn−ℓ (x0 , dρ)

(3.10.22)

so it suffices to prove that lim sup

Let Qn (x, x0 ) be the minimizer of dµ and take as g dµ trial function Qn−ℓ (x, x0 )R0 (x)/R0 (x0 ) which is 1 at x = x0 . Thus, 2 Z λn (x0 , g dρ) 1 R0 (x) ≤ g(x) [Qn−ℓ (x, x0 )]2 dρ(x) λn−ℓ (x0 , dρ) λn−ℓ (x0 , dρ) R0 (x0 ) (3.10.23) 2 Since g(x)R0 (x) is continuous, Theorem 3.10.2 implies (3.10.22). Finally, we turn to the proof of Theorem 3.10.1. We begin by stating a general inequality which is a uniform form of (3.10.2) for the free case and whose proof we defer: Proposition 3.10.6 (Nevai–Totik–Zhang [318]). For any (r, ρ) ∈ [0, ∞) × [0, ∞), θ1 , θ2 , α1 , α2 , and L = 1, 2, . . . , we have |ρei((L−1)θ1 +α1 ) − rei((L−1)θ2 +α2 ) |2 ≤

12 L

L−1 X j=0

|ρei(jθ1 +α1 ) − rei(jθ2 +α2 ) |2 (3.10.24)

Proof of Theorem 3.10.1. Let {uj }∞ j=0 solve uj+1 + uj−1 = λuj

j = 1, 2, . . .

(3.10.25)

236


for some λ ∈ [−2, 2]. Then for any k = 0, 1, . . . , 2

|uk+L−1| ≤

12 L

L−1 X j=0

|uk+j |2

(3.10.26)

To see this, note that by continuity and the fact that the constant 12/L is λ-independent, it suffices to prove this for λ ∈ (−2, 2). In that case, u has the form uj+k = aeikj + be−ikj (3.10.27) for some a, b ∈ C and 2 cos k = λ. Thus, (3.10.26) is just (3.10.24). Given ε, pick L0 so that 12 ε < (3.10.28) L0 4 For this fixed L, let T0 (j, k; λ) be the transfer matrix for (3.10.25) and T (j, k; λ; {am, bm }) for the transfer matrix of some Jacobi matrix in the Nevai class. Using the fact that T0 and L0 are fixed, we see that for any ε1 , there exists δ so that for any k, sup

sup

λ∈[−2,2] k≤m≤k+L0 −1

kT (k, m; λ; {a, b}) − T0 (k.m; λ)k < ε1

(3.10.29)

(|ak − 1| + |bk |) < δ

(3.10.30)

if sup k≤m≤k+L0 −1

and so, for some δ, (3.10.30) implies 2

|uk+L−1| ≤ ε

L−1 X j=0

|uk+j |2

(3.10.31)

for any solution of (3.2.6). Because we are in Nevai class, there is L1 so (3.10.30) holds for k > L1 . Thus, for q > L0 + L1 , q q X X 2 2 |uq | ≤ ε |uj | ≤ ε |uj |2 (3.10.32) j=q−L+1

j=0

This proves (3.10.2).

We need the following in the proof of Proposition 3.10.6: Lemma 3.10.7. For all r ∈ [0, 1], γ ∈ [0, π], and all β ∈ [ γ2 , π], we have |1 − reiβ |2 ≥ 41 |1 − reiγ |2 (3.10.33)

Remark. The worst case is r = 1, β = γ2 , and γ → 0, in which case (3.10.33) approaches equality.


237

Proof. Since |1 − reiβ |2 = 1 + r 2 − 2r cos β

(3.10.34)

is decreasing as β ∈ [0, π] decreases, we need only consider the case β = γ2 . For a > b, both in (−1, 1) and r ∈ [0, 1], d 1 + r 2 − 2ar 0)

But |1 − reiα |2 ≤ 1 + 2r + r 2 ≤ 2(1 + r 2 ) =

8 3 [ 3L 4

L(1 + r 2 )]

so (3.10.37) holds (since 83 < 12). On the other hand, if (3.10.40) holds, since L|θ| ≤ 2π, the points ei(jθ+α) are equally spaced and only fill part of the circle. If there are k points starting at j = 0 with |jθ + α| ≥ α2 , then at most 2k further points can have |jθ +α| ≤ α2 since (− α2 , α2 ) is only twice as big as ( α2 , α), that is, at least 31 points have |jθ + α| ≥ α2 . For such points, by the lemma, |1 − rei(jθ+α)| ≥ 14 |1 − reiα|2 . So, again, (3.10.37) L ). holds (since L3 41 = 12

x-ref?

Remarks and Historical Notes. For further discussion of the CD kernel, including an operator theoretic proof of the CD formula, see Simon [398]. Theorems 3.10.1–3.10.3 are from Nevai’s AMS Memoir [314] whose proofs we follow for the implications of one theorem to another. His proof of Theorem 3.10.1 is different. Our proof of Theorem 3.10.1 follows Nevai–Totik–Zhang [318] who prove (3.10.24) with a larger constant than 12 (but for | · |p not just | · |2). Theorem 3.10.1 is proven on e for asymptotically zero perturbations of a periodic Jacobi matrix with essential spectrum e by Lubinsky– Nevai [283], for all of e by Zhang [452], and for more general situations by Breuer–Last–Simon [59]. In particular, [59] has a different approach to uniform estimates motivated by Sczwarz [TK] that is illuminating. [59] provides an example of a regular measure on [−2, 2] (see Section 5.9 for the definition of regular) where Theorem 3.10.1 fails for many x’s in [−2, 2]. They also extend the theorems of this section beyond [−2, 2].

3.11. ASYMPTOTICS OF THE CD KERNEL: OPRL ON [−2, 2]

239

3.11. Asymptotics of the CD Kernel: OPRL on [−2, 2] In Sections 2.15–2.17, we studied asymptotics of the CD kernel for OPUC regular on all of ∂D with additional conditions on the weight. In this section, we will carry these over to OPRL on [−2, 2] (and in Section 5.11 to more general OPRL). Most arguments will either be a straightforward analog or the use of the Szeg˝o map of Section 1.9 to directly relate the CD kernel for OPUC to the CD kernel for OPRL. There are, however, three interesting twists: (a) When supp(dµ) was ∂D, there was no place outside to put point masses, but now the natural hypothesis is σess (dµ) = [−2, 2] and there can be pure points outside. This will require an extension, albeit a simple one, in the Nevai comparison theorem (Theorem 2.16.6). (b) For OPUC, the natural limit for the density of zeros was dθ/2π. It is not so obvious what the analog is for [−2, 2]. It is, in fact, the measure on [−2, 2]: dρ[−2,2] (x) =

1 dx √ π 4 − x2

(3.11.1)

The right way to understand this is potential theoretic, and we will defer this part of the story to Section 5.9. (c) When the Szeg˝o map is used, the CD kernel for OPUC will be related to two measures on [−1, 1]: dµ and 41 (4 − x2 ) dµ. Theorem 3.10.3 will overcome this difficulty. We begin with an analog of Theorem 2.15.1: Theorem 3.11.1. Let dµ be a measure of compact support on R. Let Kn (x, y) be its CD kernel and define dµ(N ) =

1 KN (x, x) dµ(x) N +1

(3.11.2)

and let dνn be the zero counting measure for Pn (x, dµ). Then for ℓ = 0, 1, 2, . . . , Z Z 1 1 ℓ (N ) ℓ ≤ 2ℓ x dµ (x) − x dν (x) (3.11.3) N +1 N + 1 N +1 N +1 In particular, for any subsequence N(1) < N(2) < · · · , dν∞ is a weak limit of dνN (j)+1 if and only if it is a weak limit of dµN (j) .

Proof. By Theorem 1.2.6, the zeros of PN +1 are identical to the eigenvalues of πJπ ↾ Ran π, so the proof is identical to the proof of Theorem 2.15.1.

240


Corollary 3.11.2. Let µ1 , µ2 be two (not necessarily normalized) measures of compact support on R. Suppose (1) µ1 ≥ µ2 (3.11.4) (2) For some open interval I = (a, b), µ1 ↾ (a, b) = µ2 ↾ (a, b)

(3.11.5) (j)

(3) For some subsequence N(1), N(2), . . . , and density of zeros dνn (j = 1, 2) of µj , we have (j)

Then

w

(j) dνN (k) −→ dν∞

(3.11.6)

(2) (1) dν∞ ↾ I ≥ dν∞ ↾I

(3.11.7)

Remark. An example is µ2 = µ1 ↾ (a, b). Proof. (3.11.4) ⇒ λn (x, µ2 ) ≤ λn (x, µ1 )

⇒ Kn (x, x; µ2 ) ≥ Kn (x, x; µ1 )

⇒ Kn (x, x; µ2 ) dµ2 ↾ I ≥ Kn (x, x; µ1 ) dµ1 ↾ I

⇒ (3.11.7)

by Theorem 3.11.1.

We next turn to what typical limits of dνn are. Example 3.11.3. Let dµ1 , dµ2 be given by dµ1 (x) = dρ[−2,2] (x)

(3.11.8)

with dρ[−2,2] given by (3.11.1) and dµ2(x) = 2 14 (4 − x2 ) dµ1 2√ = 4 − x2 dx π In terms of the change of variable (θ ∈ [0, π]), √

x = 2 cos θ

(3.11.9)

(3.11.10)

4 − x2 dθ, we see dθ dθ dµ1 = dµ2 = 2 sin2 θ (3.11.11) π π Thus, the normalized OPRL (essentially Chebyshev polynomials of the first and second kind) are given by √ pn (2 cos θ, dµ1) = 2 cos(nθ) (n ≥ 1) (3.11.12)

so dx = 2 sin θ dθ =


pn (2 cos θ, dµ2) =

sin((n + 1)θ) sin θ

(n ≥ 0)

241

(3.11.13)

Thus, n−1

1X dνn (x, dµ1) = δ j+1/2 n j=0 θ, n π

(3.11.14)

dνn (x, dµ2) =

(3.11.15)

n−1

In both cases, dνn →

1X δ j+1 n j=0 θ, n+1 π

dθ = dρ[−2,2] (x) 2π

(3.11.16)

Definition. A measure dµ on R is called regular for [−2, 2] if and only if σess (µ) = [−2, 2] and lim (a1 . . . an )1/n = 1

n→∞

(3.11.17)

Remark. By (3.11.12)/(3.11.13), the dµ’s of Example 3.11.3 have √ bn (dµ1 ) = 0, a1 (dµ1 ) = 2, an (dµ1 ) = 1 (n ≥ 2) and bn (dµ2) = 0, an (dµ2 ) = 1. Thus, they are regular. We will prove a generalization of the following as Theorem 5.9.2: Theorem 3.11.4. Let dµ be regular for [−2, 2]. Then (i) dνn → dρ[−2,2]

(ii) For any ε > 0, there is a δ so lim sup sup |pn (x, dµ)|1/n ≤ eε n→∞

(3.11.18) (3.11.19)

dist(x,[−2,2]) 0, dµs = dµ♯s on (x0 − δ, x0 + δ). (ii) For all ε sufficiently small, there is αε > 1, so for |x − x0 | < ε, we have αε−1 w(x) ≤ w ♯ (x) ≤ αε w(x) (3.11.36) (iii) For αε → 1 and any xn ∈ (−2, 2) with xn → x0 and every ℓ(n) with n/2 < ℓ(n) < 2n, we have that lim

n→∞

1 Kn (xℓ(n) , xℓ(n) ) = B 6= 0 n+1

Then

(3.11.37)

1 Kn♯ (xn , xn ) = B (3.11.38) n→∞ n + 1 Moreover, this is uniform in xn in the sense that if (with the same B) for all xn → x0 , there are, for any ε, a δ and an N0 so if n > N0 and |xn − x0 | < δ, then B − 1 K ♯ (xn , xn ) < ε (3.11.39) n n+1 lim

244


This is also uniform in x0 . If w and w ♯ are continuous and nonvanishing in a closed interval in (−2, 2) and we have dµs = dµ♯s in a neighborhood of I and (3.11.36) is replaced by αε−1

w(x) w ♯ (x) w(x) ≤ ♯ ≤ αε w(x0 ) w (x0 ) w(x0 )

(3.11.40)

for |x − x0 | < ε (αε independent of x0 ) and if (3.11.36) holds uniformly in x0 ∈ I where B(x0 ) is x0 -dependent, then (3.11.38) with B replaced by B(x0 )w(x0 )/w ♯(x0 ). Proof. The proof is the same as the proof of Theorem 2.16.6 with one extra step. Because we only have that σess is [−2, 2], there can be pure points for µ♯ where the regularity does not imply the polynomials pn (x, dµ) are bounded in n by eεn , so the choice Qn in (2.16.38) may not be small at those points. However, for each δ, there are only finitely δ many pure points {xj }N of µ♯ with dist(xj , [−2, 2]) > δ. Adding a QNj=1 δ multiplicative factor j=1 (x−xj )/(xn −xj ) to Qn (adjusting n(ε) to be n − m(ε) − Nδ ) kills this finite number of points. With this adjustment, the proof extends with no other change. We then have: Theorem 3.11.6 (Lubinsky [282]). Let dµ be a regular probability measure on [−2, 2] of the form dµ = w(x) dx + dµs

(3.11.41)

Suppose that, for any interval [α, β] ⊂ (−2, 2), (a) supp(dµs ) ∩ I = ∅ (b) w is “continuous” on I and nonvanishing there. Then, with ρ[−2,2] given by (3.11.20), we have (1) (Diagonal Asymptotics) For any A < ∞, uniformly in x∞ ∈ I, and sequence xn ∈ [−2, 2] with n|xn − x∞ | ≤ A for all n, we have ρ[−2,2] (x∞ ) 1 Kn (xn , xn ) → n+1 w(x∞ )

(3.11.42)

(2) (Lubinsky Universality) For any A < ∞, uniformly in x∞ ∈ I and a, b ∈ R with |a|, |b| ≤ A, we have Kn (x∞ + na , x∞ + nb ) sin(πρ[−2,2] (x∞ )(b − a)) → Kn (x∞ , x∞ ) πρ[−2,2] (x∞ )(b − a)

(3.11.43)

More generally, the limit of Kn (xn , yn )/Kn (x∞ , x∞ ) is the right side of (3.11.43) so long as |xn − x∞ | ≤ A/n, |yn − x∞ | ≤ A/n, and n(xn − yn ) → b − a.


1.

245

Remark. If b − a = 0, the right side of (3.11.43) is interpreted as

Proof. Given the improved version of the Nevai comparison theorem and the model, dµ1 , of Example 3.11.3, the proof is identical to that of Theorem 2.16.1. Theorem 3.11.7 (Máté–Nevai Upper Bound). For any measure dµ with σess (dµ) ⊂ [−2, 2] and any Lebesgue point x0 of dµ in (−2, 2), we have w(x0 ) lim sup(n + 1)λn (xn ) ≤ (3.11.44) ρ[−2,2] (x0 ) for any sequence xn ∈ (−2, 2) with supn n|xn − x0 | < ∞. Remarks. 1. Theorem 3.10.2 says that under great generality |Qn (x, x0 )|2 dµ(x)/λn (x) converge weakly as a measure to a point mass at x0 , that is, smeared with continuous functions. In essence, this proof relies on the fact that for a very nice dµ (namely, dµ1), the convergence is in a much stronger sense. 2. We emphasize that in this result w(x0 ) can be 0, in which case lim(n + 1)λn (xn ) = 0. Proof. Suppose first that σ(dµ) = [−2, 2]. Define (1) 2 Kn (x, y) −1 Fn (x, y) = ρ[−2,2] (y)[λ(1) (3.11.45) n (y)] (1) Kn (y, y) the objects associated to the dµ1 measure of Example 3.11.3. We will show that Z lim Fn (x, xn ) dµ(x) = w(x0 ) (3.11.46) n→∞

this implies (3.11.44) for this σ(dµ) = [−2, 2] case since (1) (1) Kn (x, xn )/Kn (xn , xn ) can be used as a trial polynomial in (3.10.3) showing that Z (1) −1 (n + 1)λn (xn , dµ) ≤ (n + 1)λn (xn )ρ[−2,2] (xn ) Fn (x, xn ) dµ(x) (3.11.47)

(1) 1)λn (xn )

and (n + → 1 by (3.11.24). To prove (3.11.44), we pick A (eventually, very large) and write the integral as a sum of three terms. First, the integral over |x − xn | ≥ A/n; second, what we get by taking |x − xn | < A/n (1) (1) and replacing Kn (x, xn )/Kn (xn , xn ) in F by sin(πρ[−2,2] (x0 )n(x − xn ))/πρ[−2,2] (x0 )n(x − xn ); and third, the difference between the true F and this approximate F .

246


Because of the uniform convergence in (3.11.34), the third term is bounded by A A Cnµ xn − , xn + o(1) (3.11.48) n n nµ((xn − An , xn + An )) is bounded since x0 is a Lebesgue point, so this term goes to zero for each fixed A. By the CD formula and the boundedness of pn (x; dµ1 ) on [−2, 2], |Fn (x, y)| ≤

C n|x − y|2

for y in a compact subset of (−2, 2), and thus the first term is bounded by Z dx −1 Cn = CA−1 (3.11.49) 2 |x−y|≥A/n |x − y| can be made small by taking A large. Thus, the main contribution is the second term, which we control with Lemma 2.17.8, as in the proof of Theorem 2.17.6. This completes the proof of (3.11.46) and so of (3.11.44) when σ(dµ) = [−2, 2]. This proves the result when σ(dµ) = [−2, 2]. If now σ(dµ) = [−2, 2] ∪ F with F a finite set {xj }m j=1Qoutside [−2, 2], we can set µ ˜ = µ ↾ [−2, 2] and use Pn,m (x, x0 ; d˜ µ( m j=1 (x − xj )/(x0 − xn ) as a trial function for λn (x0 , dµ) to get (3.11.44). Finally, if σess ([−2, 2]), then for any ε, σ(dµ) = [−2 − ε, 2 + ε] ∪ Fε with Fε finite. So by the above, lim sup(n + 1)λn (x0 ) ≤

w(x0 ) ρ[−2−ε,2+ε](x0 )

so taking ε ↓ 0, we obtain (3.11.44).

(3.11.50)

Theorem 3.11.8 (Simon [400]). If I = (α, β) ⊂ [−2, 2] is an open interval, if µ is regular for [−2, 2] and w(x) > 0 for a.e. x ∈ I, then Z 1 (i) (3.11.51) n + 1 Kn (x, x)w(x) − ρ[−2,2] (x0 ) dx → 0 I Z 1 (ii) Kn (x, x) dµs (x) → 0 (3.11.52) I n+1 Proof. Given Theorems 3.11.4 and 3.11.7, the proof is the same as for Theorem 2.17.7.

Theorem 3.11.9 (MNT Theorem [296]). Let µ be a regular measure for [−2, 2] which is locally Szeg˝ o on I, an open interval in [−2, 2]. Let x∞ ∈ I be a point with w(x∞ ) 6= 0 and which is a Lebesgue point for


247

both w and for the local Szeg˝ o function. Let xn ∈ (−2, 2) be a sequence with sup n|xn − x∞ | ≡ A < ∞ (3.11.53) n

Then (3.11.42) holds. The limit is uniform in all xn obeying (3.11.53). Remark. By the local Szeg˝o condition, we mean if I = (α, β) that for any ε > 0, Z β−ε

α+ε

log(w(x)) dx > −∞

(3.11.54)

Given any such w and any x∞ , we can find w˜ equal to w near x∞ with w˜ dx = d˜ µ, the image under the Szeg˝o map of a measure obeying the Szeg˝o condition on ∂D. By the local Szeg˝o function, we mean the pull back to [−2, 2] of the Szeg˝o function for this measure on ∂D. Proof. First we use the Nevai comparison theorem (Theorem 3.11.5) to reduce to a case where dµ is supported on [−2, 2] and obeys a global Szeg˝o condition. Let d˜ µ be on ∂D so that dµ = Sz(d˜ µ) ∗ and let dµ = Sz1 (d˜ µ). (Sz and Sz1 are the Szeg˝o mappings defined in Section 1.9.) By (1.9.27),

µ) = Kn (xn , xn ; dµ) + sin2 θn Kn−1 (xn , xn ; dµ♯ ) K2n (eiθn , eiθn ; d˜ (3.11.55) where xn = 2 cos(θn ) (3.11.56) 1 By a slight extension of Theorem 3.10.3 with g(x) = 4 (4 − x2 ) and the relation (1.9.14) of dµ and dµ♯ , Kn (xn xn ; dµ) →1 (3.11.57) sin θn2 Kn−1 (xn , xn ; dµ♯) Thus, (3.11.42) follows from (3.11.55) and (2.17.7). Theorem 3.11.10 (Findley’s Theorem [128]). Under the hypotheses of Theorem 3.11.9, we have (3.11.43) for each A < ∞, uniformly in a, b with |a|, |b| < A. More generally, the limit relation holds for Kn (xn , yn )/Kn (x∞ , x∞ ) for any xn , yn with |xn − x∞ | ≤ A/n, |yn − x∞ | ≤ A/n, and n(xn − yn ) → a − b.

Proof. One has a Lubinsky inequality in the real case by the same proof as for ∂D. This inequality plus the MNT theorem implies the off-diagonal result. Definition. We say the zeros of pn (x) has clock behavior at x0 with density ρ(x) if for all j, (xj+1 (x0 ) − xj (x0 ))/(2π/ρ(x0 )) → 1. If the limit is uniform for x0 ∈ I, we say there is uniform clock behavior in I.

248

x-ref?


As in the OPUC case (see Theorem TK), Lubinsky universality immediately proves: Theorem 3.11.11. Under the hypotheses of Theorem 3.11.6, one has clock behavior with density, ρ(x0 ), uniformly on I. Under the hypotheses of Theorem 3.11.9, one has uniform clock behavior at x0 .

x-ref?

Remarks and Historical Notes. TK 3.12. Asymptotics of the CD Kernel: Lubinsky’s Second Approach Our previous discussion of the slightly off-diagonal CD kernel has depended on Lubinsky’s inequality and a comparison measure. Remarkably, having revolutionized this subject with his elegant inequality, Lubinsky [281] presented an entirely different approach to universality that does not require a comparison model and that illuminates why the kernel sin(πx)/πx occurs. Here we will discuss this approach as extended by Avila–Last–Simon [30]. The main theorem is the following: Theorem 3.12.1 ([30]). Let dµ(x) = w(x) dx + dµs (x)

(3.12.1)

be a nontrivial probability measure of compact support in R. Let Σ ⊂ R be a set of positive Lebesgue measure with w(x) (a) Kn (x, x) → ρ∞ (x) ∈ (0, ∞) for a.e. x ∈ Σ (3.12.2) n+1 n 1 X (b) sup |qj (x)|2 < ∞ for a.e. x ∈ Σ (3.12.3) n n+1 j=0

Then for a.e. x0 ∈ Σ and all z, w ∈ C, we have w z Kn (x0 + n+1 , x0 + n+1 ) sin(πρ∞ (x)(z − w)) lim = (3.12.4) n→∞ Kn (x0 , x0 ) πρ∞ (x)(z − w) uniformly for z, w with |z| < A, |w| < A for any A. In particular, one has clock behavior of the zeros with density ρ∞ (x) for a.e. x ∈ Σ. Remark. (3.12.2) is a definition of ρ∞ , that is, the assertion is the existence and positivity of the limit. Thus, control of the limit in (3.12.2) and the bound of (3.12.3) imply universality, and Totik’s results on the diagonal kernel imply results for the off-diagonal. In the Notes, we will make this more precise. The sin kernel enters via the following elegant result in complex function theory:

3.12. LUBINSKY’S SECOND APPROACH

249

Theorem 3.12.2 ([281]). Let f (z) be an entire function that obeys (i) f (0) = 1 |f (x)| ≤ 1 for x ∈ R (3.12.5) (ii) Z ∞

(iii) For some C and A,

−∞

|f (x)|2 dx ≤ 1

(3.12.6)

|f (z)| ≤ CeA|z|

(3.12.7)

. . . < x−2 < x−1 < 0 < x1 < x2 < . . .

(3.12.8)

(iv) f is real on R, all the zeros of f are real, and if are the zeros, then Then

|xn | ≥ (|n| − 1)

(3.12.9)

sin(πz) (3.12.10) πz Proof. ([30]) We will prove shortly that for any ε > 0, there is Cε with |f (x + iy)| ≤ Cε e(π+ε)|y| (3.12.11) Assuming this for a moment, let us prove (3.12.10). Let fb(k) be the Fourier transform of f : Z ∞ −1/2 b f (k) = (2π) e−ikx f (x) dx (3.12.12) f (z) =

−∞

so

f (x) = (2π)

−1/2

Z

∞

−∞

b dk eikx f(k)

(3.12.13)

(where the integrals are shorthand for distributional Fourier transform). By the Paley–Wiener theorem (see the Notes), (3.12.11) implies fb(k) = 0

k 6= [−π, π]

By (3.12.6) and the Plancherel theorem, Z ∞ |fb(k)|2 dk ≤ 1

(3.12.14)

(3.12.15)

−∞

and, by (3.12.5) and (3.12.13), Z ∞ fb(k) dk = (2π)1/2 −∞

(3.12.16)

250


Therefore, Z ∞

so

−∞

b − (2π)−1/2 χ[−π,π] (k)|2 ≤ 1 + 1 − 2 = 0 |f(k) fb = (2π)−1/2 χ[−π,π]

(3.12.17)

(3.12.18)

and (3.12.10) follows from (3.12.13). Thus, we are reduced to proving (3.12.11), to which we now turn. By (3.12.7), (3.12.5) and the Hadamard factorization theorem (see the Notes), for some B real, ∞ Y z z/zj z Bz z/z−j f (z) = e 1− e 1− e (3.12.19) zj z−j j=1 from which we see for y real that ∞ Y y2 y2 2 |f (iy)| ≤ 1+ 2 1+ 2 z z−j j j=1

(3.12.20)

so, by (3.12.9), Y 2 ∞ y2 |f (iy)| ≤ (1 + Cy ) 1+ 2 n n=1 2 2 2 sinh(πy) = (1 + Cy ) πy 2

2 2

(3.12.21) (3.12.22)

by the Euler product formula (see the Notes). This implies (3.12.11) for x = 0. (3.12.6) implies (3.12.11) for y = 0, so we have (3.12.11) on the axes. Since we have (3.12.7), we can apply the Phragmén–Lindelöf principle (see the Notes) in each quadrant to get (3.12.11) for all x and y. Lubinsky used this theorem to prove the following precursor of Theorem 3.12.1: Theorem 3.12.3 ([281]). Suppose dµ has the form (3.12.1) and x0 is a Lebesgue point for dµ in the sense that Z 1 1 lim |w(x)−w(x0 )| dx → 0 µs (x0 −ε, x0 +ε) → 0 (3.12.23) ε↓0 2ε 2ε and suppose that


251

(a) For some A and C and all R < ∞, there exists N so that for n ≥ N and all z complex with |z| < R, Kn x0 + z¯ , x0 + z ≤ CeA|z| (3.12.24) n n

(b)

lim inf

1 Kn (x0 , x0 ) > 0 and w0 (x) > 0 n

(3.12.25)

(c) For all B < ∞, Kn (x0 + na , x0 + na ) lim =1 n→∞ Kn (x0 , x0 )

(3.12.26)

uniformly for real a with |a| ≤ B. Then lim

n→∞

Kn (x0 +

z , x0 nρn

+

Kn (x0 , x0 )

w ) nρn

=

sin(π(z − w)) π(z − w)

(3.12.27)

where ρn =

w(x0 ) Kn (x0 , x0 ) n

(3.12.28)

Remarks. 1. (3.12.23) holds for a.e. x with w(x) > 0 by standard harmonic analysis [363]. 2. Lubinsky does not write (3.12.24) as a hypothesis, but instead demands w(x) ≥ ε > 0 near x and then deduces (3.12.24).

3. The key is thus the hypothesis (3.12.26), which we call the Lubinsky wiggle condition. It is clearly also a key piece of Lubinsky’s other argument, but here it is the only requirement (when w is bounded strictly away from zero). Alas, Lubinsky could only prove that (3.12.26) holds in cases where his first argument also works (but see the Notes). The proof of Theorem 3.12.3 depends on a critical classical inequality (see the Notes): (n)

Proposition 3.12.4 (Markov–Stieltjes Inequality). Let xj (x0 ) be defined by requiring (n)

(n)

pn−1 (x0 )pn (xj (x0 )) − pn (x0 )pn−1 (xj (x0 )) = 0

(3.12.29)

252

3. THE KILLIP–SIMON THEOREM (n)

(n)

where xj (x0 ) < xj+1 (x0 ) and j = 1, . . . , n if pn−1 (x0 ) 6= 0 and j = 1, . . . , n if pn−1 (x0 ) 6= 0. Then X 1 ≥ µ((−∞, x0 ]) (n) (n) K (x (x ), x (x )) n−1 j 0 0 (n) j {j|xj (x0 )≤x0 }

≥ µ((−∞, x0 )) ≥

X

1

(n) (n) Kn−1 (xj (x0 ), xj (x0 )) (n) {j|xj (x0 )<x0 }

(3.12.30)

(n)

Remark. If pn (x0 ) = 0 (resp. pn−1 (x0 ) = 0), xj (x0 ) list all the zeros of pn (x) (resp. pn−1 (x)). Sketch of the Proof of Theorem 3.12.3. (See [30] for details.) Fix a real and let fn (z) be defined by fn (z) =

Kn (x0 +

a , x0 nρn

+

Kn (x0 , x0 )

a+z ) nρn

(3.12.31)

where ρn is given by (3.12.28). By hypothesis (a) and Montel’s theorem (see the Notes), {fn } is compact in the topology of uniform convergence on compact subsets of C, so we need only show that each limit point is sin(πz)/πz to conclude that the limit is sin(πz)/πz. By analyticity and the same compactness plus the Schwarz inequality for Kn (x, y) (i.e., |Kn (x, y)|2 ≤ Kn (¯ x; x)Kn (¯ y ; y)), we then get (3.12.27) for all complex z, w. So let f be a limit point of fn . We will show f obeys all the hypotheses of Theorem 3.12.2. By the Lubinsky wiggle condition and Schwarz inequality, (3.12.4) holds. By the reproducing kernel property of Kn and a change of variables and definition of ρn , Z Kn (x0 + nρan , x0 + nρan ) 2 w(x) |fn (x)| dx ≤ (3.12.32) w(x0 ) Kn (x0 , x0 ) Using that x0 is a Lebesgue point and the wiggle condition, we get (3.12.6). From (3.12.24), we get (3.12.7). Finally, the Markov–Stieltjes inequalities, the wiggle condition, and the Lebesgue point condition imply (3.12.9). Sketch of the Proof of Theorem 3.12.1. (See [30] for details.) We will establish the hypotheses of Theorem 3.12.3. (3.12.25) is immediate from (3.12.2).


253

(3.12.2), (3.12.4), and sup an < ∞, inf an > 0 (by Σac 6= ∅) imply that n+1 1 X sup kTj (x0 )k2 (3.12.33) n n+1 j=0

By Theorem 3.8.14, we conclude that

z

Tj x0 +

≤ inf (|an |)−1 kTj (x0 )k

n n+1 X j |z| exp kTk (x0 )k kTk−1(x0 )k j + 1 k=1 (3.12.34) since kTk−1 k = kTk k. By the Schwarz inequality and (3.12.33), we conclude that 2 n

1 X

Tj x0 + z

≤ C1 exp(C2 |z|) (3.12.35)

n+1 n+1 j=0

which implies (3.12.24). That leaves (3.12.26). By Egoroff’s theorem for any ε, we can find Σε ⊂ Σ, so |Σ \ Σε | < ε, and (3.12.2) holds uniformly on Σε . We will show (3.12.26) a.e. on each such Σε and thus, a.e. on Σ. Let x0 ∈ Σε be a point of density of Σε , that is, lim (2δ)−1 |(x0 − δ, x0 + δ) ∩ Σε | → 1

δ→0

(3.12.36)

and let gn (a) = LHS of (3.12.26) By the uniform convergence of fn on Σε and the implied continuity of limit, for every A < ∞, sup |gn (b) − 1| → 0

(3.12.37)

(3.12.38)

|b|≤A x0 + nb ∈Σε

By (3.12.36), sup |a|≤A

inf

|bn |≤A x0 + bnn ∈Σε

|a − b|

→0

By (3.12.24), for every A, sup |gn′ (z)| < ∞

n,|z|≤A

(3.12.39)

254


By (3.12.37), (3.12.38), and (3.12.39), sup |gn (a) − 1| → 0

|a|≤A

as n → ∞. It follows that we have proven (3.12.38) for a.e. x0 ∈ Σε . Remarks and Historical Notes. Lubinsky [281] had the wonderful idea of using Markov–Stieltjes inequalities and complex variable characterizations of the sinc (i.e., sin x/x) kernel. He used special properties of the sinc kernel (see [411]); translating these properties into a direct proof using the Paley–Wiener theorem is from [30]. Lubinsky did not directly state (3.12.24) as a hypothesis. Rather, he assumed w(x) > δ in an interval and deduced (3.12.24) using the Christoffel variational principle. He was unable to prove the Lubinsky wiggle condition except in situations where Totik and Simon had already shown how to get universality using Lubinsky’s first method. But he opened the portals to Avila–Last–Simon to handle ergodic Jacobi matrices. Let Ω be a compact metric space with probability measures, dη, and T : Ω → Ω ergodic. If A, B : Ω → R are continuous with inf ω∈Ω A(ω) > 0, one defines ergodic Jacobi matrices to be the ω-dependent matrix with Jacobi parameters {an , bn }∞ n=1 given by an (ω) = A(T n ω)

bn (ω) = B(T n ω)

(3.12.40)

[30] were able to prove for a.e. ω and a.e. x0 in the a.c. spectrum, one has universality and clock behavior with ρ∞ (x) given by the a.c. part of the density of zeros. The canonical example is the almost Mathieu equation (see Jitomirskaya [204]) where Ω = ∂D, A(ω) ≡ 1, and B(ω) = 2λ cos(παθ). If |λ| < 1 and α is irrational, the spectrum is a Cantor set with purely a.c. spectrum. For the Paley–Wiener theorem, see [354], and for the Hadamard factorization theorem, Phragmén–Lindelöf principle, Euler product formula, and Montel’s theorem, see Titchmarsh [429]. The Markov–Stieltjes inequalities are due to Markov [290] and Stieltjes [412] who consider the case where pn (x0 ) = 0. The general form is due to Freud [137]; for a proof, see this book or [398].

CHAPTER 4

Sum Rules and Consequences for Matrix Orthogonal Polynomials In this chapter, we will discuss matrix-valued orthogonal polynomials on the real line (aka MOPRL). These are based on a measure, dµ, which, instead of assigning a nonnegative number to any set, assigns a nonnegative ℓ × ℓ matrix. From the Jacobi matrix point of view, the Jacobi parameters become ℓ × ℓ matrices. 4.1. Introduction MOPRL is a strange subject. Most parts are straightforward extensions of the OPRL theory, but every so often, a subtlety arises. Fortunately, in our case of sum rules, the only subtlety concerns a possible coincidence of eigenvalues of J0 and J1 . There is another place in our considerations where a subtlety arises that we will come to shortly. The result is that the MOPRL theory is so close to the OPRL theory that much the last three sections of this chapter where we turn to sum rules will say: “Now just follow the proof from Chapter 3.” The only reason something so similar to OPRL occurs in these notes is because, remarkably, as we will see in Chapter 8, we can study perturbations of scalar periodic Jacobi matrices by relating it to a perturbation of an MOPRL with constant coefficients (indeed, An ≡ 1, Bn ≡ 0). It is for this reason that we consider MOPRL here and not MOPUC, the unit circle analog. It will turn out (see TK) that even perturbations of x-ref? periodic CMV matrices relate to MOPRL, not MOPUC. What we will be missing from our discussion is an analog of full Szeg˝o asymptotics of the matrix orthogonal polynomials—here there is a subtlety that has not yet been overcome. We will discuss this in the Notes. x-ref? Section 4.2 discusses the basic MOPRL formalism—the one surprise is that there are actually two natural families of OPs. Section 4.3 discusses coefficient stripping and contains in Theorem 4.3.3 what is not a straightforward copying of what we did in Chapter 3. Section 4.4 then proves matrix nonlocal sum rules while Sections 4.5 and 4.6 present the 255

256

4. SUM RULES AND CONSEQUENCES FOR MATRIX OPS

by now standard applications to MOPRL analogs of the Shohat–Nevai and Killip–Simon theorems. Remarks and Historical Notes. The theory of matrix orthogonal polynomials on the real line goes back to seminal papers by Krein [245] and Berezans’ki [40]. A lot of the rather large literature and survey of many of the analytic results can be found in the review article of Damanik–Pushnitski–Simon [94]. From the point of view of sum rules, the two most significant later papers are Aptekarev–Nikishin [23] and Damanik–Killip–Simon [93]. One place where there is still a lack of complete understanding is in the aymptotics of polynomials when a Szeg˝o condition holds (i.e., the subject we studied in the scalar case in Sections 2.9, 2.13, and 3.7). For MOPRL, these asymptotics were studied by Delsarte–Genin– Kamp [102]. Indeed, they developed there the approach we discuss in Section 2.13. Aptekarev–Nikishin [23] were able to handle MOPRL with no bound states by a Szeg˝o mapping and with finitely many bound states using finite coefficient stripping. The result with only a Blaschke-type condition remains open. The problem is that there is no known factorization result for matrix Herglotz functions (we will get around this by looking only at their determinants). There are factorization theorems for matrix-valued H p functions (see Potapov [346], Gohberg–Saknovich [167]), but their extension to Herglotz functions is not known. 4.2. Basics of MOPRL An ℓ × ℓ matrix-valued measure on R is the assignment of a nonnegative ℓ × ℓ matrix, µ(S), to each Borel set S ⊂ R and which is countably additive. We will usually normalize by µ(R) = 1

(4.2.1)

but in any event, suppose µ(R) is a (finite) matrix. Define a scalar measure, µt , by µt (A) = Tr(µ(A))

(4.2.2)

Then, since hϕ, Bϕi ≤ Tr(B) for B ≥ 0 and kϕk = 1, we see each measure A 7→ µ(A)ij is µt -a.c., so dµij (x) = Mij (x) dµtˆ(x)

(4.2.3)

µ positive and (4.2.2) imply that M(x) > 0

Tr(M(x)) = 1

We will postpone the definition of nontriviality of µ.

(4.2.4)

4.2. BASICS OF MOPRL

257

Throughout, we suppose µ has finite moments, that is, for all n = 0, 1, 2, . . . , Z |x|n dµt (x) < ∞ (4.2.5)

Now suppose Rf, g are ℓ × ℓ matrix-valued functions on R; we define an ℓ × ℓ matrix, f (x) dµ(x) g(x), in the obvious way, that is, Z XZ f (x) dµ(x) g(x) = f (x)ik Mkn (x)g(x)nj dµt (x) (4.2.6) ij

k,n

We define two “inner products,” that is, sesquilinear maps of the ℓ × ℓ matrix-valued functions to ℓ × ℓ matrices by (to avoid confusion with Szeg˝o dual, we use † , not ∗ , for adjoint) Z hhf, giiL = g(x) dµ(x) f (x)† (4.2.7) and hhf, giiR =

Z

f (x)† dµ(x) g(x)

(4.2.8)

initially on bounded f and g with bounded support, but eventually for suitable L2 -like spaces. The symbols L, R (for “left” and “right”) come from fact that for scalar multiplication by a matrix A, hhf, AgiiL = Ahhf, gii

hhAf, giiL = hhf, giiA†

hhf, gAiiR = hhf, giiR A

(4.2.9) (4.2.10) (4.2.11)

hhf A, giiR = A† hhf, giiR

(4.2.12)

kf kR = (Trhhf, f iiR )1/2

(4.2.13)

We will also define norms via

One has that and

kf kL = (Trhhf, f iiL )1/2

kf † kR = kf kL

hhf, gii†R = hhg, f iiR

(4.2.14) (4.2.15)

hhf, giiR = hhg † , f † iiL (4.2.16) Let P be the family of all polynomials and PR = P/{P ∈ P | kf kR = 0}, and similarly for PL . The completion of PR (resp. PL ) in k·kX we call HR (resp. HL ). If µ has bounded support, then multiplication by x is a bounded selfadjoint operator. f → f † is an antiunitary

258


map of HR to HL , leaving multiplication by x invariant. The spectrum of multiplication by x we will call σ(dµ)—it is the support in the measure theoretic sense. The following is elementary: Proposition 4.2.1. Let µ be a matrix-valued measure on R. Then the following are equivalent: (i) For every nonzero matrix-valued polynomial, f , kf kL > 0. (ii) For every nonzero matrix polynomial, f , kf kR > 0. (iii) For every n = 0, 1, 2, . . . , dim({P ∈ PL | deg(P ) ≤ n}) = ℓ2 (n + 1). (iv) For every n = 0, 1, 2, . . . , dim({P ∈ RR | deg(P ) ≤ n}) = ℓ2 (n + 1). Proof. Since dim({P ∈ P | dim(P ) ≤ n}) = ℓ2 (n + 1), we see (i) ⇔ (iii) and (ii) ⇔ (iv). By (4.2.15), (i) ⇔ (ii). Example 4.2.2. For 0 ≤ t ≤ 1, let M(t) be the orthogonal projec1 tion onto the vector −t in C2 and let dµ = χ[0,1] (t)M(t) dt

Let P (t) be the polynomial P (t) =

t t 1 1

(4.2.17)

(4.2.18)

1 ⊥ and P (t)† M(t)P (t) ≡ 0, so kP kR = 0. Thus, so Ran(P (t)) ⊂ −t dµ does not obey (i)–(iv) of the proposition. However, for any fixed 1 nonzero ϕ, hϕ, M(t)ϕi = 0 if and only if h −t , ϕi = 0, which happens at most at one t in [0, 1]. Thus, hϕ, dµ ϕi is nontrivial for any ϕ. This shows nontriviality of hϕ, dµ ϕi for all ϕ does not suffice for (i)–(iv) to hold. Definition. If (i)–(iv) of Proposition 4.2.1 hold, we say that µ is nontrivial. Henceforth we assume that µ is nontrivial. Proposition 4.2.3. A sufficient condition for µ to be nontrivial is that there is a set S with rank(M(x)) = ℓ for x ∈ S, and for any finite set, F, µt (S \ F ) 6= 0. Proof. Let P be a nonzero polynomial in P. On S, Tr(P (x)† M(x)P (x)) vanishes only at points where P (x) = 0 as a matrix—and this can only happen on the finite set where det(P (x)) = 0. Thus, by hypothesis, kP kR > 0.


259

Introduce monic MOPRL, PnR , PnL , ℓ × ℓ matrix polynomials of the form (we will use X as a generic for R or L) PnX (x) = xn + lower order in x

(4.2.19)

so that hhxj , PnX iiX = 0 for j = 0, 1, . . . , n − 1

(4.2.20)

PnX

It is easy to see these determine inductively and if µ is nontrivial that such PnX exist. Moreover, by (4.2.16), PnR (x) = PnL(¯ x )†

(4.2.21)

Indeed, we have P0X (x) = 1 and that, by (4.2.16), γn ≡ hhPnL, PnL iiL = hhPnR , PnR iiR

(4.2.22)

is nonzero if µ is nontrivial so that PnR (x)

n

=x −

n−1 X j=0

PjR (x)γj−1 hhPjR , xn iiR

(4.2.23)

To define orthonormal MOPRL, we pick ℓ × ℓ unitaries, σ0 = 1, σ1 , σ2 , . . . and τ0 = 1, τ1 , τ2 , . . . , and let R −1/2 pR σn n (x) = Pn (x)γn

pLn (x) = τn γn−1/2 PnL (x)

(4.2.24)

which obey X hhpX n , pk iiX = δnk

X n pX n (x) = κn x + lower order

κLn = τn γn−1/2

−1/2 κR σn n = γn

(4.2.25) (4.2.26) (4.2.27)

It is easy to see that if one demands p0 = 1, then pX n , obeying (4.2.25), is determined up to precisely a choice of σn , τn . In the scalar case, one picks σn ≡ 1 since it is reasonable to demand κn > 0. We will see below why one does not always demand that, but instead we associate a matrix-valued measure with an equivalence class of normalized MOPRL. Henceforth, we will always suppose that τn = σn†

(4.2.28)

so that, by (4.2.24) and (4.2.21), L pR x)† n (x) = pn (¯

(4.2.29)

Note that the pX n are an orthonormal module basis in that if f is any matrix polynomial of degree n, we have n X R f (x) = pR (4.2.30) m (x)hhpm , f iiR m=0

260


= kf k2X =

n X

m=0 n X

m=0

hhpLm , f iiL pLm (x)

(4.2.31)

† X Tr(hhpX m , f iiR hhpm , f ii)

(4.2.32)

If one completes PX in k·kX , these three formulae hold for any f in the completion if one takes n = ∞. Of course, as in the scalar case, we get a three-term recurrence relation: Theorem 4.2.4. Given a nontrivial ℓ × ℓ matrix measure µ with finite moments and choice of {σn }∞ n=0 with σn = 1, there exist ℓ × ℓ ∞ matrices {Bn }∞ and {A } with n n=1 n=1 Bn† = Bn

(4.2.33)

so that † R R R xpR n (x) = pn+1 (x)An+1 + pn (x)Bn+1 + pn−1 (x)An

(4.2.34)

xpLn (x) = An+1 pLn+1 (x) + Bn+1 pLn (x) + A†n pLn−1 (x)

(4.2.35)

and

Moreover, each An is invertible and if σ(dµ) ⊂ [−R, R]

(4.2.36)

then kAn k ≤ R

If σ ñ is another choice of the σ’s and

kBn k ≤ R

−1 un = σn−1 σ ñ−1

(4.2.37)

(4.2.38)

then en = u−1 Bn un B n

Añ = u−1 n An un+1

(4.2.39)

Conversely, for any set of ℓ × ℓ matrices {An , Bn }∞ n=1 obeying An invertible, (4.2.33), and (4.2.37), there is a matrix measure dµ obeying (4.2.36) (with a possible increase of R) and {σn }∞ n=0 with σ0 = 1 so R ∞ that the {pn }n=0 obeying (4.2.34) and pR 0 (x) = 1

pR −1 (x) = 0

(4.2.40)

en } generate the are the MOPRL for µ. Two sets of {An , Bn }, {Añ , B same dµ if and only if (4.2.39) holds.


261

Proof. Given the scalar case, this is straightforward. By (4.2.29), (4.2.34) implies (4.2.35), so we need only do the former. As usual, for j < n − 1, R R R hhpR (4.2.41) j , xpn iiR = hhxpj , pn ii = 0 since deg(xpR j ) < n. Define

R Bn+1 = hhpR n , xpn iiR

so by (4.2.27),

† Bn+1 = Bn+1

R An = hhpR n−1 , xpn iiR

R R R A†n+1 = hhxpR n+1 , pn ii = hhpn+1 , xpn ii

(4.2.42) (4.2.43)

(4.2.41)–(4.2.43) and (4.2.30) imply (4.2.34). By (4.2.42) and kpR j k = 1, (4.2.37) is immediate. By definition, R p˜R n = pn un+1

(4.2.44)

so (4.2.39) follows from (4.2.28) and (4.2.11)/(4.2.12). By (4.2.24) and (4.2.26), we have κLn = τn γn−1/2

(4.2.45)

and so is invertible. By looking at xn+1 coefficients in (4.2.34), κLn = An+1 κLn+1

(4.2.46)

so An+1 is invertible. For the converse, given {An , Bn }∞ n=1 form the block Jacobi matrix,   B1 A1 0 . . . A† B2 A2 . . .   1 (4.2.47) J =  0 A† B . . . 3   2 .. . . .. .. . . . .

acting on ℓ2 ({1, 2, . . . }, Cℓ ), which is a bounded operator with kJk ≤ (k) 3R. Let {δj }k=1,...,ℓ; j=1,2,... be the vector with a 1 in position ℓ(j − 1) + k. By the spectral theorem and multiplication, there are measures {µpk }p,k=1,...,ℓ with Z (p) m (k) hδ1 , J δ1 i = xm dµpk (x) (4.2.48) These can be put together into a matrix measure and the polynomials defined inductively by (4.2.34) are normalized MOPRL and so have the requisite form. We have thus set up a one-one correspondence between nontrivial ℓ × ℓ matrix-valued measures of compact support on R and equivalence classes of uniformly bounded Jacobi parameters under the equivalence:

262


Definition. Two sets of Jacobi parameters are called equivalent if there exist ℓ × ℓ unitaries, u0 = 1, u1 , u2 , . . . , so that (4.2.39) holds. Notice that, by (4.2.34), (4.2.35), and (4.2.26), κLn = (A1 . . . An+1 )−1

† † −1 κR n = (An+1 . . . A1 )

(4.2.49)

This partly explains why we consider equivalence classes and not a single set of Jacobi parameters. From the scalar case, one might like a choice with Aj > 0 and with κLn > 0. But since the Aj ’s may not commute, (4.2.49) shows those desires may conflict. There is actually a third natural choice: Definition. A matrix, A, is called lower triangular if Ajk = 0 for j < k, that is, it has potentially nonzero elements only on and below the main diagonal. L will denote the set of lower triangular matrices which are positive on diagonal, that is, Ajj > 0 for all j. In Section 6.3, we discuss the upper triangular matrices and the associated QR decomposition. Note that a block Jacobi matrix, J, has all An ∈ L if and only if it is 2ℓ + 1 diagonal with positive elements on the two extreme diagonals. Definition. A set of Jacobi parameters {An , Bn }∞ n=1 is said to be of type 1 if and only if, for all n, An > 0. We say it is of type 2 if and only if, for all n, A1 . . . An > 0 (equivalently, κLn = κR n > 0). We say it is of type 3 if and only if all An ∈ L. Using polar and QR decompositions, one can prove (see the Notes): Theorem 4.2.5 (Damanik–Pushnitski–Simon [94]). Each equivalence class of matrix Jacobi parameters has exactly one representative each of type 1, type 2, and type 3. Remarks and Historical Notes. We follow the notation of the review article on MOPRL and MOPUC of Damanik–Pushnitski–Simon [94] which, in particular, proves Theorem 4.2.5. 4.3. Coefficient Stripping In this section, we will define the m-function, find the coefficient stripping formula, and examine zeros and poles of det(m(z)), an object we will study in Section 4.4. Let µ be an ℓ × ℓ matrix-valued measure. We define the m-function on C+ by Z dµ(x) m(z) = (4.3.1) x−z


263

which is an ℓ × ℓ matrix ((x − z)−1 is scalar so it does not matter where we put it). We have seen that µ is associated with a Jacobi matrix, J, acting on H ≡ ℓ2 ({1, 2, . . . }, Cℓ ). Let P1 be defined from H to Cℓ by (P1 f ) = f1

(4.3.2)

which we can think of as a projection on H, but for now as a map from H to Cℓ so P1∗ (Hilbert space adjoint not just on Cℓ so we use ∗ rather than † ) takes Cℓ to H. By construction of µ from J, we have an ℓ × ℓ matrix-valued function, 1 m(z) = P1 P∗ (4.3.3) J −z 1 On C+ , we have 1 (4.3.4) Im(m(z)) ≡ (m(z) − m(z)† ) > 0 2i so m(z) is a matrix-valued Herglotz function. We will get coefficient stripping from Weyl solutions, so we start with second kind polynomials. We will only consider the R case. Define for n ≥ 0, R Z pn (x) − pR n (y) R qn (x) = dµ(y) (4.3.5) x−y and R q−1 (x) = −1 (4.3.6) Set A0 = 1 and consider solutions of (un defined for n = 0, 1, 2 . . . ) zun = un+1 A†n + un Bn + un−1An−1

(4.3.7)

for n = 1, 2, 3, . . . . We will also need the Weyl solutions, for z ∈ C+ , ψnR (z) = qnR (z) + m(z)pR n (z)

(4.3.8)

R Theorem 4.3.1. pR ·−1 (z) and q·−1 (z) both solve (4.3.7) and, thereR fore, so does ψ·−1 (z). Moreover, for fixed z ∈ C+ , we have ∞ X Tr(ψnR (z)ψnR (z)† ) < ∞ (4.3.9) n=0

Any solution of (4.3.7) obeying ∞ X Tr(un u†n ) < ∞

(4.3.10)

n=0

has the form

R un = cψn−1 (z)

for some ℓ × ℓ matrix c.

(4.3.11)

264


Proof. That pR ·−1 solves (4.3.7) is (4.2.34). From this and the R argument that led to (3.2.17), we get that q·−1 solves (4.3.7). For this it is important that in (4.3.5), dµ multiplies on the left, while in (4.3.7), A†n , Bn , An−1 multiply on the right. From (4.3.5), we get Z R ψn (z) = dµ(y)(y − z)−1 pR (4.3.12) n (y)

Since (· − z)−1 ∈ L2 (dµ), we get (4.3.9) from completeness of the polynomials and (4.2.32). We next claim that any ℓ × ℓ matrix solution, un , of (4.3.7) has the form R un = apR (4.3.13) n−1 (z) + bqn−1 (z) for some ℓ × ℓ matrices a and b. For let uñ be given by (4.3.13) with b = −u0

a = u1

(4.3.14)

Thus, uñ solves (4.3.7), and therefore, so does uñ − un = dn . But, by (4.3.14) and the initial conditions, R pR −1 (z) = q0 (z) = 0

R pR 0 (z) = −q−1 (z) = 1

(4.3.15)

d0 = d1 = 0. Since each An is invertible, the difference equation then implies dn ≡ 0. This proves (4.3.7). We next claim that if u0 solves (4.3.7) and obeys (4.3.10), then ∞ X Im(z) Tr(un u†n ) = − Im(u1 u†0 ) (4.3.16) n=1

For define

sn = Tr(un A†n−1 u†n−1 ) (4.3.17) † We multiply (4.3.7) by un on the right and sum from n = 1 to N. We get z

N X

Tr(un u†n )

n=1

since

=

N X n=1

sn+1 +

N X

Tr(un Bn u†n )

+

n=1

N X

s¯n

(4.3.18)

n=1

s¯n = Tr((un A†n−1 u†n−1 )) = Tr(un−1 An−1 u†n ) Now take imaginary parts of (4.3.18) using that

(4.3.19)

Im(Tr(un Bn u†n )) = 0 = Im(sn + s¯n )

(4.3.20)

We get Im

N X n=1

Tr(un u†n ) = − Im(s1 ) + Im(sN +1 )

(4.3.21)


265

Since (4.3.10) holds and An is bounded, sN +1 → 0 and (4.3.16) holds by taking N → ∞. One consequence of (4.3.16) is that if apR n−1 (z) obeys (4.3.10), then R a = 0. For p−1 (z) = 0 means u0 = 0, so by (4.3.16), pR 0 = 1 and † † Im(z) > 0, Tr(u1u1 ) = Tr(aa ) = 0 which implies a = 0. Now let un obey (4.3.10). By (4.3.15), R un = apR n−1 (z) + bqn−1 (z) R =a ˜ pR n−1 (z) + bψn (z)

with a ˜ = a − bm(z). Since ψnR obeys (4.3.9) and we conclude that form.

Tr(bψnR (ψnR )† b† ) ≤ kbk2 Tr(ψnR (ψnR )† )

a˜pR n−1

(4.3.22) (4.3.23)

obeys (4.3.10), so a ˜ = 0 and un has the claimed

Corollary 4.3.2. If un solves (4.3.7) and obeys (4.3.10) and u0 is invertible, then m(z) = −u−1 (4.3.24) 0 u1 Proof. By the theorem, un has the form (4.3.11), so u0 = −c

u1 = cm(z)


(4.3.25)

Theorem 4.3.3 (Aptekarev–Nikishin [23]). Let J be a bounded ℓ×ℓ block Jacobi matrix with Jacobi parameters {An , Bn }∞ n=1 and let J1 be ∞ the block matrix with parameters {An+1 , Bn+1 }n=1 . Let m(z) and m1 (z) be the m-functions for J and J1 . Then m(z) = (−z + B1 − A1 m1 (z)A†1 )−1

(4.3.26)

for all z with Im(z) > 0.

Remarks. 1. Since Im(m1 (z)) ≥ 0, Im(z − B1 + A1 m1 (z)A†1 ) ≥ Im(z)1, so the object in ( )−1 in (4.3.26) is invertible. 2. See the Notes for other proofs. Proof. Let (note n, not n − 1, in ψn ) ( ψn (z) n ≥ 1 un = A1 n=0

(4.3.27)

for n = 0, 1, . . . . Then un solves (4.3.7) for the parameters of J1 (with (1) the convention that the A0 for J1 is 1 not A1 !). Moreover, u0 = m(z) is invertible and un obeys (4.3.10). By the above corollary, −1 m1 (z) = −A−1 1 m(z) ψ1 (z)

(4.3.28)

266


By the equation (4.3.7) and initial conditions (4.3.15), we have † −1 pR 1 (z) = (z − B1 )(A1 )

q1R (z) = (A†1 )−1

(4.3.29)

so (4.3.28) becomes

A1 m1 (z) = −m(z)−1 (q1R (z) + m(z)pR 1 (z)) = −(m(z)−1 + (z − B1 ))(A†1 )−1

or which is (4.3.26).

m(z)−1 = −z + B1 − A1 m1 (z)A†1

(4.3.30)

As a final topic, we want to discuss poles and zeros of m(z). Fix x0 ∈ R \ σess (J). Let ℓ0 = dim(ker(J − x0 ))

We will prove

ℓ1 = dim(ker(J1 − x0 ))

(4.3.31)

Theorem 4.3.4. (i) m(z) has a simple pole at z = x0 and its residue is an ℓ × ℓ matrix of rank exactly ℓ0 . (ii) ℓ0 + ℓ1 ≤ ℓ (4.3.32) (iii) det(m(z)) has a pole at x0 of order ℓ0 − ℓ1 .

Remark. If ℓ0 − ℓ1 < 0, we mean det(m(z)) has a zero of order ℓ1 − ℓ0 . We need two preliminaries:

Lemma 4.3.5. If (J − x0 )u = 0 and u 6≡ 0, then u1 6= 0.

Proof. If u1 = 0, the recursion relation and A1 invertible implies u2 = 0. By induction and the second recursion relation with A2 invertible implies u ≡ 0.

Lemma 4.3.6. Let Q and P be finite-dimensional projections and suppose ϕ ∈ Ran(Q) ⇒ P ϕ 6= 0 (4.3.33) Then rank(P QP ) = rank(Q) (4.3.34) Proof. First, by (4.3.33), P maps Ran(Q) into a space of the same dimension, so rank(P Q) = rank(Q) (4.3.35) ∗ Second, if A is any operator, A Aϕ = 0 ⇔ kAϕk = 0 ⇔ Aϕ = 0, so rank(A∗ A) = rank(A) = rank(A∗ ) (4.3.36)


267

If A = QP , we conclude rank(P QP ) = rank(P Q) (4.3.35) and (4.3.37) imply (4.3.33).

(4.3.37)

Proof of Theorem 4.3.4. (i) By the spectral theorem, if Q is the projection onto the eigenspace for J with eigenvalue x0 , then Q 1 = + analytic at x0 J −z x0 − z Thus, by (4.3.3), m(z) has a simple pole at x0 with residue −P1 QP1∗ . By the lemmas, rank(P1 QP1∗ ) = rank(Q) if we think of P1 as UP , where P is the projection onto {u | un = 0 for n ≥ 1} and U is a unitary map to Cℓ . (ii), (iii) Define the ℓ × ℓ matrix-valued function G(z) = (z − x0 )m(z)

for z near x0 . This is analytic near x0 and selfadjoint for z real. It follows by eigenvalue perturbation theory (see the Notes) that the eigen˜ 1 (z), . . . , λ ˜ ℓ (z), are analytic near z = x0 . values of G(z), call them λ ˜ j (z) are nonvaBy (i), G(x0 ) is a rank ℓ0 operator, so exactly ℓ0 of λ nishing. The others vanish at least linearly in (z − x0 ) by analyticity. Thus, m(z) has eigenvalues λ1 (z), . . . , λℓ (z) near x0 and exactly ℓ0 have first-order poles at x0 and the others are analytic there. By (4.3.26), part (i), and A1 invertible, m(z)−1 has a pole of order one at x0 with residue of rank exactly ℓ1 . So, as above, exactly ℓ1 of the λj (z)−1 have poles, all first-order, at x0 . It follows that exactly ℓ1 of the λj (z) have zeros at x0 and they are order one. Clearly, a fixed λj can have either a pole or a zero but not both, so (4.3.31) holds. Moreover, det(m(z)) =

ℓ Y

λj (z)

(4.3.38)

j=1

has a product of ℓ0 simple poles, ℓ1 simple zeros, and ℓ −ℓ0 −ℓ1 nonzero regular functions, and thus has a pole of order ℓ0 − ℓ1 . Remarks and Historical Notes. Theorem 4.3.3 is from Aptekarev– Nikishin [23]. [94] has a proof relying on the method of “Schur complements” (due to Schur [371]) that for block operators (even of different square size) −1 A B (A − BD −1 C)−1 −A−1 B(D − CA−1 B)−1 = C D −D −1 C(A − BD −1 C)−1 (D − CA−1 B)−1

268


as can be checked by multiplication. One takes A = B1 − z, B = A1 , C = A†1 , D = J (1) − z. One can also use the proof in Section 10.3. Theorem 4.3.4 is from [94] whose proof we follow. It is related to the fact, due to Durán–López-Rodr´ıguez [112] and Sinap [403] (discussed also in [94]) that det(PnR (z)) has all real zeros precisely at the eigenvalues of the n × n block truncated matrix Jn;F with multiplicities of the zeros equal to the multiplicities of the eigenvalues. Matrix-valued Herglotz functions are discussed in [158, 163] and references therein. Eigenvalue perturbation for finite matrices is discussed in [210, 356]. 4.4. Step-by-Step Sum Rules of MOPRL In this section, our goal is to prove nonlocal step-by-step sum rules for det(m(z)) where m(z) is the m-function of some ℓ × ℓ nontrivial matrix-valued measure on R with σess (µ) = [−2, 2]. The zeros and poles may not interlace, so in using Theorem 3.3.2, we will not always have that N∞ = 1 as it is for the scalar case. The key will be to prove that N∞ ≤ ℓ. Lemma 4.4.1. Let J be any ℓ × ℓ block Jacobi matrix of the form (4.2.47) where An → 1 Bn → 0 (4.4.1) (called, not surprisingly, the matrix Nevai class). Let J1 be the once N± (J) stripped matrix. Let {Ej± (J)}j=1 be the eigenvalues of J in ±(2, ∞) ordered by E1− ≤ E2− ≤ · · · < −2 < 2 < · · · ≤ E2+ ≤ E1+

(4.4.2)

(counted up to multiplicity), and similarly for Ej± (J1 ). Then (i) (ii) (iii)

N± (J1 ) ≤ N± (J) Ej− (J) Ej+ (J)

≤

≥

Ej− (J1 ) Ej+ (J1 )

(4.4.3) ≤ ≥

− (J) Ej+ℓ − Ej+ℓ (J)

(4.4.4) (4.4.5)

Remark. One can prove that the inequalities in (4.4.4)/(4.4.5) are strict. Proof. If one defines Ej± for all j by setting Ej± (J) = ±2 if j > N ± (J), then one has the min-max principle (see the Notes) − (4.4.6) min hϕ, Jϕi Ej (J) = max V : dim(V )≤j−1

ϕ⊥V kϕk=1

4.4. STEP-BY-STEP SUM RULES OF MOPRL

269 (p)

Let Cℓ denote the ℓ-dimensional space spanned by {δ1 }ℓp=1 . Then for any V, min hϕ, J1 ϕi = min hϕ, Jϕi (4.4.7) ϕ⊥V kϕk=1

ϕ⊥V ⊕Cℓ kϕk=1

which immediately implies

− Ej− (J1 ) ≤ Ej+ℓ (J)

(4.4.8)

since dim(V ⊕Cℓ ) ≤ ℓ+dim(V ), so we are taking a max over a restricted set of W ’s with dim(W ) ≤ j + ℓ − 1. On the other hand, for any V with dim(V ) ≤ j − 1, if π is the projection onto Cℓ , then dim((1 − π)(V )) ≤ j − 1 and min hϕ, J1ϕi = min hϕ, Jϕi ≥ min hϕ, Jϕi ϕ⊥V kϕk=1

ϕ⊥V kϕk=1 πϕ=0

ϕ⊥(1−π)V kϕk=1 πϕ=0

(4.4.9)

leading to Ej− (J1 ) ≥ Ej− (J)

(4.4.10)

M(z) = −m(z + z −1 )

(4.4.11)

. . . ≤ z2− ≤ z1− < 0 < z1+ ≤ z2+ ≤ . . .

(4.4.12)

|zj± | → 1

(4.4.13)

The proof for E

+

is similar, using max-min rather than min-max.

Define ± Proposition 4.4.2. Let z1± , z2± , . . . and p± 1 , p2 , . . . be the zeros and poles of det(M(z)) where m( · ) is the m-function of an ℓ×ℓ block Jacobi matrix in Nevai class and M is given by (4.4.11). Do not include the zero at z = 0, which is ℓ-fold, and label so (counting multiplicity)

and similarly for (i) (ii)

p± j .

Then

|p± j | → 1

as j → ∞

+ + zj− < p− j < 0 < pj < zj

(4.4.14)

(iii) Let + Ij+ = (p+ j , zj )

σj+

Ij− = (zj− , p− j )

(4.4.15)

σj−

= −1, = 1 in the language of Theorem 3.3.2. Then for 0 < ±x < 1, 0 ≤ ∓N(x) ≤ ℓ (4.4.16) where N is given by (3.3.14), so N∞ ≤ ℓ

(4.4.17)

270


In particular, on compact subsets of C \ S where ∞ ± −1 ∞ S = {p± j }j=1 ∪ {(zj ) }j=1 ∪ {±1}

we have uniformly N b + (z) b − (z) Y zj zj j=1

b+ pj (z)

b− pj (z)

→ B∞ (z)

(4.4.18)

(4.4.19)

for a function analytic on C \ S and meromorphic on C \ {±1} with poles at S \ {±1} and zeros at S −1 \ {±1}. Moreover, on ∂D \ {±1} ⊂ C \ S, we have and in C+ ∩ D,

|B∞ (z)| = 1

(4.4.20)

arg B∞ (z) < 2πℓ

(4.4.21)

Remarks. 1. While we prove this for the m-functions that arise in our applications, a similar result holds for any function which is the determinant of M(z), which is an ℓ × ℓ matrix-valued meromorphic function of D with Im M(z) ≡ (M(z) − M(z)† )/2i > 0 on D ∩ C+ . In particular, a suitable labelling (4.4.15) holds (although ∓N(x) ≥ 0 for ±x > 0 may not). 2. arg B∞ (z) is defined with the branch defined near z = 0, with arg B∞ (0) = 0, for B∞ (0) > 0. 3. We suppose N ± (J) = ∞ with simple modification of notation if one (or both) is finite. Proof. Let p˜± ˜j± be the points in (−1, 1) with j and z −1 p˜± p± = Ej± (J) j + (˜ j )

z˜j± + (˜ zj± )−1 = Ej± (J1 )

(4.4.22)

By Lemma 4.4.1, (4.4.12) and (4.4.13) hold for the z˜j± and p˜± j , and (4.4.14) is only modified by the inequalities having ≤, not 0, and if arg(det(M(z))) is defined there to be 0 and analytically continued to C+ ∩ D, then 0 < arg(det(M(z))) < πℓ

(4.4.29)

(J − z)−1 = −z −1 + O(z −2 )

(4.4.30)

M(z) = −(J − (z + z −1 ))−1

(4.4.31)

M(z) = z1 + O(z 2 )

(4.4.32)

there. Proof. Since and we see, at z = 0, so det(M(z)) = z ℓ + O(z ℓ+1 ) (4.4.33) proving det(M(x)) > 0 for x in (0, ε) as claimed. Eigenvalue perturbation theory (see [210, 356]) implies there is a discrete set D ⊂ D \ R (i.e., the only limit points of D lie in ∂D) so, for z0 ∈ / D \ D, there are ℓ eigenvalues λ1 (z), . . . , λℓ (z) analytic near z0 which are all the eigenvalues of M(z) counting multiplicity. Continuing around at point z1 ∈ D can permute the λj ’s. If z ∈ (C+ ∩ D) \ D and for kψj k = 1, then

M(z)ψj = λj (z)ψj

(4.4.34)

Im λj (z) = Imhψj , M(z)ψj i > 0

(4.4.35)

272


by (4.4.25). Let z ∈ (C+ ∩ D) \ D and let γ(z), 0 ≤ z ≤ 1, be a simple curve with γ(0) = ε/2, γ(1) = z, and γ(t) ∈ (C+ ∩ D) \ D for t ∈ (0, 1]. By local analyticity, we can define λj (z) for z in a neighborhood of {γ(t) | 0 ≤ t ≤ 1}, and by (4.4.35), arg λj (z) ∈ (0, π). Thus, arg(det(M(z))) =

ℓ X j=1

arg λj (z) ∈ (0, ℓπ)

(4.4.36)

Since D is removable singularities of det(M(z)), we have (4.4.29).

With these preliminaries, following the proof of Theorem 3.3.6 leads directly to Theorem 4.4.4. Let M(z) be defined in (4.4.11). Let B∞ have the form (4.4.19) (and so obey (4.4.21) and the analyticity properties listed in Proposition 4.4.2). Then (i) For a.e. θ, limr↑1 det(M(reiθ )) exists and is nonzero with Z dθ |log|det(M(eiθ ))||p 0 (4.5.5) and if either (and so both) happens, then the limit is (4.5.5) exists and also ∞ X lim [k|An | − 1k2 + kBn k2 ] < ∞ (4.5.6) n→∞

n=1

Remarks. 1. If (4.5.3) and (4.5.4) hold, then the hypotheses of Theorem 4.6.1 hold, so (4.5.6) follows. Thus, in this section, we will focus entirely on (4.5.4) ⇔ (4.5.5).

274


2. det(f (x)) ≤ kf (x)kℓ , so Tr(f ) ∈ L1 implies log+ (det(f (x))) ∈ L . Thus, (4.5.4) can only diverge to −∞. 3. (4.5.4) implies f (x) is invertible for a.e. x ∈ [−2, 2], which implies that J has Σac = [−2, 2] with uniform multiplicity ℓ. ∞ ˜ j }∞ 4. If {A˜j , B j=1 is equivalent to {Aj , Bj }j=1 , then for suitable unitary A˜j = u−1 j Aj uj+1 , so 1

|A˜j | = u−1 j+1 |Aj |uj+1

and

det(|A˜j |) = det(|Aj |) so (4.5.5) is equivalence class independent.

(4.5.7) (4.5.8)

This will depend first on a step-by-step C0 sum rule: Theorem 4.5.2. If M, J, J1 are as in Theorem 4.4.5 hold, let p± j (J) be the poles of M(z; J). Then Z 2π 1 det(Im M (1) (eiθ )) − log(det(|A1 |)) = dθ log 4π 0 det(Im M(eiθ )) (4.5.9) X ± − [log(|p± (J)|) − log(|p (J )|)] 1 j j j,±

Remark. The sum in (4.5.9) is alternating with lim log(|p± j (J)|) = 0

j→∞

(4.5.10)

so the sum is at least conditionally convergent. Proof. Take z → 0 in (4.4.42) and take − log(. . . ) of both sides. det(M(z))/z ℓ → 1, so the left-hand side is − log(det(|A1 |)). Since − log(b(z, ω))|z=0 = − log(ω), the log B∞ (z)|z=0 is the sum in (4.5.9). Theorem 4.5.3 (C0 Sum Rule). Suppose (4.5.3) holds. Define Z 2π 1 sinℓ (θ) Z(µ) = log dθ (4.5.11) 4π 0 det(Im M(eiθ ))

and

E0 (J) =

X j,±

log(|zj± |)

Then, if |An | → 1 and |Bn | → 0, X n Z(µ) ≤ lim inf − log(det(|Aj |)) + E0 (J) n→∞

j=1

(4.5.12)

(4.5.13)

4.6. A KILLIP–SIMON THEOREM FOR MOPRL

X n lim sup − log(det|Aj |)) ≤ Z(µ) − E0 (J) n→∞

275

(4.5.14)

j=1

Remark. As with (3.6.8), we have for µ ˜ a pullback of µ to ∂D, 1 dθ ℓ Z(µ) = − S d˜ µ − log 2 (4.5.15) 2 2π 2

Proof. Given Theorem 4.5.2, we need only follow the proof in Section 3.6.

Proof that when (4.5.3) holds, then (4.5.4) ⇔ (4.5.5). If (4.5.4) holds, Z(µ) < ∞ by (4.5.13) and (4.5.14). The limit exists and is finite. Conversely, (4.5.5) implies X n lim inf − log(det(|Aj |)) < ∞ n→∞

j=1

so Z(µ) < ∞ by (4.5.13), and the limit exists as above.

Remarks and Historical Notes. These results are from Damanik– Killip–Simon [93]. 4.6. A Killip–Simon Theorem for MOPRL Our goal in this section is to prove a matrix analog of Theorem 3.1.1, specifically Theorem 4.6.1 ([93]). Let {An , Bn }∞ n=1 be the Jacobi parameters of an ℓ × ℓ block Jacobi matrix, J, whose matrix measure has the form (4.5.1). Then ∞ X Tr((|An | − 1)2 ) + Tr(Bn2 ) < ∞ (4.6.1) n=1

if and only if (a)

(b) The eigenvalues

σess (J) = [−2, 2] ∞ {En }j=1 ∈ / σess (J) obey ∞ X n=1

(|En | − 2)3/2 < ∞

(c) The ℓ × ℓ matrix function, f , of (4.5.1) obeys Z (4 − x2 )1/2 log(det(f (x)) dx > −∞ [−2,2]

(4.6.2)

(4.6.3)

(4.6.4)

276


Remarks. 1. As with Theorem 4.5.1, the integral in (4.6.4) can only diverge to −∞ and (4.6.4) implies Σac = [−2, 2] with uniform multiplicity ℓ. 2. Since the Hilbert–Schmidt norm on ℓ × ℓ matrices is equivalent to the operator norm, Tr( · 2) in (4.6.1) is equivalent to k · k2. 3. By (4.5.7), (4.6.1) is true for one element of the set of equivalent Jacobi parameters if and only if it is true for all. The first step in the proof is, of course, a step-by-step sum rule: Theorem 4.6.2 (Step-by-Step P2 Sum Rule for MOPRL). Define Z 2π 1 det(Im M1 (eiθ )) Q(J | J1 ) = log sin2 θ dθ (4.6.5) iθ 4π 0 det(Im M(e ))

Let F be given by (1.10.9) and G by (1.10.10). Then X 2 1 1 Tr(B ) + Tr(G(|A |)) = Q(J | J ) + [F (Ej± (J)) − F (Ej± (J1 ))] 1 1 1 4 2 j,±

(4.6.6)

Proof. By (4.3.26) and m1 (z) = − z1 + O(z −2 ), we have −1 M(z) = 1 − B1 z − (A∗1 A1 − 1)z 2 + O(z 3 ) (4.6.7) z Since det(C) = exp(Tr(log(C))) (4.6.8) if kC − 1k < 1, we have M(z) log det = Tr(B1 )z+Tr{[A∗1 A1 −1]+ 21 B12 }z+O(z 3 ) (4.6.9) z Moreover, A∗1 A1 = |A1 |2 (4.6.10) and |det(A1 )| = det(|A1 |) (4.6.11) Given the analog of (3.4.26) and Theorem 4.4.5, we get (4.6.6) by following the proof of Theorem 3.4.6 and Corollary 3.4.7. We can now follow the argument in Section 3.5 to obtain the following, which immediately implies Theorem 4.6.1: Theorem 4.6.3 (P2 Sum Rule for MOPRL). Let J be a block Jacobi matrix with σess (J) = [−2, 2]. Let dµ be its spectral measure and Z 2π 1 sinℓ θ Q(µ) = log sin2 θ dθ (4.6.12) 4π 0 det(Im M(eiθ ))

4.6. A KILLIP–SIMON THEOREM FOR MOPRL

277

Then Q(µ) +

X

E ∈σ / ess (J)

F (E) =

∞ X n=1

[ 14 Tr(Bn2 ) + 12 Tr(G(|An |))]

(4.6.13)

As a final topic, we want to note that for the type 3 case (i.e., An ∈ L, the lower triangular matrices), we can replace Tr((|An | − 1)2 ) in (4.6.1) by Tr((An − 1)2 ). Lemma 4.6.4. Let Cn ∈ L and suppose |Cn | → 1. Then Cn → 1. (1)

(ℓ)

Proof. Since |Cn | → 1, Cn∗ Cn → 1. Let xn , . . . , xn be the rows of Cn . Then Cn∗ Cn → 1 implies (k) hx(j) n , xn i → δjk

(4.6.14)

(1)

Since L is lower triangular, xn has only its first component nonzero. (1) Since this component is positive, (4.6.14) says xn → δ1 = (1 0 . . . 0). Orthogonality then implies the first column of Cn goes to (1 0 . . . 0)t . (2) Thus, by (4.6.14) for j = k0 = 2, xn → δ2 = (0 1 0 . . . 0). Repeating this shows that Cn → 1. Lemma 4.6.5. The map from L to strictly positive matrices given by A 7→ |A† | is a smooth diffeomorphism. Proof. A ∈ L√means det(A) > 0, so A is invertible. On strictly positive matrices, · is smooth, so √ A → |A† | = AA† (4.6.15)

is smooth. For the converse, given B strictly positive, the QR factorization of Section 6.3 implies that we can write B = QR

(4.6.16)

with Q unitary and R upper triangular with Rjj > 0 and the map B → R is smooth from invertible B’s by construction. Then L = R† is lower triangular, and since B is Hermitian, B = LQ−1

(4.6.17)

so B 2 = LQ−1 QL† = LL† and B = |L† |

(4.6.18)

278


so the smoothness QR algorithm shows the smoothness of the inverse map. Lemma 4.6.6. For any invertible A, Tr((|A† | − 1)2 ) = Tr((|A| − 1)2 )

(4.6.19)

Proof. There is a unitary with |A† | = U|A|U −1

so

|A† | − 1 = U(|A† | − 1)U −1 from which (4.6.19) is immediate.

(4.6.20) (4.6.21)

Theorem 4.6.7. If {An , Bn }∞ n=1 are type 3 Jacobi parameters, then ∞ X Tr[(|An | − 1)2 + |Bn |2 ] < ∞ (4.6.22) n=1

if and only if

∞ X n=1

Tr((An − 1)2 + Bn2 ) < ∞

Proof. By Lemma 4.6.6, (4.6.22) is equivalent to ∞ X Tr[(|A†n | − 1)2 + |Bn |2 ] < ∞

(4.6.23)

(4.6.24)

n=1

If only of the three conditions holds by Lemma 4.6.4, An → 1, so by Lemma 4.6.5, for n large and some c0 , c1 and all large n, c0 k|A∗n | − 1k ≤ kAn − 1k ≤ c1 k|A∗n | − 1k

which shows (4.6.24) is equivalent to (4.6.23).

Remarks and Historical Notes. These results are from Damanik– Killip–Simon [93]. If one permutes the rows and columns of a matrix in L under (1 2 3 . . . n) → (n n − 1 . . . 2 1), one gets a matrix in R and vice-versa. Thus, one can find an analog of the QR algorithm so B = QL with L lower triangular and so show the map A 7→ |A| on L is a diffeomorphism. This allows a slightly more direct proof of Theorem 4.6.7, which is what [93] do. We use QR since we need it again in Chapter 6.

CHAPTER 5

Periodic OPRL 5.1. Overview Thus far we have been looking at perturbations of OPUC and OPRL with constant Jacobi parameters; specifically, we looked at perturbations of a(0) b(0) (5.1.1) n = a n = b where b ∈ R, a ∈ (0, ∞). By scaling and translation covariance, we focused on a = 1, b = 0. In this chapter, we will study the periodic case where (0) (0) an+p = a(0) bn+p = b(0) (5.1.2) n n for all n and some fixed p. (5.1.1) is, of course, p = 1. The perturbation theory will be the focus of Chapters 8 and 9; this chapter will study the surprisingly rich unperturbed case. We will drop (0) henceforth in this chapter since we are restricting to the periodic case. For (5.1.1), the spectrum is e = [b − 2a, b + 2a] and is purely a.c. In the period p case, generically, the essential spectrum of the Jacobi matrix associated to {an , bn }∞ n=1 will be p closed intervals with

e = [α1 , β1 ] ∪ · · · ∪ [αp , βp ]

(5.1.3)

α1 < β1 < α2 < · · · < βp (5.1.4) Naively, the parameter counting seems simple. For p = 1 (i.e., (5.1.1)), every interval [α1 , β1 ] occurs (take b = 12 (α1 + β1 ), a = 14 (β1 − α1 )) and the map of (0, ∞) × R to e’s is one-one and onto. For period p, there are 2p free Jacobi parameters since {an , bn }pn=1 and periodicity determine all Jacobi parameters, and there are 2p free {αj , βj }pj=1, so the simple expectation is that all e’s of the form (5.1.3)/(5.1.4) are allowed and the map is one-one or it might be finite-to-one. In fact, this naive expectation is wrong! The set of e’s that occurs as essential spectrum of period p is not of dimension 2p but only a small subset of dimension p + 1. Not surprisingly, given this, the inverse image of a single e is a manifold of dimension p − 1. The reason for this is a natural set of p + 1-dimensional objects lies between {an , bn }pn=1 and e. 279

280

5. PERIODIC OPRL

Let Tn (λ) be the transfer matrix of (3.2.3). By periodicity, for any k = 1, 2, . . . , Tkp (λ) = Tp (λ)k (5.1.5) Since det(Tn ) = 1, all solutions will be bounded if and only if the eigenvalues of Tp have magnitude 1 and Tp is diagonalizable. Since Tp (λ) has determinant 1, its eigenvalues are completely determined by ∆(λ) = Tr(Tp (λ)) (5.1.6) called the discriminant. Since each factor in T is linear in λ and Tp has p factors, ∆(λ) is a polynomial of degree p. Since det(Tp ) = 1, its eigenvalues are distinct and of magnitude one if and only if they are e±iθ , 0 < θ < π, in which case ∆(λ) = 2 cos θ. It is thus not surprising that we will show e = ∆−1 ([−2, 2])

(5.1.7)

The parameter counting is now clearer. ∆ as a real degree p polynomial has p + 1 free parameters, so rather than think of we should think of

{an , bn }pn=1 → e

(5.1.8)

{an , bn }pn=1 → ∆ → e (5.1.9) p Since ∆ has only p+1 parameters, the set of {an , bn }n=1 with a given ∆ should be a set of dimension p − 1 (= 2p − (p + 1)). And indeed, in the generic case, when e has p pieces, the set will be a torus of dimension p − 1. For the other piece of the map, {αj , βj }pj=1 will be the points where ∆(λ) = ±2. Indeed, ∆(λ) = 2 at βj , αp−1, βp−2 , αp−3 , . . . , and −2 at αp , βp−1 , αp−2 , βp−3, . . . . Clearly, ∆ is determined by the p points where it is +2 and one of the points where it is −2, showing the rigidity in possibilities of e. There are two other big themes in the analysis of this chapter: quadratic equations and potential theory. It is an eighteenth century result (see the Notes to Section 5.2) that a real number x has a continued fraction expansion with ξn+p (x) = ξn (x) for some p and all n ≥ N0 for some N0 if and only if x obeys a quadratic equation with integral coefficients. Given this and the fact that the Jacobi parameters appear in a continued fraction expansion for m(z), it should not be surprising that if a Jacobi matrix has periodic Jacobi parameters, then its m-function obeys a quadratic equation with polynomial (in z) coefficients. This in turn implies m(z) has a natural continuation to a two-sheeted Riemann surface. This surface will play a major role, especially in Sections 5.12 and 5.13.

5.1. OVERVIEW

281

One can ask for a simple intrinsic criterion that determines whether a set e of the form (5.1.3)/(5.1.4) is the essential spectrum of a periodic Jacobi matrix. Given any compact e ∈ C which is not too small, e supports measures, ν, with Z E(ν) = log|x − y|−1 dν(x)dν(y) < ∞ (5.1.10)

For example, if e has the form of (5.1.3), Lebesgue measure restricted to e has E(ν) < ∞. If there is at least one ν with E(ν) < ∞, there is a unique probability measure, ρe, on e, called the equilibrium measure for e, that minimizes E(ν) among all probability measures ν on e. Remarkably, if e comes from a period p problem and has p disjoint pieces, then 1 ρe([αj , βj ]) = (5.1.11) p for all j, and conversely. Via potential theory, (5.1.11) provides the desired intrinsic criteria. There are two extensions of the sketch so far to keep in mind. First, while a period p problem generically has an essential spectrum, e, with p connected components, it can have fewer—indeed, any number ℓ + 1 from 1 to p. We use ℓ for the number of gaps and ℓ + 1 for the number of components. What is happening is that ∆−1 ((−2, 2)) always has p disjoint components, but the boundaries of these sets (i.e., ∆−1 ({−2, 2})), while generically distinct, can overlap (essentially, if ∆2 − 4 has a double zero). The set between the closures of the components of ∆−1 ((−2, 2)) are called “gaps,” and when there are fewer than p − 1 gaps, we say some gaps are closed. We thus consider sets e = [α1 , β1 ] ∪ · · · ∪ [αℓ+1 , βℓ+1 ] α1 < β1 < α2 < · · · < βℓ+1

(5.1.12) (5.1.13)

The condition that a set e be the essential spectrum of a period p Jacobi matrix with perhaps some gaps closed is that there are integers, k1 , . . . , kℓ+1, so that kj ρe([αj , βj ]) = (5.1.14) p Thus, e is the spectrum of some periodic problem if and only if each ρe([αj , βj ]) is rational. The other extension that will appear in Sections 5.12 and 5.13 is that if some ρe([αj , βj ]) is irrational, then there are almost periodic

282

5. PERIODIC OPRL

Jacobi matrices whose spectrum is e. These sets will be studied further in Chapter 9. Section 5.2 will discuss quadratic equations for m. Sections 5.3 and 5.4 will discuss ∆ and related structures. Section 5.5 will provide background on potential theory and its relevance to periodic Jacobi matrices and, in particular, prove (5.1.14). Sections 5.12 and 5.13 will explore the Riemann surface associated to e and its function theory to prove that if e has ℓ gaps, then the family of {(an , bn )}pn=1 with essential spectrum e is an ℓ-dimensional torus, called appropriately the isospectral torus. Sections 5.6–5.11 are a grand aside that approximate general compact sets in R by ones that are spectra of periodic problems, and use this as a tool to complete the discussion of the CD kernel begun in Sections 2.14–2.17 and 3.11. Remarks and Historical Notes. The issue of a period p problem having p bands generically will not be discussed formally, so let us make a few remarks. As we will see in Theorem 5.3.4, closed gaps are equivalent to degenerate eigenvalues for J(θ = 0) or J(θ = π), the p × p truncated Jacobi matrices with periodic and antiperiodic boundary conditions. By using degenerate eigenvalue perturbation theory (see Reed–Simon [356]), it is easy to see that if these operators have a degenerate eigenvalue and a single bj is changed slightly (with all other parameters fixed), then there are no degenerate eigenvalues. That implies the set of {αj , βj }pj=1 with any closed gaps is of codimension 1 at most. In fact, using ideas of Wigner–von Neumann [442], it is to be expected that the codimension is there, but I am not aware of any proof of this ([442] consider all n × n real matrices, not the Jacobi ones with a single corner matrix element added). In his work in the 1880’s on the stability of the moon’s orbit, Hill was led to look at the −u′′ (z) + V (z)u(z) = λu(z) where V is periodic, so the continuum analog of periodic Jacobi matrices is called Hill’s equation. Many of the ideas of this chapter are analogs of ideas developed there. Along the way, we will relate to these continuum forebears. 5.2. m-Functions and Quadratic Irrationalities If one iterates the Stieltjes expansion (3.2.28), one sees that m and the n-times stripped m-function, mn , are related by m=

Amn + B Cmn + D

(5.2.1)

5.2. m-FUNCTIONS AND QUADRATIC IRRATIONALITIES

283

where A, B, C, D are polynomials in z. But if the original Jacobi matrix is periodic, that is, obeys (1.11.1), then Jp = J and so mp = m, and (5.2.1) becomes a quadratic equation for m. This allows m to be meromorphically continued in z to a compact Riemann surface, which will play a big role later. Our goal in this section is to make this precise and find the relation of the coefficients A, B, C, D to OPs. In fact, we will go through the inverse of (5.2.1) which we have found as (3.7.23). Theorem 5.2.1. Let {an , bn }∞ n=1 obey (1.11.1). function obeys α(z)m(z)2 + β(z)m(z) + γ(z) = 0

Then the m(5.2.2)

where α(z) = ap pp−1 (z)

β(z) = pp (z) + ap qp−1 (z) γ(z) = qp (z)

(5.2.3)

The quadratic equation “discriminant” is given by β 2 − 4αγ = ∆(z)2 − 4

where

∆(z) = pp (z) − ap qp−1 (z) is called the discriminant.

(5.2.4) (5.2.5)

Remarks. 1. It is an unfortunate terminology clash that in analogy with an object in the study of Hill’s equation, ∆, given by (5.2.5), is called the discriminant, so the object in (5.2.4), which is usually called the discriminant of the quadratic equation, can’t be given that name! 2. We will see (see (5.4.5)) that ∆(z) is the trace of a transfer matrix. Proof. Given (3.2.23) and (3.7.23), we have that if Jp = J (implied by (1.11.1)), then mpp + qp (5.2.6) m=− ap (mpp−1 + qp−1 ) which implies (5.2.2)/(5.2.3). By (5.2.3), β 2 − 4αγ = ∆2 − 4[ap (qp pp−1 − pp qp−1 )]

and, by (3.2.22),

which proves (5.2.4).

ap (qp pp−1 − pp qp−1 ) = 1

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5. PERIODIC OPRL

As a quadratic equation, (5.2.2) has a second solution, and the remarkable fact is that the other solution is also related to an m-function. To describe it, we need to extend {an , bn }∞ n=1 to a two-sided sequence {an , bn }∞ by requiring that (1.11.1) holds for all n. The two-sided n=−∞ sequence generates a two-sided Jacobi matrix which acts on ℓ2 (Z) by (Ju)n = an un+1 + bn un + an−1 un−1 so the matrix is



   J =  

..

.

..

..

. . a−2 b−1 a−1 a−1 b0 a0

a0 b1 a1 .. .. .. . . .

(5.2.7)       

(5.2.8)

If we replace aℓ by zero, the matrix breaks into a direct sum via ℓ2 ({j}ℓj=−∞) ⊕ ℓ2 ({j})∞ j=ℓ+1 ). For ℓ ≥ 0, the second summand is the Jacobi matrix we have called Jℓ (i.e., J0 is the original Jacobi matrix; Jℓ is ℓ-times stripped) and will now call Jℓ+ . We will use Jℓ− for the Jacobi matrix obtained from the other half turned around to be a conventional Jacobi matrix. Thus, Jℓ+ = J({an+ℓ , bn+ℓ }∞ n=1 )

Jℓ− = J({aℓ−n , bℓ+1−n }∞ n=1 )

(5.2.9)

We will use m(z; Jℓ± ), Pk (z; Jℓ± ), α(z; Jℓ± ), etc. when we want to emphasize the J dependence; so, for example, m(z; J0+ ) solves α(z; J0+ )m2 + β(z; J0+ )m + γ(z; J0+ ) = 0

(5.2.10)

The second solution is given by Theorem 5.2.2. The second solution of (5.2.10) for z ∈ C \ R is given by m♯ (z) ≡ (a2p m(z; J0− ))−1 (5.2.11) As we discuss in the Notes, we will give several proofs of this theorem—not so much for their own sakes as they give different ways of looking at the result. Our proof here depends on relations of OPs among the various Jℓ± . Recall that P are monic and p normalized. Lemma 5.2.3. We have that (i)

± qk (z; Jℓ± ) = (aℓ±1 . . . aℓ±k )−1 Pk−1 (z; Jℓ±1 )

(5.2.12)

For k = 0, 1, 2, . . . , p − 1 and any ℓ, (ii)

− Pp−k (z; Jℓ+ ) = Pp−k (z; Jℓ−k )

(5.2.13)

(iii)

qp (z; J0± ) = (ap )−1 pp−1 (z; J0∓ )

(5.2.14)

5.2. m-FUNCTIONS AND QUADRATIC IRRATIONALITIES

285

(iv)

pp (z; J0+ ) = pp (z; J0− )

(5.2.15)

(v)

qp−1 (z; J0+ ) = qp−1 (z; J0− )

(5.2.16)

Proof. (i) This is just a restatement of (3.2.16). (ii) Take first ℓ = 0. By Theorem 1.2.10, Pp−k (z; J0+ ) is the characteristic polynomial for   b1 a1 ..   . a1 b2    . . .. ..   ap−k−1 bp−k

− and Pp−k (z; J−k ) for



b−k

a−k−1

 a−k−1 b−k−1  ..  .

..

.

..

. a1−p



  ..  . b1−p

By periodicity, these matrices are obtained from each other by inverting the order of rows and columns. The general ℓ case follows by translation covariance. ± (iii) By (5.2.12), qp (z; J0± ) = (a1 . . . ap )−1 Pp−1 (z; J±1 ), and by ± ∓ (5.2.13) for k = 1, Pp−1 (z; J±1 ) = Pp−1 (z; J0 ), from which (5.2.14) follows. (iv) This follows from (5.2.13) for k = 0. (v) Since a1 . . . ap−1 = a−1 . . . a−(p−1) , this is equivalent, by (5.2.12), − to Pp−2 (z; J1+ ) = Pp−2 (z; J−1 ), which is (5.2.13) for k = 2, ℓ = 1. Proof of Theorem 5.2.2. By (5.2.3), (5.2.14), (5.2.15), and (5.2.14), we have + γ(z; J0− ) = a−2 p α(z; J0 )

α(z; J0− ) = a2p γ(z; J0+ )

β(z; J0− ) = β(z; J0+ )

(5.2.17)

Use ˜ for the J0− objects and no ˜ for the J0+ objects. This means a2p γ m ˜ 2 + βm ˜ + a−2 p α = 0 or multiplying by a−2 ˜ −2 , p m α(ap−2 m ˜ −1 )2 + β(a−2 ˜ −1 ) + γ = 0 p m which says m♯ given by (5.2.11) obeys (5.2.10). That m♯ is distinct from m on C \ R and so the second solution is immediate if we notice on C+ that Im m > 0 while Im m♯ < 0.

286

5. PERIODIC OPRL

By the quadratic equation formula and (5.2.3)/(5.2.4), the solutions of (5.2.2) are p β(z) ± ∆(z)2 − 4 m(z) = − (5.2.18) 2ap pp−1 (z) p where one takes the branch of square root with ∆(z)2 − 4 = ∆(z) + O(1/∆(z)) near z = ∞. As a check, we see this leads to 2ap qp−k (z) + O( z12 ) m(z) = − 2ap pp−1 (z) (5.2.19) = − 1z + O( z12 )

near infinity. We will see below that (see Theorem 5.4.2 and 5.4.15) (i) ∆(z)2 = 4 has all its solutions on R. (ii) pp−1 (z) (whose roots are all simple and real) has zeros in ∆−1 ([−2, 2]) only at those points where ∆(z) ∓ 2 has a double zero and at such points β(z) = 0 also. This implies m(z) has continuous boundary values on ∆−1 ([−2, 2]), is real off that set and poles at some of the points where pp−1 (z) (some because the numerator might also vanish). Thus, by Proposition 2.3.12, Theorem 5.2.4. The Jacobi matrix associated to a sequence of Jacobi parameters obeying (1.11.1) has purely a.c. spectrum on ∆−1 ([−2, 2]) and at most p − 1 additional pure points off that set and no other spectrum. The quadratic equation (5.2.2) defines √ a two-sheeted branched cover of C ∪ {∞}, the Riemann surface of ∆2 − 4. This will be the major theme of Sections 5.12–5.13. Since m♯ will define the second sheet and − zeros of m(z; J0− ) are poles of m(z; J−1 ), we will see that poles of m on ± the two-sheeted surface are precisely the eigenvalues of J±1 .

Remarks and Historical Notes. The link between continued fractions, periodicity, and quadratic equations goes back to the study of continued fraction expansions of reals like (2.5.10). Euler noted that if the ξj in (2.5.10) are periodic (i.e., ξj+p = ξj for fixed p and all j > 0), then x obeys a quadratic equation with integral coefficients (the proof of Theorem 5.2.1 is essentially his proof). Legendre proved continued fractions of x’s obeying a quadratic equation with integral coefficients are eventually periodic (i.e., ξj+p ≡ ξj for some p and j ≥ J for some J). Galois specified the set with strictly periodic ξ’s. This is discussed, for example, in Koch [227] and Lang [259]. We will see several other proofs of Theorem 5.2.2 later. In Section 5.4, we will see its close relation to reality of the Green’s function

5.3. REAL FLOQUET THEORY AND DIRECT INTEGRALS

287

for the whole-line problem. Finally, in Section 5.13, we will see its relation to reflectionless operators. 5.3. Real Floquet Theory and Direct Integrals In the last section, we saw that two-sided periodic Jacobi matrices as defined in (5.2.8) are naturally associated to periodic Jacobi parameters. We will concentrate on these two-sided matrices in this section (and the next), although we will briefly return to the one-sided case in the next section. While we will do spectral analysis of J as an operator on ℓ2 (Z), it is useful to allow Ju to be defined via (5.2.7) for any sequence {un }∞ n=−∞ . Periodicity of the Jacobi parameters implies that J has a large commutant. Define S by (Su)n = un+1 (5.3.1) so by periodicity, JS p = S p J

(5.3.2)

thought of as operators on ℓ2 (Z) or on all sequences or on any ℓp (Z) including ℓ∞ (Z). A physicist would say (5.3.2) means J and S p can be simultaneously diagonalized, that is, have a common complete set of eigenvectors. Of course, we have to be prepared to consider “continuum eigenvectors” which the general theory (see the Notes to Section 5.4) says are polynomially bounded eigenvectors of Ju = λu

(5.3.3)

Since S p is unitary, its eigenvalues should lie in ∂D, so we look for solutions obeying un+p = eiθ un (5.3.4) for all n and some real θ. Such solutions are called Floquet solutions. In this section, we consider θ real to do a spectral resolution. In the next, to consider Weyl solutions, we will consider θ complex. While this paragraph is motivation, it will not be used directly below. We will need the following result about (5.3.3): Proposition 5.3.1. Let λ ∈ C. The set of solutions of (5.3.3) among all two-sided sequences is at most two-dimensional. Proof. If the dimension is more than 2, there is a nonzero solution with u0 = u1 = 0, but then (5.3.3) and an 6= 0 for all n implies u ≡ 0.

288

5. PERIODIC OPRL

To study solutions obeying (5.3.4) for θ ∈ [0, 2π), we define ℓ∞ θ = {u | u obeys (5.3.4)}

(5.3.5)

p−1 As the notation suggests, such u’s lie in ℓ∞ . Since u ∈ ℓ∞ θ → {un }n=0 is a bijection, dim(ℓ∞ (5.3.6) θ ) = p (j) Indeed, if we define δ (θ) for j = 1, 2, . . . , p by

[δ (j) (θ)]n+pℓ = eiθℓ δjn then

{δ (j) (θ)}pj=1

n = 1, . . . , p

(5.3.7)

is a basis for ℓ∞ θ . We have

∞ Proposition 5.3.2. J leaves ℓ∞ θ invariant. Its restriction to ℓθ p (call it J(θ)) in the {δ (j) (θ)}j=1 basis has the matrix   b1 a1 0 . . . . . . e−iθ ap  a1 b2 a2 . . . . . . 0      .. .  0 a2 b3 0   J(θ) =  (5.3.8) .. .. .. ..   . . . .     .. ..  0 . . bp−1 ap−1  eiθ ap . . . . . . . . . ap−1 bp

∞ p Proof. By (5.3.2), J takes ℓ∞ θ to itself since ℓθ = {u | S u = (1) e u}. The extra corner pieces come from (J(θ)δ (θ))0 = a0 , so by (1) iθ definition of ℓ∞ θ and periodicity of a, (J(θ)δ (θ))p = e ap . iθ

We will need the following below: Lemma 5.3.3. If u(j) ∈ ℓ∞ θj for j = 1, . . . , q are nonzero with the θj (j) q distinct, then {u }j=1 are linearly independent in ℓ∞ . Proof. For each j, n, because the θj are distinct, lim

L→∞

Thus, if

Pq

so γj = 0.

j=0

γj u

(j)

γj u(j) n

L X 1 (k) e−iθj ℓ un+ℓp = δjk u(k) n 2L + 1 ℓ=−L

= 0, then

X q L X 1 −iθj ℓ (k) = e γk u n =0 2L + 1 ℓ=−L k=1

J(θ) is a selfadjoint p × p matrix, so it has (counting multiplicity) p eigenvalues, e1 (θ) ≤ e2 (θ) ≤ · · · ≤ ep (θ) (5.3.9)


289

Theorem 5.3.4. (i) ej (2π − θ) = ej (θ)

for θ ∈ (0, π)

(5.3.10)

(ii) For eiθ 6= ±1, the ej (θ) are simple, that is, J(θ) has simple spectrum for θ ∈ (0, π) ∪ (π, 2π). Each ej (θ) is real analytic on (0, π). (iii) For θ 6= θ′ , J(θ) and J(θ′ ) have disjoint spectra. (iv) We have ep (0) > ep (π) ≥ ep−1 (π) > ep−1 (0) ≥ . . .

(5.3.11)

(v) On (0, π), (−1)p−j ej (θ) is strictly monotone decreasing. Remark. For now we will prove strict monotonicity of ej (θ) in (0, π). Eventually (see Theorem 5.4.2), we will prove (−1)p−j e′j (θ) < 0. Proof. (i) If M means the matrix with complex conjugates, then J(θ) = J(2π−θ) which, given that the eigenvalues are real, immediately implies (5.3.10). (ii) If J(θ) has a degenerate eigenvalue, say λ, then u(1) , u(2) ∈ ℓ∞ θ are linearly independent. By (i), λ is also an eigenvalue of ℓ∞ , and 2π−θ so there is an eigenvector, u(3) , of J(2π − θ) (could be chosen as u(1) ). By Lemma 5.3.3, u(1) , u(2) , u(3) are linearly independent, so there is a violation of Proposition 5.3.1. The ej (θ) are analytic as simple roots of a polynomial with analytic coefficients. (iii) By the same argument in (ii), if ej (θ) = eℓ (θ′ ) and θ ∈ (0, π), then there are at least three linearly independent eigenvectors, violating Proposition 5.3.1. This handles all cases but {θ, θ′ } = {0, π}, which follows from part (iv). (0)

(0)

(iv) Let an ≡ 1, bn ≡ 0, so the solutions of J (0) u = λu are un = eikn with λ = 2 cos(k). un+p = eiθ un with θ = kp (0) (mod 2π) and k is real and in [−π, π). It follows that ep (0) = 2, (0) (0) (0) (0) ep−1 (0) = ep−2 (0) = 2 cos(± 2π ), ep−3 (0) = ep−4 (0) = 2 cos(± 4π ); p p (0)

(0)

ep (π) = ep−1 (π) = 2 cos(± πp ), etc. We thus have (5.3.11) for J (0) . Since eigenvalues are continuous in θ and we have proven nondegeneracy for θ ∈ (0, π), then ej (θ) 6= ej (0 or π). We also have that (v) holds for J (0) . For y ∈ [0, 1], let J (y) = (1 − y)J (0) + J. The J (y) (θ) are continuous in y and θ. There is no way for an eigenvalue of J (y) (0) to cross an eigenvalue of J (y) (π) as y varies without going past the J (y) (θ) eigenvalues, which cannot happen by the proof of (iii) we have given. Thus (5.3.11) still holds at y = 1. (v) As noted in the proof of (iv), (iii) + (iv) implies (v).

290

5. PERIODIC OPRL

We can now define the important notions of bands and gaps. We define ej = Ran(ej (θ) | θ ∈ [0, 2π)) as the bands with e=

p [

j = 1, . . . , p

ej

(5.3.12)

(5.3.13)

j=1

By Theorem 5.3.4, we have eint j = ej [(0, π)]

int eint j ∩ ej = ∅ for j 6= k

(5.3.14)

so the ej can only intersect in their endpoints. Thus, ej = [αj , βj ] with α0 < β0 ≤ α1 < β1 ≤ · · · ≤ αp < βp

(5.3.15)

ℓ≤p−1

(5.3.16)

a rewriting of (5.3.11). The gaps are the sets (βj , αj+1) (or sometimes those of these sets that are nonempty). If βj = αj+1, we say the jth gap is closed; otherwise, we say it is open. We use ℓ for the number of open gaps, so

We will see later that ℓ = p − 1 generically and that ℓ is the genus of the Riemann surface defined by m. We are heading towards a proof that for the full-line Jacobi matrix, σ(J) = e

(5.3.17)

and that the spectrum is purely a.c. of multiplicity 2. We begin by putting the usual Fourier transform into a mod p setting. We define dθ p 2 2 F : ℓ (Z) → L ∂D, ; C (5.3.18) 2π

the L2 functions with values in Cp by (n = 0, 1, . . . , p − 1) (F u)n (θ) =

∞ X

un+ℓp e−iℓθ

(5.3.19)

ℓ=−∞

where, as usual with Fourier transform, we define this for u ∈ ℓ1 and extend by using Z dθ X kF u·(θ)k2 = |un |2 (5.3.20) 2π ∂D n


291

dθ 2 since {eiℓθ }∞ ℓ=−∞ is a basis for L (∂D, 2π ). Of course, we have the inverse dθ p −1 2 ; C → ℓ2 (Z) F : L ∂D, 2π

by

(F

−1

f )n+ℓp =

Z

eiℓθ fn (θ)

dθ 2π

(5.3.21)

for ℓ ∈ Z and n = 0, 1, . . . , p − 1. By the spectral theorem for finite matrices, there exist unitaries U(θ) : Cp → Cp so   e1 (θ) ..   .   U(θ)J(θ)U(θ)−1 =  (5.3.22)  . ..   ep (θ)

It is easy to see that U can be picked measurably and, not much harder, using the simplicity to see that it can be chosen continuously on (0, π)∪ (π, 2π). We fix U(θ) once and for all measurable so that (5.3.22) holds. dθ ; Cp ) to itself by We define U : L2 (∂D, 2π (Uf )(θ) = U(θ)f (θ)

(5.3.23)

Theorem 5.3.5. Let J be a two-sided periodic Jacobi matrix. Then dθ (a) As operators on L2 (∂D, 2π ; Cp ), [(F JF −1)f ]n (θ) = (J(θ)f )n (θ)

(5.3.24)

[(UF )J(UF )−1 f ]n (θ) = en (θ)fn (θ)

(5.3.25)

(b) (ℓ)

Proof. (a) Let δn ∈ ℓ2 (Z) be a delta function at n ∈ Z and let fn for ℓ ∈ Z, n ∈ {0, . . . , p − 1}, be the function with nonzero component (ℓ) n and value e−iℓθ . Then F (δn+ℓp ) = fn by (5.3.19). (5.3.24) is then an easy calculation. (b) is immediate from (5.3.22), (5.3.23), and (5.3.24). Lemma 5.3.6. Let F be strictly monotone and continuous on [a, b] and let A be the selfadjoint operator (Af )(x) = F (x)f (x)

(5.3.26)

on L2 ([a, b], dx). Then A is unitarily equivalent to (Bg)(y) = yg(y) on L2 ([F (a), F (b)], dF −1).

(5.3.27)

292

5. PERIODIC OPRL

Remark. F −1 is also a continuous and strictly monotone function, and dF −1 means its Stieltjes measure. Here F −1 is the functional inverse (not 1/F ). Proof. Let V : L2 ([F (a), F (b)], dF −1) → L2 ((a, b), dx) by (V g)(x) = g(F (x)) Then V is unitary and VBV −1 = A.

(5.3.28)

This lemma and Theorem 5.3.5 immediately imply Theorem 5.3.7. Let J be a two-sided period p periodic Jacobi matrix with bands {ej }pj=1. Then σ(J) = e and the spectrum is purely absolutely continuous with multiplicity 2. Proof. We get multiplicity 2 by separately considering ej in (0, π) and (π, 2π). Since ej (θ) is real analytic, its inverse is real analytic after a discrete set is removed, and so de−1 j is an absolutely continuous measure. There is another way of writing this more explicitly. The proof just follows the various mappings above, so we will only provide a sketch. Let ˜e = ∪pj=1 eint e, there is a unique θ ∈ (0, π) and j so λ = ej (θ). j . If λ ∈ ˜ We write θ(λ). There are solutions ϕ± n (λ) of (J − λ)ϕ± (λ) = 0

(5.3.29)

±ikθ(λ) ± ϕn (λ) ϕ± n+kp (λ) = e

(5.3.30)

with ±

We can normalize ϕ by requiring ϕ± 0 (λ)

>0

p−1 X j=0

2 |ϕ± j | = 1

(5.3.31)

− + ϕ+ 0 (λ) cannot be zero since then ϕ0 (λ) = ϕ0 (λ) is also zero and there is a linear combination vanishing at 0 and 1, violating Lemma 5.3.3. Thus, the normalization in (5.3.31) is possible. With this normalization,

ϕ− (λ) = ϕ+ (λ)

(5.3.32)

We define for {un }∞ n=−∞ of finite support ±

u b (λ) =

∞ X

n=−∞

ϕ± n (λ) un

(5.3.33)


293

We define the measure dν on ˜e by

Then:

1 dθ dλ dν(λ) = (λ) pπ dλ

(5.3.34)

Theorem 5.3.8. b extends to a unitary map of ℓ2 (Z) to L (e, dν(λ); C2 ) with inverse Z p + − − ˇ [ϕ+ (5.3.35) (f)n = n (λ)f (λ) + ϕn (λ)f (λ)] dν(λ) 2 Moreover, c ± (λ) = λb Ju u± (λ) (5.3.36) 2

Remarks. 1. In (5.3.35), we use f ± (λ) for the two components of C2 -valued function f ∈ L2 (e, dν(λ); C2 ). 2. dν will be the density of states discussed in Proposition 5.4.6. 3. The normalization of dν which requires a p/2 in (5.3.35) is made so is a probability measure. For θ′ has a fixed sign on each ej so R dν dθ | | dλ = π as θ runs from 0 to π or π to 0. Thus, ej dλ Z p Z X 1 dθ 1 dν = dλ = p π=1 pπ dλ pπ j=1 ej

ℓ−1 Sketch. ϕ˜+ ≡ {ϕ+ n }n=0 is an eigenvector of J(θ) normalized because of (5.3.31), so if λ1 , . . . , λp are the λ’s with a given θ, {ϕ˜+ (λj )}pj=1 is an orthogonal basis for Cp and unitarity of b follows from that for F . (5.3.35) comes from the fact that the inverse of b is its adjoint. (5.3.36) comes from (5.3.29).

by

Example 5.3.9. Let an ≡ 1, bn ≡ 0, and p = 1. Then θ(λ) is given

λ = 2 cos(θ(λ)) for θ ∈ (0, π) and λ ∈ (−2, 2). We have

±inθ(λ) ϕ± n (λ) = e

and

and

dλ dθ

(5.3.37) (5.3.38)

= 2 sin(θ(λ)), so dθ 1 =√ dλ 4 − λ2 dν =

1 1 √ dλ 2π 4 − λ2

(5.3.39)

(5.3.40)

294

5. PERIODIC OPRL

the free density of states. (5.3.33) is just the ordinary Fourier transform. dθ Remarks and Historical Notes. The space L2 (∂D, 2π ; Cp ) is often written as a direct integral and this is the language used in discussing eigenfunction expansions for periodic Schrödinger operators in arbitrary dimension. This section is essentially a discrete version of that theory specialized to one dimension. The ideas originated in the physics literature (as Bloch waves) and were expressed mathematically by Gel’fand [145]; see the historical background and exposition in Reed–Simon [356].

5.4. The Discriminant and Complex Floquet Theory In this section, we mainly discuss periodic full-line Jacobi matrices, J, although some results will hold for general full-line matrices (with bounded Jacobi parameters). We will also say something about the half-line operators Jℓ± of (5.2.9). Except for the fact that we will use that J(θ) has only real eigenvalues (see the Notes for a way to avoid this), the discussion in this section will not use results from the last section although it will illuminate them. We will be interested in solutions of (J − λ)u = 0

(5.4.1)

where λ ∈ C and u is an arbitrary sequence. We focus on solutions that obey un+p = ηun (5.4.2) for some η, all n (and p the period of J). Unlike the previous section, η need not be in ∂D. η is called the Floquet index and u a Floquet solution. When we want to focus on the solutions of the last section where |η| = 1, we speak of Floquet plane waves. A major role will be played by the transfer matrix (3.2.19) over p units pp (λ) −qp (λ) Tp (λ) = (5.4.3) ap pp−1 (λ) −ap qp−1 (λ) Notice that (3.2.28) says that

det(Tp (λ)) = 1

(5.4.4)

We will define the discriminant ∆(λ) = Tr(Tp (λ)) = pp (λ) − ap qp−1 (λ)

(5.4.5)

5.4. THE DISCRIMINANT AND COMPLEX FLOQUET THEORY

295

the object defined already in (5.2.5). Recall that if u solves (5.4.1), then (since ap = a0 ) u1 up+1 Tp (λ) = (5.4.6) a0 u0 a0 up Thus,

Theorem 5.4.1. There is a Floquet solution of (5.4.1) with Floquet index η if and only if η is an eigenvalue of Tp (λ) and the Floquet solution has (u1 a0 u0 )t as eigenvector. In particular, (i) If η is a Floquet index, so is η −1 . (ii) If ∆(λ) 6= ±2, there are exactly two Floquet solutions (up to constant multiples). (iii) We have that for θ ∈ [0, 2π], det(λ − J(θ)) = (a1 . . . ap )[∆(λ) − 2 cos θ]

(5.4.7)

∆(ej (θ)) = 2 cos θ

(5.4.8)

(iv) The eigenvalues, ej (θ), of J(θ) solve

Remarks. 1. Since deg(∆(λ)) = p, ∆(λ) = +2, and ∆(λ) = −2, each has at most p solutions. So there are two Floquet solutions except for at most 2p points. 2. We explore below (see Proposition 5.4.3) when there are two Floquet solutions and when only one if ∆(λ) = ±2. 3. J(θ) in (5.4.7) is given by (5.3.8). It is an interesting exercise to expand det(λ − J(θ)) in minors to get (5.4.7) using Theorem 1.2.10 and the definition (5.4.5) in terms of orthogonal polynomials. 4. By the spectral theorem for Hermitean matrices like J(θ), (5.4.8) immediately implies ∆(J(θ)) = (2 cos θ)1 (5.4.9) Proof. If u obeys (5.4.1) and (5.4.2), then up+1 u1 =η a0 up a0 u0

(5.4.10)

so η is an eigenvalue of Tp (λ). Conversely, if (u1 a0 u0 )t is an eigenvector, (5.4.10) holds, which means by periodicity of {an , bn } that (5.4.2) holds for the solution of (5.4.1) with (u1 , u0 ) initial conditions. This verifies the first statement in the theorem. To prove (i), we note det(Tp (λ)) = 1 says that if η is an eigenvalue, so is η −1 . (ii) then follows since if η 6= ±1, then η −1 6= η, and there are two eigenvalues. But since the algebraic eigenvalues have product 1, η = ±1 if and only if ∆(λ) = Tr(Tp (λ)) = ±2.

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5. PERIODIC OPRL

To get (iii), suppose first that θ 6= 0, π. We note λ is an eigenvalue of J(θ) if and only if η = eiθ is a Floquet index, and that happens if and only if ∆(λ) = η + η −1 = 2 cos θ. It follows that the two sides of (5.4.7) have the same zeros. Since both are monic polynomials, they must be equal. θ = 0, π then follows by continuity. To obtain (iv), note that if λ = ej (θ), then by the Hamiltonian– Jacobi theorem, det(λ − J(θ))|λ=ej (θ) = 0, so by (5.4.10), ∆(ej (θ)) = 2 cos θ. We note that, conversely, (5.4.10) shows any solution of ∆(λ) = 2 cos θ is an eigenvalue of J(θ). We can now analyze ∆ rather completely: Theorem 5.4.2. ∆ has the following properties: (i) ∆−1 ([−2, 2]) ⊂ R ± ± (ii) Let x± 1 ≤ x2 ≤ · · · ≤ xp be the zeros (counting multiplicity) of ∆(λ) ∓ 2. Then − − + + − x+ p > xp ≥ xp−1 > xp−1 ≥ xp−2 > xp−2 ≥ . . .

(5.4.11)

+ (iii) ∆(λ) is strictly monotone on each interval (x− p−2j , xp−2j ) and − ′ (x+ p−1−2j , xp−1−2j ), j = 0, 1, 2, . . . . Indeed, ∆ (λ) > 0 on intervals of the first type and ∆′ (λ) < 0 on intervals of the second type. (iv) If ej (θ) are the eigenvalues of J(θ) for θ ∈ (0, π), then (−1)p−j e′j (θ) > 0.

Remark. (5.4.11) is equivalent to (5.3.11). Proof. (i) If ∆(λ) = 2 cos θ, then e±iθ are Floquet indices, so λ is an eigenvalue of J(θ) which is selfadjoint. Thus λ is real. (ii), (iii) We first claim that if ∆(λ0 ) ∈ (−2, 2), then ∆′ (λ0 ) 6= 0, for if ∆′ (λ0 ) 6= 0, then λ 7→ ∆(λ) is many to one near λ = λ0 in C which implies, by the implicit function for analytic functions, that there are nonreal λ’s near λ0 with ∆(λ) ∈ (−2, 2), violating (i). This means that when ∆(λ) varies in (−2, 2), it is strictly monotone. Similarly, we see that if ∆(λ0 ) = ±2 and ∆′ (λ0 ) = 0, then ±∆′′ (λ0 ) < 0 to avoid nonreal solutions of ∆(λ) ∈ (−2, 2). Since ∆(λ) = (a1 . . . ap )−1 λp + lower order, ∆(λ) > 2 near +∞. 2 + Thus, the first zero, x+ p , of ∆(λ) − 4 has ∆(λp ) = 2. By the result on points where ∆′ = 0, we have ∆′ (λ+ p ) 6= 0. Thus, as λ decreases, ∆(λ) 2 runs from 2 down to −2. Either ∆ − 4 has a double zero at this point or else ∆(λ) < −2 just below this point. As λ decreases, ∆ must turn − − around (for ∆2 − 4 to have 2p zeros), and so we see x+ p > xp ≥ xp−1 . Repeating this analysis leads to the full string (5.4.11) and proves (iii) at the same time.


297

(iv) Since ∆(ej (θ)) = 2 cos θ, ∆′ (ej (θ))e′j (θ) = 2 sin θ

(5.4.12)

proving the result.

By deg(qℓ ) = ℓ − 1, we see, by (1.2.13), that

∆(x) = pp (λ) + O(λp−2) X p −1 p p−1 = (a1 . . . ap ) λ + bj λ + O(λp−2)

(5.4.13)

j=1

As in the last section, we define bands + ep = [x− p , xp ]

− ep−1 = [x+ p−1 , xp−1 ]

...

(5.4.14)

and gaps. If ∆(λ) 6= ±2, there are two Floquet solutions since the eigenvalues of Tp (λ) are distinct. As for points where ∆(λ) = ±2: Proposition 5.4.3. Suppose ∆(λ0 ) = ±2. Then the following are equivalent: (i) All solutions of (5.4.1) at λ0 are periodic (if ∆(λ0 ) = 2) or antiperiodic (if ∆(λ0 ) = −2). (ii) Tp (λ0 ) = ±1 (iii) J(θ = 0) (if ∆(λ0 ) = 2) or J(θ = π) (if ∆(λ0 ) = −2) has an eigenvalue of multiplicity 2. (iv) ∆′ (λ0 ) = 0 (v) The gap at λ0 is closed. Remarks. 1. Antiperiodic means un+p = −un . 2. If (i) fails, there is a unique Floquet solution (up to a constant). 1 3. If (ii) fails, Tp (λ0 ) has ±1 0 ±1 as Jordan normal form, which implies any solution of (5.4.1) independent of the (anti-)periodic solution grows so there are a c1 n upper bound and c2 n lower bound on |un |. 4. If the gap is open at the edges where ∆(λ0 ) = ±2, there is a unique (up to a constant) periodic (if ∆(λ0 ) = 2) or antiperiodic (if ∆(λ0 ) = −2) solution. Proof. (i) ⇔ (ii) is immediate from (5.4.6). (i) ⇔ (iii) Eigenvectors of J(θ = 0) with eigenvalue λ0 are precisely periodic solutions of (5.4.1) for λ = λ0 . Since the set of potential solutions is two-dimensional, (i) is equivalent to there being a twodimensional family of eigenvectors. (iii) ⇔ (iv) ∆′ (λ0 ) = 0 if and only if ∆(λ) ∓ 2 has a double zero at λ = λ0 . By (5.4.7), this is true if and only if det(λ − J(θ)) has a

298

5. PERIODIC OPRL

double zero at λ = λ0 for θ = 0 (or π). Since J(θ) is selfadjoint, the order of the zero is the multiplicity of the eigenvector. (iv) ⇔ (v) A gap is closed if and only if ∆(λ) ∓ 2 has a double zero, which happens if and only if ∆′ (λ0 ) = 0. Recall that the measure, dν, in the spectral representation (5.3.35) has the form (5.3.34). The formula (5.4.12) lets us compute dν in terms of ∆: Theorem 5.4.4. The measure dν of (5.3.3) can be written 1 |∆′ (λ)| p dλ pπ 4 − ∆2 (λ) 1 d ∆(λ) = arccos dλ pπ dλ 2

dν(λ) =

(5.4.15) (5.4.16)

Remark. Again, we see (via (5.4.16)) that ν(ej ) = 1/p since ∆(λ)/2 runs from 1 to −1 or −1 to 1, and so arccos from 0 to π or π to 0. Proof. (5.4.12) can be rewritten −1 q dθ ′ ∆ (λ) = 2 sin θ = 2 1 − dλ

∆′ (λ) 2 2

so (5.4.15) follows from (5.3.34). (5.4.16) is a direct calculation of the derivative of arccos. Since (5.4.15) is explicit, we see that dν/dλ is real analytic on e with square root diverges at the edges: Corollary 5.4.5. The Radon–Nikodym derivative dν/dλ of ν is real analytic on eint and obeys c1 dist(λ, R \ e)−1/2 ≤

dν ≤ c2 dist(λ, R \ e)−1/2 dλ

(5.4.17)

dν Remarks. 1. In fact, dist(λ, R \ e)−1/2 dλ has nonzero limits as one approaches an open gap edge.

2. There is an “explicit” formula for dν/dλ in Corollary 5.4.19 that immediately shows the bounds in (5.4.17) are exact. Proof. Except for points in eint where ∆ = ±2, this is obvious from (5.4.15). Such points occur at closed gaps,√λ, where ∆′ (λ0 ) has a simple zero and 4−∆2 (λ0 ) a double zero, so ∆′ / 4 − ∆2 is regular.


299

(5.4.16) allows us to reinterpret dν as a density of states (aka density of zeros). Let Jm;F be the truncated transfer matrix associated to (p) {an , bn }m n=1 (actually, am does not enter) and let Jm;F be the matrix with periodic boundary conditions (i.e., (5.3.8) with p replaced by m (m) and eiθ = 1). The eigenvalues {λj }m j=1 of Jm;F are the zeros of Pm (z) pi (m) m (p) by (1.2.31). We will let {λj }j=1 be the eigenvalues of Jm;F (which may be degenerate, so we count multiplicities). Define the normalized counting measures m 1 X dνm (λ) = δ (m) (5.4.18) m j=1 λ,λj m

(p) dνm (λ)

1 X = δ p (m) m j=1 λ,λj i

(5.4.19)

Proposition 5.4.6. Suppose {an , bn }∞ n=−∞ is periodic. Then as (p) m → ∞, the measures dνm and dνm converge weakly to the same measure, dν, called the density of states or density of zeros. Remarks. 1. We use the same symbol, dν, since we will prove shortly that it is the dν defined in (5.3.34). 2. We will identify this limit as a potential theoretic equilibrium density in Theorem 5.5.17. 3. The same proof works in more general situations; see TK. x-ref? (p)

Proof. Since kJm;F k and kJm;F k are uniformly bounded, the dνm (p) and dνm are supported on a fixed interval [−A, A], so it suffices to R R (p) prove for all ℓ that λℓ dνm (λ) and λℓ dνm (λ) converge to a limit, and the limit is the same (for polynomials are dense in C([−A, A])). Note that Z 1 λℓ dνm (λ) = Tr((Jm;F )ℓ ) (5.4.20) m (p)

(and similarly for dνm ). It is easy to see that for ℓ < m and ℓ < j < (p) m − ℓ, the jj matrix element of (Jm;F )ℓ and (Jm;F )ℓ are equal and are independent of m (for m > ℓ + j) and periodic in j. From this, the existence and equality of the limits follow. Theorem 5.4.7. The measure dν of (5.3.34) and (5.4.15) is the density of states. (p)

Proof. Consider Jrp;F where m is a multiple of p. As we have seen, its eigenvalues are connected with when Trp (λ) has eigenvalue 1.

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5. PERIODIC OPRL

But Trp (λ) = Tp (λ)r by periodicity of the a’s and b’s, so we want to know when Tp (λ) has an eigenvalue, η, with η r = 1, that is, η = e2πj/r , (p) j = 0, 1, . . . , r − 1. Thus the eigenvalues of Jrp;F are precisely the solutions of 2jπ ∆(λ) = 2 cos j = 0, 1, . . . , r − 1 (5.4.21) r

Except perhaps when r = 0 or r/2 (if r is even), these zeros are all simple but involve (except for those values of r) a doubling of j and r − j. The doubling cancels the 2 in (2π)−1 . The normalized counting measure thus converges to ∆(λ) −1 −1 d p 2(2π) arccos dλ dλ 2 which is (5.4.16).

The following can be viewed as a whole-line analog of Theorem 2.15.1: Theorem 5.4.8. Let f be a continuous function on e = σ(J), the spectrum of a full-line period p periodic Jacobi matrix, J. Let f (J)nm be the matrix elements of f (J) in the standard basis. Then f (J)nm is periodic, that is, f (J)n+p m+p = f (J)nm (5.4.22) and Z p 1X f (J)nn = f (λ) dν(λ) (5.4.23) p n=1 where ν is the density of states.

Proof. As usual, we need only prove this for f (λ) = λℓ , ℓ = 0, 1, 2, . . . . As in the proof of Theorem 2.15.1, we have n 1 X ℓ 1 ℓ lim (J )jj − Tr(Jn;F ) → 0 n→∞ n n j=1

so by Proposition 5.4.6,

n

1X ℓ lim (J )jj = n→∞ n j=1

By (5.4.22), kp

Z

λℓ dν(λ)

1 X ℓ (J )jj = RHS of (5.4.23) for f (λ) = λℓ kp j=1

(5.4.24)


proving (5.4.23).

301

Next we turn to the Lyapunov exponent: Theorem 5.4.9. For λ ∈ C, 1 lim log kTn (λ)k = γ(λ) n→∞ n exists and is given by ∆(λ) q ∆(λ) 2 1 + − 1 γ(λ) = log 2 p 2

(5.4.25)

(5.4.26)

√ Remarks. 1. (5.4.26) requires one to specify which branch of is intended. We place branch cuts on e ⊂ R ⊂ C and take the branch √ which is ∆(λ) + O(λ−p ) near λ = ∞. There is a discontinuity of 2 across e, but since |. . .| = 1 there and the two branches are complex conjugates, the function in (5.4.26) is continuous there. 2. We will place the existence of the limit in (5.4.25) into a more general framework in TK. x-ref? 3. γ is called the Lyapunov exponent. 4. If ∆(λ) ∈ [−2, 2], the square root in (5.4.26) is pure imaginary and |. . .| = 1. Thus, on e, γ(λ) = 0. Proof. Since

Tnp+j = Tj (Tp )r on account of periodicity, and since {kTj k, kTj−1k}p−1 j=0 are bounded, it is easy to see that it suffices to establish the limit exists for n = rp and to note that limit is just limr→∞ p1 log kTp (λ)r k1/r , which exists by the spectral radius formula. Thus, γ exists and 1 γ(λ) = log max{|η| | η an eigenvalue of Tp (λ)} (5.4.27) p Thus, eigenvalues are the solutions of η 2 − 2∆(λ)η + 1 = 0 √ so (with the branch of given in Remark 1 above) q ∆(λ) ∆(λ) 2 η± (λ) ≡ ± −1 2 2 η± are analytic in C \ e and nonvanishing, so t(λ) ≡ |η+ (λ)| − |η− (λ)|

(5.4.28)

is harmonic. t → ∞ as λ → ∞ (since |η+ | = O(|λ|p) and |η− | = O(|λ|−p) and t(λ) → 0 as λ → e. Thus, by the minimum principle, t > 0 on C \ e, that is, |η+ | > |η− |, so (5.4.27) is (5.4.26).

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5. PERIODIC OPRL

Next we turn to a remarkable relation between the density of states, dν, and the Lyapunov exponent, γ. We first need a lemma: Lemma 5.4.10. ∆′ (λ) has exactly one zero in each gap (including closed gaps) and no other zeros. Proof. We first prove there is at least one zero in each gap. If a gap is closed, ∆(λ) has a double zero at the location λ0 of this closed gap so ∆′ (λ0 ) = 0. In an open gap (λ0 , λ1 ), we have ∆(λ0 ) = ∆(λ1 ), so ∆′ has a zero in (λ0 , λ1 ) by Snell’s theorem. Thus each gap has at least one zero. There are p − 1 gaps (counting closed gaps) and ∆′ is a polynomial of degree p − 1, so this accounts for all the zeros: one per gap and no others.

x-ref?

Theorem 5.4.11. For any λ in C, Z 1 γ(λ) = − log(a1 . . . a0 ) + log|λ − x| dν(x) p

(5.4.29)

Proof. Consider the function on C+ , q 1 ∆(λ) ∆(λ) 2 g(λ) = log + −1 2 p 2

(5.4.30)

Remarks. 1. (5.4.29) is called the Thouless formula. We will provide a proof in a more general context TK. 2. This formula also plays a role in the potential theoretic analysis; see Section 5.5 and its Notes.

Pick the branch of the square root which, near i∞, is ∆(λ) + O(λ−p) 2 and the branch of log that, near i∞, has log(λp ) = p log|λ| + ipπ . As 2 usual, we put a branch cut of the square root on e so the quantity in [. . . ] in (5.4.30) is analytic in C+ . Since q q ∆(λ) ∆(λ) ∆(λ) 2 ∆(λ) −1 −1 =1 + − 2 2 2 2

the expression in [. . . ] is nonvanishing. So g(λ) is analytic in C+ . Then on C+ , 1 ∆′ (λ) ∆(λ) ′ p g (λ) = 1+ p (5.4.31) p ∆(λ) + ∆′ (λ)2 − 4 ∆2 (λ)2 − 4 1 ∆′ (λ) = p (5.4.32) p ∆(λ)2 − 4 √ d since dx (x + x2 − 4) = 1 + √xx2 −4 .


303

g ′ (λ) is thus analytic in C+ with boundary values on R\{λ | ∆(λ) = ±2}. ∆′ is real on R, positive above the top band, and so also on the top p band. By the lemma, it alternates sign from one band to the next. ∆2 (λ) − 4 is real in the gaps and above and below the bands. Every time it moves from above a zero of ∆2 − 4 to below, its argument p increases by 12 π, so ( ∆2 (λ) − 4)−1 is pure imaginary on each band with negative imaginary part on the top band, positive on the next, etc. Taking into account that ∆′ also alternates sign, we see, by (5.4.15), Im g ′(λ + i0) ≤ 0 λ∈e (5.4.33) 1 Im g ′(λ + i0) dλ = −dν(λ) (5.4.34) π Near λ = ∞, g ′(λ) ∼ λ−1 , so Im g ′ (λ) < 0 near λ = ∞ in C+ . Since Im g is harmonic, Im g ′ ≤ 0 on all of C+ , and thus − Im g ′ is an m-function. So, by (5.4.34), Z dν(x) ′ g (λ) = − (5.4.35) x−λ Therefore,

g(λ) = c + for some constant c, and so

Z

log(x − λ) dν(x)

(5.4.36)

Z

(5.4.37)

γ(λ) = Re g(λ) = Re c +

log|λ − x| dν(x)

Since log|λ − x| = log|λ| + log|1 − λx |, near λ = i∞, 1 RHS of (5.4.37) = log|λ| + Re c + O |λ|

By (5.4.26) and

∆(λ) = (a1 . . . ap )−1 λp + lower order

(5.4.38)

we have γ(λ) =

1 [p log|λ| + log|a1 . . . ap |−1 + O(|λ|−1)] p

which implies (5.4.29).

Next we turn to considering the connection of Floquet solutions and the spectral theorist’s Green’s function, aka matrix elements of the resolvent. Our first two results hold for any bounded two-sided Jacobi matrices.

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5. PERIODIC OPRL

Theorem 5.4.12. Let J be a two-sided bounded Jacobi matrix. For 2 any λ ∈ C+ , there are solutions u± n (λ) of (5.4.1) which are ℓ at ±∞ unique up to constants. Their Wronskian, − + − W (λ) = an (u+ n+1 (λ)un (λ) − un (λ)un+1 (λ))

is n-independent, and for n ≥ m,

hδn , (J − λ)−1 δm i =

− u+ n (λ)um (λ) W (λ)

(5.4.39)

(5.4.40)

Moreover, if pn (λ) are the orthonormal polynomials associated to J0+ , we have for n ≥ m, hδn , (J0+ − λ)−1 δm i = where

u+ n (λ)pm−1 (λ) f(λ) W

(5.4.41)

f(λ) = an (u+ (λ)pn−1 (λ) − u+ (λ)pn (λ)) W n+1 n

(5.4.42)

m(λ, Jn+ ) = hδn+1 , (Jn+ − λ)−1 δn+1 i

(5.4.43)

which is n-independent. In particular, if

m(λ, Jn− )

=

then Gnn (λ) = −

hδn , (Jn−

−1

− λ) δn i

a2n m(λ, Jn+ )

1 − m(λ, Jn− )−1

(5.4.44) (5.4.45)

Remarks. 1. We will normally normalize u± by requiring u± n=0 = 1. Normalization changes drop out of (5.4.40). 2. Since J and J0+ are symmetric and real, hδn , (J − λ)−1 δm i = hδm , (J − λ)−1 δn i, so (5.4.40)/(5.4.41) determine the full resolvent. 3. (5.4.40) is usually called the Green’s function by spectral theorists. 4. In (5.4.41), it we take n = m = 1 and note that (since p0 = 1, p−1 = 0) f(λ) = −a0 u+ W 0 we have u+ (λ) + −1 hδ1 , (J0 − λ) δ1 i = − 1 + (5.4.46) a0 u0 (λ) which is essentially (3.2.33) for n = 1. So (5.4.41) generalizes (3.2.33). + 5. By (3.2.23) and (3.2.25), u+ n normalized by un=0 = 1 has the form u+ (5.4.47) n = −qn−1 (λ) − m(λ)pn−1 (λ)


305

for n ≥ 1. 6. (5.4.45) has a disconcerting asymmetry in J + and J − . There are two ways of restoring the symmetry. One is to note (by the symmetry or by mimicking the proof) Gnn (λ) = −

1 + − m(λ, Jn−1 )−1

− a2n−1 m(λ, Jn−1 )

(5.4.48)

The other is to use coefficient stripping

− ) −m(λ, Jn− )−1 = z − bn + a2n−1 m(λ, Jn−1

(5.4.49)

to get from (5.4.45) that Gnn (λ) = −

1 − z − bn + a2n−1 m(λ, Jn−1 ) + a2n m(λ, Jn+ )

(5.4.50)

Proof. By using Theorem 3.2.1 on J0± , we find solutions u± n for 2 ±n ≥ 1 which are ℓ at ±∞ and unique up to a constant. But any solution on (1, ∞) can be uniquely extended to (−∞, ∞), so we get u± n . Independence in n of (5.4.39) follows by the same argument, using determinants of transfer matrices, that led to (3.2.21). Define Gmn (λ) = hδm , (J − λ)−1 δn i (5.4.51) Fix m and note that

{(J − λ)[Gm· ]}n = δmn

(5.4.52)

+ Gmn (λ) = c− m un

(5.4.53)

and Gm· is ℓ2 at +∞. So for n ≥ m,

By the symmetry in m and n and looking at −∞, for m ≤ n, − Gmn (λ) = c+ n um

(5.4.54)

It follows that + Gmn (λ) = c u− m un Evaluating (5.4.52) if n = m shows

Since

+ − + − + c[am−1 u− m−1 um + (bm − λ)um um + am um um+1 ] = 1 + + (bm − λ)u+ m + am um+1 = −am−1 um−1

(5.4.55) (5.4.56)

this says cW (λ) = 1, which proves (5.4.40). + If we note that any {vj }∞ j=1 obeying [(J0 −λ)v]j = 0, j = 1, 2, . . . , n, has vj = cpj−1 for j = 1, 2, . . . , n + 1 (see Proposition 1.3.1), the proof of (5.4.41) is identical to the proof of (5.4.40).

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5. PERIODIC OPRL

To prove (5.4.45), we note that, by (5.4.63), u+ n+1 (λ) an u+ n (λ) − un (λ) m(λ, Jn− ) = − an u− n+1 (λ) m(λ, Jn+ ) = −

and, by (5.4.40), Gnn (λ)

−1

= an

u+ u− n+1 (λ) n+1 (λ) − + un (λ) u− n (λ)


(5.4.57) (5.4.58)

(5.4.59)

− + 2 Define GD mn (λ) to be the resolvent of J−1 ⊕∞⊕J0 on ℓ (−∞, −1)⊕ ℓ2 ({0}) ⊕ ℓ2 (1, ∞) (i.e., set b0 = a−1 = a0 = 0). Thus,  + −1  hδm , (J0 − λ) δn i if m, n ≥ 1 − (5.4.60) GD hδm , (J−1 − λ)−1 δn i if m, n ≤ −1 mn (λ) =  0 otherwise

Theorem 5.4.13. For any whole-line Jacobi matrix and all n, m and λ ∈ C+ , G00 (λ) 6= 0 and −1 GD nm (λ) = Gnm (λ) − G0n (λ)G0m (λ)[G00 (λ)]

(5.4.61)

Remarks. 1. If one considers J(α) which is J with b0 replaced by b0 + α, as α → ∞, then as α → ∞, Gnm (λ; J(α)) → GD nm (λ). (5.4.61) can be viewed in terms of rank one perturbations at infinite coupling; see [160]. 2. Since GD nm = 0 for n ≤ 0 ≤ m, we see that n ≤ k ≤ m ⇒ Gnm (λ)Gkk (λ) = Gnk (λ)Gkm(λ)

(5.4.62)

which follows directly from (5.4.40). 3. If (a, b) ⊂ R \ σ(J), (5.4.61) extends to all λ in (a, b) with G00 (λ) 6= 0. Points with G00 (λ) = 0 are (as we will see below) eigen− values of J−1 ⊕ J0+ , and so poles of GD nm (λ) for suitable n and m. Proof. We have Imhδ0 , (J − λ)−1 δ0 i ≥ |Im λ|−1

− ± so G00 (λ) 6= 0. Thus, u+ 0 6= 0 6= u0 . Let n ≥ m ≥ 0. Then, with u ± normalized by u0 = 1, − + + RHS of (5.4.61) = W −1 (u+ n um − un um ) − + = W −1 (u+ n (um − um ))

(5.4.63)


307

+ u− m − um is a solution of (5.4.1) vanishing at m = 0, so a multiple of pm−1 . Since the Wronskian of u+ and u− is the Wronskian of u+ and u− − u+ , RHS of (5.4.62) = GD nm by (5.4.41).

We now specialize to the periodic case. We define the integrated density of states, N(λ), for λ ∈ R by Z λ N(λ) = dν(x) (5.4.64) −∞

We also define

k(λ) = ig(λ)

(5.4.65)

where g is given by (5.4.30). Thus, e−ik(λ) is a pth root of the eigenvalues, η, of Tp (λ) of larger magnitude. So, in particular, γ(λ)1/p = |e−ik(λ) |

(5.4.66)

We state a result about zeros of G00 in the next theorem but defer its proof slightly: Theorem 5.4.14. Let J be a two-sided periodic Jacobi matrix. Then G00 (λ) vanishes precisely once in each gap at points we label p−1 {µj }p−1 j=1 and nowhere else. For each λ ∈ C \ e ∪ {µj }j=1 , there exist 2 solutions u± n (λ) of (5.4.1) which are ℓ at ±∞ and obey u± 0 (λ) = 1

(5.4.67)

± They are analytic on C \ e ∪ {µj }p−1 j=1 . In addition, un (λ) have limits as λ → eint from above and below u± n (λ ± i0). Moreover,

(i)

± ¯ u± n (λ) = un (λ)

(ii)

− u+ n (λ + i0) = un (λ + i0)

(iii)

u+ n (λ

(iv)

k(λ) = π[n(λ) − 1]

+ i0) =

u− n (λ

− i0)

all λ ∈ C \ e ∪ {µj }p−1 j=1

(5.4.68)

all λ ∈ e

(5.4.69)

for λ ∈ e

(5.4.71)

all λ ∈ e

(5.4.70)

We have ±ink(λ) ± u± vn n = e

(5.4.72)

with vn± (λ) periodic, that is, ± vn+p = vn±

(5.4.73)

Remarks. 1. Except for a different normalization, {u± (λ + i0) | λ ∈ e} are the plane wave solutions discussed in (5.3.30). 2. For λ ∈ e, (5.4.72) shows that u± n (λ + i0) are almost periodic in n (unless N(λ) is rational, in which case they are periodic).

308

5. PERIODIC OPRL

3. There is a slight misstatement in the theorem. It can happen that G00 (λ) has no zero in some gap—that’s the case where G00 (λ) → 0 so one approaches the edge of a gap. This point will be explained below. Proof. The existence of u± is Theorem 5.4.12 supplemented by the discussion of G on gaps in σ(J). (i) is immediate from Tp (λ) = ¯ (ii) follows by noting that since λ ∈ e has Tp (e) real, if Tp s = ηs, Tp (λ). then Tp s¯ = η¯s¯ = η −1 s¯ since |η| = 1. (iii) follows from (i) and (ii). (iv) follows from (5.4.35) which implies on R 1 dν Im g ′ (λ) = − π dλ

(5.4.74)

Since g(λ) ∼ log(λ) near λ = i∞, Im g(λ) = π for λ ∈ R near −∞, so Im g(λ) = π[1 − n(λ)] Since Re g(λ) = 0 on e (i.e., |η| = 1 there), we find (5.4.71). Since ∓ ± e±ik(λ)p = η ∓1 and u± n+p = (η un , we get (5.4.72)/(5.4.73). We next turn to when G00 (λ) = 0. Theorem 5.4.15. Let J be a two-sided periodic Jacobi matrix. Then the p − 1 zeros of pp−1 (λ) lie one in each gap. At each such zero, λ0 , exactly one of the following holds: (i) it is an eigenvalue of J0+ , in which case λ0 is in the interior of a gap; − (ii) it is an eigenvalue of J−1 , in which case λ0 is in the interior of a gap; (iii) λ0 is at a gap edge, in which case there is a periodic or antiperiodic solution of Ju = λ0 u which vanishes at n = 0. The zeros of G00 (λ) in C \ e are precisely the points in C \ e where pp−1(λ) = 0. x-ref?

Remarks. 1. There are many proofs that pp−1 has one zero per gap besides the one we will give here; see TK. 2. If pp−1 has a zero at a boundary point of an open gap, we say that J0+ has a resonance at λ0 . 3. In a sense we will make precise below (see Theorem 5.4.18), resonances are also zeros of G00 ; we will prove if λ0 is the edge of an open gap, then lim G00 (λ) = 0 λ→λ0 λ∈e /

if λ0 is a resonance and ∞ if it is not.


309

Proof. We first analyze the zeros of pp−1 . We use the same device used in the proof of (iv) in Theorem 5.3.4. Define J0+ (µ), 0 ≤ µ ≤ 1, to be the half-line periodic Jacobi matrix with Jacobi parameters an (µ) = (1 − µ) + µan bn (µ) = µbn

(5.4.75) (5.4.76)

which interpolates between the free Jacobi matrix and J0+ . Let pp−1(λ, µ) be the associated orthogonal polynomials. Since pp−1 is a multiple of an orthogonal polynomial, its zeros are all real. We will show in a moment that it cannot have zeros in any eint j . At µ = 0, pp−1 is a Chebyshev polynomial of the second kind, that is, sin(pθ) pp−1 (2 cos θ, µ = 0) = c sin θ which vanishes precisely at the points 2 cos( jπ ), j = 1, 2, . . . , p − 1 p with each zero simple. These are the locations of the closed gaps of J0+ (µ = 0) which, viewed as a period p Jacobi matrix, has all gaps closed. As µ varies, the zeros move continuously, must stay on R and cannot go into the interiors of bands. Thus, they stay trapped, one in each gap. This concludes the proof of the first statement in the theorem. By (5.4.2), 1 pp (λ) Tp (λ) = 0 ap pp−1 (λ) so 1 pp−1 (λ0 ) = 0 ⇔ is an eigenvector of Tp (λ0 ) (5.4.77) 0 First of all, this means λ0 ∈ / eint j , for there Tp (λ0 ) has eigenvalues ±iθ e (θ ∈(0, π)), and so linearly independent nonreal eigenvectors. If 10 is an eigenvector, let η be the eigenvalue. If η = ±1, then ∆(λ0 ) = ±2, and we are at a band edge. If |η| < 1, Tn (λ) 10 defines a solution of (5.4.1) that goes to zero like |η|n/p as n → +∞, and so is an eigenvector of J0+ . Similarly, if |η| > 1, Tn (λ) 10 defines a solution − decaying as |η|−|n|/p as n → −∞, and so is an eigenvector of J−1 . This proves the second assertion of the theorem. Finally, G00 (λ0 ) = 0 for λ0 ∈ C \ e if and only if either u˜+ 0 = 0 or − ± ± u˜0 = 0 where u˜ are normalized by u˜1 = 1. As we have seen, that happens if and only if 10 is an eigenvector with eigenvalue |η| < 1 or |η| > 1, and so is a zero of pp−1 (λ).

310

5. PERIODIC OPRL

Next we turn to the significance of (5.4.69): Theorem 5.4.16. For any λ ∈ eint and any n, limε↓0 Gnn (λ + iε) ≡ Gnn (λ + i0) exists and Re[Gnn (λ + i0)] = 0

(5.4.78)

Remarks. 1. For reasons we discuss in the Notes, either (5.4.69) or (5.4.78) is described by saying that J is reflectionless. 2. Reflectionless Jacobi matrices will be a major theme in Sections 7.5 and 7.6. Proof. By (5.4.69), − + − W = a0 (u+ 1 u0 − u0 u1 ) = −W

so W is pure imaginary. Thus, using (5.4.69) again and (5.4.40), Gnn =

2 |u+ n| ∈ iR W

(5.4.79)

We have just seen that (5.4.69) implies (5.4.78). enough, the converse holds:

Interestingly

Theorem 5.4.17 (Gesztesy–Krishna–Teschl [159]; Sodin–Yuditskii [404]). Suppose J is a two-sided Jacobi matrix, and that for some λ0 ∈ R, we have ± (a) limε↓0 u± n (λ0 + iε) = un exists for all n + + (b) w(λ0 + i0) ≡ an (un+1u− n − un un+1 ) 6= 0 (c) Re Gnn (λ0 + i0) = 0 for n = 0, −1, 1 ± ± (d) u± 0 6= 0, u1 6= 0, u−1 6= 0, b0 6= λ Then + u− (5.4.80) n = un ± Proof. Since u± 0 6= 0, we can normalize so u0 = 1. Then (c) is equivalent to − Im(u+ n un ) = 0 for n = ±1

Re w = 0

(5.4.81)

± ± Define v1± = a1 u± 1 , v−1 = a0 u1 . Then (5.4.81) plus (5.4.1) imply

Re v1+ = Re v1− + v± ±

+

+ v−1

=

v1−

+

+ − Im v±1 v±1 = 0

− v−1

= λ − b0

(5.4.82) (5.4.83)

Writing vj± = |vj± |eiϕj , the second equation in (5.4.82) and u± 1 6= 0 ± u−1 6= 0 implies + + ϕ− ϕ− (5.4.84) 1 = −ϕ1 −1 = −ϕ−1


311

By (5.4.83) and b0 − λ 6= 0, one of v1+ or v1− has nonzero real part, in which case (5.4.84) and the first equation in (5.4.82) implies either + v1− = v1+ or v−1 = v1− . In either case, this plus u− 0 = u0 implies (5.4.80). One consequence of the fact that Gnn is purely imaginary is a remarkable explicit formula of Craig [91]: Theorem 5.4.18 (Craig [91]). Suppose α1 < β1 < α2 < β2 < · · · < αℓ+1 < βℓ+1

are distinct real numbers. Suppose that G(z) is analytic on C \ ∪ℓ+1 j=1 [αj , βj ] with (a) Im G(z) > 0 for Im z > 0 (b) G(¯ z ) = G(z) (5.4.85) (c) For a.e. x ∈ ∪ℓ+1 j=1 (αj , βj ), Re G(x + iε) =0 Im G(x + iε)

(5.4.86)

1 1 G(z) = − + o z z

(5.4.87)

lim ε↓0

(d) Near ∞,

Then there exist xj ∈ [βj , αj+1] for j = 1, 2, . . . , ℓ so that Y −1/2 ℓ ℓ+1 Y G(z) = − (z − xj ) (z − αj )(z − βj ) j=1

(5.4.88)

j=1

where the branch of square root which is O(z ℓ+1 ) near ∞ is taken. The only zeros of G on C \ ∪ℓ+1 j=1 [αj , βj ] are at those xj in (βj , αj+1 ). If xj = βj or αj+1 , then G(x) → 0 as x → xj from (βj , αj+1). If xj ∈ {βj , αj+1}, then G(x) → ∞ as x → xj in (βj , αj+1). Remarks. 1. We emphasize that this theorem is not specific to the periodic case—the intervals are arbitrary disjoint closed intervals. We will call the set e = ∪ℓ+1 j=1 [αj , βj ] a finite gap set, or sometimes an ℓ-gap set. In Section 5.13, we will see when such intervals arise as the intervals associated to an almost periodic problem. 2. In the periodic case, only open gaps contribute. In fact, if we added both the edges and zero for a closed gap, they would cancel in (5.4.88) in any event.

312

5. PERIODIC OPRL

Proof. By the Herglotz representation theorem (Theorem 2.3.6), there is a measure dη on ∪ℓ+1 j=1 [αj , βj ] so that Z dη(x) G(z) = (5.4.89) x−z In particular, G(z) < 0 on (βℓ+1 , ∞)

G(z) > 0 on (−∞, α1 ) On any gap (βj , αj+1),

(5.4.90)

Z

dη(x) >0 (5.4.91) (x − z)2 so G is strictly monotone. In particular, there is at most a single zero in each gap, say at xj . If there is no zero in (βj , αj+1 ), we set xj = βj if G(x) > 0 on (βj , αj+1) and xj = αj+1 if G(x) < 0 on (βj , αj+1). Define for z ∈ C+ , H(z) = log G(z) (5.4.92) with the branch of log picked so that near ∞, ′

G (z) =

H(reiθ ) = − log r + i(π − θ) + O(r −1)

(5.4.93)

H is a Herglotz function since arg G ∈ (0, π) on C+ . For a.e. x ∈ R by definition of xj , by (5.4.90) and (5.4.86), π on ∪ℓ+1  j=1 (αj , βj ) 2     0 on (−∞, α1 ) Im H(x + i0) = π on (βℓ+1 , ∞) (5.4.94)    0 on each (xj , αj+1)    π on each (βj , xj ) e Let H(z) be defined on C+ by

e H(z) = log[RHS of (5.4.88)]

with the branch chosen so e iθ ) = − log r + i(π − θ) + O(r −1) H(re

(5.4.95)

e also obeys (5.4.94). It follows by the It is easy to see that Im H general Herglotz representation theorem discussed in the Notes to Section 2.3 that for some A, B, e H(z) = H(z) + Az + B

and (5.4.92)/(5.4.94) then imply A = B = 0. The assertions about zeros follow from the definition of xj and about the behavior of G as x ↓ βj or x ↑ αj+1 from the explicit form of G.


313

As a first consequence, we get an “explicit” formula for the density of states. Corollary 5.4.19. Let dν be a density of states for a periodic Jacobi matrix, J, whose essential spectrum is ∪ℓ+1 j=1 [αj , βj ] with βj < αj+1 (j = 1, 2, . . . , ℓ). Then for suitable xj ∈ (βj , αj+1 ), we have Y 1/2 ℓ ℓ+1 1Y 1 dν(x) = |x − xj | χ∪ℓ+1 [αj ,βj ] (x) dx j=1 π j=1 |x − αj | |x − βj | j=1

(5.4.96)

Proof. By the theorem, we need only prove that in this case xj is = −G(x) so, since γ = 0 at in the interior of the jth gap. In a gap, ∂γ ∂x both ends, Z αj+1

G(x) dx = 0

(5.4.97)

βj

which implies G does not have a definite sign in (βj , αj+1) and so it must have a zero.

Remarks. 1. Basically, the xj are determined by (5.4.97). That there is a solution follows from the existence of a density of states (see TK for the case of general intervals). Uniqueness as well as a direct x-ref? proof of existence of solutions to (5.4.97) will be proven in Proposition 5.5.21. 2. This corollary also follows from (5.4.15). The xj are precisely the zeros of ∆′ (λ); see Lemma 5.4.10. Another consequence of Theorem 5.4.20 is Theorem 5.4.20 (Borg–Hochstadt Theorem). Let J be a periodic Jacobi matrix all of whose gaps are closed. Then for some α and β, an ≡ α and bn ≡ β. Remarks. 1. Periodicity is not used in this proof—only that it is reflectionless, that is, Re Gnn (λ) = 0 on σ(J) which is assumed to be an interval. 2. There are many other proofs of this theorem; see Section 11.14 of [391] for the OPUC analog. Also see Corollary 5.13.9 below. Proof. Since all gaps are closed, σ(J) = [γ, δ] for some γ, δ. By replacing J by κJ + λ for suitable κ, λ, we can arrange that σ(J) = [−2, 2], which we do henceforth. By Theorem 5.4.18 (with ℓ = 0), for each n, Gnn (λ) = −(λ2 − 4)−1/2

(5.4.98)

314

5. PERIODIC OPRL

If J (0) is the Jacobi matrix with an ≡ 1, bn ≡ 0, we see hδn , (J − λ)−1 δn i = hδn , (J (0) − λ)−1 δn i

(5.4.99)

hδn , J ℓ δn i = hδn , (J (0) )ℓ δn i

(5.4.100)

bn = 0

(5.4.101)

for all n and all λ ∈ C \ [−2, 2]. Looking at the Taylor series about λ = ∞, we see for all n and ℓ = 0, 1, 2, . . . that Taking ℓ = 1, we find for all n,

Then, using ℓ = 2, hδn , J 2 δn i = kJδn k2 and Jδn = an−1 δn−1 + an δn+1 , we find a2n + a2n−1 = 2 (5.4.102) which, given an > 0, implies a2n = a0

a2n+1 = a1

(5.4.103)

Now take ℓ = 4, n = 0 in (5.4.100) and use (5.4.103) plus hδ0 , J 4 δ0 i = kJ 2 δ0 k and J 2 δ0 = a0 a1 (δ−2 + δ2 ) + (a20 + a21 )δ0 to find 2(a0 a1 )2 + (a20 + a21 )2 = 6

(5.4.104)

Using (5.4.102), we see (a0 a1 )2 = 1, so (a0 − a1 ) = a20 + a21 − 2a0 a1 = 0

and thus a0 = a1 = 1, that is, J = J (0) .

Our final topic is to provide a second proof of Theorem 5.2.2 based on (5.4.69) and the fact that for λ ∈ eint , m(λ +

i0, J0+ )

u+ (λ + i0) = − 1+ a0 u0 (λ + i0)

(5.4.105)

by taking limits of (3.7.23). Similarly, m(λ +

i0, J0− )

u− (λ + i0) = − 0− a0 u1 (λ + i0)

(5.4.106)

Second Proof of Theorem 5.2.2. By (5.4.69), (5.4.105), and (5.4.106), we have (with m♯ given by (5.2.11)) that for λ ∈ eint j , m♯ (λ + i0) = m(λ + i0, J0+ )

(5.4.107)

Since (5.2.10) has real coefficients for λ ∈ e, m♯ also solves (5.2.10) and so, by analyticity, it solves (5.2.10) for all λ. + ♯ For a.e. λ ∈ eint j , Im m(λ + i0, J0 ) 6= 0, so m is distinct so it is the second solution.


315

Remarks and Historical Notes. As explained in the Notes to Section 5.1, much of the theory of periodic ODEs goes back to Hill’s equation. In particular, the use of discriminants goes back to Lyapunov [285], Hamel [186], Haupt [188], and Kramers [240]. Magnus–Winkler [286] and Eastham [115] provide monograph presentations. The discussion for Jacobi and/or discrete Schrödinger equations can be found in Hochstadt [195], van Moerbeke [438], Toda [430], Last [263], and Teschl [425]. Instead of using selfadjointness of J(θ) to conclude ∆−1 ([2, 2]) ⊂ R, one can proceed as follows: If ∆(λ) ⊂ (−2, 2), all solutions of (5.4.1) are bounded, and by cutting off a bounded solution, one gets wn ∈ ℓ2 , so k(J − λ)wn k/kwn k → 0, implying λ ∈ σ(J). Thus λ is real. By continuity and analyticity, ∆−1 ([−2, 2]) = ∆−1 ((−2, 2)). There is a more “physical” meaning to “reflectionless.” It can be proven (in analogy with the Schrödinger case discussed by Davies– Simon [97]) that if Hac is the range of the projection onto the a.c. subspace for J, an arbitrary bounded two-sided Jacobi matrix, then ± there exist spaces Hℓ,r so Hac = Hℓ+ ⊕ Hr+ = Hℓ− ⊕ Hr− so that ϕ ∈ Hℓ± ⇔ for all n ∈ Z, limt→∓∞ kχ[n,∞)e−itJ ϕk = 0 (and similarly for Hr± with χ[n,∞) replaced by χ(−∞,n] ). Thus, for example, Hℓ+ is the set ϕ for which e−itJ ϕ move to −∞ (the left) as t → −∞. For this point of view, reflectionless means Hℓ+ = Hr−

(5.4.108)

so that there is no reflection back from where e−itJ ϕ came from! In many cases, including ours, one can use stationary phase estimates to prove that in terms of (5.3.35), Hℓ+ is the span of u+ (λ + i0) (in the sense that in (5.3.35), f − = 0) and Hr+ of u− (λ + i0). On the other hand, since e−itJ = e+itJ , Hℓ+ = Hℓ− and Hr+ = Hr− . Therefore, (5.4.108) is equivalent to Hr+ = Hℓ+ , which is equivalent to (5.4.69)— which is what we have called reflectionless. TK on Remling See the Notes to TK for discussion of the history of the density of states, Lyapunov exponent, and the Thouless formula. TK on continuum eigenfunctions and polynomially bounded solutions. Theorem 5.4.18 is due to Craig [91] who considered some situations with infinitely many gaps. His proof (and ours) depends on an exponential Herglotz representation (i.e., passing to the log and then writing

x-ref?

x-ref? x-ref?

x-ref?

316

5. PERIODIC OPRL

down a Herglotz representation), first emphasized by Akhiezer–Krein [15] and used extensively by Aronszajn–Donoghue [27]. The continuum analog of what we have called the Borg–Hochstadt theorem is due to Borg [54]. The Jacobi matrix analog is due to Hochstadt [195]; see also Flaschka [131]. The proof we give here is closely related to a proof of Clark et al. [86]. 5.5. Potential Theory, Equilibrium Measures, the DOS, and the Lyapunov Exponent Because of (5.4.29) and γ(z) = 0 on e, there is a close connection between potential theory and the fundamental objects of the periodic theory—the density of states will be the potential theoretic equilibrium measure, γ will be the potential theoretic Green’s function, and (a1 . . . ap )1/p will be the logarithmic capacity. This realization shows that dν is intrinsic to e and will be important when we discuss other finite gap situations in Chapter 9. We begin this section with a brief minicourse on two-dimensional potential theory. Define on C, G0 (z) = log(|z|−1 ) (5.5.1) If µ is a measure on C of compact support, its logarithmic potential is defined by Z Φµ (z) = G0 (z − w) dµ(w) (5.5.2)

This integral converges if z ∈ / supp(dµ), and since dµ has compact support, G0 (z −w) is uniformly bounded below for (z, w) ∈ supp(dµ)× supp(dµ), so the integral for each z ∈ supp(dµ) either converges or diverges to +∞, in which case we set Φµ (z) = +∞. The same semiboundedness lets us use Fubini’s theorem to conclude that for any two (positive) measures of compact support, Z Z Φµ (z) dν(z) = Φν (z) dµ(z) (5.5.3)

Potentials enter naturally in studying growth of polynomials as n → ∞. For if n Y (n) Pn (x) = (x − xj ) (5.5.4) j=1

then

1 log|Pn (x)| = −Φνn (x) n

(5.5.5)

5.5. POTENTIAL THEORY AND HARMONIC MEASURE

where

317

n

νn =

1X δ (n) n j=1 xj

is the counting measure for the zeros. So if νn converges to ν∞ , one can hope that root asymptotics of Pn (i.e., the limiting behavior of |Pn (x)|1/n ) is connected to the potential of ν∞ . Φµ (z) is bounded below on supp(dµ), so Z E(µ) = Φµ (z) dµ(z) (5.5.6) Z = log(|z − w|−1) dµ(z)dµ(w) (5.5.7)

is either finite or diverges to +∞. E(µ) is called the potential energy of µ or, for short, the energy of µ. Given a compact set e ⊂ C, we consider all probability measures, M+,1(e), on e. We say e has capacity zero if and only if E(µ) = ∞ for all µ ∈ M+,1(e). Otherwise, we define the capacity, C(e), of e by C(e) = exp(− inf(E(µ) | µ ∈ M+,1(e)))

(5.5.8)

and we say e has positive capacity.

Remark. The use of exp in (5.5.8) is as an inverse for log. We will eventually show (with [a, b] ⊂ R a closed interval; see Example 5.5.20) C([a, b]) = 14 (b − a)

(5.5.9)

We are heading towards a proof of Theorem 5.5.1. Let e ⊂ C be a compact set with positive capacity. Then there is a unique measure, ρe, in M+,1(e) (called the equilibrium measure for e) so that E(ρe) =

min

µ∈M+,1 (e)

E(µ) = log(C(e)−1 )

(5.5.10)

Lemma 5.5.2. (i) G0 is harmonic on C \ {0}. (ii) We have, as a tempered distribution, (∆G0 )(x) = −2πδ(x)

(iii) For any x0 , and r, Z 2π dθ h(r, x0 ) ≡ G0 (x0 + reiθ ) = G0 (x0 ) 2π 0 ≤ G0 (x0 )

(5.5.11)

r ≤ |x0 | r > |x0 |

(5.5.12)

(iv) For x0 fixed, h(r, x0 ) is monotone decreasing as r increases.

318

5. PERIODIC OPRL

(v) If j ∈ C0∞ (R2 ), j(Rx) = j(x) for rotations R about 0, j ≥ 0, Z j(x) d2 x = 1 (5.5.13) and

jε (x) = ε−1 j(ε−1 x)

(5.5.14)

and if (ε)

(ε)

then G0 is C ∞ ,

G0 (x) = (jε ∗ jε ∗ G0 )(x) (ε)

G0 ≤ G0

lim ε↓0

(ε) G0 (x)

= G0 (x)

(5.5.15) (5.5.16) (5.5.17)

Indeed, for any r > 0, there is A > 0 so (ε)

|x| > r and ε < A ⇒ G0 (x) = G0 (x)

(5.5.18)

E(µ) = lim E(µ ∗ jε )

(5.5.19)

(vi) For any (positive) measure µ,

ε↓0

(vii) For any (positive) measure µ, Z (ε) E(µ) = lim G0 (x − y) dµ(x)dµ(y) ε↓0

(5.5.20)

Remark. The proof provides an explicit formula for h(r, x0 ). Proof. (i), (ii) Since ∆ in polar coordinates is given by ∆f =

1 ∂ ∂ 1 ∂2 r f+ 2 2f r ∂r ∂r r ∂θ

(5.5.21)

we see ∆G0 = 0 (5.5.22) for z 6= 0, first classically and then as distributions. For any f ∈ C0∞ , say f (z) = 0 if |z| ≥ R, Z Z 2 G0 (∆f ) d x = lim [(G0 ∆f ) − (f ∆G0 )] d2x by (5.5.22) ε↓0 R>|r|>ε Z ~ − f ∇G ~ 0 ] d2 x = lim div[G0 ∇f ε↓0 R>|r|>ε Z ~ )(z)G0 (z) rdθ = lim f (z)[− 1r ] − (∇f ε↓0

|r|=ε

= −2πf (0)


319

by Gauss’s theorem, continuity of f and |z|G0 → 0 as z → 0. This proves (5.5.11). (iii), (iv) (5.5.11) and Gauss’s theorem and r 6= x0 , Z 2π ∂h 1 r = ∇G0 (x + reiθ ) · n b dσ ∂r 2π 0 Z 1 = ∆G0 d2 x 2π |y−x|≤r ( 0 r < |x0 | = (5.5.23) −1 r > |x0 | Since h is continuous at r = x0 and h(r, x0 ) → G0 (x0 ) as r ↓ 0, we get (5.5.12), and monotonicity by (5.5.23). (5.5.23) and h(r, x0 ) = G0 (x0 ) at r = 0 leads to the explicit formula h(r, x0 ) = log(min(|x0 |, r)−1)

(5.5.24)

The analog of this for potentials on R3 goes back to Newton! (v) By (5.5.12), if supp(j(x)) ⊂ {x | |x| ≤ ρ0 } obeys (5.5.13), j ≥ 0, and j(Rx) = j(x) for rotations, then j ∗ G0 obeys (j ∗ G0 )(x) = G0 (x)

≤ G0 (x)

|x| > ρ0

|x| ≤ ρ0

(5.5.25)

Moreover, if j is C ∞ , so is j ∗ G0 by general results on convolutions of distributions. (5.5.25) implies (jε ∗ G0 )(x) = G0 (x) if x ≥ 2ερ0 where ρ0 is such that supp(j) ⊂ {x | |x| ≤ ρ0 }. (vi), (vii) This follows from (5.5.16) and (5.5.17). If E(µ) < ∞, then dominated convergence implies (5.5.19) since Z E(µ ∗ jε ) = (G0 ∗ jε ∗ jε )(x − y) dµ(x)dµ(y) (5.5.26) If E(µ) = ∞, it is obvious, by (5.5.17), that for any ρ > 0, Z lim inf E(µ ∗ jε ) ≥ G0 (x − y) dµ(x)dµ(y) |x−y|≥ρ

Taking ρ ↓ 0 and using monotone convergence, we see E(µ ∗ jε ) → ∞. As a consequence of this lemma:

Theorem 5.5.3. (i) For any measure µ of compact support in C, Φµ (z) is lower semicontinuous in z and superharmonic. On C \ supp(µ), Φµ is harmonic. (ii) For fixed z, Φµ (z) is weakly lower semicontinuous in µ.

320

5. PERIODIC OPRL

(iii) µ 7→ E(µ) is weakly lower semicontinuous. Remarks. 1. Lower semicontinuity in (iii) means µn → µ ⇒ lim inf E(µn ) ≥ E(µ) (mnemonic: the value at the limit can be lower). Equivalently, E −1 ((−∞, a]) is closed for all a. Equivalently, E −1 ((a, ∞]) is open for all a. 2. g, taking values in (−∞, ∞], is called superharmonic if it is lower R dθ semicontinuous, |g(z0 + reiθ )| 2π < ∞ for all z0 ∈ C, r > 0, and if Z dθ g(z0 + reiθ ) ≤ g(z0 ) (5.5.27) 2π

This implies (one inequality comes from (5.5.27) and the other from lower semicontinuity) Z dθ = g(z0 ) (5.5.28) lim g(z0 + reiθ ) r↓0 2π 3. g is harmonic if it is continuous and equality holds in (5.5.27); equivalently, if g is C ∞ with ∆g = 0. Proof. Let jε be as in the lemma and Φµ(ε) = jε ∗ Φµ

Eε (µ) = E(µ ∗ jε )

(5.5.29)

(ε)

Then Φµ (z) is jointly continuous in µ and z. By the lemma and monotone convergence, Φµ (z) = sup Φ(ε) µ (z) ε

E(µ) = sup Eε (µ)

(5.5.30)

ε

which implies the claimed semicontinuity results (if g = supn gn , then g −1((a, ∞)) = ∪n gn ((a, ∞))). The mean inequalities are immediate from (5.5.12) and averaging in x0 . Proposition 5.5.4. (a) Let f ∈ C0∞ (R2 ) with Z f (x) d2 x = 0

(5.5.31)

Then 2 Z Z Z 1 f (y) 2 −1 f (x)f (y) log(|x − y| ) dxdy = d y d2 x 2π |x − y| (5.5.32) (b) Under the hypothesis of (a), Z b 2 |f(k)| 2 LHS of (5.5.32) = 2π dk (5.5.33) |k|2


(c) Let µ be a (positve) measure of compact support. Then Z |b µ(k)|2 2 E(µ) < ∞ ⇔ d k 0

2

d2 x

(5.5.36)

(5.5.37) (5.5.38)

(5.5.39)

Remarks. 1. f ∈ C0∞ implies the integral on the left side of (5.5.32) is absolutely convergent. Since R Z f (y) d2y 1 f (y) d2y = +O |x − y| |x| |x|2

the integral on the right side is finite if and only if (5.5.31) holds. So (5.5.32) only holds if (5.5.31) does. 2. If µ has compact support, µ b(k) is defined by Z −1 µ b(k) = (2π) e−ik·x dµ(x) (5.5.40)

and is an entire function of ~k. 3. Because µ, ν have compact support, the integral in (5.5.35) is either convergent or it diverges to +∞. 4. B(µ, ν) may not be positive. For example, if dµ is the probability measure uniformly distributed in {z | |z| ≤ 2}, then B(µ, µ) = − log 2. 5. (5.5.39) is called “strict conditional positive definiteness.” 6. One can understand parts of this proposition in terms of the distribution G0 (x) = log|x|−1 . Since (5.5.11) holds, b0 (k) = 1 k2 G

(5.5.41)

322

5. PERIODIC OPRL

b0 (k) = 1/k 2 as a distribution because 1/k 2 is not This does not imply G b 0 (k) is a distribution, a distribution since it is not L1 at k = 0. Rather G 2 2 which is a regularization of 1/k . If h ∈ S(R ) and h(0) = 0, then Z b0 (h) = h(k)k −2 d2 k G (5.5.42) which explains (5.5.33). Proof. (a) Let hα (x) = |x|−1−α (5.5.43) for α > 0. Then, by rotation invariance and scale covariance, (hα ∗ hα )(x) = Cα |x|−2α

where

Cα =

Z

(5.5.44)

1 1 d2 y 1+α |y| |y − (1, 0)|1+α (1)

(5.5.45) (2)

(3)

We can write Cα as a sum of three terms: Cα , Cα , Cα , where the first is the integral over |y| < 2, the second the integral over |y| > 2 with integrand 1 1 1 − (5.5.46) |y|1+α |y − (1, 0)|1+α |y|1+α and Z 2π 1 (3) Cα = d2 y = (2)−2α (5.5.47) 2+2α |y| 2α |y|>2 (1)

(2)

Since Cα and Cα have finite limits as α ↓ 0, we see lim αCα = π α↓0

By (5.5.44) and (5.5.31), 2 Z Z f (y) 2 2 d y d x = Cα f (x)f (y)[|x − y|−2α − 1] |x − y|1+α

(5.5.48)

(5.5.49)

Take α ↓ 0 in each side of (5.5.49). On the left side, since (5.5.31) holds, Z f (y) d2 y = O(|x|−2−α ) (5.5.50) |x − y|1+α uniformly in α. So the integral converges to (2π)× RHS of (5.5.32). On the other hand, (2α)−1(|x − y|−2α − 1) → log|x − y|−1

(5.5.51)

as α ↓ 0. So by dominated convergence and (5.5.48), the right side of (5.5.49) converges to (2π)× LHS of (5.5.32).


323

(b) By rotation and scale invariance of h, b h0 = ch0 and c = 1 since b b h0 = h0 . Thus, by f[ ∗ g = (2π)fbb g , we have if Z f (x) 2 g(x) = dy (5.5.52) |x − y| then

gb(k) = (2π)|k|−1fb(k) and so, by the Plancherel theorem, Z Z b 2 1 |f(k)| 2 2 2 |g(x)| d x = 2π dk 2π k2 proving (5.5.33). (c) Let f0 be a fixed function in C0∞ (R2 ) with Z f0 (x) d2 x = 1

(5.5.53)

(5.5.54)

(5.5.55)

Let jε be as in Lemma 5.5.2 and let fn be defined by By (5.5.19),

j1/n ∗ dµ = fn (x) d2 x

(5.5.56)

E(µ) < ∞ ⇔ lim E(fn d2 x) < ∞

(5.5.57)

n→∞

Define B by (5.5.35) and E((fn − f1 )d2 x) by (5.5.36). Of course, fn − f1 obeys (5.5.31) and E((fn − f1 )d2 x) is given by (5.5.32) with f = fn − f1 . E(f d2 x) < ∞ and fixed and Z 2 2 B(fn d x, f1 d x) = fn (x)Φf1 (x) d2 x (5.5.58) Z → Φf1 (x) dµ(x) (5.5.59) w

since fn d2 x −→ dµ. Thus, (5.5.57) becomes

E(µ) < ∞ ⇔ lim E((fn − f1 )d2 x) < ∞ n→∞ Z b |fn − fb1 |2 2 ⇔ lim d k (log k)4/5 . Since N 2 N X 1 X 2 ρk,n ≥ ρk,n N n=1 n=1

(5.5.85)


327

we have (log k)1/5

X

(log k)ρ2k,n

n=1

≥ (log k)

4/5

(logX k)1/5

ρk,n

n=1

2

(5.5.86)

On the other hand, X 2 X X ∞ ∞ ∞ 6 2 −6 ρk,n ≤ n ρk,n n n=N

n=N

n=N

≤ C −1 N −5 so

∞ X

n6 ρ2k,n

n=(log k)1/5

≥ C(log k)

∞ X

n6 ρ2k,n

n=N ∞ X

ρk,n

n=(log k)1/5

P P(log k)1/5 Since ∞ ρk,n ≥ n=1 ρk,n implies either n=1 1 , we see, by (5.5.85), that 2 Thus, by (5.5.80), Thus for k large,

1 2

2 or

(5.5.87) P∞

(log k)1/5

ρk,n ≥

log(C(ek )−1 ) ≥ C(log k)4/5

(5.5.88)

Φk ( n1 ) ≥ C(log k)4/5

(5.5.89)

lim Φk ( n1 ) > Φk (0) and Φk , which is bounded, is discontinuous at x = 0 ∈ ek .

(5.5.90)

Thus, continuity properties of potentials are not automatic and we need to prove something in nonpathological cases. Because we are interested in e ⊂ R and this case as some simplifications, we will study that case but mention the general situation in the Notes. Theorem 5.5.6. Let supp(µ) ⊂ R. Then Φµ is continuous on C if and only if Φµ ↾ supp(µ) is continuous on supp(µ). Proof. We will prove the contrapositive, that is, if Φµ is discontinuous on C, its restriction to e ≡ supp(µ) is discontinuous. Since Φµ is lower semicontinuous, if it is discontinuous, there exist zn → z∞ , so lim Φµ (zn ) = a > Φµ (z∞ )

n→∞

Φµ is harmonic, hence continuous off e, so z∞ ∈ e. If x, y, w are real, |x + iy − w|−1 ≤ |x − w|−1 , so e ⊂ R ⇒ Φµ (x + iy) ≤ Φµ (x)

(5.5.91)

(5.5.92)

328

5. PERIODIC OPRL

and thus, Re zn → z∞ and

lim inf Φµ (Re zn ) ≥ a > Φµ (z∞ ) n→∞

(5.5.93)

Thus, by passing to a subsequence, we can suppose (5.5.91) holds and zn ∈ R with either zn > z∞ or zn < z∞ for all n. For notational simplicity, we will suppose zn > z∞ for all n. Suppose (α, β) ⊂ R \ e with α, β ∈ e. Since x → log|x|−1 is convex on (0, ∞) and we can use monotone convergence at the endpoints, Φµ is convex and continuous on [α, β] (continuous in the extended sense that ∞ is an allowed value at α or β). By convexity, sup Φµ (x) = max(Φµ (α), Φµ (β))

(5.5.94)

x∈[α,β]

The above continuity plus (5.5.91) implies z∞ is not the lower end of an open interval in R \ e. Thus, there are zn± ∈ e, with zn ∈ [zn− , zn+ ] and zn+ → z∞ . By (5.5.94), lim inf max(Φµ (zn+ ), Φµ (zn− )) > Φµ (z∞ )

so, since zn+ → z∞ , Φµ ↾ e is not continuous.

Theorem 5.5.7. If e ⊂ R and C(e) > 0, there exists ν ∈ M+,1 (e), so Φν is continuous on C. Proof. Pick µ in M+,1(e) with E(µ) < ∞. Then Φµ ∈ L1 (dµ), so by Lusin’s theorem (see the Notes), there are Kn ⊂ e compact with µ(Kn ) → 1 and Φµ ↾ Kn continuous. Pick Kn0 with µ(Kn0 ) > 0 and let η = µ ↾ Kn 0 (5.5.95) By the choice, Φµ is continuous on Kn0 and so, Φη = Φµ − Φµ−η

(5.5.96)

is upper semicontinuous on Kn0 . Of course, it is lower semicontinuous there, so Φη is continuous on Kn0 , and so on supp(η). Thus, Φη is continuous by Theorem 5.5.6. By µ(Kn0 ) > 0, η 6= 0, so ν = η/η(e) ∈ M+,1 (e) with a continuous potential. For any Borel subset, X ⊂ C, we define

C(X) =

sup C(e)

(5.5.97)

e⊂X e compact

= exp(inf{E(µ) | supp(µ) compact, supp(µ) ⊂ X, µ(C) = 1}) (5.5.98)


329

Thus C(X) = 0 if and only if E(µ) = ∞ for any measure µ with compact support in X. If an event depends on z and fails on a Borel subset of capacity zero, we say the event holds quasi-everywhere (q.e.). Corollary 5.5.8. For any measure, µ, of compact support, Φµ (z) < ∞ q.e. In fact, {z | Φµ (z) = ∞} is a Gδ of capacity zero. Remark. It can be shown (see Landkof [258]) that if X is any bounded Gδ of capacity zero, there is a measure, µ, of compact support so that X = {z | Φµ (z) = ∞}. Proof. Since X = {z | Φµ (z) = ∞} =

∞ \

n=1

{z | Φµ (z) > n}

(5.5.99)

the set is a Gδ . Suppose X has positive capacity. Then it contains a compact K with C(K) > 0. Let ν ∈ M+,1 (K) so Φν is continuous. Then Φν is uniformly bounded on supp(dµ), so Z Φν (z) dµ(z) < ∞ (5.5.100) But Φµ (z) = ∞ on K, so

Z

Φµ (z) dν(z) = ∞

This contradicts (5.5.3), so C(X) = 0.

(5.5.101)

Proposition 5.5.9. Let η be a measure of compact support with E(η) < ∞, and let X ⊂ C with C(X) = 0. Then η(X) = 0. Proof. Since any measure is inner regular, it suffices to prove this result when X is compact. If η(X) 6= 0, η ↾ X (i.e., A 7→ η(X ∩ A)) is a nonzero measure. Moreover, if r = supx,y∈supp(η) |x − y|, then Z Z −1 log(r|x − y| ) dη(x)dη(y) ≤ log(r|x − y|−1) dη(x)dη(y) X×X

C×C

= log r[η(C)2] + E(η) < ∞

so E(η ↾ X) < ∞, showing C(X) > 0. Thus, C(X) = 0 ⇒ η ↾ X = 0 ⇒ η(X) = 0. A second major theorem in potential theory is Theorem 5.5.10 (Upper Envelope Theorem). Let e ⊂ R be comw pact and let νn , ν∞ ∈ M+,1 (e) with νn −→ ν∞ . Then

330

5. PERIODIC OPRL

(i)

(ii)

Φνn (z) → Φν∞ (z)

(5.5.102)

lim inf Φνn (z) ≥ Φν∞ (z)

(5.5.103)

for all z ∈ C \ e.

for all z ∈ e. (iii) Equality holds in (5.5.103) for q.e. z ∈ e.

n

w

Remark. If νn gives weight 2nj+1 to { 2jn }2j=0 , dνn −→ dx, Lebesgue measure. At any dyadic rational lim inf Φνn (x) = ∞ but Φν∞ (x) < ∞. So equality in (5.5.103) may not hold everywhere. Proof. (i) For z ∈ C \ e, log|z − w|−1 is continuous in w ∈ e so (5.5.102) follows from the weak convergence. (ii) Let a < ∞ and Z (a) Φν (x) = log(min(a, |x − y|−1)) dν(y) (a)

(a)

Since Φν ≤ Φν and Φν (x) is weakly continuous in ν, (a) Φ(a) ν∞ (x) = lim Φνn ≤ lim inf Φνn (x)

(5.5.104)

(a)

Taking a → ∞, using Φν → Φν (by monotone convergence), we obtain (5.5.103). (iii) Let X ⊂ e be the set where strict inequality holds in (5.5.103). If C(X) > 0, use Theorem 5.5.7 to find η ∈ M+,1 (e) with supp(η) ⊂ X so that Φη is continuous. Then Z Φη (x) dν∞ (x) Z = lim Φη (x) dνn (x) Z = lim Φνn (x) dη(x) (by (5.5.3)) Z ≥ lim inf Φνn (x) dη(x) (by Fatou’s lemma) Z > Φν∞ (x) dη(x) (by definition of X and supp(η) ⊂ X) Z = Φη (x) dν∞ (x) (by (5.5.3))

The strict inequality is a contradiction, so C(X) = 0 and equality holds q.e.


331

A third major theorem in potential theory is Theorem 5.5.11 (Frostman’s Theorem). Let e ⊂ R be a compact set and let ρe be its equilibrium measure. Then (i) For all z ∈ C, Φρe (z) ≤ log(C(e)−1 ) (5.5.105) (ii) Equality holds in (5.5.105) for q.e. z ∈ e. (iii) Strict inequality holds in (5.5.105) on C \ supp(dρe). Remark. Equality may not hold everywhere on e. For example, if e = [−1, 1] ∪ {2} and ˜e = [−1, 1], then ρe = ρ˜e, so Φρe (2) = Φρ˜e (2) > log(C(e)−1 ) by (iii). R Proof. (i) Let f be a bounded Borel function on e so f dρe = 0. Then for ε real with |ε| small, (1 + εf ) dρe is a probability measure, so Z d E((1 + εf ) dρe) = 2 f (x)Φρe (x) dρe(x) = 0 dε This implies that there is a constant c so Φρe (x) = c Thus, c=

Z

dρe-a.e. x

Φρe (x) dρe(x) = E(ρe) = log(C(e)−1 )

(5.5.106)

(5.5.107)

By lower semicontinuity, (5.5.105) holds everywhere on supp(dρe). As noted in the proof of Theorem 5.5.6, Φρe (z) is convex and continuous on any interval [α, β] ⊂ R with (α, β) ∩ e = ∅ which, together with lim|z|→∞ Φρe (z) = −∞, implies that (5.5.105) holds on R, and then, by (5.5.105), on all of C. (ii) Let X = {x ∈ e | Φρe (x) < log(C(e)−1 )}

(5.5.108)

We need to prove C(X) = 0. If not, there is a measure dη concentrated on X with E(η) < ∞. In particular, E(tη + (1 − t)ρe) is finite for all t and is a quadratic function of t with Z d E(tη + (1 − t)ρe) = 2 Φρe (x)[dη − dρe] dt t=0 Z = 2 [Φρe (x) − log(C(e)−1 )] dη(x) (5.5.109) 0, then ρe(X) > 0. Proof. If ρe(X) = 0, then X ⊂ C \ supp(dρe) (since X is open), and thus the inequality in (5.5.105) is strict. But (5.5.105) holds q.e. on e, so C(X ∩ e) = 0. A closed set e ⊂ R is called potentially perfect if for all x0 ∈ e and ε > 0, C((x0 − ε, x0 + ε) ∩ e) > 0. It is easy to see that any compact e in R can be decomposed e = e1 ∪ e2 where e1 is potentially perfect and C(e2 ) = 0. The last corollary immediately implies: Corollary 5.5.13. Let e ⊂ R be compact. Then supp(dρe) = e if and only if e is potentially perfect. For purposes of solving the Dirichlet problem, one often defines the potential theorist’s Green’s function by Ge(z) = log(C(e)−1) − Φρe (z)

(5.5.111)

It is the unique function harmonic on C \ e, subharmonic on C with as |z| → ∞

Ge(z) = log(|z|) + O(1)

for q.e. x ∈ e

Ge(x) = 0

(5.5.112) (5.5.113)

It is unfortunate that spectral theorists use the term Green’s function for a different object (namely, (5.4.40)) than Ge, which is why we add “potential theorist’s”! Notice that as |z| → ∞, Ge(z) = log|z| − log(C(e)) + O( z1 )

(5.5.114)

Theorem 5.5.14 (Bernstein–Walsh Lemma). Let qn (x) be a polynomial of degree x and let kqn ke = sup |qn (x)| x∈e

for any compact e ⊂ R. Then, for all z, |qn (z)| ≤ kqn ke exp(nGe(z))

(5.5.115)


333

Proof. Fix ε > 0. Let gε (z) = log|qn (z)| − log kqn ke − (n + ε)Ge (z)

gε is harmonic on C \ e ∪ {zj }nj=1 where zj are the zeros of q. By ε > 0 and (5.5.114), gε (z) ∼ −ε log|z| → −∞ at ∞ and gε (z) → −∞ at the zj . Thus, for any δ > 0 and dist(z, e) > δ, we have gε (z) ≤

max

dist(z,e)=δ

gε (z)

By Frostman’s theorem, Ge(z) ≥ 0 for all z, so max

dist(z,e)=δ

gε (z) ≤

max |qn (z)| − max |qn (z)| z∈e

dist(z,e)=δ

→0

as δ ↓ 0. Thus,

gε (z) ≤ 0 R dθ first on C \ e and then, by gε (z) = limδ↓0 gε (z + δeiθ ) 2π , on e. Taking ε ↓ 0, we obtain (5.5.16). For applications of potential theory to periodic Jacobi matrices, we state a converse of Frostman’s theorem whose hypotheses can be weakened. Theorem 5.5.15. Let e ⊂ R be compact. Suppose η ∈ M+,1 (e) obeys Φη (x) = α for all x ∈ e (5.5.116) for some α. Then α = log(C(e)−1 ) (5.5.117) and η = ρe (5.5.118) Remark. By Remark 4 after Theorem 5.4.9, the potential theorist’s Green’s function, Ge, obeys Ge(x) = 0

for x ∈ e

(5.5.119)

if e is the spectrum of a two-sided Jacobi matrix. Proof. By (5.5.116), E(η) =

Z

Φη (x) dη(x) = α < ∞

(5.5.120)

so, by Proposition 5.5.9, η gives zero weight to the subset of e where equality fails in (5.5.105), that is, Φρe (x) = log(C(e)−1 )

a.e. dη

(5.5.121)

334

5. PERIODIC OPRL

Thus, by (5.5.3), −1

log(C(e) ) = =

Z

Z

=α

Φρe (x) dη(x) Φη (x) dρe(x) (by (5.5.116))

proving (5.5.117). Therefore, by (5.5.120), E(η) = log(C(e)−1 ), so (5.5.118) holds by uniqueness of minimizers. Theorem 5.5.16. Let µ, ν be two probability measures of compact support in R. Suppose Φµ (z) ≤ Φν (z) near infinity. Then µ = ν. In particular, if Φη (z) ≤ Φρe (z) for all z ∈ C+ or Φη (z) ≥ Φρe (z) for all z ∈ C+ (where supp(η) ⊂ R, e ⊂ R), then η = ρe. Proof. Since log|x − z|

−1

= log|z|

−1

|x| +O |z|

Φµ − Φν is harmonic at infinity and vanishes there. By the maximum principle, it is either identically zero off R or takes both signs near infinity. If it is identically zero off R, by averaging, it is zero on R and then µ = ν since ∆Φµ = −2πµ as distributions (by (5.5.11)). This completes our minicourse on potential theory, and we return to periodic Jacobi matrices: Theorem 5.5.17. Let e = ∪pj=1 ej be the spectrum of a two-sided Jacobi matrix, J, of period p. Let ∆ be its discriminant, let dν be given by (5.3.34) (or (5.4.15)), and let γ(z) be the Lyapunov exponent (given by (5.4.25) and (5.4.26). Then (i) dν is dρe, the equilibrium measure of e. (ii) C(e) is the capacity of e given by C(e) = (a1 . . . ap )1/p

(5.5.122)

(iii) γ(z) is the potential theorist’s Green’s function for e; equivalently, −γ(λ) − p−1 log(a1 . . . ap ) is the equilibrium potential for e. Proof. By (5.4.29) and (5.4.26) (which says γ(λ) = 0 for λ ∈ e), we have for x ∈ e, Φν (x) = − p1 log(a1 . . . ap ) (5.5.123)


335

By Theorem 5.5.15, ν = ρe and log(C(e)−1 ) = − 1p log(a1 . . . ap ), proving (5.5.122). By (5.4.29), γ(z) is the potential theorist’s Green’s function. This has two immediate corollaries about periodic problems: Corollary 5.5.18. If two two-sided periodic Jacobi matrices of period p have the same spectra, they have the same ∆, the same dν, and the same γ. Proof. Theorem 5.5.17 shows that ν and γ are intrinsic to e = spec(J). dν determines ∆ by (5.4.16) or γ by (5.4.26). Corollary 5.5.19. If e = ˜e1 ∪· · ·∪˜eℓ is the spectrum of a two-sided periodic Jacobi matrix, J, with ˜ej the connected components of e, then the harmonic measure of each ˜ej is rational. Remark. We will discuss the converse of this shortly. Proof. Each band ek has harmonic measure 1/p (see the remark after Theorem 5.4.4), so ˜ej , which is a union of ek ’s, has harmonic measure nj /p, which is rational. Example 5.5.20. Let e = [α, β]. This is the spectrum of the twosided Jacobi matrix with constant parameters bn = 12 (α + β) Thus,

an = 41 (β − α)

C([α, β]) = 41 (β − α) By translation and scaling (5.3.39), we see 1 1 dx dρ[α,β] (x) = π [(x − α)(β − x)]1/2 consistent with (5.4.96).

(5.5.124) (5.5.125)

(5.5.126)

(5.4.96) thus gives a formula for the equilibrium measure (with {λj }ℓj=1 determined by (5.4.97)) of the essential spectrum of periodic Jacobi matrices. Our next immediate goal is to extend this to general finite gap sets e = [α1 , β1 ] ∪ · · · ∪ [αℓ+1 , βℓ+1 ] (5.5.127) where α1 < β1 < α2 < · · · < αℓ+1 < βℓ+1 (5.5.128) The function ℓ+1 Y R(z) = (z − αj )(z − βj ) (5.5.129) j=1

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5. PERIODIC OPRL

will play a critical role here and later (see Section 5.12). Notice that each factor in the product is positive on R \ (αj , βj ) and negative on (αj , βj ) so R(x) > 0 if x ∈ R \ e R(x) ≤ 0 x ∈ e (5.5.130) √ We want to define R as an analytic function on C \ e, the branch with p R(x) > 0 if x > βℓ+1 (5.5.131) This implies p R(x) < 0 (βℓ , αℓ+1 ) ∪ (βℓ−2 , αℓ−1 ) ∪ . . . (5.5.132) p R(x) > 0 (βℓ−1 , αℓ ) ∪ (βℓ−3 , αℓ−2 ) ∪ . . . (5.5.133) p (−1)ℓ−1 R(x) > 0 on (−∞, α1 ) (5.5.134) p p and ( R(x + i0) means limε↓0 ( R(x + iε)) p (−i) R(x + i0) > 0 on (αℓ+1 , βℓ+1 ) ∪ (αℓ−1 , βℓ−1 ) ∪ . . . (5.5.135) p i R(x + i0) > 0 on (αℓ , βℓ ) ∪ (αℓ−2 , βℓ−2 ) ∪ . . . (5.5.136)

Following (5.4.96)/(5.4.97), we are interested in solutions of Z αj+1 P (x) p dx = 0 (5.5.137) |R(x)| βj

where P is a monic polynomial of degree ℓ:

Proposition 5.5.21. (a) If P is a nonzero polynomial of degree ℓ −1 or less, it cannot happen that (5.5.137) holds for j = 1, . . . , ℓ. (b) There is a unique monic polynomial, P, of exact degree ℓ so that (5.5.137) holds for j = 1, . . . , ℓ. This P has all its zeros in the gaps, one each and simple in each (βj , αj+1), j = 1, . . . , ℓ. Remark. (a) assures us the ℓ × ℓ matrix Z αj+1 Yjk = xk−1 |R(x)|−1/2 dx 1 ≤ j, k ≤ ℓ

(5.5.138)

βj

is invertible, and then the coefficients of P can be explicitly written in terms of the inverse of this matrix and the vector Yjk=ℓ+1. Proof. (a) For any real polynomial, if (5.5.137) holds for some j0 , P must change sign on (βj , αj+1) so have a zero there. Since deg(P ) ≤ ℓ − 1 means P has ℓ − 1 zeros, it cannot have a zero in each gap, so (5.5.137) cannot hold for all j = 1, . . . , ℓ. Thus, there is no solution with P real. But if P is any nonzero solution, both P (z) + P (¯ z ) and


337

i(P (z) − P (¯ z )) solve the same equations and are real, and at least one must be nonzero. (b) (5.5.137) for j = 1, . . . , ℓ and deg(P ) ≤ ℓ (not necessarily monic) represents ℓ linear conditions on ℓ + 1 parameters, so there is always a solution. By (a), the solution must have a nonzero xℓ term, so there is a monic solution. If there were two monic solutions, their difference would violate (a), so this solution is unique. As in (a), P must have at least one and so exactly one zero in each of the ℓ gaps. Henceforth, we will use P (z) or P (z; α1 , β1 , . . . , αℓ+1 , βℓ+1 ) or j=1 (z − zj ) where zj ∈ (βj , αj+1 ). With the function R above and branch of square root, we define, initially on C \ e, Qℓ

P (z) H(z) = − p R(z)

which is clearly analytic there, and at infinity where 1 1 H(z) = − + O 2 z z

(5.5.139)

(5.5.140)

Since R(z) is entire and nonvanishing on eint , H(x ± i0) exist (and are complex conjugate). We prove: Theorem 5.5.22. (i) H(x) is real on R \ e and H(x + i0) is pure imaginary with strictly positive imaginary part on eint . (ii) H(z) is a Herglotz function on C+ so that Z dν(x) H(z) = (5.5.141) x−z

for a probability measure on e. (iii) dν is a purely a.c. measure with density given by (5.4.96). (iv) dν is the equilibrium measure for e. (v) The potential Φν is given, for z ∈ C+ , by Z 1 −Φν (z) − log|z| = Re H(w) + dw (5.5.142) w z where the curve is the straight line from z to z +i∞. In particular, for any x ∈ e, x 6= 0, Z ∞ 1 −1 log(C(e) ) = log|x| − Im H(x + iy) + dy (5.5.143) x + iy 0

Remark. In fact, (5.5.142) can have any curve in C \ e ∪ {0}. The imaginary part is curve dependent, but not the real part.

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5. PERIODIC OPRL

Proof. (i) P is a monic polynomial with real zeros, hence real coefficients. Reality in the gaps is thus immediate by (5.5.132), (5.5.133), p and (5.5.134). By (5.5.135), Im(−1/ R(x + i0)) > 0 on (αℓ+1 , βℓ+1 ). Since P is monic with all zeros below αℓ+1 , P is positive on that interval, so Im(H(x + i0)) p > 0 on (αℓ+1 , βℓ+1 ). The sign of Im(1/ R(x + i0)) shifts from band to band, but because it has a single zero, so does the sign of P , so H is pure imaginary with Im(H(x + i0)) > 0 on each band. (ii) Fix ε > 0. Then since |H(z)| = O(1/|z|) near infinity, Im(H(z) + iε) > 0 near infinity. On R, Im(H(x + i0)) ≥ 0. So, by the maximum principle and the fact that Im(H(z)) is harmonic on C+ , continuous on C+ ∪ {0} ∪ R, we see Im(H(x + i0) + iε) > 0. Since ε is arbitrary, H is Herglotz. Thus, by Theorems 2.3.6 and 2.3.7, (5.5.141) holds. Since H(z) = − z1 + O( z12 ) at infinity, ν is a probability measure.

(iii) H(z) is bounded and continuous on R \ {αj , βj }ℓj=1 so on that set, dν is a.c. with density given by π1 Im(H(x+i0)), that is, by (5.4.96). The only potential singular measure is on {αj , βj }ℓj=1 which, as a finite set, can only support a pure point piece. Since limε↓0 ε|H(x + iε)| = 0 for all x ∈ R, ν has no pure points by Proposition 2.3.12. (iv),(v) Define Φν by (5.5.142). We claim Z Φν (z) = − log|z − x| dν(x)

(5.5.144)

for both sides have the same derivative (by (5.5.141)) and both are − log|z| + o(1) at infinity, so their difference goes to zero. Φν is conR dν(z) tinuous on C \ {αj , βj }ℓj=1 with derivative Re (x−z) −1 off R and with ℓ continuous boundary values on e \ {αj , βj }j=1 . Thus, Φν (x) is constant on each band, since the derivative is 0 there. By (5.5.137), the integral of the derivative across each gap is 0, so Φν (x) is constant on e. It follows by Theorem 5.5.15 that ν is the equilibrium measure for e and that the constant value of Φν on e, given by the right side of (5.5.143), is log(C(e)−1 ), proving (5.5.143). let

Proposition 5.5.23. Let e1 ⊂ (−∞, 0], e2 ⊂ [0, ∞). For a ≥ 0, e(a) = e1 ∪ (e2 + a)

(5.5.145)

Then C(a) is monotone increasing as a increases. Remark. This is an expression of the repulsive nature of the Coulomb force.


339

Proof. Let Ma be M+,1 (e(a)). Map Ma to Ma′ by Qa′ ,a (µ) ↾ e1 = µ ↾ e1 Qa′ ,a (µ) ↾ e2 + a′ = (µ ↾ e2 + a) + (a′ − a) If a′ > a, |x − y + a′ − a|−1 < |x − y|−1 for x ∈ e1 , y ∈ e2 + a, so E(Qa′ ,a (µ)) < E(µ)

(5.5.146)

log(C(a′ ))−1 ≤ log(C(a))−1

(5.5.147)

Since Qa′ ,a is a bijection,

Thus, C(a) ≤ C(a′ ).

Theorem 5.5.24. For any e ⊂ R, C(e) ≥ 41 |e|

(5.5.148)

Proof. By the last proposition, C(e) decreases as gaps are shrunk to zero, leaving an interval ˜e with |e| = |˜e| and C(e) ≥ C(˜e) = 14 |˜e| by (5.5.19). Finally, we want to discuss the converse of Corollary 5.5.19. Here is the key theorem, part of which will not be proven until later: Theorem 5.5.25. Let e = ∪ℓ+1 ej be a union of ℓ + 1 disjoint closed j=1˜ intervals in R. Then the following are equivalent: (i) There is a two-sided Jacobi matrix, J, of period p so that σ(J) = e. (ii) Each ˜e has rational harmonic measure. (iii) There is a polynomial ∆ with real coefficients and leading positive coefficients so (a) All zeros of ∆ lie on R and are simple. (b) All zeros of ∆′ lie in R and ∆′ (x0 ) = 0 ⇒ |∆(x0 )| ≥ 2

(5.5.149)

e = ∆−1 ([−2, 2])

(5.5.150)

(c) Remarks. 1. The proof shows that the minimal p in (i) is the minimal integer, p, with pρe(˜ej ) ∈ Z.

2. The proof also shows that the minimal degree of the ∆ in (iii) is the minimal p in (i). 3. The analog of (iii) ⇒ (i) for OPUC is dubbed the “Quacks like a discriminant” theorem in [391].

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We will prove this result as Theorem 5.13.8 later. (i) ⇒ (iii) is a combination of Theorems 5.3.7 and 5.4.2. That (ii) ⇔ (iii) is sometimes called Aptekarev’s theorem, after its discoverer [21]. Toda maybe TK. x-ref? The condition (5.5.150) is intended as a map of R to R. However, (a)–(c) are equivalent to (5.5.150) as a complex result. Proposition 5.5.26. (a)–(c) in Theorem 5.5.25 where (5.5.150) is intended in the sense e = {x ∈ R | ∆(x) ∈ [−2, 2]}

(5.5.151)

are equivalent to (5.5.150) in the sense that e = {z ∈ C | ∆(z) ∈ [−2, 2]}

(5.5.152)

Proof. Suppose deg(∆) = p. If (a)–(c) hold, then as we have seen, ∆ ((−2, 2)) ∩ e is p disjoint intervals on each of which ∆ is one-one onto (−2, 2). Thus, if λ ∈ (−2, 2), ∆(z) − λ = 0 has p roots in e so, since deg(∆) = p, all roots. Thus, −1

e ⊃ {z ∈ C | ∆(z) ∈ (−2, 2)}

(5.5.153)

so, by continuity, (5.5.152) holds. Conversely, suppose (5.5.152) holds. Then ∆(¯ z ) = ∆(z) for z ∈ e so, by polynomial continuation for all z, so ∆ is real. Clearly, all roots are real. If f is an analytic function with f ′ (x) = 0 for some x0 in R and f (x0 ) real, there are nonreal z near x0 , so f (z) is real and near f (x0 ) (by writing f (z) = f (x0 ) + c(z − x0 )ℓ + . . . with ℓ ≥ 2), so (5.5.152) implies (5.5.149) for real solutions of ∆′ (x0 ) and, in particular, all zeros of ∆ are simple. Since ∆ has p zeros on R, by Snell’s theorem, ∆′ has all its zeros on R also. Remarks and Historical Notes The use of potential theory in the study of orthogonal polynomials goes back to work of Faber [120] and Szeg˝o [422] about 1920 and was rediscovered in the physics literature fifty years later [194, 427]. After important contributions by Erd˝os– Turán [118], Widom [447], and Ullman [436], it was raised to high art by Stahl–Totik [408]. Applications of potential theory to OPs are reviewed in [395]. For expositions of the mathematics of two-dimensional potential theory, see especially Ransford [352], Landkof [258], the appendices of [395], and also [19, 191, 292, 369, 408, 435]. In particular, [257, 352, 408] discuss the theory for e ⊂ C rather than just e ⊂ R. The result mentioned after Corollary 5.5.8 that {z | Φµ (z) = ∞} can be an arbitrary bounded Gδ of capacity zero is proven in [258].

5.6. APPROXIMATION BY PERIODIC SPECTRA, I. FINITE GAP SETS 341

Theorem 5.5.15 is true if (5.5.116) is assumed to hold for dρe-a.e. x; see the appendix to [395]. It is easy to prove Lusin’s theorem that we need in the proof of Theorem 5.5.7, namely, if µ is a measure on a compact set, E, and f ∈ L1 (E, dµ), then there are compact K with µ(E \ K) arbitrarily small and f ↾ K continuous. For pick fn continuous with kf −fn k1 ≤ 2−n and kfn+1 − fn k1 ≤ 2−n . Let Un be the open set where |fn+1 (x) − fn (x)| ≥ 2−n/2 so µ(Un ) ≤ 2−n/2 . If Km = E \ ∪∞ n=m Un , then µ(E P P∞\ Km ) ≤ ∞ −n/2 2 can be made arbitrarily small, and on K , f + m 1 n=m n=1 fn+1 − f1 is uniformly convergent, hence continuous. One can easily go from this case (f ∈ L1 ) to the general case (f measurable and finite almost everywhere). Our discussion of the equilibrium measure for arbitrary finite gap sets, that is, Proposition 5.5.21, follows Totik [433]. Theorem 5.5.17 is a well-known fact associated with work of Widom [448] and Aptekarev [21]. 5.6. Approximation by Periodic Spectra, I. Finite Gap Sets The next five sections are a grand aside from the main subject of this chapter, periodic Jacobi matrices, and represent an application of this theory. In this section and Section 5.8, we approximate general compact subsets of R by periodic spectra in two stages: finite gap sets here and general sets in Section 5.8. Our main result in this section is: Theorem 5.6.1 (Bogatyrëv–Peherstorfer–Totik Theorem). Let e = ∪ℓ+1 j=1 ej be an ℓ-gap set of the form (5.5.127). Then for all m large, there (m) exist ℓ-gap sets e(m) = ∪ℓ+1 with j=1 ej (i) (m) ej ⊂ ej (5.6.1) (m)

(m)

(ii) Each ej has has harmonic measure in e(m) equal to kj /m with (m) kj ∈ {1, 2, . . . }. (iii) For some C1 , C2 , (m)

|ej

C(e) ≤ C(e

\ ej | ≤ C1 m−1

(m)

(5.6.2) −1

) ≤ C(e) + C2 m (m)

(5.6.3) (m)

Remarks. 1. We will construct ej so ej 6⊂ (ej )int (only the (m) right endpoints will move and eℓ+1 = eℓ ), but as we will explain in the (m) Notes, it is easy to arrange that ej ⊂ (ej )int .

342

5. PERIODIC OPRL

2. Only (i) and (ii) are in [49], [329], [433]. (iii) is a later refinement of Totik [434]. Because of our explicit construction in Theorem 5.5.22, we can prove regularity of harmonic measures and capacities in {αj , βj }ℓ+1 j=1 . ℓ+1 The key will then be to prove that if we fix {αj }j=1 and βℓ+1 and only vary β1 , . . . , βℓ , the map from these ℓ variables to (µe([α1 , β1 ]), . . . , µe([αℓ , βℓ ]) is nonsingular, hence invertible. First the regularity: Proposition 5.6.2. Let e be given by (5.5.127) where {αj , βj }ℓ+1 j=1 obey (5.5.128). Then (1) The coefficients of the monic polynomial P of degree ℓ obeying (5.5.137) are real analytic functions of {αj , βj }ℓ+1 j=1 in the region (5.5.128). (2) Each of the ℓ + 1 measures µe([αj , βj ]) is a real analytic function of {αj , βj }ℓ+1 j=1 in the region (5.5.128). (3) The capacity C(e) is a real analytic function of {αj , βj }ℓ+1 j=1 in the region (5.5.128). Proof. (1) Let Yjk for j = 1, . . . , ℓ; k = 1, . . . , ℓ + 1 be given by (5.5.138). We will show this is real analytic in {αj , βj }ℓ+1 j=1 . For j = j0 , analyticity of Yj0 k in {αj }j6=j0+1 and {βj }j6=j0 is immediate since h is real analytic in these parameters uniformly on each (βj + ε, αj+1 − ε) with uniform O(ε−1/2) integrable bounds on derivatives. But there appears to be an issue with ∂Yj0 k /∂βj0 since |h(x)|−1/2 = ∞ at βj0 and with ∂h/∂βj0 which is not integrable at βj ! These problems actually cancel. To see this, change variables from x to y = (x − βj0 )/(αj0+1 − βj0 ) so the integral goes over [0, 1]. There is no endpoint variation and all derivatives in any αj or βj is bounded by |y(1 − y)|−1/2 . Put more succinctly, h(x(y))x−1 (1 − x)−1 is real analytic in {αj , βj }ℓ+1 j=1 and nonvanishing uniformly in a neighborhood of y ∈ [0, 1]. Once we have analyticity of Y, the fact that det((Yjk )j,k=1,...,ℓ ) 6= 0 and the resulting explicit formula for P in terms of Y yields the required analyticity. (2) We have 1 µe([αj0 , βj0 ]) = π

Z

βj0

αj0

|P (x)| p dx |h(x)|

(5.6.4)


By the change of variables, y=

x − αj0 βj0 − αj0

the region of integration becomes one over [0, 1] and, as above, is real analytic. (3) This follows from (1) and (5.5.143). We now turn to monotonicity properties of the harmonic measures, heading towards a proof that for k 6= j, ∂µe([αk , βk ])/∂βj < 0. Proposition 5.6.3. If e, e′ are two ℓ-gap sets with e ⊂ e′ , then for x ∈ e, ρe(x) ≥ ρe′ (x) (5.6.5) Remarks. 1. We will prove this is a more general context in Theorem 5.8.6. We will also see below (see (5.6.9)) that the inequality is strict. 2. This is saying that if an extra material is added to a perfect conductor, charge flows out into the extra material, decreasing the charge density everywhere in the original conductor. Proof. Let Ge be the potential theorist’s Green’s function given by (5.5.127). We claim first that for all z ∈ C, we have Ge(z) ≥ Ge′ (z)

(5.6.6)

For Ge − Ge′ is harmonic on (C ∪ {∞}) \ e′ continuous on C ∪ {∞}. Thus, it suffices to prove the result on e′ by the maximum principle. On e, Ge = Ge′ = 0, so (5.6.6) is trivial. On e′ \ e, Ge′ = 0 ≤ Ge, since (5.5.105) holds. We have thus proven (5.6.6). In the case at hand where Ge is real analytic in a neighborhood of e and Ge(x) = 0 for x ∈ e, we have for x ∈ e that ρe(x) =

Ge(x + iε) 1 lim π ε↓0 ε

so (5.6.6) implies (5.6.5).

(5.6.7)

Proposition 5.6.4. Let e be given in the form (5.5.127). Fix j0 and let e(β) for αj0 < β < αj0 +1 (or infinity if j0 = ℓ + 1) be the set with βj0 changed and the other parameters fixed. Then for x ∈ eint , ∂ρe(β) (x) 0 ∂βk ∂βk j=1

(5.6.20)

by (5.6.9). ∂f ∂fk Also by (5.6.9), ∂βjk < 0 for j 6= k, and thus, ∂β > 0 by (5.6.20). k It follows that ℓ ∂fk X ∂fj X ∂fj − = >0 (5.6.21) ∂βk ∂βk ∂βk j=1 j6=k j≤ℓ

by (5.6.20). So the derivative of F is diagonally dominant and so invertible. Thus, F is a locally invertible C 1 (indeed, real analytic) map with 1 C local inverse. Therefore, for any fixed initial set e with parameters α(0) , β (0) , those (0) (0) β1 , . . . βℓ in Rℓ+ near β1 , . . . βℓ map to a set S which contains the intersection of an open ball about F (β (0) ) and an open cone with vertex F (β (0) ). Such an S for all large n contains √ balls with center sn obeying (0) |sn − F (β )| ≤ K1 /n and radius rn = ℓ/n. Such balls contain points of the form ( pn1 , . . . , pnℓ ) for integral pj , so since F −1 is C 1 , we obtain β1 , . . . , βℓ . Hence, (0)

(0)

βj ≤ βj ≤ βj +

C1 n

(5.6.22)

with Fj (β) = pj /n for j = 1, . . . , ℓ+1 (ℓ+1 can be included by (5.6.19)). Thus, we have (i), (ii), and (5.6.2). (5.6.3) then follows from the fact (0) (0) that C(·) is a C 1 function of (β1 , . . . , βℓ ) near (β1 , . . . , βℓ ). As an application of Theorem 5.6.1, we study: Definition. Let e be a compact subset of C. The Chebyshev constants, tn (e), are defined by tn (e) = min{kQn ke | Qn monic of degree n}

(5.6.23)

where kf ke = sup |f (z)|

(5.6.24)

z∈e

There are minimizing Q’s, the Chebyshev polynomials studied in the next section. The Chebyshev constants are relevant to the theory of orthogonal polynomials because:


Theorem 5.6.6. If µ is a measure supported by a compact set, e ⊂ C, and Xn (z, dµ) are the monic OPs for µ, then kXn kL2 (C,dµ) ≤ tn (e)µ(e)1/2

(5.6.25)

In particular, if e ⊂ R and {an , bn }∞ n=1 are the Jacobi parameters for µ, then a1 . . . an ≤ tn (e) (5.6.26) Proof. Clearly, for any Qn , kQn k2L2 ≤ kQn k2e µ(e)

so minimizing using we get (5.6.25).

(5.6.27)

minkQn kL2 (C,dµ) = kXn kL2 (C,dµ)

Theorem 5.6.7 (Totik–Widom Theorem). Let e be a finite gap set in R. Then there exists a constant w so tn (e) ≤ wC(e)n

In particular, if supp(µ) ⊂ e and µ(e) = 1, then a1 . . . an ≤w C(e)n

(5.6.28)

(5.6.29)

Remarks. 1. To put this in context, we note we will prove that for any e ⊂ C, one has (see Theorem 5.7.8) tn (e) ≥ C(e)n

and

lim tn (e)1/n = C(e)

n→∞

(5.6.30)

and for e ⊂ R, one has (see Corollary 5.7.7) tn (e) ≥ 2C(e)n

2. We show later (see Example 5.7.3) that the polynomial Tem below is actually the minimizer of kQm ke(m) , so one has equality in (5.6.35).

Proof. Pick M so for m ≥ M, we have sets e(m) obeying the conclusions of Theorem 5.6.1. Since e(m) is the spectrum of a periodic problem of period m (see Theorem 5.5.25), there are Jacobi parameters {aj , bj }∞ j=1 of period m, so (see (5.5.122)) a1 . . . am = C(e(m) )m

(5.6.31)

and discriminant ∆m (x) for this Jacobi matrix. Since e(m) = ∆−1 m ([−2, 2]) and ∆m (x) = (a1 . . . am )−1 xm + · · ·

(5.6.32)

348

5. PERIODIC OPRL

we have that

Tem (x) = (a1 . . . am )∆m (x) is a monic polynomial with kTem ke(m) = 2C(e(m) )m

(5.6.33) (5.6.34)

which implies

so

tm (e(m) ) ≤ 2C(e(m) )m (5.6.35) (m) (m) But trivially tm (e) ≤ tm (e ) since e ⊂ e and by (5.6.3), m C2 (m) m m tm (e) ≤ 2C(e ) ≤ 2C(e) 1 + (5.6.36) C(e)m lim sup

tm (e) ≤ 2 exp(C2 C([e])−1 ) < ∞ C(e)m

proving (5.6.28).

x-ref?

(5.6.37)

Remarks and Historical Notes. Theorem 5.6.1(i), (ii) were obtained with very different proofs by Bogatyrëv [49] (using conformal mapping techniques), Peherstorfer [329] (using Chebyshev polynomials; see the Notes to TK), and Totik [433] (using methods close to ours here). Totik then noted (iii) (with a different proof) in [434]. The argument we use to get (iii) is new here. Theorem 5.6.7 follows from a theorem of Widom [448] who proved tn (e)/C(e)n is a bounded almost periodic function. The much simpler approach we use here is due to Totik [434]. As noted, our (e(m) )int does not include all of e. However, one can first increase all β’s by O(1/m) and decrease all α’s by O(1/m) and then use our construction on this larger set to get new e(m) ’s that also obey e ⊂ (e(m) )int . 5.7. Chebyshev Polynomials Chebyshev polynomials are everywhere dense in numerical analysis. – Mason and Handscomb [293], who say it is well known and might be due to Phillip Davis or to George Forsythe.

In an aside on our asides, we study in more detail the minimizers in the definition of Chebyshev constants. Definition. Given a compact set e ⊂ R, the Chebyshev polyno(e) mials, Tn (or Tn (x) if we need to make e explicit), are the monic polynomials of degree n that minimize kf ke ≡ sup |f (x)| x∈e

(5.7.1)

5.7. CHEBYSHEV POLYNOMIALS

349

that is, tn (e) ≡ kTn ke = min{kQn ke | Qn monic of degree n}

(5.7.2)

We will prove below (see Corollary 5.7.6) that if e is infinite, Tn is (0) unique. To see there is a minimum, suppose e is infinite, pick any Qn , (0) and note {Qn | kQn ke ≤ kQn ke} is a nonempty set compact in the topology of convergence of coefficients. Qn → kQn ke is continuous in this topology, so the minimum value is taken. Since e is infinite, kQn ke is never zero for a monic Qn , so kTn k > 0. One can make this definition for any compact e ⊂ C, and occasionally we will indicate results for that case. Theorem 5.7.1 (Alternation Principle). If Qn is a monic polynomial with n simple zeros in e so that each zero zj lies in an interval (zj− , zj+ ) ⊂ e where Q′n (x) 6= 0 and |Qn (zj± )| = kQn ke

(5.7.3)

then Qn is a Chebyshev polynomial for e of degree n. Proof. Suppose there is a monic polynomial Tn with kTn ke < kQn ke

(5.7.4)

sgn(Qn (zj± ) − Tn (zj± )) = sgn(Qn (zj± ))

(5.7.5)

Then |Tn (zj± )| < |Qn (zj± )|, so

Since Q′n (x) 6= 0 on (zj− , zj+ ), Qn (zj+ ) = −Qn (zj− ) and thus Qn − Tn has different signs at zj+ and zj− , and so a zero in (zj− , zj+ ). Since Q′n (x) has a zero between any two zeros of Qn , these intervals are disjoint, so Qn − Tn has at least n zeros. But Qn and Tn are distinct monic polynomials, so Qn − Tn has at most n − 1 zeros. This contradiction shows (5.7.4) cannot occur, so Qn has minimum norm. Example 5.7.2. Recall that the classical Chebyshev polynomials of the first kind are defined by pn (cos θ) = cos(nθ)

(5.7.6)

Since cos(nθ) = (einθ + e−inθ )/2 and cosn θ = [(eiθ + e−iθ )/2]n = 2−n einθ + lower order, pn is not monic. Rather, pn (x) = 2n−1xn + lower order

(5.7.7)

Tn (x) = 2−(n−1) pn (x)

(5.7.8)

We claim that

350

5. PERIODIC OPRL

are the Chebyshev polynomials for [−1, 1]. They are monic and the 2π(ℓ+ 1 ) zeros of pn (x) occur at xℓ = cos( n 2 ) for ℓ = 0, 1, . . . , n − 1 and + ± −(n−1) each xℓ lies in an interval [x− = kTn k[−1,1] ℓ , xℓ ] where |pn (xℓ )| = 2 − + and pn (x) is monotone on (xℓ , xℓ ). This proves (5.7.8) is indeed the Chebyshev polynomial for the set. Notice that (with C( · ) = capacity) kTn k1/n = 2−(n−1)/n →

1 2

= C([−1, 1])

(5.7.9)

Example 5.7.3. Let e = ∪ℓ+1 j=1 ej be an ℓ gap set which is the spectrum of a two-sided Jacobi matrix J of period p. Let ∆(x) = (a1 . . . ap )−1 xp + lower order

(5.7.10)

be its discriminant. Let ˜e1 , . . . , ˜ep be the closed bands. Each has a zero zj ∈ ˜ej of ∆, supx∈e|∆(x)| = 2 since e = ∆−1 ([−2, 2]), and every ˜ej is precisely the kind of interval required in Theorem 5.7.1. It follows that Tp (x) = (a1 . . . ap )∆(x)

(5.7.11)

is the Chebyshev polynomial of e and kTp ke = 2(a1 . . . ap )

(5.7.12)

For each k = 1, 2, . . . , we can consider J as a matrix of period kp with discriminant ∆(k) . Indeed, if pk (cos θ) = cos(kθ), then ∆(k) (x) = 2pk ( 12 ∆(x)). As above, Tpk (x) = (a1 . . . ap )k ∆(p) (x)

(5.7.13)

and kTpk ke = 2(a1 . . . ap )k

In particular, by (5.5.15),

kTpk k1/pk → (a1 . . . ap ) = C(e) e

(5.7.14)

Notice that if Tn (z) = z n + an−1 z n−1 + . . . , Ten (z) = z n + Re(an−1 )z n+1 + . . . has Ten (x) = Re Tn (x) on e, so kTen ke ≤ kTn ke and thus we can suppose Tn is a real polynomial, which we henceforth do. Lemma 5.7.4. Let qm (x) be a real polynomial and a ≤ b. For ε > − a), let

− 21 (b

(ε)

pm+2 (x) = (x − (b + ε))(x − (a − ε))qm (x)


351

Then for ε > 0 and any compact K ⊂ R \ [a − ε, b + ε], (ε)

(0)

sup |pm+2 (x)| < sup |pm+2 (x)|

x∈K

(5.7.15)

x∈K

and for ε < 0 and any compact K ⊂ R \ [a, b], (ε)

(0)

(5.7.16)

(0)

(5.7.17)

sup |pm+2 (x)| > sup |pm+2 (x)|

x∈K

x∈K

For ε > 0 and any compact K ⊂ (a, b), (ε)

sup |pm+2 (x)| > sup |pm+2 (x)|

x∈K

x∈K

and for ε < 0 and any compact K ⊂ (a − ε, b + ε), (ε)

(0)

sup |pm+2 (x)| < sup |pm+2 (x)|

x∈K

(5.7.18)

x∈K

Remark. In other words, if a pair of zeros are moved symmetrically apart, it decreases |p| outside the zeros and increases it inside, and viceversa if they are symmetrically moved together. Proof. Without loss, we can suppose a = −b with b > 0. Then x2 − (b + ε)2 is strictly decreasing as ε increases. So |x2 − (b + ε)2 | strictly increases in |x| < (b + ε) and strictly decreases in |x| > (b + ε). This holds for all x so remains true if we multiply by |qm (x)|.

Theorem 5.7.5. Let e ⊂ R be compact and let Tn be the Chebyshev polynomials for e. Then (i) All zeros of Tn lie in R. (ii) All zeros of Tn are simple. (iii) All zeros of Tn lie in cvh(e), the convex hull of e. (iv) If (a, b) ∩ e = ∅, then Tn has at most one zero in (a, b). (v) If xj < xj+1 are two successive zeros of Tn , then Tn′ (y) has exactly one zero yj in [xj , xj+1 ] and |Tj (yj )| ≥ kTn ke

(5.7.19)

with equality if yj ∈ e. (vi) Moreover, there is wj ∈ (xj , xj+1 ) so wj ∈ e and |T (wj )| = kTn ke . Similarly, there is w0 , wn ∈ e, w0 ∈ (−∞, x0 ), and wn ∈ (xn , ∞) so that |Tn (w0 )| = |Tn (wn )| = kTn ke.

Proof. (i) As noted above, we can suppose Tn is real on R, so Tn (¯ z ) = Tn (z), and if a + ib is a zero, so is a − ib. Since |(x − (a − ib))(x − (a + ib))| = (x − a)2 + b2

(5.7.20)

Tn (x) would be decreased for all x if we replace b 6= 0 by b = 0. By the minimum norm definition, no zero can have b 6= 0.

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5. PERIODIC OPRL

(ii) By the lemma, if x0 is a double zero, replace (x − x0 )2 by (x − (x0 + ε))(x − (x0 − ε)) and decrease kTn k on e \ (x0 − ε, x0 + ε). Since for ε small, (x − (x0 − ε))(x − (x0 + ε)) is small on [x0 − ε, x0 + ε], we see we can decrease kTn ke. Thus, Tn cannot have double zeros. (iii) If a = inf e and x0 < a, then |x − (x0 + ε)| < |x − x0 | for all x > a. Thus, we can decrease kTn ke by moving a zero below a upward. By the minimum definition, there can be no zeros on (−∞, a). (iv) By the lemma, if (a, b) has two zeros x0 < x1 , we can decrease kTn ke by moving them slightly apart, violating the minimum property. (v) By Snell’s theorem, Tn′ has at least one zero in each (xj , xj+1). Since Tn has n distinct zeros on R, this accounts for n − 1 zeros of Tn′ and so for all the zeros, so there is exactly one in each (xj , xj+1 ). If |Tn (yj )| < kTn ke, sup[xj ,xj+1 ] |Tn (x)| < kTn ke . Moving xj , xj+1 apart, we decrease |Tn (x)| on e \ [xj , xj+1 ] and the increase on [xj , xj+1 ] can be kept so small that we remain strictly less than kTn ke there. This would decrease kTn ke, violating the minimum definition. Thus, (5.7.19) holds. Clearly, if yj ∈ e, |Tn (yj )| ≤ kTn ke. (vi) As in the proof of (iv), if supw∈[xj ,xj+1]∩e|Tn (w)| < kTn ke, we can move the zeros slightly apart, so the new polynomial is still strictly less than kTn ke on [xj , xj+1 ] and the sup is decreased off [xj , xj+1 ]. Similarly, if supw∈(−∞,x1]∩e|Tn (w)| < kTn ke, we can move x1 up and decrease kTn ke. Corollary 5.7.6. The Tn minimizing kTn ke is unique. Proof. Suppose Pn and Qn are two distinct minimizers and let Tn = 21 (Pn + Qn ). Since kTn ke ≤ max(kPn ke, kQn ke), Tn is also a minimizer. Let x1 < · · · < xn be its simple zeros (by (ii) of the theorem) and let xj < wj < xj+1 and w0 ∈ (−∞, x1 ), wn ∈ (xn , ∞) be such that wj ∈ e and kTn (wj )k = kTn ke (which exist by (vi) of the last theorem). Since |Pn (wj )| ≤ kTn ke, |Qn (wj )| ≤ kTn ke, and 12 |Pn (wj ) + Qn (wj )| = |Tn (wj )| = kTn ke, we have Qn (wj ) = Pn (wj ) = Tn (wj ). Thus, Pn − Qn has at least n + 1 zeros! Since deg(Pn − Qn ) ≤ n − 1, Pn = Qn . Corollary 5.7.7 (Schiefermayr’s Theorem). We always have tn (e) ≥ 2C(e)n

(5.7.21)

where C(e) is the capacity of e.

Remarks. 1. By Example 5.7.2, one has equality in (5.7.21) for all n if e = [−1, 1]. Thus, the number 2 in (5.7.21) cannot be increased. 2. For e = ∂D, Tn (z) = z n and tn (∂D) = 1, so (5.7.21) only holds for e ⊂ R, not all e ⊂ C.


Proof. Let ∆n (x) = and let

353

2Tn (x) kTn ke

(5.7.22)

en = ∆−1 (5.7.23) n ([−2, 2]) By Theorem 5.7.5, ∆n has all its zeros on R, they are simple, and ∆′ (x0 ) = 0 ⇒ |∆(x0 )| ≥ 2. Thus, by Theorem 5.5.25, en is the spectrum of a Jacobi matrix of period n and ∆n is its discriminant. (e ) By Example 5.7.3, Tn n is the monic multiple of ∆n , and so Tn = (e ) (e ) Tn n and kTn ke = kTn n ken = 2C(en )n ≥ 2C(e)n since e ⊂ en , proving (5.7.21). We are heading towards generalizing (5.7.9) and (5.7.14) and show1/n ing kTn ke → C(e) for all e ⊂ R, a result that holds for all e ⊂ C essentially by the same proof. It will be useful to have an additional notion: (0)

(0)

Definition. Let e ⊂ R. An n-point Fekete set is x1 , . . . , xn ∈ e so that if Y qn (x1 , . . . , xn ) = |xj − xy | (5.7.24) i6=j

then

(0)

qn (x1 , . . . , x(0) n ) =

sup

qn (x1 , . . . , xn )

(5.7.25)

(x1 ,...,xn )∈e

We set

(0)

1/n(n−1) ζn (e) = qn (x1 , . . . , x(0) n )

(5.7.26)

Remark. The number of i 6= j in (5.7.24) is n(n − 1), explaining the power in (5.7.26). Notice that (0) (0) qn+1 (x1 , . . . , xn+1 )n−1

=

n+1 Y j=1

(0)

(0)

(0)

(0)

(0)

qn (x1 , . . . , x bj , . . . , xn+1 )

(5.7.27)

(where x bj means dropping xj ) since each pair (i, j) occurs on the right of (5.7.3) n − 1 times. Thus, (n+1)n n−1

[ζn+1

or and

]

≤ [ζn(n−1)n ]n+1

ζn+1 (e) ≤ ζn (e)

(5.7.28)

ζ∞ (e) = lim ζn (e)

(5.7.29)

n→∞

exists. It is called the transfinite diameter of e.

354

5. PERIODIC OPRL

Theorem 5.7.8 (Faber–Fekete–Szeg˝o Theorem). Let e ⊂ R be compact. Then, for all n, C(e) ≤ kTn k1/n ≤ ζn+1 e

(5.7.30)

lim kTn k1/n = C(e) e

(5.7.31)

Moreover, (i) The normalized counting measure for Fekete sets converges to dρe, the equilibrium measure for e. (ii) ζ∞ (e) = C(e), so n→∞

(iii) If e is potentially perfect, then the zero counting measure for Tn converges to dρe, the equilibrium measure of e. Remark. We proved in (5.7.21) a stronger statement than the first inequality in (5.7.30). We include (5.7.30) here because, unlike (5.7.21), it holds for all e ⊂ C. Proof. Let Qn be any monic polynomial. By the Bernstein–Walsh lemma (5.5.115), |Qn (z)| ≤ kQn ke exp(n[Ge(z) − log(|z|)]) |z|n

(5.7.32)

Take |z| → ∞, |Qn (z)|/|z|n → 1 by the fact that Qn is monic. By (5.5.114), Ge(z) − log(|z|) → − log(C(e)). Thus, (5.7.32) becomes or

1 ≤ kQn ke exp(−n log(C(e)))

kQn k1/n ≥ C(e) e which implies the first inequality in (5.7.30). For j = 1, . . . , n + 1, let Y (0) Qj (z) = (z − xk )

(5.7.33)

(5.7.34)

(5.7.35)

k6=j

for an (n + 1)-point Fekete set. By (5.7.25), Y (0) (0) sup |Qj (x)| = |xj − xk | x∈e

so

k6=j

n+1 Y j=1

By kQj ke ≥ kTn ke, we have

n(n+1)

kQj ke = [ζn+1

n(n+1)

kTn kn+1 ≤ ζn+1 e

]

(5.7.36)


355

which is the second inequality in (5.7.30). (i), (ii) Let ν∞ be a limit point of νn(j) , where νn(j) is a normalized counting measure for Fekete sets with n(j) points. Fix m > 0. Then, n(j) if {xj }j=1 are the Fekete points, Z Y 1/n 1/n2 −1 −1 m |xj −xk | ≤ exp − log[min(m , |x−y| )] dνn(j) (x)dνn(j) (y) j6=k

(5.7.37)

since −1 ,|x−y|−1 )]

e− log[min(m

= max(|x − y|, m) ≥ |x − y|

(5.7.38)

and we can use m for the n-terms with x = y in (5.7.38). In that inequality, take n → ∞, m1/n → 1, and since n(n − 1)/n2 → 1, we get ζ∞ (e) ≤ exp(−Em−1 (ν∞ ))

where Ea (ν) =

Z

log(min(a, |x − y|−1 )) dν(x)dν(y)

(5.7.39)

(5.7.40)

and we used ν → Ea (ν) is weakly continuous. By the monotone convergence theorem, lima→∞ Ea (ν) = E(ν), so taking m → 0 in (5.7.39), we obtain ζ∞ (e) ≤ exp(−E(ν∞ ))

(5.7.41)

But, by (5.7.30), C(e) ≤ ζ∞ (e), so

E(ν∞ ) ≤ log(C(e)−1 ) = E(ρe)

By the minimization property of ρe, ν∞ = ρe, proving convergence of νn to ρe (by compactness of M+,1 (e)), and by (5.7.41), ζ∞ (e) ≤ C(e)

proving ζ∞ (e) = C(e). (iii) By the Bernstein–Walsh lemma (5.5.115), 1/n 1 kTn ke 1 log|Tn (z)| ≤ log − Φρe (z) n n C(e)

(5.7.42)

(5.7.43)

If νn(j) → ν∞ with νn the counting measures for zeros of Tn , then for z ∈ / H = cvh(e), (5.7.43), (5.5.52), (5.5.5), and (5.5.102) (see Theorem 5.5.10) imply that 1/n

−Φν∞ (z) ≤ −Φρe (z)

since kTn ke → C(e) by (5.7.31). By Theorem 5.5.16, we conclude that ν∞ = ρe.

356

5. PERIODIC OPRL

Remarks and Historical Notes. Classical Chebyshev polynomials were introduced by him in two papers [76, 77], neither of which used the relation to cos nθ! He noted that they minimized kpn k[−1,1] among all other polynomials with the same top order coefficients. Classical Chebyshev polynomials have many applications to numerical analysis (see Mason–Handscomb [293] and Rivlin [361]). The Faber–Fekete–Szeg˝o theorem is named after their papers [120, 127, 422]. For other discussions of general Chebyshev polynomials, see [19, 175, 369, 435]. For a single interval, the Fekete points are known to be the zeros of a certain Jacobi polynomial; see Szeg˝o [424, p. 382, prob. 37]. Corollary 5.7.7 is due to Schiefermayr [370]. Peherstorfer’s proof [329] of Theorem 5.6.1(i)/(ii) looks at en defined by (5.7.22)/(5.7.23) which, in general, has ℓ bands containing e (if e is an ℓ gap set) and ℓ − 1 tiny bands around the at most ℓ − 1 zeros in gaps of e. He showed one could remove all tiny bands by slightly enlarging e. 5.8. Approximation by Periodic Spectra, II. General Sets If e is the spectrum of a two-sided periodic Jacobi matrix, we have several nice properties. We have Floquet solutions and we know Ge(x) is zero on e. In this section, we want to approximate any compact e ⊂ R from the outside by periodic spectra and use this in one way and in Section 5.11 in a deeper way. The question, of course, is what we mean by approximate. While there are weaker notions, we will find approximants in the following strong sense: and

e ⊂ · · · ⊂ en+1 ⊂ en ⊂ · · · ⊂ R \

en = e

(5.8.1) (5.8.2)

n

and each en is the spectrum of a periodic problem. Define ˜en = {x ∈ R | dist(x, e) ≤ n1 }. These will obey (5.8.1) and (5.8.2) and we will prove each is a finite union of disjoint intervals, that is, an ℓ-gap set. Since ˜en ⊂ (˜en−1 )int , we will be able to use Theorem 5.6.1 to find en a periodic spectrum with ˜en ⊂ en ⊂ (˜en−1 )int , and so find the required en . First, we give a few preliminaries. To carry over Ge = 0 in intervals, I, in e, we will want the following: Proposition 5.8.1. Let I = (a, b) ⊂ e ⊂ R with e compact. Suppose we know Ge(x) = 0 on I. Then (i) Ge is the real part of a function analytic in a neighborhood of I.

5.8. APPROXIMATION BY PERIODIC SPECTRA, II. GENERAL SETS

357

(ii) dρe ↾ I = ρe(x) dx where ρe is a real analytic function of x. (iii) For each k = 0, 1, 2, . . . and ε > 0, k d ρe(x) sup dxk

(5.8.3)

(5.8.4)

x∈[a+ε,b−ε]

is bounded with bounds depending only on ε, a, b, and diam(e).

Remarks. 1. We will eventually see (Corollary 5.8.5) that Ge(x) = 0 on I always holds. 2. Let J = cvh(e) so I ⊂ e ⊂ J. We will eventually show (see Corollary 5.8.7) that on I, ρJ (x) ≤ ρe(x) ≤ ρI (x)

(5.8.5)

where ρI , ρJ are the equilibrium density for an interval given by (5.5.126). This will imply bounds on (5.8.4) depending only on I and not on diam(e). Proof. (i) Let zn → z∞ ∈ I. By upper semicontinuity of Ge, Ge(z∞ ) ≥ lim sup Ge(zn ). But Ge(z∞ ) = 0 and Ge(zn ) ≥ 0 (by Frostman’s theorem), so lim inf Ge(zn ) ≥ Ge(z∞ ). Thus, Ge is continuous on C+ ∪ I. Since Ge is harmonic on C+ , there is an analytic function ˜ e(z) with Re G ˜ e(z) = Ge (z). By the Schwarz reflection principle (in G the strong form that only requires continuity of Re f ; see Ahlfors [7, ˜ e has an analytic continuation to C+ ∪ C− ∪ I with x-ref? Theorem 4.24], G ˜ e(¯ ˜ e(z) G z ) = −G

(5.8.6)

(ii) By the formula for the potential, Z ˜ e(z) dG dρe(w) =− (5.8.7) dz w−z since Z d 1 Ge(x + iy) = dρe(w) Re (5.8.8) dx x + iy − w From Propositions 2.3.11 and 2.3.12, ρe is absolutely continuous on I, and for x ∈ I, 1 ∂Ge(x + iy) ρe(x) = lim (5.8.9) π y↓0 ∂y y=0

1 Ge(x + iy) = lim π y↓0 y

(5.8.10)

358

5. PERIODIC OPRL

(iii) A Cauchy estimate shows that for any function f analytic in a neighborhood of {z | |z − z0 | ≤ ε}, we have f for ℓ = 1, 2, . . . , |f (ℓ) (z0 )| ≤ 2ε−ℓ sup |Re f (z)|

(5.8.11)

|z−z0 |=ε

This follows from (ℓ)

−1 −ℓ

f (z0 ) = (2π) ε and

Z

Z

e−iℓθ f (z0 + εeiθ ) dθ

e−iℓθ f (z0 + εeiθ ) dθ = 0

This, in turn, implies (using (5.8.6) and (5.8.9)) that k d ρe 2 sup k ≤ ε−k−1 sup |Ge(x + iy)| π x∈[a+ε,b−ε] dx x∈[a,b]

(5.8.12)

0≤y≤ε

1 |b − a| 4

and log|x + iy − w| ≤ log(diam(e) + ε) Since C(e) ≥ C(I) = for 0 ≤ y ≤ ε and x ∈ I, w ∈ e, we find 2 −ℓ−1 4 (5.8.4) ≤ ε log + log(ε + diam(e)) (5.8.13) π |b − a|

We first turn to approximations in the strong sense: Proposition 5.8.2. Let e be compact so (5.8.1) and (5.8.2) hold for compact e1 , e2 , . . . . Then (i) w ρen −→ ρe (5.8.14) (ii)

C(en ) ↓ C(e)

(5.8.15)

(iii) If I = (a, b) is an interval in e so that Gen = 0 on I, then Ge = 0 on I and the densities ρen (x) converge uniformly to ρe(x) on each (a + ε, b − ε). Proof. (i),(ii) Let ρ∞ be a weak limit of ρen(j) . By hypothesis, ρ∞ ∈ M+,1 (e). By the obvious C(e) ≤ C(en(j) ) and lower semicontinuity of the Coulomb energy, E,

log(C(e)−1 ) ≤ E(ρ∞ ) ≤ lim inf E(ρn ) = lim(log(C(en )−1 )) ≤ log(C(e)−1 ) It follows that lim C(en ) = C(e) and ρ∞ = ρe, so by compactness of w M+,1(e), ρen −→ ρe.

5.8. APPROXIMATION BY PERIODIC SPECTRA, II. GENERAL SETS

(iii) By Proposition 5.8.1, we have uniform bounds on

359

dρe n dx [a+ε,b−ε]

so equicontinuity of ρen (x), so compactness in the topology of uniform convergence. But if ρen (x) → f (x) uniformly, dρen ↾ [a + ε, b − ε] → f (x) dx, so dρe = f (x) dx determining f , and so proving uniform convergence. RThe uniform bounds on ρen (x) and ρe(x) imply uniform convergence of |x−x0|ε log|x − x0 |−1 dρen (x) R to |x−x0|>ε log|x − x0 |−1 dρ(x) implies Gen (x0 ) → Ge(x0 ) on (a, b), so Ge(x0 ) = 0. Proposition 5.8.3. Let e ⊂ R be compact. Let ˜en = {x ∈ R | dist(x, e) ≤ n1 }

(5.8.16)

Then (i) ˜en obey (5.8.1) and (5.8.2). (ii) Each ˜en is a finite union of disjoint closed (positive measure) intervals. Proof. (i) ˜en+1 ⊂ ˜en is trivial, and ∩˜en = ˜e by the compactness of e (for x ∈ ∩˜en implies there are xn ∈ e with dist(xn , x) ≤ n1 ). (ii) R\e is an open set, so a disjoint union of maximal open intervals, PN two unbounded and the others {Jk }N in cvh(e). Thus, k=1 k=1 |Jk | < ∞, so for each n, #{k | |Jk | > n2 } is finite. Thus, all but finitely many Jk lie in a given ˜en , showing R \ ˜en has finitely many open intervals. It is easy to see that each of the finite disjoint closed intervals in ˜en must have positive measure. By combining this with Theorem 5.6.1, we get: Theorem 5.8.4. Let e ⊂ R be compact. Then there exist en so that (5.8.1) and (5.8.2) hold, and moreover, en ⊂ eint n−1

(5.8.17)

ρen (x) → ρe(x)

(5.8.18)

and each en is a finite gap set with rational harmonic measures, that is, each en is the spectrum of some two-sided periodic Jacobi matrix. Moreover, w (i) ρen −→ ρe (ii) C(en ) → C(e) (iii) If I = (a, b) is an interval in e, then Ge = 0 on I and uniformly on each [a + δ, b − δ].

360

5. PERIODIC OPRL

Proof. Let ˜en be given by (5.8.16). Since x ∈ ˜en and |x − y| ≤ [n(n − 1)]−1 implies y ∈ ˜en−1 , we see ˜en ⊂ (˜en−1 )int

By Proposition 5.8.3, ˜en is a finite gap set, so by Theorem 5.6.1, we can find en a periodic spectrum with ˜en ⊂ en ⊂ (˜en−1 )int

This implies (5.8.17), and (5.8.2) for ˜en implies it for en . (i)–(iii) are immediate from Proposition 5.8.2 and the fact that Ge(x) = 0 on e for periodic spectra. Corollary 5.8.5. Let I = (a, b) ⊂ e ⊂ R. Then Ge = 0 on I and ρe ↾ I is absolutely continuous with real analytic ρe(x). As an application of the approximation theorem, we can prove various comparison theorems: Theorem 5.8.6. Let e ⊂ e′ be compact subsets of R. Then: (i) For all z ∈ C, Ge′ (z) ≤ Ge(z) (5.8.19) (ii) dρe′ ↾ e ≤ dρe (5.8.20) (iii) If I = (a, b) ⊂ e ⊂ e′ , then x ∈ I ⇒ ρe′ (x) ≤ ρe(x)

(5.8.21)

Proof. Since our periodic approximations obey e ⊂ (en )int and 1 }, it is easy to see we can find periodic en ⊂ {x | dist(e, x) < n−1 approximations en , e′n of e, e′ with en ⊂ e′n . By the convergence results in Theorem 5.8.4, it suffices to prove this theorem in case e, e′ are finite gap sets. We did this in Proposition 5.6.3 (the statement of that proposition required the set have the same number of gaps, but all that was used in the proof was continuity of Ge on C and absolute continuity of dρe). Corollary 5.8.7. Let I ⊂ e ⊂ cvh(e) = J ⊂ R. Then, on I, ρJ (x) ≤ ρe(x) ≤ ρI (x)

(5.8.22)

Remarks and Historical Notes. Totik [432, 433] emphasized the approximation of general compact e ⊂ R by periodic spectra as a tool for extending not only results on CD kernels (we follow him in part in Section 5.10) but also classical polynomial inequalities like the Markov inequality.

5.9. REGULARITY: AN ASIDE

361

For Totik, the periodic spectrum did not play a big role—rather, he exploited the existence of a polynomial ∆ of the type in (c) of Theorem 5.7.5 with ∆−1 ([−2, 2]) = e. Objects like Floquet solutions are never used. His use of polynomial inverse images was motivated by Geronimo–Van Assche [151]. Standard work on potential theory [191, 258, 352] develops a “theory of barriers” to prove (a, b) = I ⊂ e ⊂ R implies Ge is continuous and vanishing on I. 5.9. Regularity: An Aside This section has nothing to do with periodic Jacobi matrices— rather it provides a tool needed in Section 5.11 which has the deepest application of periodic approximations. We will address the issue of root asymptotics mentioned in Sections 2.9, 2.15, and 3.11. Definition. A measure µ with compact support e ⊂ R is called regular if and only if its Jacobi parameters obey lim (a1 . . . an )1/n = C(e)

(5.9.1)

n→∞

To partly motivate this notion, we note Proposition 5.9.1. For any measure µ of compact support e ⊂ R, we have lim sup (a1 . . . an )1/n ≤ C(e) (5.9.2) n

Remark. Below (see the remark after the proof of Theorem 5.9.2) we will provide a second proof of (5.9.2). Proof. By (1.2.13) (assuming µ(R) = 1), 1/n

lim sup (a1 . . . an )1/n = lim sup kPn ( · , dµ)kL2(dµ) n

(5.9.3)

n

while, by (5.6.6), kPn ( · , dµ)kL2(dµ) ≤ kTn kL2 (dµ)

≤ µ(R)1/2 kTn ke

1/n

Since kTn ke (5.9.2).

(5.9.4)

→ C(e) (by Theorem 5.7.8), (5.9.3) and (5.9.4) imply

Here is the main result on regular measures: Theorem 5.9.2. Let µ be a measure supported by a compact set e ⊂ R so that µ is regular. Then

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5. PERIODIC OPRL

(i) The zero counting measures, νn , for the OPRL obey w

νn −→ ρe

(5.9.5)

lim |pn (z, dµ)|1/n = exp(Ge(z))

(5.9.6)

the equilibrium measure for e. (ii) For any z ∈ / cvh(e), the convex hull of e, n→∞

(iii) For any z ∈ cvh(e),

lim sup |pn (z, dµ)|1/n ≤ exp(Ge(z))

(5.9.7)

n→∞

and for q.e. z ∈ e,

lim sup |pn (z, dµ)| = 1

(5.9.8)

n→∞

We need one preliminary: Lemma 5.9.3. For z ∈ C+ and any measure µ of compact support in R, we have lim inf |pn (z, dµ)|1/n ≥ 1 (5.9.9) Proof. If not, there exist n(j) → ∞ with lim |pn(j) (z)| = 0

(5.9.10)

since a ∈ [0, 1) implies an → 0. Pn(j)−1 Let ϕj (x) = k=0 pk (x)pk (z). By the recursion relation for the p’s and Jpk = ak+1 pk+1 + bk+1 pk + ak pk−1 , we have

((J − z)ϕj )(x) = an(j) (pn(j) (x)pn(j)−1 (z) − pn(j)−1 (x)pn(j) (z)) (5.9.11)

(essentially the CD formula (3.10.7)). Thus,

hϕj , (J − z)ϕj i = −an(j) pn(j) (z) pn(j)−1 (z) Pn(j)−1 which implies, using kϕj k2 = k=1 |pk (z)|2 ≥ |pn(j)−1 (z)|2 , |hϕj , (J − z)ϕj i| ≤ an(j) kϕj k |pn(j)(z)|

(5.9.12)

(5.9.13)

Since kϕj k ≥ 1 (from the p0 term) and supn |an | < ∞, this implies, given (5.9.10), that lim

j→∞

But

|hϕj , (J − z)ϕj i| =0 kϕj k2

(5.9.14)

|hϕj , (J − z)ϕj i| ≥ (Im z)kϕj k2 (5.9.15) This contradiction shows that (5.9.10) cannot happen, so (5.9.9) holds.

5.9. REGULARITY: AN ASIDE

363

Proof of Theorem 5.9.2. (i) Suppose that νn(j) → ν∞ . By (5.5.37) and (5.5.5), |Pn(j) (z, dµ)|1/n(j) → exp(−Φν∞ (z))

(5.9.16)

e lim |pn(j) (z, dµ)|1/n(j) = exp(G(z))

(5.9.17)

for z ∈ C \ cvh(e). By (5.9.1), j→∞

where By (5.9.9),

e G(z) = −Φν∞ (z) + log(C(e)−1 )

(5.9.18)

Φν∞ (z) ≤ log(C(e)−1 )

(5.9.19)

E(ν∞ ) ≤ log(C(e)−1 ) = E(ρe)

(5.9.20)

for Im z 6= 0. By (5.5.28), (5.9.19) holds for z ∈ R also. Integrating dν∞ , we find

Since νn has at most weight 1/n in any gap of e, ν∞ is supported on e. Thus, by (5.9.20), ν∞ = ρe, that is, ρe is the only limit point of w νn . By compactness of M+,1 (e), νn −→ ρe. e = Ge and thus, (5.9.17) is (5.9.6). (ii) ν∞ = ρe implies G (iii) This is immediate from (ii) and (iii) of Theorem 5.5.10.

Remark. If νn(j) → ν∞ and (a1 . . . an(j) )1/n(j) → A, (5.9.20) becomes E(ν∞ ) ≤ log(A−1 ) (5.9.21) so log(A−1 ) ≥ log(C(e)−1 ) (5.9.22) that is, A ≤ C(e). This provides the promised second proof of (5.9.2). There is a converse to part of Theorem 5.9.2 that we will need: Theorem 5.9.4. Let e ⊂ R be compact and regular for the Dirichlet problem and let µ be a measure with σess (µ) = e. Suppose w

νn −→ ρe

(5.9.23)

Then either µ is regular for e or else there exists K of capacity zero so that µ(R \ K) = 0. 1/n(j)

Proof. Let (a1 . . . an(j) ) → A for some A. By the argument leading to (5.9.16) and the upper envelope theorem (Theorem 5.5.10), there is a set K of capacity zero so that for z ∈ C \ K, |pn(j)(x, dµ)|1/n(j) → A−1 exp(−Φρe (z))

(5.9.24)

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5. PERIODIC OPRL

In particular, for all x ∈ e \ K,

C(e) A = 1, we have

|pn(j) (x, dµ)|1/n(j) →

(5.9.25)

On the other hand, since kpn kL2 (dµ) Z X ∞ (n + 1)−2 |pn (x)|2 dµ < ∞ n=0

so for µ a.e. x and an x-dependent constant, B(x), |pn (x)| ≤ B(x)(n + 1)

(5.9.26)

If A < C(e), the object on the right of (5.9.25) is larger than 1 and this is inconsistent with (5.9.26)! Thus, either A ≥ C(e) or else µ is supported on the set where (5.9.25) fails, that is, µ(R \ K) = 0. Since A ≤ C(e) always, we see if the first case holds, then C(e) is the only limit point (and µ is regular). Definition. A set e is called regular for the Dirichlet problem if and only if Ge(x) = 0 for all x ∈ e.

Corollary 5.9.5. Let e ∈ R be a potentially perfect set which is regular for the Dirichlet problem. Let µ be a measure on R with σess (µ) = e. Then for any δ > 0, there exists a neighborhood Kδ of e and constant Cδ so that for all n, sup |pn (z, dµ)| ≤ Cδ eδn

(5.9.27)

z∈Kδ

Proof. By hypothesis, Ge is continuous on e and so on C and vanishing on e. Let 1 Kδ = G−1 (5.9.28) e ([0, 2 δ)) ∂Kδ is compact, disjoint from e, and Ge = 21 δ on ∂Kδ . Thus, uniformly in z ∈ Kδ , 1 lim |pn (z, dµ)|1/n = e 2 δ (5.9.29) It follows that we can find Cδ so (5.9.27) holds for z ∈ ∂Kδ , and thus, by the maximum principle, on Kδ . The following, which we state without proof (but see the Notes), provide criteria for regularity: Theorem 5.9.6. Let e ⊂ R be potentially perfect and let µ obey σess (µ) = e and dµ(x) = f (x) dρe(x) + dµs (x) (5.9.30) where f (x) > 0 for dρe-a.e. x. Thus, µ is regular.

5.10. THE CD KERNEL FOR PERIODIC JACOBI MATRICES

365

Theorem 5.9.7. Let e be a finite union of disjoint closed intervals. Let µ have σess (µ) = e. Suppose for every η > 0, lim |{x ∈ e | µ([x − n1 , x + n1 ]) ≤ e−nη }| = 0

n→∞

(5.9.31)

where | · | is Lebesgue measure. Then µ is regular. We will prove a special case of Theorem 5.9.6 (when e has a large interior so dρe is dx-a.c.) later (see Theorem 5.11.3). Remarks and Historical Notes. For e = [−1, 1], the relation of kPn k1/n → 12 , of the convergence of dνn to π −1 (1 − x2 )−1/2 dx and positivity of the weight (i.e., the hypothesis of Theorem 5.9.6 in this case) go back to a 1940 paper of Erd˝os–Turán [118]. Systematic study of regularity on [−1, 1] was begun by Ullman [436] (see the references in [395]). The general theory was initiated and brought to fruition in a remarkable book of Stahl–Totik [408] who, in particular, prove Theorems 5.9.2 and 5.9.7. Theorem 5.9.6 appears implicitly in Widom [447] and explicitly in Van Assche [437]. Simon [395] has a review of the theory, including proofs of Theorems 5.9.6 and 5.9.7. We also note the following due to Stahl–Totik [408] and proven also in [395]. Theorem 5.9.8. Let e = e1 ∪ · · · ∪ eℓ be a union of ℓ disjoint closed intervals. Let µ be a measure with σess (µ) = e and let µj = µ ↾ ej . Then µ is regular for e if and only if each µj is regular for ej . 5.10. The CD Kernel for Periodic Jacobi Matrices As we’ve seen in the our analysis of CD kernel asymptotics in Sections 2.15–2.17 and 3.11, a key role is played by one example that dθ we can analyze completely. For OPUC, this was 2π , and for OPRL on [−2, 2], it was the measures dµ1 , dµ2 of Example 3.11.3. In this section, as preparation for the next, we will study in detail the asymptotics of the CD kernel associated to the spectral measure of a periodic Jacobi matrix. (This is the analog of dµ2 in Example 3.11.3; while we used dµ1 more extensively in that section, we could have used dµ2.) Throughout this section, all results refer to a fixed periodic Jacobi matrix. We let {an , bn }∞ n=−∞ be the Jacobi parameters of the half-line Jacobi + matrix J0 extended by periodicity. e = e1 ∪ · · · ∪ eℓ+1 is the essential spectrum of J0+ . By Theorem 5.4.14, for λ ∈ eint , we can define solutions ± u± n (λ) ≡ un (λ + i0)

(5.10.1)

366

5. PERIODIC OPRL

of (5.4.1)/(5.2.7) with + u− n (λ) = un (λ)

u± n=0 (λ) = 1

(5.10.2)

and (by (5.4.105) and Proposition 5.10.2(ii) below) Im u+ 1 (λ) 6= 0

(5.10.3)

which implies u± n are linearly independent. Moreover, ±imθ(λ) ± u± un (λ) (5.10.4) n+mp (λ) = e and θ is related to ρe, the density of the equilibrium measure, dρe, by 1 dθ (5.10.5) ρe(λ) = pπ dλ

by (5.3.34). Since p·−1 also solves (5.2.7), we have pn−1 (λ) =

− [u+ n (λ) − un (λ)] − [u+ 1 (λ) − u1 (λ)]

(5.10.6)

since equality holds at n = 0, 1. Define

I(λ) = −2 Im u+ 1 (λ)

(5.10.7)

Theorem 5.10.1. Let J0+ be a periodic Jacobi matrix. (i) The weight w(x) of the spectral measure for J0+ is given by I (5.10.8) 2a0 π (ii) The density, ρe(x), of the equilibrium measure for e is given by w(x) =

p 1 X + ρe(x) = |un (λ)|2 a0 pπI n=1

(5.10.9)

Proof. (i) By (5.4.41) and u+ 0 (λ) = 1,

u+ (λ) m(λ) = hδ1 , (J0+ − λ)−1 δ1 i = − 1 a0 R −1 Since m(λ) = dµ(x)(x − λ) , we have 1 Im m(λ + iε) ε↓0 π 1 =− (2 Im u+ 1 (λ)) 2πa0 I = 2a0 π

w(λ) = lim

(5.10.10)

(5.10.11) (5.10.12)


367

(ii) By Theorems 5.4.8 and 5.5.17,

so as above,

Z p 1 X ρe(x) dx −1 hδn , (J − λ) δn i = p n=1 x−λ

(5.10.13)

p

X 1 ρe(λ) = lim Im(Gnn (λ + iε)) πp ε↓0 n=1 Pp + 2 1 n=1 |un (λ)| =− πp Im(W (λ))

(5.10.14) (5.10.15)

by (5.4.79). Here

− − + W (λ) = a0 (u+ 1 (λ)u0 (λ) − u1 (λ)u0 (λ)

= −a0 I

(5.10.16)

(5.10.15) and (5.10.16) imply (5.10.9).

When we square (5.10.6), |pn−1 (λ)|2 will have a cross term + 2 − u+ n (λ) un (λ) = un (λ) and a key role will be played by the fact that uniformly on compact subsets of eint , one has lim

N →∞

N 1 X + 2 u (λ) = 0 N j=1 j

(5.10.17)

This is more subtle than it might appear at first. By (5.10.4), what is relevant is M X

e2imθ(λ) =

n=1

e2i(M +1)θ(λ) e2iθ(λ) − 1

(5.10.18)

which easily yields (5.10.17) pointwise if 2θ(λ) 6= 2πk

(5.10.19)

for an integer k. But if (5.10.19) fails, there is an issue and uniformity fails in (5.10.18) as θ(λ) → some πk. Points in eint where (5.10.19) fails are precisely closed gaps, so we will need to look closely at what happens there. The key will be that at a closed gap, p X 2 u+ (5.10.20) j (λ) = 0 j=1

As a warmup:

Proposition 5.10.2. Let J0+ be a periodic Jacobi matrix. (i) At any closed gap, λ0 , ρe(λ) is continuous and nonvanishing.

368

5. PERIODIC OPRL

(ii) At a closed gap, λ0 , w(λ) is continuous and nonvanishing. Proof. (i), first proof. By Craig’s formula (5.4.86), ρe is continuous and strictly positive on any compact subset of eint . dθ (i), second proof. By (5.10.5), we need to show dλ remains smooth a nonzero at a closed gap. θ solves 2 cos(θ(λ)) = ∆(λ)

(5.10.21)

where ∆ is the discriminant, (5.4.6) (see Theorem 5.4.1). At a closed gap, λ0 , 2 ± ∆(λ) = c(λ − λ0 )2 with c > 0 (see Proposition 5.4.3). So, by (5.10.18), we have that is,

(θ(λ) − θ(λ0 ))2 = d(λ − λ0 )2 + O((λ − λ0 )3 )

dθ dλ λ=λ0

6= 0.

(ii) Let λ0 be the gap edge and let α, β, γ be the coefficients of (5.2.2). By Theorem 5.4.15 and (5.2.3), α(λ) has a simple zero at λ0 . By Proposition 5.4.3, ∆2 − 4 has a double zero at λ0 so, by (5.2.4), β vanishes at λ0 . Since γ, like α, has simple zeros, (5.2.4) √ implies β and α have simple zeros at λ0 . At λ0 , β/α is real and ∆2 − 4/α is pure imaginary in (λ0 − ε, λ0 + ε)/{λ0}, and so nonvanishing and imaginary at λ0 . Thus Im m is continuous and nonvanishing near λ0 . Theorem 5.10.3. At any closed gap, λ0 , we have p X 2 u+ j (λ0 ) = 0

(5.10.22)

j=1

Proof. We consider the case that ∆(λ0 ) = 2. The case ∆(λ0 ) = −2 is similar. Thus θ(λ0 ) = 0. J(θ = 0) given by (5.3.8) thus has a doubly generated eigenvalue at λ0 by Proposition 5.4.3. Let θ be small and positive. Then J(θ) has two eigenvalues e+ (θ) > λ0 > e− (θ) near λ0 . By eigenvalue perturbation theory (see the Notes), the corresponding eigenvalues have limits. Since e± (θ) are distinct, the eigenvectors are orthogonal, so the limits are orthogonal. For θ 6= 0, π, the only possible Floquet eigenfunctions are u± (e(θ)), so either u± (e± (θ)) or u∓ (e± (θ)) are the eigenvectors. It cannot be that u+ (e+ (θ)) and u+ (e− (θ)) are the eigenvectors for θ > 0 since u+ is continuous and the limits are orthogonal. Since the limits are orthogonal, p X + u− (5.10.23) j (λ0 ) uj (λ0 ) = 0 j=1


which is (5.10.22) by (5.4.69).

369

Here is one of two main results of this section: Theorem 5.10.4. Fix a periodic Jacobi matrix with σess (J) = e. Let I = [α, β] be a closed interval in eint . Then uniformly in I: (i) For any A > 0 and uniformly for all λn → λ0 ∈ I with n|λn −λ0 | ≤ A, we have lim

n→∞

ρe(λ0 ) 1 Kn (λn , λn ) = n+1 w(λ0)

(5.10.24)

(ii) Under the same conditions as (i) for all such λn and |a| ≤ A, |b| ≤ A, Kn (λn + na , λn + nb ) sin(πρe(λ0 )(b − a)) lim = n→∞ Kn (λ0 , λ0 ) πρe(λ0 )(b − a)

(5.10.25)

Remark. See the Notes for an alternate proof of this theorem. Proof. (i) We first claim that uniformly for λ ∈ I, n 1 X + (u (λ))2 → 0 n j=1 j

(5.10.26)

+ + Since |u+ j+1 (λ)| = |uj (λ)| and each uj is continuous on I, we have

sup |u+ j (λ)| < ∞

(5.10.27)

j,λ∈I

That implies it suffices to prove (5.10.26) for n = kp, k = 1, 2, . . . . But then, by (5.10.24), k−1 X kp p 1 X + 1 X 2iℓθ(λ) 2 + 2 (u (λ)) = e (uj (λ)) kp j=1 j kp j=1 ℓ=0 2ikℓθ(λ) X p 1 e −1 2 = (u+ j (λ)) 2iθ(λ) kp e − 1 j=1

(5.10.28)

2iθ(λ) where we interpret [. . . ] as Pkp if e+ 2= 1. By (5.10.23), R(λ) ≡ j=1 uj (λ) vanishes at each λ in eint where e2iθ(λ) = 1 (since eiθ(λ) = ±1 and we are at a closed band edge). By eigenvalue perturbation theory, u+ 1 (λ) is analytic in θ(λ), including at closed band edges, and θ is invertible and so real analytic in λ on eint . + By the recursion relation (and u+ 0 (λ) = 1), uj (λ) is analytic for each j,

370

5. PERIODIC OPRL

and so R(λ) is real analytic on eint . It follows for any compact K ⊂ eint that p X 2iθ(λ) −1 + 2 sup (e − 1) uj (λ) ≡ RK < ∞ (5.10.29) λ∈K j=1

so

LHS of (5.10.28) ≤

2 DK → 0 kp

(5.10.30)

as k → ∞, that is, (5.10.26) holds. 2 − 2 Thus, squaring (5.10.6), the (u+ n ) and (un ) converge to zero and we see, by (5.10.6), n+1 X 2 1 Kn (λ0 , λ0 ) = |u+ (λ0 )|2 n+1 I(λ0 )2 (n + 1) j=1 j

2πa0 ρe(x) I(λ0 ) ρe(x) = w(x)

→

by (5.10.9) (5.10.31)

by (5.10.8). The above shows the convergence is uniform, and by going through the above, it is easy to accommodate the λn → λ0 extension. (ii) Suppose first that a 6= b and λn ≡ λ0 . Let n = kp + j

j = 0, 1, . . . , p − 1

(5.10.32)

+ + Then, since u+ 0 , u1 , . . . , up−1 are real analytic near λ0 as is θ(λ), we have by (5.10.4) that, for ℓ = 1, 2, 1 a ik(θ(λ0 )+θ ′ (λ0 )a/n)+O(1/n2 ) + + un+ℓ λ0 + =e uj+ℓ (λ0 ) + O n n (5.10.33) Plugging this into (5.10.6) yields a b b a an+1 pn+1 λ + pn λ 0 + − pn+1 λ0 + pn λ 0 + n n n n 1 1 ′ = W 2i sin +O θ (λ0 )(a − b) p n (5.10.34)

where + + W = an+1 [u+ n+2 (λ0 ) un+1 (λ0 ) − un+2 (λ0 ) un+1 (λ0 )]

= a0 [2i Im u+ 1 (λ0 )]

(5.10.35) (5.10.36)


371

+ (5.10.34) used nk = 1p + O( n1 ), (5.10.35) that u− n (λ0 ) = un (λ0 ), and (5.10.36) the constancy of the Wronskian and u+ 0 (λ0 ) = 1. The left side of (5.10.6) enters in the CD formula for Kn (λ0 + na , λ0 + b ), so we obtain, using the definition of I, (5.10.7), n a b 2a0 1 lim Km λ 0 + , λ 0 + = sin[πρe(λ0 )(a − b)] (5.10.37) n→∞ n + 1 n n I

on account of (5.10.5). Using (5.10.8) and (5.10.24), we obtain (5.10.25) for the case (a 6= b, λn ≡ λ0 ). Next, we return to (5.10.24) and note it holds if Kn (λn , λn ) ˜ n ) so long as n(λn −λ ˜ n ) → 0 with n|λn −λ0 | ≤ A. is replaced by Kn (λn , λ ′ For uniformly in n, pn (λ) near λ0 is O(n) by (5.10.33) and (5.10.6), which in the CD formula controls the change of n1 Kn . With this in place, one can easily control a = b and λn → λ0 in (5.10.25). Finally, as preparation for extending the Máté–Nevai bounds to general sets in R, we note the following pair of results: Theorem 5.10.5. Fix a periodic Jacobi matrix with σess (J) = e. For any compact set K ⊂ eint , we have sup |pn (λ)| < ∞

(5.10.38)

n,λ∈K

Remark. This also follows from an analysis of transfer matrices. Proof. By (5.10.28), (5.10.8), and Proposition 5.10.2(ii), we get a + + −1 uniform bound on |u+ n (λ)| and on |u1 (λ) − u1 (λ) | . By (5.10.6), we obtain (5.10.38). Theorem 5.10.6. Fix a periodic Jacobi matrix with σess (J) = e. (a) For any compact K ⊂ eint , we have Kn (x, y) C ≤ Kn (x, x) |x − y|

for all x, y ∈ K

(5.10.39)

(b) For any A and ε, there is N so for n > N and all x, y in the region |x − y| ≤ A/n, x, y ∈ K, we have Kn (x, y) sin(πρe(x)(x − y)n) (5.10.40) Kn (x, x) − nπρe(x)(x − y) < ε

Proof. (a) follows from Theorem 5.10.5 and the CD formula. (b) follows from the uniformity of the convergence in Theorem 5.10.4.

372

x-ref?

5. PERIODIC OPRL

Remarks and Historical Notes. The use of Floquet solutions to study asymptotics of the CD kernel is due to Simon [399] who used a different approach which has the advantage of also working for almost periodic isospectral tori of the type studied in TK. Because it is illuminating how the other proof uses the magic of the CD formula to avoid the need to prove (5.10.22), we sketch that approach here. Actually, we go slightly further than [399]. That paper did not compute directly a constant that we compute below, but instead relied on Theorems 3.11.1 and 3.11.4. Define for λ ∈ eint fn (λ) = e−inθ(λ)/p u+ n (λ)

(5.10.41)

By (5.10.4), fn has period p in n and it is real analytic on eint . So, for any compact K ⊂ eint , dfn =B 0 on I, then Z 1 (5.11.3) n + 1 Kn (x, x)w(x) − ρe(x) dx → 0 I Z 1 Kn (x, x) dµs (x) → 0 (5.11.4) I n+1

374

5. PERIODIC OPRL

Theorem 5.11.3. Let e ⊂ R be compact so that e \ eint has capacity zero; in particular, e can be a finite gap set. Let µ be a measure with σess (µ) = e and dµ(x) = w(x) dx + dµs (x) where w(x) > 0 for Lebesgue a.e. x. Then µ is regular. Remark. This is a special case of Theorem 5.9.6. Proof. By Theorem 5.9.4, it suffices to show that the zero density dνn → dρe (since µ is obviously not supported on a set of capacity zero). 1 1 Suppose n(j)+1 Kn(j) (x, x)w(x) dx → dκ1 and n(j)+1 Kn(j) (x, x) dµs → dκ2 . By the argument in (2.17.38), R

R

dκ1 ≥ ρe(x) dx = dρe

But dκ1 + dκ2 = 1 = dρe, so dκ2 = 0 and dκ1 = dρe. 1 By compactness, n+1 Kn (x, x) dµ → dρe. So by Theorem 3.11.1, dνn → dρe. To get lower bounds, we need a one-sided but extended Nevai comparison theorem (see Theorems 2.16.6 and 3.11.5): Theorem 5.11.4 (Nevai Comparison Theorem). Let e be a compact subset of R which is regular for the Dirichlet problem. Let I ⊂ eint be a closed interval. For every ε, there is a δ so that if e′ ⊂ {x | dist(x, e) < δ} and µ, µ′ are any measures on R with µ regular for e and σess (µ′ ) ≡ e′

(5.11.5)

(n + 1)λn (x0 , dµ′) → C > 0

(5.11.6)

µ ↾ I = µ′ ↾ I

(5.11.7)

lim inf(n + 1)λn (x0 , dµ) ≥ C(1 − ε)

(5.11.8)

σess (µ) = e and for some x0 ,

and then Moreover, these results are unchanged if x0 in (5.11.6) and (5.11.8) are replaced by xn obeying xn → x0 , and if (5.11.6) (with x-dependent C) holds uniformly in I, then so does (5.11.8). Proof. Pick D so that for dist(x, e) < 1 and x0 ∈ I, we have |x − x0 | >0 (5.11.9) 1− D

5.11. ASYMPTOTICS OF THE CD KERNEL: OPRL ON GENERAL SETS 375

and let Q be defined by

sup

x∈e / x0 ∈I dist(x,e) 1, so for |x − x0 | < ε, we have αε−1 w(x) ≤ w ♯ (x) ≤ αε w(x) (5.11.14) (iii) That αε → 1 and any xn ∈ eint with xn → x0 and every ℓ(n) with n/2 < ℓ(n) < 2n, we have that lim

n→∞

1 Kn (xℓ(n) , xℓ(n) ) = B 6= 0 n+1

(5.11.15)

1 Kn♯ (xn , xn ) = B n+1

(5.11.16)

Then lim

n→∞

376

5. PERIODIC OPRL

Moreover, this is uniform in xn in the sense that if (with the same B) for all xn → x0 , there are, for any ε, a δ and an N0 so if n > N0 and |xn − x0 | < δ, then 1 ♯ B − Kn (xn , xn ) < ε (5.11.17) n+1

This is also uniform in x0 . If w and w ♯ are continuous and nonvanishing in a closed interval in eint and we have dµs = dµ♯s in a neighborhood of I and (5.11.14) is replaced by αε−1

w ♯ (x) w(x) w(x) ≤ ♯ ≤ αε w(x0 ) w (x0 ) w(x0 )

(5.11.18)

for |x − x0 | < ε (αε independent of x0 ) and if (5.11.14) holds uniformly in x0 ∈ I where B(x0 ) is x0 -dependent, then (5.11.16) with B replaced by B(x0 )w(x0 )/w ♯(x0 ). Proof. Identical to the proof of Theorem 3.11.5.

Next we generalize Lubinsky’s theorem (Theorem 3.11.6): Theorem 5.11.6. Let e be a compact subset of R regular for the Dirichlet problem. Let dµ be a regular probability measure on e of the form dµ = w(x) dx + dµs (5.11.19) Suppose that, for some interval [α, β] ⊂ eint , (a) supp(dµs ) ∩ I = ∅ (b) w is “continuous” on I and nonvanishing there. Then, with ρe given by the equilibrium measure for e, we have (1) (Diagonal Asymptotics) For any A < ∞, uniformly in x∞ ∈ I, and sequence xn ∈ e with n|xn − x∞ | ≤ A for all n, we have 1 ρe(x∞ ) Kn (xn , xn ) → n+1 w(x∞ )

(5.11.20)

(2) (Lubinsky Universality) For any A < ∞, uniformly in x∞ ∈ I and a, b ∈ R with |a|, |b| ≤ A, we have Kn (x∞ + na , x∞ + nb ) sin(πρe(x∞ )(b − a)) → Kn (x∞ , x∞ ) πρe(x∞ )(b − a)

(5.11.21)

More generally, the limit of Kn (xn , yn )/Kn (x∞ , x∞ ) is the right side of (5.11.21) so long as |xn − x∞ | ≤ A/n, |yn − x∞ | ≤ A/n, and n(xn − yn ) → b − a.


Proof. If e is the spectrum of a two-sided periodic Jacobi matrix, the proof follows that of Theorems 3.11.6 and 2.16.1; the upper bound comes from Theorem 5.11.1 and the lower bound uses Theorems 5.10.4 and 5.11.5. In the general case, we approximate e using Theorem 5.8.4. By Theorem 5.11.1, we see that lim inf m→∞

ρe(x0 ) 1 Km (xm , xm ) ≥ m+1 w(x0 )

(5.11.22)

and by the approximation, the special case above, and Theorem 5.11.4, we see that for each n, lim sup m

1 ρe (x0 ) Km (xm , xm ) ≤ (1 − εn )−1 n m+1 w(x0 )

(5.11.23)

where εn → 0 as n → ∞. Taking n → ∞ using (5.8.18) yields (5.11.20). To get (5.11.21), we compare µ with a measure µn which is µ on I and max(µ, ρen ) off I. This is regular for en (see the Notes). Putting x-ref? this into Lubinsky’s inequality and using (5.10.25) shows the absolute e n (x∞ )(b−a)) value of the difference of the LHS of (5.11.21) for µ and sin(πρ πρe (x∞ )(b−a) is asymptotically less than as n → ∞ by (5.8.18).

ρe (x∞ ) ρe (x∞ )−ρe n (x∞ ) | |, ρe n (x∞ ) ρe n (x∞ )

n

which goes to zero

Finally, we turn to results on locally Szeg˝o weights. By using approximation by periodic spectra, the key will be the extension to weights on periodic spectra. Here we will use the discriminant, ∆, to map e to [−2, 2] and we will be able to relate the Christoffel variational problems for such weights to ones on [−2, 2]. We will be able to do this initially for weights with a symmetry between bands. The localization intrinsic in Nevai trial functions will let us then go to nonsymmetric weights. We begin by studying the symmetry between bands, that is, solutions of ∆(x) = λ ∈ (−2, 2). We take a polynomial, Q(z), which we will eventually specialize to ∆. Suppose deg(Q) = N. For any λ, we will look at the solutions of Q(z) = λ

(5.11.24)

Q has a double (or higher-order) root at z0 if and only if Q′ (z0 ) = 0, so this occurs at a maximum of N − 1 points. The corresponding values of Q are a set, Λ, of at most N − 1 points, the critical values of Q. If λ ∈ / Λ, (5.11.24) has N solutions z1 (λ), . . . , zn (λ) which can be chosen analytically in the neighborhood of any λ ∈ / Λ. One cannot make a global choice since points in Λ are branch points—following a path around them will permute the zj (λ). Indeed, if Q is irreducible

378

5. PERIODIC OPRL

by following some path in the region C \ Λ, one can go from any zj to any zk . However, analytic symmetric functions of {zj (λ)}N j=1 will be analytic and singular-valued on C\Λ. Typically, there will be removable singularities at points in Λ. The following describes the special case of symmetrized polynomials: Theorem 5.11.7. Let ℓ = 0, 1, 2, . . . . For we have that

ℓN ≤ k < (ℓ + 1)N N X

(zj (λ))k = Rk (λ)

(5.11.25)

(5.11.26)

j=1

where Rk is a polynomial with deg(Rk ) = ℓ

(5.11.27)

Proof. We use induction in ℓ starting with ℓ = 0 (which will be the most subtle case!). For ℓ = 0, we need to show the sum is constant. Rk is continuous in λ so it suffices to prove constancy for λ ∈ / Λ. Thus, zj (λ) is locally analytic and dQ(zj (λ)) dzj (λ) 1 =1⇒ = ′ dλ dλ Q (zj (λ))

(5.11.28)

Therefore, N

X zj (λ)k−1 dRλ =0⇔k =0 dλ Q′ (zj (λ)) j=1

(5.11.29)

and we need only prove the right equality in (5.11.29). Fix λ ∈ / Λ. Consider a circle, Γ, about zeros of radius R so large that sup |zj (λ)| < R (5.11.30) j

and look at

Z 1 kz k−1 dz (5.11.31) 2πi |z|=R Q(z) − λ Since Q − λ has no zeros outside Γ, we can take R → ∞. This integral is bounded by −1 1 (2πR)kRk−1 inf |Q(z) − λ| (5.11.32) Γ 2π and the inf goes like R−N . The quantity in (5.11.32) is bounded for large R by Rk−N → 0 since ℓ = 0 (so k < N). This means that the integral in (5.11.31) is zero. It can also be evaluated in terms of the


residues inside the circle which is the sum on the right of (5.11.29). This completes the proof for ℓ = 0. For general ℓ, we use induction in k, assuming k ≥ N. Then zjk can be written using (Q(zj ) − λ)zjk−N = 0 as a sum of constants times k−N {zjk−m }N , and so write Rk (λ) as a sum m=1 plus a constant times λzj N of {Rk−m }m=1 and λRk−N . This proves the result inductively. Corollary 5.11.8. Let P be any polynomial of exact degree k obeying (5.11.25). Then there is a polynomial R of exact degree ℓ so that for z obeying Q(w) = Q(z) ⇒ Q′ (w) 6= 0, X P (w) = R(Q(z)) (5.11.33) {w|Q(w)=Q(z)}

Remark. (5.11.33) holds (by continuity) at points where Q′ (w) = 0 for some root so long as we count multiplicity in the sum. Proof. Immediate given the theorem which handles monomials. Given any measure µ on [−2, 2] and ∆, the discriminant from e to [−2, 2], we define a measure Sµ on e as follows: Write e = ∪pj=1 ej the closed bands, that is, ej is the closure of one of the connected components of ∆−1 ((−2, 2)). If A ⊂ eint j for some j, then Sµ(A) =

1 p

µ(∆(A))

(5.11.34)

If x0 ∈ ∆−1 ({−2, 2}), we set ( 1 µ(∆(x0 )) if x0 is an open gap edge Sµ({x0 }) = p2 µ(∆(x0 )) if x0 is a closed gap edge p This definition is such that for any f : [−2, 2] → R, Z Z f (∆(x)) d(Sµ)(x) = f (x) dµ(x)

(5.11.35)

Moreover, if Xjk : ej → ek is defined by demanding ∆(Xjk (x)) = ∆(x)

then for any function g : ek → R and any j, k, Z Z g(Xjk (x)) d(Sµ)(x) = g(x) d(Sµ)(x) ej

(5.11.36)

(5.11.37)

ek

It is not hard to see that (5.11.35)/(5.11.36) uniquely characterize Sµ.

380

5. PERIODIC OPRL

Proposition 5.11.9. (a) If dµ = w dx + dµs and d(Sµ) ≡ w˜ dx + d˜ µs , then µ ˜s = Sµs

w(x) ˜ =

d∆ 1 w(∆(x)) p dx

(5.11.38)

(b) The equilibrium measures and their densities on e and [−2, 2] are related by Sρ[−2,2] = ρe

(5.11.39)

ρe(x) = 1p ρ[−2,2] (∆(x))

(5.11.40)

Proof. (a) The formula for w˜ is a standard change of variables, and µ ˜s = Sµs follows from |∆(A)| = 0 ⇔ |A| = 0 where | · | is Lebesgue measure. (b) (5.11.40) is equivalent to (5.11.39), given (5.11.38). To see (5.11.40), we use the explicit formulae (5.4.15) and (5.5.126): ρe(x) =

1 ∆′ (x) p pπ 4 − ∆2 (x)

ρ[−2,2] (x) =

1 1 √ π 4 − x2

and (5.11.38).

(5.11.41)

√ Remark. Lest (b) seem like a miracle, if E(z) = 12 [z + z 2 − 4] is the conformal map of (C∪{∞}\[−2, 2]) bijectively to (C∪{∞})\D (the inverse of z → z + z1 ), then E ◦ ∆ maps (C ∪ {∞}) \ e to (C ∪ {∞}) \ D conformally and bijectively, and since deg(∆) = p, log|E − ∆(x)| ∼ p log(z) at infinity, so the potential theorist’s Green’s functions are related by Ge (z) = 1p G[−2,2] (∆(z0 )), which leads to another proof of (5.11.39)/(5.11.40). Theorem 5.11.10. Suppose µ is a measure on [−2, 2] so that Sµ is regular and x0 ∈ ∪j eint j . Then lim sup nλn (x0 , Sµ) ≤ lim sup nλn (∆(x0 ), µ)

(5.11.42)

lim inf nλn (x0 , Sµ) ≥ lim inf nλn (∆(x0 ), µ)

(5.11.43)

lim nλn (x0 , Sµ) = lim nλn (∆(x0 ), µ)

(5.11.44)

and In particular if limn→∞ nλn (∆(x0 ), µ) exists, then n→∞

n→∞

Moreover, (5.11.42)/ (5.11.43) hold if all x0 ’s are replaced by xn → x0 , and for each A > 0, this is uniform in xn ’s with supn n|xn −x0 | ≤ A.


Proof. We suppose throughout that xn = x0 . The accommodations for xn → x0 are straightforward. We first prove (5.11.42). Since λn (x0 ; Sµ) is an inf, we will use a trial function built from the optimizers Qℓ (x, ∆(x0 ); µ) for µ. One might first try Spℓ (x) = Qℓ (∆(x), ∆(x0 ); µ)

(5.11.45)

This certainly obeys Spℓ (x0 ) = 1 and deg(Spℓ ) = pℓ. By (5.11.35), Z Z 2 |Spℓ (x)| d(Sµ)(x) = |Qℓ (x, ∆(x0 ))|2 dµ(x) = λℓ (∆(x0 ); µ)

(5.11.46)

so λpℓ (x0 , Sµ) ≤ λℓ (x0 , µ)

(5.11.47)

This is terrible! It will not give (5.11.42) but only an inequality with the right multiplied by p. The problem is that S is symmetric in the sense of (5.11.37) and that makes the integral too large because Spℓ (x) is 1 not only at x0 but at all of the p elements of ∆−1 (∆(x0 )). To kill the contributions from the other points, we use the localization idea behind Nevai trial functions. Let B = sup{|x − x0 | | x ∈ e} and let ℓ, k be positive integers. Let 2 x − x0 L(x) = 1 − (5.11.48) B and Tℓp+k (x) = L(x)k Qℓ (∆(x), ∆(x0 ); µ)

(5.11.49)

which has degree ℓp + k and has Tℓp+k (x0 ) = 1. Let x0 ∈ eint j 2 x − x0 D = sup 1 − 0,

|Rℓ (∆(z)) − Xn+k (z)| ≤ Ce−K(δ)n

with n positive, so Z Z 2 Rℓ (x) dµ(x) = Rℓ (∆(x))2 d(Sµ)(x) [−2,2]

ej

(5.11.57)

(5.11.58)

−K(δ)n

≤ λn (x0 , Sµ) + C1 e

so by (5.11.35) and (5.11.37), Z Rℓ (x)2 dµ(x) ≤ p[λn (x0 , Sµ) + C1 e−K(δ)n ]

(5.11.59)

[−2,2]

Thus,

λℓ (∆(x0 ), µ) ≤ Rℓ (∆(x0 ))−2 p[λn (x0 , Sµ) + C1 e−K(δ)n ]

(5.11.60)

Picking n(j) so n(j)λn(j) (x0 , Sµ) goes to lim inf and using ℓp/n → 1 + δ, we get (5.11.43) with an extra (1 + δ) on the left. Since δ is arbitrary, (5.11.43) follows. To apply this to general measures, we need Proposition 5.11.11. Let e = ∪pj=1 ej be the essential spectrum of a period p Jacobi matrix with discriminant ∆. Let x0 ∈ eint for j int some j and let J ⊂ ej be a closed interval containing x0 . Let µ be a measure with σess (µ) = e which is locally Szeg˝ o on J. Then there exists a measure ν on [2, 2] so that (i) Sν ↾ J = µ ↾ J (ii) ν is regular for [−2, 2] and Sν is regular for e. Remark. It can be proven that any measure ν on [−2, 2] is regular for [−2, 2] if and only if Sν is regular for e (see Totik [432]), but we will not use this.


Proof. Let w be the weight for µ. Since µ is locally Szeg˝o on J, w > 0 for a.e. x in J. Let ν1 be a measure on ej so that ν1 ↾ J = µ ↾ J and dν1 /dx > 0 for a.e. x in ej . This is possible by the positivity of w on J. Let ν be the unique measure on [−2, 2] so that Sν ↾ ej = ν1 (ν is made by mapping ν1 to [−2, 2] using ∆ and then mapping to Sν using ∆−1 ). (i) holds by construction. ν and Sν are regular by Theorem 5.9.6 (or Theorem 5.11.13). Theorem 5.11.12. Let e = ∪pj=1 ej be the spectrum of a periodic Jacobi matrix. Let x0 ∈ eint j and let µ be a measure with σess (µ) = e, µ regular for e, and locally Szeg˝ o near x0 . Suppose x0 is a Lebesgue point for µ with w(x0 ) 6= 0 and for the locally Szeg˝ o function. Let xn be a sequence with sup n|xn − x∞ | ≡ A < ∞ (5.11.61) n

Then 1 ρe(x0 ) Kn (xn , xn ) = n→∞ n + 1 w(x0 ) and the limit is uniform in xn ’s obeying (5.11.61). lim

(5.11.62)

Proof. Let dν be the measure on [−2, 2] given by Proposition 5.11.11 and w˜ its weight. ∆(x0 ) is a Lebesgue point for w˜ and for its local Szeg˝o function (since ∆ is real analytic near x0 with ∆′ (x0 ) 6= 0) so Theorem 3.11.9 is applicable. Thus, lim

n→∞

ρ[−2,2] (∆(x0 )) 1 Kn (∆(xn ), ∆(xn ); ν) = n+1 w(∆(x ˜ 0 ))

(5.11.63)

By (5.11.38) and (5.11.40), ρ[−2,2] (∆(x0 )) ρe(x0 ) = w(∆(x ˜ w(x0 ) 0 ))

(5.11.64)

and by Theorem 5.11.10, 1 1 Kn (∆(xn ), ∆(xn ); µ) = lim Kn (xn , xn ; Sν) n→∞ n + 1 n→∞ n + 1 (5.11.65) Finally, by the Nevai comparison theorem, Theorem 5.11.5, since µ ↾ J = Sν ↾ J, lim

lim

n→∞

1 1 Kn (xn , xn ; µ) = lim Kn (xn , xn ; Sν) n→∞ n+1 n+1

(5.11.66)

384

5. PERIODIC OPRL

(5.11.61) follows from (5.11.63)–(5.11.66). Uniformity follows from the uniformity in Theorem 3.11.9. Once we have this result and the general Máté–Nevai upper bound (Theorem 5.11.1), by following the proof of Theorem 5.11.6, we get Theorem 5.11.13. Let e be a compact subset of R regular for the Dirichlet problem. Let dµ be a regular probability measure on e of the form dµ = w(x) dx + dµs (5.11.67) int Suppose that, for some closed interval I ⊂ e , w obeys a local Szeg˝ o condition on I. Then for a.e. x∞ ∈ I, with ρe given by the equilibrium measure for e, we have (1) (Diagonal Asymptotics) For any A < ∞ and sequence xn ∈ e with n|xn − x∞ | ≤ A for all n, we have 1 ρe(x∞ ) Kn (xn , xn ) → n+1 w(x∞ )

(5.11.68)

(2) (Lubinsky Universality) For any A < ∞ and a, b ∈ R with |a|, |b| ≤ A, we have Kn (x∞ + na , x∞ + nb ) sin(πρe(x∞ )(b − a)) → Kn (x∞ , x∞ ) πρe(x∞ )(b − a)

(5.11.69)

More generally, the limit of Kn (xn , yn )/Kn (x∞ , x∞ ) is the right side of (5.11.21) so long as |xn − x∞ | ≤ A/n, |yn − x∞ | ≤ A/n, and n(xn − yn ) → b − a. As in the case e = [−2, 2] (see Theorem 3.11.11), Theorems 5.11.6 and 5.11.13 imply clock behavior for zeros. x-ref?

Remarks and Historical Notes. TK from proof of Theorem 5.11.6. 5.12. Meromorphic Functions on Hyperelliptic Surfaces As explained in the overview section, the map from {an , bn }pn=1 to ∆, a polynomial of degree p, maps R2p to Rp+1 so inverse images of points are generically of dimension p − 1 and, in all cases, turn out to be a torus of dimension ℓ, the number of gaps. Our proof of this in the next two sections will involve a two-step process. We’ve already seen (see Theorem 5.4.15) that each periodic Jacobi matrix has an m-function with exactly one pole in each gap, although it may lie on either sheet. There will also be a pole at ∞ on the second sheet. Thus, m will be a function meromorphic on the √ two-sheeted Riemann surface associated to ∆2 − 4 with exactly ℓ + 1 poles (which we will see is minimal among all “nontrivial” meromorphic

5.12. MEROMORPHIC FUNCTIONS ON HYPERELLIPTIC SURFACES

385

functions). We will prove that such minimal Herglotz functions (normalized to be − z1 + O(1) at ∞ on the first sheet) are exactly in one-one correspondence to ℓ-tuples of points, one on each gap (on either part of the two-sheeted set associated to a gap) and thus to a point on an ℓ-dimensional torus. This part of the argument, which shows the set of meromorphic m-functions is an ℓ-torus, will be discussed in the next section. The second step will be to show that each such m-function is associated to a period p Jacobi matrix. This will involve coefficient stripping. Since the poles of the once stripped m-function are the zeros of the unstripped m-function, we will care about the relation of zeros and poles of this meromorphic function. This is the subject of this section where we will also formally construct the Riemann surface that we study. We can do this in the context of general ℓ-gap sets, which is what we will do. So e ⊂ R has the form with

e = [α1 , β1 ] ∪ · · · ∪ [αℓ+1 , βℓ+1 ]

(5.12.1)

α1 < β1 < α2 < · · · < βℓ < αℓ+1 < βℓ+1

(5.12.2)

Basic to what we do is the Riemann surface S, which we will sometimes write as Se to emphasize the set e. We start with an informal description: Take two copies, S+ and S− , of the Riemann sphere with e removed, that is, (C ∪ {∞}) \ e. Include the set e as “top edges.” S+ and S− are glued together by the rule that when one passes through e starting on C+ ∩ S+ , one winds up on C− ∩ S− and from C+ ∩ S− to C− ∩ S+ . Two spheres with one cut, glued in this way, is topologically a sphere, two cuts are a torus (see Figure TK), . . . , ℓ + 1 cuts are a x-ref? sphere with ℓ handles, so S is an orientable manifold of genus ℓ. More formally, we begin without the points of infinity and think of S ⊂ C2 as those points hz, wi with w 2 = R(z) ≡ Notice that x ∈ e ⇒ R(x) ≤ 0

ℓ+1 Y (z − αj )(z − βj )

(5.12.3)

j=1

x ∈ R \ e ⇒ R(x) > 0

(5.12.4)

In case e is the essential spectrum of a periodic Jacobi matrix with all gaps open, R(z) = (a1 . . . ap )2 [∆2 (z) − 4] (5.12.5)

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but double zeros are dropped if some gaps are closed and there is no ∆ in general (i.e., if some band has irrational harmonic measure). w 2 − R(z) = 0 defines a Riemann surface (one-dimensional complex manifold) since ∇(w 2 − R(z)) 6= 0 for all hw, zi ∈ S. If z ∈ / {αj , βj }ℓ+1 j=1 , ∂ 2 then ∂w (w −R(z)) = 2w 6= 0, so w is a smooth function of z and we can ∂ 2 ′ use z as a local coordinate. If z ∈ {αj , βj }ℓ+1 j=1 , ∂z (w − R(z)) = −R (z) is nonzero, and we can use w as a local coordinate, but not z. This means functions defined on S near hz0 , w0 i ∈ S are “analytic” if and only if they have convergent power series in z − z0 if z0 ∈ / {αj , βj }ℓ+1 j=1 , ℓ+1 and if z0 ∈ {αj , βj }j=1, we only need convergent power series in w, equivalently in (z − z0 )1/2 . Removing {hz, wi | z ∈ e} breaks S into two pieces S+ and S− . To get the points at infinity, one passes to a two-dimensional projective space. The reader unfamiliar with the view of the Riemann sphere as CP1 , the one-dimensional complex projective space should consult Section 9.2. Here we will use CP2 , the space of lines in C3 \ {0}, that in, in C3 \ {0}, we say hz, w, ui ∼ hz ′ , w ′, u′ i if and only if there is λ ∈ C so hz, w, ui = λhz ′ , w ′, u′ i and CP2 is the set of equivalence classes. In C3 \ {0}, we consider triples obeying 2ℓ

2

u w =u

2ℓ+2

Y ℓ+1 z P = (z − αj u)(z − βj u) u j=1

If some point lies in the set, so do all equivalent points and S is the set of equivalence classes. With this “proper” definition behind us, we will shift back to the hz, wi picture with the understanding there are two extra points at ∞, one, called ∞+ , with w ∼ z ℓ+1 , and the other, called ∞− , with w ∼ −z ℓ+1 . We use 1/z as a local coordinate near ∞. The map π : hz, wi → z is a two-to-one map over C ∪ {∞} \ e. For any point z ∈ C ∪ {∞} \ e, we use z+ and z− for the two points with w > 0 for z+ and z ∈ (βℓ+1 , ∞). We have labelled the two points at infinity ∞+ and ∞− . We define τ: S →S by τ (z+ ) = z− , where τ (z) = z if π(z) ∈ {αj , βj }ℓ+1 j=1 . We call this latter set branch points. We will be interested in meromorphic function f on S, that is, maps from S to SR , the Riemann sphere, that are locally “analytic” as SR -valued maps, that is, locally meromorphic in the conventional sense.


387

We recall that if f is a meromorphic function (defined as being locally meromorphic at every point) on the entire Riemann sphere, then p(z) f (z) = (5.12.6) a(z) for polynomials p and a. For f has only finitely many poles by compactness. Take a to have zeros at the finite poles of order equal to the order of those poles. Then f (z)a(z) is an entire function with a finite order pole at infinity, so a polynomial. Proposition 5.12.1. Every meromorphic function, f , on S has the form p(z) + q(z)w (5.12.7) f (z) = a(z) where p, q, a are polynomials with no common zeros and with a 6≡ 0, and conversely. Remarks. 1. We will start writing √ p±q R f= a

(5.12.8)

2. Here no common zeros means zeros of all three of p, q, a—not of just two. Proof. If p, q, a have a common zero, we can factor it out, so we will ignore that condition henceforth. Define fs (z) = 21 (f (z) + f (τ z)) (5.12.9) f is symmetric under τ , so fs is a function of π(z) only which, by an abuse of notation, we will write as fs (z) also. fs is obviously meromorphic in z at any nonbranch point since f (z) and f (τ z) are. At a branch point z0 , f (z) = f (τ z) =

∞ X n=0 ∞ X n=0

an (z − z0 )n/2 an (−1)n (z − z)n/2

so fs is also analytic in z. Thus, by the remark before the theorem, fs (z) =

p1 (z) a1 (z)

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Similarly, we define f (z) − f (τ z) w and see it is also entire meromorphic, so q1 (z)/a2 (z). Pick a(z) = a1 (z)a2 (z) and get the required form after pulling out common zeros. fa (z) =

We want to note that meromorphic functions on S are all the solutions of quadratic equations. Proposition 5.12.2. Let f be a meromorphic function on S so that f has at most first-order poles at branch points and if z ∈ S is a pole but not a branch point, then τ z is not a pole. Then the two values of f are the two solutions of α(z)f (z)2 + β(z)f (z) + γ(z) = 0

(5.12.10)

Indeed, in terms of (5.12.7), α(z) = a(z)

β(z) = −2p(z)

γ(z) =

p2 (z) − q 2 (z)R(z) a(z) (5.12.11)

Remark. We claim γ given by (5.12.11) is a polynomial. Proof. Clearly, (5.12.7) is equivalent to (af − p)2 = q 2 R

(5.12.12)

which is (5.12.10) if we prove γ is a polynomial. If a(z) has a zero of order k at z0 , not a √ branch point, √ then by hypothesis, as an analytic function either p +√q R or p√ − q R has a zero of order at least k. So 2 2 p − q R = (p + q R)(p − q R) as an analytic function, and so as a polynomial has a zero of order at least k. At branch points, z0 , if a(z0 ), then for f to have a simple pole (given that a, p, q have no common zeros), we must have that a has a simple zero, p(z) = 0, q(z) 6= 0. So p2 − q 2 R has a simple zero. Thus, γ is a polynomial, as claimed. We count orders of zeros and poles in terms of local analytic coordinates. Thus, if z0 is a branch point, we must use w or (z − z0 )1/2 as local coordinates so f (z) = z − z0 has a second-order zero at such a z0 . Associated to any zero or pole, z0 , we associate a single integer N(f ; z0 ) which is the order of the zero if z0 is a zero and the negative of the order of the pole if a pole. That is, if ζ is a local analytic coordinate near z0 with ζ(z0 ) = 0, then f (z) = ζ N (f ;z) (c + O(ζ))

(5.12.13)


389

with c 6= 0. Theorem 5.12.3. Let f be a meromorphic function on S with zeros/poles at {zj }m j=1 . Then m X

N(f ; zj ) = 0

(5.12.14)

j=1

Remark. This is usually stated by saying: “The number of zeros is equal to the number of poles.” In case where all zeros and poles are simple (i.e., |N(f ; zj )| = 1 for all zj ), this is literally true. Otherwise, one needs to count with multiplicities. Proof. Let Γ+ be the curve in S+ that goes clockwise around a cut from α1 to βℓ+1 , say, a distance ε around from the cut, and let Γ− be the same curve on S− . We will consider Z Z 1 f ′ (z) 1 f ′ (z) ξ= dz + dz (5.12.15) 2πi Γ+ f (z) 2πi Γ− f (z) Suppose first that f has no zero or pole with π(zj ) ∈ [α1 , βℓ+1 ]. In that case, we claim ξ=0 (5.12.16) For by taking ε ↓ 0, the contributions of the gaps cancel individually in Γ+ and Γ− (since the contours go in opposite directions). Along the bands [αj , βj ], f ′ /f is “continuous” across the band if we jump from S + to S − , so the top piece of the Γ+ contour cancels the bottom piece of the Γ− contour, and we get (5.12.16). On the other hand, one can evaluate the integrals by looking at residues at poles of f ′ /f (since infinity is either a regular point or a simple pole of f ′ /f ) and get Z X 1 f ′ (z) dz = − N(f ; zj ) (5.12.17) 2πi Γ± f (z) ± {zj |zj ∈S \[α1 ,βℓ+1 ]}

In this case, (5.12.16) and (5.12.17) yield (5.12.14). If there are zeros zj with π(zj ) ∈ [α1 , βℓ+1], we claim X ξ= N(f ; zj ) (5.12.18) {zj |π(zj )∈[α1 ,βℓ+1 ]}

so, taking into account that (5.12.17) is always true, we get (5.12.14) in general. For zeros zj ∈ [α1 , βℓ+1 ] \ {αj , βj }ℓ+1 j=1 , (5.12.18) is immediate, since for zeros in gaps, the noncancelling parts of Γ+ or Γ− precisely surround

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the poles of f ′ /f , and for poles in eint , the noncancelling parts of the contours that cancel surround the poles on S. If zj is a branch point, arg(f ) changes by πN(f ; zj ) (rather than 2πN(f ; zj )) because of how orders are defined. But there are contributions for both Γ+ and Γ− , yielding a total change of 2πN(f ; zj ), which proves (5.12.18). More generally, one can define an order of a value of f at any point as follows:   if f (z0 ) 6= a 0 n(f ; z0 , a) = N(f − a; z0 ) if f (z0 ) = a 6= ∞ (5.12.19)  −N(f ; z ) if f (z0 ) = a = ∞ 0 so n(f ; z0 , a) ≥ 0 and is nonzero at only finitely many points. Corollary 5.12.4. For any meromorphic f on S, X deg(f ) ≡ n(f ; z, a)

(5.12.20)

{z|n(f ;z,a)>0}

is independent of a.

Proof. Call the right side of (5.12.20) d(f ; a). Then X d(f ; a) − d(f ; ∞) = N(f − a; zj ) {zj |f (zj )=a or f (zj )=∞}

=0

by Theorem 5.12.3, which proves the a-independence.

The number deg(f ) is called the degree of f . As we will discuss in the Notes, degree and the formula (5.12.20) have a topological interpretation. Definition. A meromorphic function, f , is called root free if f (τ z) = f (z). Equivalently, f (z) =

p(z) a(z)

(5.12.21)

for polynomials p and a. Theorem 5.12.5. (a) Every root-free function has even order and all nonnegative even integers 2, 4, 6, . . . occur. Indeed, if f has the form (5.12.21), where p, a have no common zeros, then deg(f ) = 2 max(Deg(p), Deg(a)) where Deg(·) is the conventional degree of a polynomial.

(5.12.22)


391

(b) If f is not root free, it has degree at least ℓ + 1, and every degree larger than that occurs. In addition, if f has the form (5.12.8), deg(f ) ≥ max(Deg(a), ℓ + 1 + Deg(q))

(5.12.23)

Proof. (a) On the Riemann sphere, if f has the form (5.12.21) and Deg(p) ≥ Deg(a), f has a pole or nonzero value at ∞ and zeros (including multiplicity) at the zeros of p, so degR.S. (f ) = Deg(p) (where degR.S. means degree as a function on the Riemann sphere). If Deg(a) > Deg(p), there are Deg(p) zeros on C and ∞ is a zero of degree Deg(a) − Deg(p), so degR.S. (f ) = Deg(p) + Deg(a) − Deg(p) = Deg(a). Thus, degR.S. (f ) = max(Deg(p), Deg(a))

This degree is doubled on S since nonbranch point values occur at both z+ and z− and branch point orders are doubled because of the change to (z − z0 )1/2 counting. This proves (5.12.22), and that implies the allowed degrees of such functions are 2, 4, . . . . (b) We first prove (5.12.23). Let z1 , . . . , zA be the zeros of a (where A = Deg(a)). If zjp is not a branch point, p R(zj ) 6= 0 and so at least one of p(zj ) + q(zj ) R(zj ) or p(zj ) − q(zj ) R(zj ) is nonzero. (Note: If q(zj ) = 0, p(zj ) 6= 0 and both are nonzero.) f has a pole of order at least the order of the zeros zj in a at either (zj )+ or (zj )− . Ifpzj is a branch point, one of p(zj ) or q(zj ) is nonzero, so p(z) + q(z) R(z) is either O(1) or O((z − zj )1/2 ), in which case if zj is a zero of a of order nj , f has a pole of order 2nj or 2nj − 1 ≥ nj . We conclude √

deg(f ) ≥ Deg(a)

√ Q+ℓ+1 If Q = Deg(q), q R ∼ c z near ∞ . p can cancel q R or ± √ −q R, but not both, that is, f has a pole of order Deg(q) + ℓ + 1 − Deg(a) (if positive), at at least one of ∞+ or ∞− . So we have Deg(q) + ℓ + 1 − Deg(a) + Deg(a) poles, that is, deg(f ) ≥ Deg(q) + ℓ + 1

(5.12.24)

deg(f ) ≥ ℓ + 1

(5.12.25)

This proves (5.12.23) and shows that

To see that every integer larger than or equal to ℓ = 1 occurs, proceed as follows: Let m ≥ 0 be an integer and define p g(z) = z m R(z) (5.12.26)

where p g is meromorphic near ∞ in C ∪ {∞}. If we take the value of R(zj ) which is positive on (βℓ+1 , ∞), it has a pole of order m + ℓ + 1. Let p(z) be the “negative” order terms in the Laurent series at

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infinity—negative in z −1 . So p(z) is that unique polynomial (or degree ℓ + 1 + m) with g(z) = p(z) + o(1) (5.12.27) near infinity. Now let p (5.12.28) f (z) = p(z) ± z m R(z) Clearly, f is meromorphic on S. At finite points of S, f is finite and, by (5.12.27), f (z) = o(1) near ∞− , and so f has a zero there. Its only pole is at ∞+ and there, f (z) = 2z m+ℓ+1 + O(z m+ℓ ). Thus, the pole is of order m + ℓ + 1 and deg(f ) = m + ℓ + 1 proving the claim.

(5.12.29)

A compact Riemann surface which has meromorphic functions of degree 1 is conformally equivalent to the Riemann sphere since f is one-one and onto that sphere. A compact Riemann surface which has meromorphic functions of degree 2 is called hyperelliptic and this last √ theorem tells us that S, the Riemann surface of R, is hyperelliptic. We now turn to the question of what sets can be the zeros/poles Np z of a function on S. By Theorem 5.12.3, if {zj }N j=1 and {pj }j=1 are the zeros and poles of a meromorphic function (counting multiplicity and with no zj equal to any pk ), then Nz = Np

(5.12.30)

so we will henceforth use N. For the Riemann sphere, (5.12.30) is the only restriction on the zeros and poles. But we recall the situation for classical elliptic functions, that is, meromorphic functions on C which obey f (z + 1) = f (z)

f (z + τ ) = f (z)

(5.12.31)

for some τ ∈ / R; by replacing τ by −τ , we can suppose Im τ > 0. Let Lτ = {n + mτ | n, m ∈ Z}

(5.12.32)

which is a discrete lattice in C, and so let C Sτ = (5.12.33) Lτ equivalence classes in C mod Lτ . It can be shown Sτ is conformal to Sτ ′ if and only if for c ∈ C\ {0}, Lτ = cLτ ′ if and only if there exists ′ an A ∈ SL(2, Z) with A τ1 = c τ1 for c ∈ C \ {0}. Moreover, every Riemann surface which is topologically a torus is conformal to some


393

Sτ . In particular, our S is an Sτ if ℓ = 2 (with τ pure imaginary a function of (β2 − α2 )/(β1 − α1 ) and (α2 − β1 )/(β1 − α1 )). Meromorphic functions on Sτ are precisely the same as f ’s on C obeying (5.12.31). Liouville’s second theorem on elliptic functions ((5.12.30) is his first theorem on elliptic functions) says that N X j=1

zj − pj ∈ Lτ

(5.12.34)

where, for example, one normalizes zj , pj by putting them in the fundamental region F = {a + bτ | 0 ≤ a < 1, 0 ≤ b < 1}. To prove (5.12.34), one takes a contour, Γ, which is shown in Figure TK, that goes clockwise around the parallelogram with sides x-ref? Γ1 = {(a, 0) | 0 ≤ a < 1}, Γ2 = {(1, bτ ) | 0 ≤ b < 1}, Γ3 = {(a, τ ) | 0 ≤ a < 1}, Γ4 = {(0, bτ ) | 0 ≤ b < 1}, and assuming f has no zeros or poles on Γ, one looks at Z 1 f′ ξ= z dz (5.12.35) 2πi Γ f On the one hand, ξ is the left side of (5.12.34) by the residue calculus. On the other hand, since f ′ /f is the Γ1 and Γ3 contriR periodic, f′ 1 butions partially cancel to give − 2πi Γ1 τ f dz, and the Γ2 and Γ4 to R f′ 1 give 2πi dz. But by the argument principle and periodicity again, R f ′ Γ2 f 1 dz is an integer, so ξ = n1 τ + n2 ∈ Lτ . 2πi Γ1 f In this argument, there are two main players: the function z and the contours Γ1 and Γ2 . z enters because dz is an analytic one-form and z is its integral. In the torus Sτ , Γ1 and Γ2 are precisely homology generators for the homology of Sτ , closed curves that loop once about the two “holes” of Sτ . Returning to our hyperelliptic surface, S, its homology group has 2ℓ generators which loop about the two holes of each of the ℓ handles. We can realize these generators explicitly. For j = 1, . . . , ℓ, let G+ j be + − the line on S+ from βj to αj+1 and Gj the same on S− . Gj = Gj − G− j is a closed curve on S called Γ(Gj ). For j = 1, . . . , ℓ + 1, let Γ(Bj ) be the closed curve that goes from αj to βj on S+ just below the cut and then returns from βj to αj just above the cut. Γ(B1 ) + Γ(B2 ) + · · · + Γ(Bℓ+1 ) is homologous to the curve Γ+ used in the proof of Theorem 5.12.3 and that curve is even homotopic to 0 by “pulling it through ∞+ .” Thus, {Γ(Bj )}ℓ+1 j=1 are not independent in homology, but {Γ(Bj )}ℓj=1 are, and {Γ(Gj )}ℓj=1 ∪ {Γ(Bj )}ℓj=1 are a set of homology generators.

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As for that other player, analytic one-forms, consider p −1 ω1 = R(z) dz

(5.12.36)

that is, w −1 dz in hw, zi coordinates. Since R vanishes at each branch point, one might think w is singular there, but recall that the proper local coordinate there is w and w ∼ c0 (z − z0 )1/2 , that is, dz ∼ c1 w dw and w −1 dz = c1 dw is nonsingular at z0 . Near ∞, we need to shift from z to ζ = z −1 and dz = −z 2 dζ is singular. But since R(z) ∼ O(z ℓ+1 ) if ℓ ≥ 1, ω1 is regular at ∞± also. More generally, if P (z) is a polynomial, then p ωP = P (z) R(z)−1 dz (5.12.37)

is regular at all finite points and is regular at infinity so long as deg(P ) ≤ (ℓ + 1) − 2 = ℓ − 1

(5.12.38)

We thus get an ℓ-dimensional family of analytic one-forms and, by deRham’s theorem and the fact that the homology is dimension 2ℓ, this is all of them (the 2ℓ-dimensional deRham cohomology is spanned by ℓ analytic and ℓ anti-analytic forms). It is natural to evaluate the cohomology elements on homology generators, and so define for j = 1, . . . , ℓ, Z π(P ; Bj ) = ωP (5.12.39) Γ(Bj ) Z π(P ; CG ) = ωP (5.12.40) Γ(Gj )

called the periods of the one-form ωP . The following is basic:

Theorem 5.12.6. For P a real polynomial of degree at most ℓ − 1, define vectors in Rℓ by B(P )j = −iπ(P ; Bj )

(5.12.41)

G(P )j = π(P ; Gj )

Then B and G are bijections of real polynomials to Rℓ . Proof. Since the polynomials of degree at most ℓ − 1pare an ℓdimensional space, it suffices to prove ker B = ker G = 0. i R(z) has a definite sign on the top of each Bj and the contour has the opposite direction on the bottom and opposite signs, so if P has a definite sign on Bj , then −iπ(P ; Bj ) 6= 0. It follows that if B(P ) = 0, the P has a zero on each of the ℓ sets B1 , . . . , Bℓ . But if P is nonzero, it can only have ℓ − 1 zeros. Thus, ker B = 0. A similar argument proves that ker G = 0.


395

Since G is a bijection, we can find polynomials P1 , . . . , Pℓ so that π(Pk ; Gj ) = δkj

(5.12.42)

which we call the canonical basis. The periods, τkj ∈ R, of S are defined by π(Pk ; Bj ) = iτkj (5.12.43) In Cℓ , we define the lattice LS of S by

LS = {~n + iτ m ~ | n, m ∈ Zℓ }

where (τ m) ~ k=

X

τkj mj

(5.12.44) (5.12.45)

By the theorem, the vectors τk· are independent, so LS is a discrete lattice in Cℓ , which means that the Jacobi variety, JS =

Cℓ LS

(5.12.46)

is a torus of real dimension 2ℓ. Given any rectifiable (not necessarily closed) contour, Γ, on S, define A(Γ) ∈ Cℓ by Z A(Γ)k = ωPk Γ

If Γ is closed and homologous to zero, Cauchy’s theorem implies A(Γ) = 0. More generally, since {Γ(Bj }ℓj=1 and {Γ(Gj )}ℓj=1 are generators of homology, we have Γ closed ⇒ A(Γ) ∈ LS

That means that if x, y ∈ S is fixed and Γxy is any curve from x to y, A(Γxy ) has a value whose ambiguity is an element in LS , and thus, [A(Γxy )]LS ≡ Ax (y) is an element of JS . Thus, once we fix a base point x in S, we have a map Ax : S → JS (5.12.47)

called Abel’s map. LS is an abelian group, and it is easy to see that the change of base point is given by Ax1 (y) = Ax0 (y) + Ax1 (x0 )

(5.12.48)

= Ax0 (y) − Ax0 (x1 )

(5.12.49)

While Ax0 depends on the base point x0 , we will often just use A with some fixed x0 in mind. The fundamental results about zeros and poles and meromorphic functions are:

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Theorem 5.12.7 (Abel’s Theorem, First Half). Let f be a meroNp z morphic function on S and let {zj }N j=1 and {pj }j=1 be its zeros and poles counting multiplicity. Then (a) Nz = Np (b) We have Np Nz X X A(zj ) = A(pj ) (5.12.50) j=1

j=1

z Theorem 5.12.8 (Abel’s Theorem, Second Half). Let {zj }N j=1 and Np {pj }j=1 be points on S with no zj equal to a pk (although z’s or p’s can be repeated). Then there is a meromorphic f on S with zeros precisely at the zj and poles precisely at the pj if and only if (a) and (b) of Theorem 5.12.7 hold.

Remarks. 1. This is a single result which we state as two because we will prove and extensively use the first half below (and in the next section). We will only prove a special case of the second half in Section 9.11 and will use it once below in the first proof of a theorem (Theorem 5.12.10), for which we also provide a second proof below that does not use the second half. 2. Our use of Abel’s theorem only requires the existence of a map U from S to JS with the required properties. Indeed, in Section 9.11, our U will map ∪ℓj=1 Gj ∪ {∞± } to a natural torus group (∂D)ℓ and we will shift from additive notation for the group action to multiplicative. 3. Because of (5.12.49) and Nz = Np , the equality (5.12.50) is base point independent. 4. The sum in (5.12.50) is in the abelian group JS . 5. We emphasize that the sets in these theorems are really sets z with multiplicity, and a zero of order k appears k times in {zj }N j=1 , and similarly for poles. As a preliminary for the proof and because it is useful in further developments, we want to describe a specific realization of A in Cℓ for S with suitable cuts. Remove from both S+ and S− the intervals [β1 , βℓ+1 ]. The two halves are still connected by crossing (α1 , β1 ), and the reader can convince himself/herself that the resulting set with ∞± included is simply connected. So taking the base point as α1 for definiteness, one gets a single-valued map A♯ with values in Cℓ . In each gap (βj , αj+1) on either sheet, A♯ is discontinuous across the gap but only by a period (i.e., element of LS ) which, by discreteness, has to be constant on each gap. Similarly, for each band [αj , βj ], j =


397

2, . . . , ℓ+1, A♯ is discontinuous if we approach x ∈ (αj , βj ) from C+ ∩S+ or from C− ∩ S− (which are the same point in S), and again we get a constant which is a period. For Theorem 5.12.7, all we need are these facts, but for Theorem 5.12.12 below, we need the precise constant: Proposition 5.12.9. (a) If x ∈ (βj , αj+1), then X j ♯ ♯ A (x± + i0) − A (x± − i0) = ±i ~τ· m

(5.12.51)

m=1

(b) If x ∈ (αj , βj ), j = 2, . . . , ℓ + 1, then

A♯ (x± + i0) − A♯ (x∓ − i0) = ±

j−1 X

~δ· m

(5.12.52)

m=1

Remark. ~τ· m is the vector whose components are τjm . Similarly, ~δ· m has components δjm , that is, j−1 X

m=1

~δ· m = 1, . . . , 1, 0, . . . , 0 | {z } | {z } j−1

(5.12.53)

ℓ−(j−1)

Proof. (a) A curve that goes from α1 in S+ in the upper halfplane of S+ to x ∈ (βj , αj+1) and returns in α1 in the lower half-plane of S+ is homologous to Γ(B1 ) + · · · + Γ(Bj ), so (5.12.51) for + is just (5.12.43). The minus sign is immediate if we √ √ note that with a base point α1 , the periods flip sign from R to − R in going from S+ to S− . (b) Consider first j = 2. To get to x+ + i0, we go above (α1 , β1 ), then follow (β1 , α2 ) in S+ and then go to x + i0. To get to x− + i0, we do the same, but follow (β1 , α2 ) in S− . The difference is just Γ(G1 ). For general j, the difference is Γ(G1 ) + · · · + Γ(Gj−1 ). This leads to (5.12.52). Proof of Theorem 5.12.7. (b), let Γ± be the contours used let Z 1 f ′ (z) ♯ ξA = A (z) dz + 2πi Γ+ f (z)

(a) is Theorem 5.12.3. To prove in the proof of that theorem and 1 2πi

Z

Γ−

f ′ (z) ♯ A (z) dz f (z)

(5.12.54)

We will suppose no zeros or poles lie inside Γ± —the change when some do is as with the proof of Theorem 5.12.3. In that earlier theorem, the residues of f ′ /f outside Γ± at z0 are just #(zj = z0 ) − #(pj = z0 ). Now they are multiplied by A♯ (z0 ).

398

5. PERIODIC OPRL

Thus, ξA = −

X

zj ,pj ∈S ± \[αj ,βj+1 ]

(A♯ (zj ) − A♯ (pj ))

(5.12.55)

On the other hand, there is not the compete cancellation that caused ξ = 0 in (5.12.17) because A♯ is discontinuous across cancelling curves. Rather, since A♯ is constant on each Bj or Gj , we get Z j ℓ X X 1 f′ ~τ· m ξΓ = i dz 2πi Γ(Gj ) f m=1 j=1 (5.12.56) Z j−1 ℓ+1 X ′ X 1 f ~δ· m + dz 2πi f Γ(B ) j m=1 j=2 ′

Since 1i ff = d(arg f ) (plus a change of log|f | which integrates to 0) which is a 2πi integer, so for integers nj and mj , Z Z 1 f′ 1 f′ dz = nj dz = mj 2πi Γ(Gj ) f 2πi Γ(Bj ) f and so, ξA ∈ LS . Thus, the sum in (5.12.55) is 0 in Cℓ /LS .

Next, we want to prove a result about sums of the type in Abel’s theorem being one-one on certain special sets, whose relevance to mfunctions of periodic problems should be evident. Let Gj be the set which is the range of Γ(Gj ), that is, Gj = π −1 ([βj , αj+1]) which is a circle formed from two lines between two branch points. Let Te = G1 × · · · × Gℓ

(5.12.57)

Theorem 5.12.10. Map Te to JS by

e is one-one. Then A

e 1 , . . . , zℓ ) = A(z

ℓ X

A(zj )

(5.12.58)

j=1

First Proof. If not, we can find (z1 , . . . , zℓ ) and (p1 , . . . , pℓ ) in Te, so ℓ ℓ X X A(zj ) = A(pj ) (5.12.59) j=1

j=1


399

Drop those z’s equal to p’s, so we find {zj }j∈J and {pj }j∈J , all distinct with |J| ≤ ℓ, so X X A(zj ) = A(pj ) j∈J

j∈J

By the second half of Abel’s theorem, there is a meromorphic function, f , with those zeros and poles. Clearly, deg(f ) = |J| ≤ ℓ

(5.12.60)

So, by Theorem 5.12.5, f is root free. But every such root free function has zeros at +/− pairs or double zeros at branch points, and this f does not. This is a contradiction. Second Proof. Let (5.12.59) hold. Then for integers n1 , . . . , nℓ and m1 , . . . , mℓ , j = 1, . . . , ℓ, and k = 1, . . . , ℓ, XZ ωk = nj δkj + imj τkj (5.12.61) j

Γj

where Γj is that contour on Γ(Gj ) that goes clockwise from zj to pj . Since ωk is real on Γ(Cj ) and the Rτk are linearly independent, we see mj = 0 for all j. Moreover, since Γ(Gj ) ωk = δjk , we can subtract nj

copies of Γ(Gj ) to Γj and get a contour Γ♯j from zj to pj . So for all j, k, XZ ωk = 0 (5.12.62) j

Γ♯j

Since {Pk }ℓk=1 is a basis, we conclude for any polynomial P of degree ˜ j and σj = ±1, at most ℓ − 1 and suitable Γ Z X p −1 σj P (x) R(x) dx = 0 (5.12.63) j

˜j Γ

˜ j is either Γ♯ or Γ♯ run backwards, and the choice is made so where Γ j j that Z p −1 R(x) dx ≥ 0 (5.12.64) ˜j Γ

Here σj are picked to accommodate the change of direction Γ♯j if needed. By multiplying all σj by −1 if necessary, we suppose σ1 = 1. Pick P plus or minus a monic with zeros one in each band (αj+1 , βj+1) where σj σj+1 = −1 and so P is positive on (β1 , α2 ). Thus, σj P (x) > 0 on each (βj , αj+1) (5.12.65) and all terms in (5.12.63) are nonnegative and can only sum to zero if ˜ j is a single point, which implies zj = pj for all j. each Γ

400

5. PERIODIC OPRL

It is a consequence of degree theory (see the Notes) that any oneone map between compact orientable manifolds of the same dimension e − A(z e 0 ) is real, that is, is a bijection. Note that if z0 , z ∈ Te, then A(z) e e 0) ∈ A(z) − A(z

the standard ℓ torus Rℓ /Zℓ . Thus:

Rℓ = Tℓ L S ∩ Rℓ

(5.12.66)

Corollary 5.12.11. Fix z0 ∈ Te. then e e 0) z 7→ A(z) − A(z

is a bijection of Te and Tℓ .

Finally, we want to find an explicit formula for A(∞+ ) − A(∞− ) in terms of harmonic measure. The analytic one-forms, ωP , with deg(P ) ≤ ℓ − 1 played a critical role in defining A. If deg(P ) = ℓ, then ωP is no longer analytic at ±∞ but has simple poles at ±∞, so Z 1 ξH = H(z)A♯ (z) dz (5.12.67) 2πi Γ+ ∪Γ− will pick up A♯ at the poles, that is, at ±∞. Here H is given by (5.5.139), that is, P (z) H(z) = − p (5.12.68) R(z) where P is the unique monic polynomial of degree ℓ with Z αj+1 P (x) p dx = 0 (5.12.69) |R(x)| βj and (see Theorem 5.5.22)

H(z) =

Z

−

dρe(x) x−z

(5.12.70)

Theorem 5.12.12. Let ej = [αj , βj ]. Then A(∞) − A(−∞) = (ρe(e1 ), ρe(e1 ∪ e2 ), . . . , ρe(e1 ∪ · · · ∪ eℓ )) (5.12.71) Proof. H(z) outside Γ has poles only at ∞± with residue 1 at ∞+ and −1 at ∞− (note if w = 1/z, −dz/z = dw/w, so −dz/z has residue +1 at ∞!). Thus, ξH = A♯ (−∞) − A♯ (∞) (5.12.72) On the other hand, H(z) dz is regular inside Γ+ and Γ− , so there would be complete cancellation between the pieces if A♯ was not there. Because A♯ is discontinuous, these cancellations give constants times


401

integrals of boundary values of H over each band and gap. The contributions over the gaps cancel and, by (5.12.52), (αj , βj ) contributes (note H(x+ + i0) = H(x− + i0)) X Z βj j−1 2 ~δ· m Im H(x+ + i0) dx (5.12.73) 2π αj m=1

where the 2 comes from Γ+ and Γ− both contributing. But dρe/dx = 1 Im H(x+ + i0), so π (ξH )k =

ℓ+1 X

ρe(ej )

j=1

=

ℓ+1 X

j=k+1

j−1 X

δkm

m=1

ρe(ej ) = 1 −

k X

(5.12.74) ρe(ej )

j=1

which, given the fact that A is measured modulo integers, yields (5.12.71). Remarks and Historical Notes. The theory of elliptic and hyperelliptic functions was a major theme in nineteenth century mathematics, with critical contributions by Abel, Liouville, Jacobi, and Riemann. For the basic theory of meromorphic functions on Riemann surfaces, see Farkas–Kra [121], Griffiths–Harris [180], and Miranda [300]. The earliest realization that elliptic functions are connected to twoband problems is due to Akhiezer [12]. For finite gap Hill equations (a continuum analog of the Jacobi case), the relevance of hyperelliptic functions is a discovery of Dubrovin–Matveev–Novikov [110] and McKean–van Moerbeke [298] in the context of studying the KdV equation. Our use of Pj ’s obeying (5.12.42) and the resulting proof of Theorem 5.12.10 (given as the second proof) is motivated by Levitan’s discussion [272] for the Hill equation. The development of these ideas for Jacobi matrices is due to Flaschka–McLaughlin [132], Krichever [248, 249], and van Moerbeke [438]. For a list of the vast related literature, see [391]. Theorem 5.12.12 is motivated by the analogous result in [391], found following suggestions of Peherstorfer–Yuditskii. Our proof there is somewhat more complicated because it uses the potential R R log|z − x| dρe(x) rather than (x − z)−1 dρe(x) and so has an extra logarithmic cut to cope with. The degree theory result needed to obtain Corollary 5.12.11 runs as follows: Let M, N be two C ∞ orientable compact manifolds of the same dimension n so H n (M) = H n (N) = Z for the homology groups.

402

5. PERIODIC OPRL

Any continuous f : M → N induces a map H n (f ) : H n (M) → H n (N) which is a group homomorphism, and so of the form k → Dk for some D ∈ Z, called the degree, deg(f ), of f . Now let f be a C ∞ map. A point m ∈ M is called regular point if dfm , the derivative of f at m, is nonsingular. A point n ∈ N is called a regular value if each point in f −1 (n) is a regular point. In particular, if f −1 (n) is empty, n is regular. By compactness and the inverse function theorem, each regular value has f −1 (n), a finite set. Sard’s theorem asserts the set of regular values is the complement a set of measure zero. If m is a regular value, the signature of f at m, Sm (f ), is the sign of det(f ). (In general, this requires one to pick orientations on M and N as does determining the sign of deg(f ); if M = N, making the two orientations the same fixes signs.) The fundamental theorem of degree theory says that for any regular value, n, X Sm (f ) = deg(f ) (5.12.75) m∈f −1 (n)

In particular, if f −1 (n) is empty, deg(f ) = 0, and then regular points with f −1 (n) 6= ∅ must have an even number of points to get the sum of ±1 to be 0. So if f is one-one, the degree is ±1, and so f is onto, as claimed. In the case studied in this section for f meromorphic on S, f maps S to SR , the Riemann sphere, and the topological degree is the degree as we have defined it. Analytic functions, f , where nonsingular, are conformal and so have signature +1 and (5.12.75) and (5.12.20) agree at points, a, for which n(f ; z, a) = 0 or 1 for all z. For expositions of degree theory for smooth maps, see Fonseca– Gangbo [133], Guillemin–Pollack [184], Krawcewicz–Wu [241], Lloyd [277], Milnor [299], and Spivak [407]. 5.13. Minimal Herglotz Functions and Isospectral Tori In Section 5.2, we saw the m-function, m(z), for a periodic Jacobi matrix, J, with essential spectrum an ℓ-gap set, e, has a meromorphic continuation to Se. From the point of view of the last section, we will see m has some simple properties. And it will turn out that the study of all J’s that lead to a fixed e is related to the study of functions with these properties. Theorem 5.13.1. m is a meromorphic function on Se with the following properties:

5.13. MINIMAL HERGLOTZ FUNCTIONS AND ISOSPECTRAL TORI

403

(i) m is Herglotz in the sense that if Im z > 0, Im m(z+ ) > 0

(5.13.1)

that is, Im m > 0 on S+ ∩ C+ . (ii) On S+ near ∞+ ,

1 1 m(z) = − + O 2 z z

(5.13.2)

(iii) m has degree ℓ + 1. (iv) m has one zero and one pole on each set {Gj }ℓj=1 and, moreover, a zero at ∞+ and a pole at ∞− . Proof. (i) and (ii) hold for any m-function; see Example 2.3.1 and (2.3.10). By Theorem 5.2.1, m(z) obeys the quadratic equation α(z)m(z)2 + β(z)m(z) + γ(z) = 0

(5.13.3)

where α(z) = ap pp−1 (z) and the discriminant is ∆(z)2 − 4. Thus, p β(z) ± ∆2 (z) − 4 m(z) = − 2α(z)

(5.13.4)

(5.13.5)

√ m(z) clearly has a meromorphic continuation to all of S since ∆2 − 4 has branch points precisely at the edges of open gaps (the double zeros of ∆2 − 4 at closed gaps are not branch points) with the only possible poles at ∞± and at the zeros of pp−1(x). These zeros are analyzed in Theorem 5.4.15: one occurs in each gap. If the gap is closed, ∆2 − 4 has a double zero, and since that means β 2 − αγ = 0 and α = 0, we have β = 0. So, in (5.13.5), α has a simple zero and the numerator is also zero. So (as also remarked in Proposition 5.10.2), m has neither zero nor pole at the closed gaps. If a gap is not closed and the zero is at the interior point of the gap, z0 , then α(z) has a simple zero at z0 . Since ∆(z0 ) 6= ±2, β 2 − αγ 6= 0, so β(z0 ) 6= 0. Thus, p −β(z) ± β 2 (z) − α(z)γ(z)

vanishes at one of (z0 )± and is nonzero (indeed, −2β(z0 )) at the other point. So m has a single pole on one sheet or the other, but not both. If the zero is at a resonance, that is, at an edge, z0 , of a closed gap, 2 2 ∆ −4 has a simple zero at z0 and β(z0 )2 = α(z√ 0 )γ(z0 )+(∆ (z0 )−4) = 0. 2 Thus, β(z0 ) = c(z − z0 ) + O((z − z0 ) ) while ∆2 − 4 = c(z − z0 )1/2 + O((z − z0 )3/2 ) and m(z) = c(z − z0 )−1/2 + O(1), so by the way poles

404

5. PERIODIC OPRL

are counted at branch points, m has a simple pole at z0 . We have thus proven m(z) has exactly one pole in each Gj , j = 1, . . . , ℓ. By coefficient stripping (see (3.2.28)), m(z)−1 = b1 − z − a21 m1 (z)

(5.13.6)

Since m1 is also the m-function of a periodic Jacobi matrix, m1 has one pole in each gap, and so m has exactly one zero in each two-sheeted gap. Besides zeros of α, the only other possible poles of m(z) are at ∞± . At ∞+ , m is zero by (5.13.2). Thus, since α(z) ∼ c1 z p−1 , β(z) ∼ c√2 z p , and ∆2 (z) ∼ z 2p , we must have β(z) cancelling the z p growth of ∆2 − 4 at ∞+ . That means at ∞− , the numerator is −2c2 z p +O(z p−1) and so, m(z) has a simple pole at ∞− . We have thus proven m has exactly ℓ + 1 simple poles, so m has degree ℓ + 1. Since we have accounted for ℓ + 1 zeros of m, we have them all. This leads to a natural definition in the context of general finite gap sets, not just those which are periodic spectra. Definition. Let e be a finite gap subset of R and let Se be the associated Riemann surface. A minimal Herglotz function on Se is a meromorphic function m on Se obeying: (i) m is Herglotz in the sense that (5.13.1) holds for z ∈ S+ ∩ C+ and Im m(x+ + i0) has compact support. (ii) m obeys (5.13.2) (so m is a discrete m-function in the sense of Section 2.3). (iii) deg(m) = ℓ + 1. (iv) m has a pole at ∞− . Remark. The word minimal is used because m has minimal degree among non-square root free functions. The set of all minimal Herglotz functions on Se will be denoted by Me. We will show first that Me is a torus of dimension ℓ; indeed, naturally associated to the torus Te of (5.12.57). We will then study the Jacobi matrix associated to an m in Me and prove, for general e, it is almost periodic, and if e comes from one periodic Jacobi matrix, then all the minimal Herglotz functions associated to e have associated periodic Jacobi matrices and have the same ∆. This will provide the promised proof that the set of periodic J’s with a given ∆ is a torus. Here is the general structure of minimal Herglotz functions:


405

Theorem 5.13.2. Every minimal Herglotz function, m, in Me has the form p p(z) ± R(z) m(z) = (5.13.7) a(z) where

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

Deg(a) = ℓ

(5.13.8)

Deg(p) = ℓ + 1

(5.13.9)

−p is monic. Moreover, p and a are real polynomials. a has one simple zero in each gap. m has exactly one simple pole in each gap plus the pole at ∞− . m has exactly one simple zero in each gap plus the zero at ∞+ .

Remarks. 1. A polynomial is called real if all its coefficients are real. 2. In the periodic case with closed gaps, a is not the 2α of (5.13.5) but it has zeros at closed gaps that occur in the numerator removed. In addition, even if all gaps are open and ∆2 − 4 has simple zeros, it is not R, but rather (a1 . . . ap )−2 R. Proof. As a rational function on S, m has the form p p(z) ± q(z) R(z) m(z) = (5.13.10) a(z) By (5.12.23) and deg(m) = ℓ + 1, we see deg(q) = 0, so we can take q = 1. Also by (5.12.23), deg(a) ≤ ℓ + 1. Since (5.13.2) holds, and on S+ , p + R(z) = z ℓ+1 + O(z ℓ ) (5.13.11) near ∞+ , we must have that p(z) = −z ℓ+1 + O(z ℓ )

(5.13.12)

(since deg(a) ≤ ℓ + 1 means the z ℓ+1 term in the numerator must cancel). Thus, p −p is monic p and (5.13.9) holds. Since − R(z) (i.e., R(z) on S− ) has the opposite sign, near ∞− , p p(z) ± R(z) = −2z ℓ+1 (5.13.13) so to have a pole at ∞− , we must have

Deg(a) ≤ ℓ (5.13.14) p Since m(z) is real on (βℓ+1 , ∞) and R(z) is real there, p(z)/a(z) is real there. So, by analyticity, all its zeros and poles come in conjugate pairs or lie on R. Since −p is monic, we see p and then a is real.

406

5. PERIODIC OPRL

On each band, p/a is real, so Im m(x+ + i0) =

Im

p

R(x+ + i0) a(x + i0)

(5.13.15)

p Since R(x) changes sign from one band to the next, a must change sign to keep Im m(x+ + i0) ≥ 0. Thus, a has an odd number of zeros in each gap. Since there are ℓ gaps and, by (5.13.14), at most ℓ zeros, we conclude each gap has precisely one zero and (5.13.8) holds. As in the analysis in the proof of Theorem 5.13.1, if a has a zero at a point, z0 , in the interior of a gap where R(z0 ) 6= 0, m must have a pole at either p (z0 )+ or (z0 )− (or both), and if a has a zero at a band edge, z0 , p(z) ± R(z) vanishes at (z −z0 )1/2 or approaches a constant. Thus, in that case also, m has a pole at z0 . Thus, m has at least one pole in each gap, and so since ∞− is a pole and there are only ℓ + 1 poles, we see each gap has exactly one simple pole. Define m1 by (5.13.6) where b1 , a1 are picked so m1 (z) obeys (5.13.2). By coefficient stripping, m1 is a Herglotz function and clearly, m1 is meromorphic on S. m1 has a pole at each finite zero of m and, by deg(m) = ℓ + 1 and the fact that ∞− is not a zero, and by (5.13.2), ∞+ is a simple zero, we know m has an ℓ finite zeros. Thus, m1 has ℓ poles in S \ {∞± }. At ∞+ , m1 has a zero and, by (5.13.6) and m(z)−1 → 0 at ∞− , we see m1 has a simple pole at ∞− . Thus, deg(m1 ) = ℓ + 1 and ∞− is a pole, so m1 is also in Me. By the analysis above, m1 has exactly one simple pole in each gap so, by (5.13.6), m(z) has exactly one simple zero in each gap. Along the way, we have also proven: Corollary 5.13.3. If m ∈ Me, the coefficient stripped m1 defined by (5.13.16) also lies in Me. Remark. The proof of this corollary did not use that m had a pole at ∞− , only that m did not have a zero at ∞− . Example 5.13.4. This example shows that property (iv) in the definition of minimal Herglotz functions is not automatic. Let J be a periodic Jacobi matrix, and for y ∈ R, let Jy be the matrix where only b1 is changed from b1 to b1 +y. Let my (z) be the associated m-function. By (5.13.6) and the fact that Jy and J once stripped are the same, we see my (z)−1 = y + m(z)−1

(5.13.16)


407

Thus, my is also a meromorphic function of degree ℓ + 1 and so obeys (i)–(iii) of the definition of Me. But, by (5.13.16), my (∞− ) = y −1

(5.13.17)

so my fails to obey condition (iv) of the definition. my still has a pole in each gap, but instead of a pole at ∞− , there is one additional pole on (−∞, α1 ] ∪ [βℓ+1 , ∞) whose location and sheet depend on the sign and magnitude of y. Also, now deg(a) = ℓ + 1 rather than deg(a) = ℓ. Changing a1 from the periodic value changes the degree of m. There is a natural map, D, from Me to Te, the torus described in (5.12.57). Namely, each f ∈ Me has ℓ poles other than at ∞− , one each in G1 , G2 , . . . , Gℓ . The set of these poles describes a point (z1 , . . . , zℓ ) ∈ Te. This is called the Dirichlet data for f . D is called the Dirichlet map. The reason for this name will be explained in the Notes. Theorem 5.13.5. D is a one-one continuous map of Me onto Te. In particular, Me is topologically a torus. Remark. Here Me is topologized using the topology of uniform convergence (uniform as SR -valued functions). Proof. We will describe a point in Te with coordinates D(f ) = (z1 , δ1 ; z2 , δ2 ; . . . )

(5.13.18)

where zj ∈ [βj , αj+1 ] and δj is ±1, with the convention that we take δj = −1 if zj is at a band edge. Any f ∈ Me has the form Z X g(x) dx wj f (z) = + (5.13.19) x − zj e x−z {j|δj =1}

where

1 Im f (x+ + i0) π wj = lim (iε)f ((xj )+ + iε)

g(x) =

ε↓0

(5.13.20) (5.13.21)

This is just (2.3.7), (2.3.41), (2.3.54), and (2.3.58) where only the poles on S+ are relevant, since the measure is limε↓0 π1 Im f (x+ + iε) dx. Poles at branch points do not enter the sum because they only have |x − zj |−1/2 singularities. (They will affect g; at nonresonant gap edges, g vanishes as (x − z0 )1/2 , while at resonance edges, g diverges as (x − z0 )−1/2 .)

408

5. PERIODIC OPRL

We know f has the form

p p(z) + R(z) f (z) = a(z)

(5.13.22)

a has zeros at precisely the points {zj }ℓj=1, so ℓ Y a(z) = A (z − zj )

(5.13.23)

j=1

p

Since all zj < αℓ+1 and Im( R(x+ + i0)) > 0 on [αℓ+1 , βℓ+1 ] (from p R(x+ + i0) > 0 on (βℓ+1 , ∞) and the branch of (z − βℓ+1 )1/2 which is positive on (βℓ+1 , ∞) + i0 has positive imaginary part on (−∞, βℓ+1 ) + i0), we have A > 0. Thus, by (5.13.20), in (5.13.19) for x ∈ e, p |R(x)| 1 g(x) = (5.13.24) Qℓ π A j=1|x − zj | while, by (5.13.21),

p 2 |R(zj )| wj = Q A k6=j |zk − zj |

(5.13.25)

p for to avoid a pole on S− , we must have p(zj ) − R(zj ) = 0, which yields to 2 in the numerator. The normalization condition f (z) = −z −1 + O(z −2) is equivalent to Z X g(x) dx + wj = 1 (5.13.26) e

{j|δj =1}

which determines A. Thus, knowing D(f ) determines A and then g and wj , and then f which proves the map is one-one. Conversely, given a set of Dirichlet data (i.e., a point in Te), define a(z) by (5.13.23) p where A is determined by (5.13.26), determine p(z) by (since (p(z) + R(z))/a(z) is O(z −1 )) p p(z) + R(z) = O(z ℓ−1 ) (5.13.27)

near ∞+ (which determines the top two coefficients of p(z)) and the conditions (since m has no pole at (zj ; δj )) q p(zj ) ∓ δj R(zj ) = 0 (5.13.28)

This defines f by (5.12.7). Tracking signs of a proves Im f (x+ +i0) ≥ 0 on e and that the residues of poles on S+ are positive. Thus, the Cauchy


integral formula proves in C+ ∩ S+ Z f (w) f (z) = dw Γ+ w − z

409

(5.13.29)

and then (5.13.19) which shows Im f > 0 on S+ ∩ C+ . In (5.13.29), Γ+ is contour in the proof of Theorem 5.12.3 and the fact that constructed f has O(|z|−1 ) at ∞+ means the contour at ∞+ in the full Cauchy integral formula vanishes. This proves existence. Each f ∈ Me is an m-function, so the m-function of a unique Jacobi matrix, P Jf , which is determined either from the spectral measure g(x) dx + {j|δj =1} wj δzj or from the continued fraction expansion at ∞+ . The topology on Me is equivalent to the topology of pointwise convergence on the parameters in Jf (once we prove Jf is periodic or almost periodic, this will be the same as uniform convergence in n). Note that f determines a1 , b1 directly by f (z)−1 = −z + b1 + a21 z −1 + O(z −2 )

(5.13.30)

at ∞+ . We will study the n-dependence of the Jacobi parameters by studying the impact of coefficient stripping. We proved in Corollary 5.13.3 that f → f1 , coefficient stripping given by (5.13.30) and (5.13.6) is a map of Me to Me. We will also need a map of e : Me → Tℓ A

the canonical ℓ-torus, Rℓ /Zℓ , by mapping Te to Tℓ by Corollary 5.12.11, and composing this with D, that is, if D(f ) = (z1 , . . . , zℓ )

(zj ∈ Gj )

then

(0)

e )= A(f

ℓ X j=1

(0)

A(zj ) − A(zj )

(5.13.31)

(0)

where zj is some convenient point, say zj = αj . We can prove uniform (over the isospectral torus) bounds on the weight. Theorem 5.13.6. There are positive constants C, D so that uniformly over Te, one has for all x ∈ e, DR(x)1/2 ≤ g(x) ≤ CR(x)−1/2

(5.13.32)

410

5. PERIODIC OPRL

Proof. We have dist(x, R \

e) min ( 12 j=1,...,ℓ+1

|βj − αj |)

ℓ−1

ℓ Y ≤ |x − zj | ≤ |βℓ+2 − α1 |ℓ j=1

(5.13.33)

so, by (5.13.24), for some C1 , D1 , D1 A−1 R(x)1/2 ≤ g(x) ≤ C1 A−1 R(x)−1/2

(5.13.34)

Also, we have, by (5.13.25), where

0 ≤ wj ≤ A−1 C2

(5.13.35)

C2 = 2|βℓ+1 − α1 |ℓ+1 (min|βj − αj |)−ℓ+1 (5.13.36) (5.13.26) and these bounds provide uniform (in Te) upper and strictly positive lower bounds on A and then (5.13.34) implies (5.13.32). e is a bijection of Me to Tℓ . Theorem 5.13.7. (a) A (b) Coefficient stripping f → f1 obeys e 1 ) − A(f e ) = A(∞− ) − A(∞) A(f

(5.13.37)

e is the composition of D and the map of CorolProof. (a) A lary 5.12.11, each of which is a continuous bijection. (b) f has poles at the points in D(f ) plus at ∞− and, by (5.13.6) (other that is at ∞± ), zeros of f are precisely poles of f1 plus the zeros at ∞+ . Thus, by the first half of Abel’s theorem (Theorem 5.12.7), which is (5.13.37).

e ) + A(∞− ) = A(f e 1 ) + A(∞+ ) A(f

This is truly a remarkable theorem: f → f1 is a map of a torus to itself. In general, iterating maps on a torus is complicated, but if the map is just addition by a fixed group element, iteration n times is just adding n times that element! x → x + nx0 is an affine map (on Rℓ ), so (5.13.37) is sometimes summarized by the phrase: “Abel’s map linearizes coefficient stripping.” With this in place, we get some immediate consequences (they are corollaries, but so significant that we call them theorems!): Theorem 5.13.8. Let e ⊂ R be a finite gap set. Let p ∈ {1, 2, . . . }. The following are equivalent: (i) One Jacobi matrix, Jf , associated to one f ∈ Me is periodic of period p.


411

(ii) All Jacobi matrices, Jf , associated to all f ∈ Me are of period p. (iii) Each harmonic measure, ρe(ej ) (where ej = [αj , βj ]) is rational with pρe(ej ) ∈ Z (5.13.38) (iv) There is a polynomial of degree p with ∆−1 ([−2, 2]) = e

(5.13.39)

(inverse as a map from C). Proof. Consider the statement p(A(∞− ) − A(∞+ )) = 0

(5.13.40)

that is, p times the element of the torus is the identity. By (5.13.37), if f1 , f2 , . . . are what we get by coefficient stripping, (5.13.40) is equivalent to e p ) − A(f e )=0 A(f (5.13.41) e is a bijection, this is equivalent to fp = f , for one f or for all f ! Since A that is, J is itself after stripping p times, that is, J is periodic! By (5.12.71), (5.13.40) holds if and only if p

k X j=1

ρe(ej ) ∈ Z

for k = 1, 2, . . . , ℓ, which is equivalent to (5.13.38). Finally, we note that (i) ⇒ (iv); just take ∆ to be the discriminant. Conversely, (iv) implies (5.13.40). For let p F (z) = −∆(z) ± ∆2 (z) − 4

Since ∆−1 ([−2, 2]) = e, ∆2 − 4 has double roots at internal points of e and single roots at edges of e, so F is meromorphic on Se. Since √ ± ∆2 − 4 = ±(∆(z) + O(∆(z)−1 )) (5.13.42) we see at ∞+ , F has a zero of order p and at ∞− a pole of order p. It thus has degree p (since there are no other poles) and so no √ other zeros 1 −1 (as can also be seen by noting that F (z) = 4 (−∆(z) ∓ ∆2 − 4)). Thus (5.13.40) is just the first part of Abel’s theorem for F .

Notice that Theorem 5.13.8 implies Theorem 5.5.25 (given Proposition 5.5.26) and provides a proof of that theorem. Our proof of Aptekarev’s theorem (i.e., (ii) ⇒ (iii) in Theorem 5.5.25) is indirect: Rational harmonic measure implies (5.13.40) by the calculation in (5.12.71) and that implies there is a periodic J and then ∆ is its discriminant. Peherstorfer’s proof [331] is via a direct construction—its OPUC analog appears as Theorem 11.4.8 in [391].

412

5. PERIODIC OPRL

The following generalizes the Borg–Hochstadt theorem (Theorem 5.4.20): Corollary 5.13.9. Let {an , bn }∞ n=1 be a set of Jacobi parameters obeying an+p = an bn+p = bn (5.13.43) where p = kq with k and q integral. Suppose all the gaps Gj are closed for j 6= k, 2k, . . . , (q − 1)k. Then, a, b are periodic at period q, that is, an+q = an

bn+q = bn

(5.13.44)

Remark. The Borg–Hochstadt theorem is the case q = 1. Proof. Each band has harmonic measure m/q.

For general finite gap sets, the Jacobi matrices are quasiperiodic: Theorem 5.13.10. Let e be a finite gap set and Jf a Jacobi matrix whose m-functions is a minimal Herglotz function in Me. Then its Jacobi parameters are almost periodic. To be totally explicit, there are real analytic functions Ae and Be on Tℓ , the standard ℓ torus with values in (0, ∞) and R, respectively, so that for every such Jf , we have t0 ∈ Tℓ so that an = Ae(t0 − nω)

bn = Be(t0 − nω)

(5.13.45)

where ω is given in terms of the harmonic measures of e by (5.12.71). ee on Me by Proof. Define A˜e and B ee(f ) + A˜e(f )2 z −1 + O(z −2 ) f (z)−1 = −z + B

(5.13.46)

which are clearly real analytic on Me. Define e −1 e −1 ee ◦ A Ae = A˜e ◦ A Be = B e is the bijection of Me to Tℓ of Theorem 5.13.7. Then (5.13.45) where A is just (5.13.37) iterated. One can naturally use (5.13.45) to define (an , bn ) for all n ∈ Z and so get natural two-sided Jacobi matrices for any e. The set of such two-sided matrices is called the isospectral torus, Te, for e. In the periodic case, it is precisely the set of periodic J’s with a given ∆. Just as Chapter 3 is the theory of special classes of perturbations of Te for e = [−2, 2], we want to understand the analogous perturbations for general e. For the rational harmonic measure case, this will be the subject of Chapter 8 and for general e’s, of Chapter 9. Finally, we use these ideas to find another proof of (5.2.11) and show that for the general finite gap situation, the whole-line Jacobi matrices are reflectionless. (i.e., have purely imaginary Green’s functions).


413

Theorem 5.13.11. Let e be a finite gap set, m a minimal Herglotz function on Se, and J the two-sided Jacobi matrix given by (5.13.45) for n ∈ Z, so that m(z) = m(z; J0+ ) (5.13.47) Then m(z; J0− ) = (a20 m(τ (z)))−1 (5.13.48) −1 that is, one can recover m(z; J0 ) from the second sheet values of m. Remark. In the periodic case, this provides another proof of (5.2.11). Proof. By the fact that m(z) has a pole at ∞− and by (5.13.7), −2 we see that m1 (z) − (−a−2 1 z + a1 b1 ) has a zero at ∞− , so near ∞− , −2 −1 m1 (z) = −a−2 1 z + a1 b1 + O(z )

(5.13.49)

In particular, near ∞− on C+ ∩ S+ , Im m1 (τ (z)) ≤ 0. On the other hand, on e, m1 (τ (x + i0)) = m(x + i0) also has a negative imaginary part. Finally, the same argument that showed poles on S+ have positive residues shows they have p p negative residues on S− (for on S− , p(z) + R(z) = 0 and −2 R(z)/a(z) has positive sign). Thus, by the maximum principle for harmonic functions, Im m1 (τ (z)) ≤ 0 on S+ ∩ C+ . It follows that (a21 m1 (τ (z))−1 is a discrete m-function. Similarly, if we let m+,n (z) = m(z; Jn+ ) (5.13.50) then m−,n (z) ≡ (a2n m+,n (τ (z))−1 (5.13.51) is a discrete m-function. With this definition, the recursion relation m+,n (z)−1 = bn+1 − z − a2n+1 m+,n+1 (z)

(5.13.52)

a2n m−,n (z) = bn+1 − z − (m−,n+1 (z))−1

(5.13.53)

which initially holds on S+ ∩ C+ extends by analytic continuation, and since τ (z) = z implies which shows inductively that the Jacobi parameters associated to m−,n − are {aj−2+n , bj−1+n }∞ j=1 , that is, Jn . Thus, m−,n (z) = m(z; Jn− )

which for n = 0 is (5.13.48).

(5.13.54)

Theorem 5.13.12. Let J be a two-sided Jacobi matrix in Te where e is a finite gap set. Then,

414

5. PERIODIC OPRL

(i) The diagonal Green’s function, Gnn (z), is real for z = x + i0 with x ∈ e. Thus, J is reflectionless on e. (ii) σ(J) = e and the spectrum is purely absolutely continuous of uniform multiplicity 2. Proof. (i) By (5.4.45), Gnn (z) = −

a2n m(z; Jn+ )

On e,

1 − m(z; Jn− )−1

(5.13.55)

m(x + i0, Jn− ) = m(τ (x − i0), Jn− ) = m(x + i0, Jn− )

(5.13.56)

so, by translates of (5.13.48), m(x + i0, Jn− )−1 = a2n m(x + i0, Jn )

(5.13.57)

and, by (5.13.55), Gnn is pure imaginary. (ii) By (5.13.55) and (5.13.48), (−Gnn (z))−1 = a2n [m(z; Jn+ ) − m(τ (z); Jn+ )]

(5.13.58)

for all z ∈ C \ e. √ Consider a gap [βj , αj+1 ]. Writing m in the form (p ± R)/a, we see p 2a2n R(z) −1 (−Gnn (z)) = a(z) where a(z) has a single zero in [βj , αj+1]. Suppose first that zero is in (βj , αj+1). Then (−Gnn (z))−1 vanishes at βj and αj+1. Moreover, on R \ σ(J),

d d Gnn (x) > 0 ⇒ (−Gnn (x))−1 > 0 dx dx away from the zero of a. Thus, by monotonicity, (−Gnn (z))−1 has no zero in (βj , αj+1). If (a(z)) has a zero at βj , then (−Gnn (βj ))−1 = ∞, (−Gnn (αj+1 )) = 0, and (−G)−1 is finite and monotone in all of (βj , αj+1 ), so always strictly negative. Similarly, if a(z) has a zero at αj , (−Gnn (z))−1 is strictly positive on (βj , αj+1 ). In all cases, (−Gnn (z))−1 is nonvanishing on (βj , αj+1), so no Gnn (z) has a pole in those intervals, so σ(J) ⊂ e. By the fact that Gnn (x+i0) is pure imaginary, Craig’s theorem (Theorem 5.4.18) implies the spectrum is purely a.c. Since Im(a2n m(x+i0, Jn+ )) = Im((−m(x+i0, Jn ))−1 ) =

1 2

Im((−Gnn (x+i0))−1 )

APPENDIX TO SECTION 5.13

415

we see that the a.c. spectrum is of multiplicity two.

Remarks and Historical Notes. This is the second half of the theory developed by Flaschka–McLaughlin–Krichever–van Moerbeke quoted (with background) in the Notes to the last section. By the discussion in Example 5.13.4 and the remark after Corollary 5.13.3, if m obeys all the conditions for a function in Me, except it is finite and nonzero at ∞− rather than a pole, then the once stripped m1 is in Me. So every such Jacobi matrix is an almost periodic one with b1 modified. In the periodic case, the Dirichlet data points are the roots of pp−1(z) which are eigenvalues of the truncated matrix Jp−1;F , so associated to solutions of (J − λ)u = 0 with un=0 = un=p = 0, thus Dirichlet eigenvalues, which is the reason for the name. Alternatively, in terms of the operators J0± of the truncated full-line problem, Dirichlet data in the interior of a gap are eigenvalues of J0+ if in S+ and of J0− if in S− . There are basically two ways of thinking of the isospectral torus, Te: a set of whole-line Jacobi matrices or as their restrictions to the half-line (which, by almost periodicity, determine the whole-line matrix). The half-line objects are defined as the set of minimal Herglotz functions. The whole-line objects are the set of reflectionless whole-line J’s with σess (J) = Σac (J) = e. That every such object lies in the isospectral torus, as we have defined it, will be the major theme in Section 7.5 which will also discuss the history of this point of view. Among all almost periodic Jacobi matrices, the finite gap ones are unusual in that, generically, one expects infinitely many gaps and Cantor spectrum. For results on such generic Cantor spectrum, see [28, 29, 116, 168]. Appendix to Section 5.13: A Child’s Garden of Almost Periodic Functions As we have seen, Jacobi parameters induced by the minimal Herglotz functions associated to a general finite gap set are quasiperiodic, and so almost periodic. In this appendix, we discuss the general definition of quasiperiodic and almost periodic. Given a function, f , on Z and n ∈ Z, we define fn on Z by fn (m) = f (n + m)

(5.13A.1)

Given a bounded function, f , on Z, we define kf k∞ = sup |f (n)| n

(5.13A.2)

416

5. PERIODIC OPRL

and let C(Z) be the set of all bounded functions in this norm. Definition. A function, f , from Z to C is called almost periodic (in Bochner sense) if and only if f is bounded and {fn }n∈Z has compact closure in k · k∞. Definition. A Bohr almost periodic function on Z is a bounded function, f , so that for any ε, there is an L so that for all m ∈ Z, there is an n so that |n − m| ≤ L and kfn − f k∞ < ε

(5.13A.3)

Let T1 be the circle ∂D = {z | |z| = 1}, Tn = ×nj=1 T1 , the ndimensional torus, and T∞ , the countably infinite product. We will think of Tn as ∂Dn and use (z1 , . . . , zn ) as coordinates. Notice that we use additive notation for Z but multiplication for T. The main theorem at the center of the theory is: Theorem 5.13A.1. Let f be a bounded function on Z. The following are equivalent: (1) f is (Bochner) almost periodic. (2) f is Bohr almost periodic. (3) f is a uniform limit of finite sums of the form gN (n) =

N X

(N)

aj e2πiαj

n

(5.13A.4)

j=1

(N )

for α1 , . . . , αN ∈ R/Z. ∞ (4) There exists a continuous function F on T∞ and {zj }∞ j=1 in T so that f (n) = F (z n ) (5.13A.5) where (z n )j = zjn . Remarks. 1. If F depends on only finitely many variables (equivalently, F can be viewed as a function of a finite-dimensional torus), f is called quasiperiodic. 2. In Theorem 5.13.10, we have functions of the form (5.13A.5) on a finite-dimensional torus, but only for n ≥ 0. So the question comes up how to define almost periodic functions on n ≥ 0. The answer is as restrictions to n ≥ 0 of functions almost periodic on Z. There is at most one such extension, for if there were two, their difference would be an almost periodic function vanishing for n ≥ 0 and, by the Bohr definition, such a function is identically zero.


417

It is natural to prove this result in the general context of locally compact abelian groups. Let G be such a group, µ Haar measure, and b the set of characters, that is, continuous homomorphisms of G to G ∂D. Besides Z, the example to think about is R. Let C(G) stand for bounded continuous functions on G with k · k∞. For f ∈ C(G) and g ∈ G, define fg by fg (x) = f (x + g)

(5.13A.6)

f is (Bochner) almost periodic if {fg }g∈G has compact closure in k · k∞. f is called Bohr almost periodic if and only if for all ε, there is a compact set K so that for all g, there is h in g + K so that kfn − f k∞ ≤ ε

(5.13A.7)

The general form of Theorem 5.13A.1 is:

Theorem 5.13A.2. Let G be a separable compact abelian group. Let f ∈ C(G). Then the following are equivalent: (1) f is (Bochner) almost periodic. (2) f is Bohr almost periodic. (3) f is a uniform limit of finite sums of the form gN (x) =

N X

(N )

aj χj (x)

(5.13A.8)

j=1

(N )

b with χj ∈ G. (4) There exists a continuous function F on T∞ to C and a homomorphism ζ : G → T∞ so f (x) = F (ζ(x))

(5.13A.9)

Theorem 5.13A.2 ⇒ Theorem 5.13A.1. Only parts (2) and (4) look a little different. For (2), note compacts sets in Z are finite and so contained in intervals. As for (4), note for G = Z, homomorphisms ζ : G → T∞ are given precisely by ζ(1) since ζ(n) = ζ(1)n (using a product rather than additive notation for T). (4) ⇒ (3) in Theorem 5.13A.2. Let z1 , z2 , . . . be coordinates on T∞ . Let χj : G → ∂D be zj ◦ ϕ. Then χj is a character on G and thus, so is any finite product of χj ’s. By the Stone–Weierstrass theorem, polynomials in the zj are dense in C(T∞ ), and so F is a uniform limit in polynomials in zj . Thus, F ◦ ϕ is a uniform limit of finite linear combinations of characters. (3) ⇒ (1) in Theorem 5.13A.2. A set Q in a complete metric space, X, has compact closure if and only if for all ε, there are finitely

418

5. PERIODIC OPRL

many q1 , . . . , qℓ in X so that ∪ℓj=1 {q | ρ(q, qℓ ) < ε} contains Q. If f is a limit of fN ’s of the form (5.13A.8), given ε, pick ε/2 so kf −fN k∞ < ε/2. Since N X (fN )g = aj χj (g)χj (5.13A.10) PN

j=1

{(fN )g } ⊂ { j=1 aj zj χj | |zj | = 1} is compact, and so covered by finitely many ε/2 balls. Thus, since kfg − (fN )g k∞ = kf − fN k∞ , {fg } is covered by finitely many ε balls.

(1) ⇒ (2) in Theorem 5.13A.2. Given ε, pick g1 , . . . , gN in G so every fg is within ε of some fgj . Let K = {−g1 , . . . , −gN } which is finite, and so compact. If kfg − fgj k∞ < ε, then kfg−gj − f k∞ < ε and h = g − gj ∈ g + K. Remark. Once we have (2) ⇒ (1), this implies the compact K in Bohr almost periodic can be taken as a finite set! Lemma 5.13A.3. If f is Bohr almost periodic, then f is uniformly compact, that is, for any ε, there is a neighborhood N of the identity e ∈ G so that if x − y ∈ N, then |f (x) − f (y)| < ε. Proof. Each fy is continuous at e, so given ε, there is Ny , a neighborhood of e, so that w ∈ Ny ⇒ |fy (w) − fy (e)| < ε/4, so if w, w ′ ∈ Ny , then |fy (w) − fy (w ′ )| < ε/2. By continuity of addition, we can find My , a neighborhood of e, so My + My ⊂ Ny . Thus, if ε w, w ′, w ′′ ∈ My ⇒ |fy+w′′ (w ′ ) − fy+w′′ (w)| < (5.13A.11) 2 If K is compact, we have K ⊂ ∪y∈K (y + My ), so pick y1 , . . . , yℓ so K ⊂ ∪ℓj=1 (yj + Myj ) and MK = ∩ℓj=1 Myj . Thus, by (5.13A.11), ε y ∈ K, w, w ′ ∈ MK ⇒ |fy (w) − fy (w ′)| < (5.13A.12) 2 Given ε, let K compact be chosen so (5.13A.7) holds for ε/4 and pick MK as above. Suppose x − y ∈ MK . By Bohr almost periodicity, there is h ∈ K so that kfh−y − f k∞ < ε/4. Thus, kfh − fy k∞ < ε/4, so by (5.13A.12), w, w ′ ∈ MK ⇒ |fy (w) − fy (w ′ )| < ε

(5.13A.13)

x − y ∈ MK ⇒ |f (x) − f (y)| < ε

(5.13A.14)

Taking w = x − y and w ′ = e, we see which is uniform continuity.


419

(2) ⇒ (1) in Theorem 5.13A.2. By Lemma 5.13A.3, f is uniformly continuous, which implies x → fx is continuous as a map of G to C(G). Given ε, let K be the compact set so that (5.13A.7) holds for ε/2. Since x → fx is continuous, {fx }x∈K is compact, so we can find x1 , . . . , xℓ in K whose ε/2 balls cover this set of f ’s. Given any y ∈ G, there is x ∈ K so kf−y+x − f k∞ < ε/2, so kfy − fx k < ε/2 and fy is within ε of some fxj . Thus, {fy }y∈G is covered by finitely many ε balls. Since ε is arbitrary, f is (Bochner) almost periodic. (1) ⇒ (4) in Theorem 5.13A.2. This final step is the most elaborate and elegant. Let H ⊂ C(G) be the closure of {fx }x∈G . H is called the hull of f . Define ϕ0 : G → H by ϕ0 (x) = fx

(5.13A.15)

Since (1) ⇒ (2) ⇒ f is uniformly continuous, ϕ0 is continuous. Since kpx − qx k∞ = kp − qk∞ , we see that kfx+y − fx′ +y′ k∞ ≤ kfx − fx′ k∞ + kfy − fy′ k∞

(5.13A.16)

that is, kϕ0 (x + y) − ϕ0 (x′ + y ′)k ≤ kϕ0 (x) − ϕ0 (x′ )k + kϕ0 (y) − ϕ0 (y ′)k (5.13A.17) Let h, h′ ∈ H. Picking xn , yn ∈ G so ϕ(xn ) → h, ϕ(yn ) → h′ , we see, by (5.13A.17), that ϕ(xn + yn ) is Cauchy, which allows us to define h + h′ (“+” is map of H × H to H, not to be confused with adding the functions!). It is easy to see this turns H into a compact group. Since H is a metric space, compactness implies separability. By definition, ϕ is a homomorphism. Now we need a fact about compact separable abelian groups (see the Notes): Such groups have characters that separate points, and by b separability, there is a countable family, {χj }∞ j=1 ⊂ H, that separates ∞ points. Let Q : H → T by Q(h)j = χj (h) and ϕ : G → T∞ by ϕ = Q ◦ ϕ. e Q is an injective map since {χj } separates points. ϕ is a group homomorphism. Since H is compact, Q[H] is closed in T∞ . Define Fe : H → C by Fe(h) = h(e). Then F is continuous and Fe(ϕ(x)) = Fe(fx ) = fx (e) = f (x)

(5.13A.18)

that is, Fe ◦ ϕ = f . Since Q is one-one, we can define a function F on Q[H] so F ◦ Q = Fe (5.13A.19)

420

5. PERIODIC OPRL

Since Q[H] is closed, F has an extension to T∞ by the Tieztze extension theorem. We will still use F for this extension. Clearly, (5.13A.19) remains true; F : T∞ → C and F ◦ϕ=F ◦Q◦ϕ e = Fe ◦ ϕ e=f

by (5.13A.18).

(5.13A.20)

Remarks and Historical Notes. The definition of almost periodic functions on R and their properties is due to Harald Bohr [50, 51], using the definition we gave for Bohr almost periodic on Z (but for R). The Bochner property (which we codified in the Bochner definition) is due to Bochner [46, 48]. Sometimes what we call “almost periodic” is called “uniformly almost periodic” since there are also Besicovitch almost periodic or L2 almost periodic functions, which we will define below. For book treatments of the theory, see Besicovitch [43], Bohr [52], Corduneanu [90], and Levitan–Zhikov [273]. We used the fact that any abelian separable compact group, G, has enough characters to separate points. This is essentially the Peter– Weyl theorem for such groups (see, e.g., Simon [385]); here is a sketch of the argument explicitly. Let f be a function on G with f (−x) = f (x). Define T : L2 (G) → L2 (G) by Z (T h)(x) = f (x − y)h(y) dµ(y) where dµ is Haar measure. T is Hilbert–Schmidt (so compact) and selfadjoint. Moreover, if Ux : L2 → L2 by (Ux f )(y) = f (y−x), then T commutes with {Ux }. Thus, {Ux } leave each eigenspace invariant. If V is such an eigenspace and is finite-dimensional, the Ux are commuting unitaries on V, so they have a common eigenvector χ e(x). Thus, χ e(x + y) = (Ux χ e)(y) = λx χ e(y)

and Ux+y = Ux Uy implies λx+y = λx λy . Since x → Ux is continuous, this shows χ e is continuous and everywhere nonzero: χ(x) = χ e(x)/e χ(e) is thus a (continuous) character. So the characters span Ran(T ). Since we can find fn so Tfn → 1, we see the characters χ span L2 , which implies they separate points. Further developments depend on the notion of the average of an almost periodic function. Given an almost periodic function, f , let H be its hull, Fe the function in (5.13A.18), and dν normalized Haar on


H. We define Av(f ) =

Z

H

For R or Z, one can prove that

(or

PT

1 Av(f ) = lim T →∞ 2T

Z

Fe(x) dν(x)

(5.13A.21)

T

f (x) dx

(5.13A.22)

−T

f (n) for Z). b by One defines the Fourier coefficients of f for χ ∈ G 1 2T +1

421

−T

fb(χ) = Av(χf ¯ )

(5.13A.23)

b noting that χf ¯ is also almost periodic. It is not hard to see that f(χ) is nonzero for only countably many χ’s. Indeed, one has a Plancherel theorem X |fb(χ)|2 = Av(|f |2) (5.13A.24) b χ∈G

One also has an L2 convergence of Fourier series; if {χj }∞ j=1 is a numb bering of those χ’s with f (χ) 6= 0, then 2 N X b j )χj → 0 Av f − f(χ (5.13A.25) j=1

These results are all easy to prove by using the fact that if H is the b that is, hull, fb(χ) 6= 0 implies χ ∈ H, χ=χ e◦ϕ e

(5.13A.26)

where χ e is a character of H. (5.13A.24) and (5.13A.25) are then expressions of the fact that characters of H are a basis of L2 (H, dν). For R, one defines Besicovitch almost periodic functions as functions on R, for which there exists, for any z, a finite sum fN = PN (N ) iw(N) x j with j=1 aj e Z T 1 |fn − fN (x)|2 dx ≤ ε (5.13A.27) lim sup 2T T →∞ −T The frequency model of f , an almost periodic function, is the set of characters of G that comes from H, the hull, via (5.13A.26). It is a b b It is generated by {χ | f(χ) countable subgroup of G. 6= 0}. A function is called almost periodic if it is a uniform limit of periodic functions. Such functions are obviously almost periodic. A typical

422

5. PERIODIC OPRL

example is f (x) =

∞ X

2−n cos(2π2−n x)

(5.13A.28)

n=1

We note that the term quasiperiodic is sometimes used for a very different notion from our use and that those quasiperiodic functions are not almost periodic. The set of all almost periodic functions in k · k is Banach algebra. Its Gel’fand spectrum (see [146] for the theory of commutative Banach algebras) is called the Bohr compactification of G. It is huge, containing b and putting every hull as a subgroup. One can construct it by taking G the discrete topology in it and taking the dual of that. 5.14. Periodic OPUC

We have discussed OPRL with periodic Jacobi matrices in much of this chapter. The theory of OPUC whose Verblunsky coefficients obey αn+p = αn

(5.14.1)

for all n and some fixed p is the subject of Chapter 11 of [391]. Our goal in this section is to sketch some parts of this theory, emphasizing the differences to the OPRL theory. A major difference is that the transfer matrix for OPRL has determinant 1 since 1 z − b −1 det =1 (5.14.2) a2 0 a while in the OPUC case, the m step transfer matrix has determinant z m since 1 z −¯ α =z (5.14.3) det ρ −αz 1 (see (2.4.3)). The natural discriminant is thus ∆(z) = z −p/2 Tr(Tp (z))

(5.14.4)

For this reason, it is natural to restrict to the case p even and control p odd by other means (e.g., by viewing it as period p instead as period 2p). We shall do this henceforth. ∆(z) is thus a Laurent polynomial (i.e., polynomial in z and z −1 ). It is real on ∂D, and one can show the associated measure is purely absolutely continuous on e = ∆−1 ([−2, 2]) ⊂ ∂D with potentially one pure point per gap. The Carathéodory function obeys a quadratic equation and extends to a two-sheeted Riemann surface with branch points at the edges of connected components of e.

5.14. PERIODIC OPUC

423

The most significant difference from OPRL comes from the following: If e has ℓ + 1 connected components, in the OPRL case, there are ℓ significant gaps—the gap on C \ e that goes from βℓ+1 to ∞ and then −∞ to α1 is not considered for Dirichlet data. In some sense, the pole at −∞ and zero at ∞+ are fixed and only the zeros and poles in the finite gaps vary. But if e has ℓ + 1 components, there are, on ∂D, ℓ + 1 gaps and none is distinguished. The natural Dirichlet data is a torus of dimension ℓ + 1, one for each gap. On the other hand, S is e from Mℓ to Tℓ maps an still of genus ℓ, so the analog of our map A ℓ + 1-dimensional torus to an ℓ torus and is no longer one-one. Instead, e −1 (x) of a fixed point in Tℓ is a circle. Indeed, in the inverse image A ′ p−1 e the periodic case, {αn }p−1 n=0 and {αn }n=0 have the same image under A ′ iθ if and only if αn = e αn for some fixed θ. This means that the natural result of Abel’s theorem is to show only that elements of Te obey αn+p = eiθ αn for some θ and, more generally, are almost periodic up to phase. Controlling this phase turns out to be simple in the periodic case and very involved in the almost periodic case. Another significant difference is the function to be used in Abel’s theorem. In the OPRL case, the m-function itself realized coefficient stripping, that is, the poles of the once stripped Jacobi matrix were exactly the zeros of m. One might hope the Carathéodory function had this property, but that is not true. The zeros of the Carathéodory function associated to {αn }∞ eodory function n=0 are the poles for the Carath´ associated to {−αn }∞ . n=0 Instead, one needs to use the function z(δ0 D)(z) of (2.6.9). It has poles at the poles of the Carathéodory function for {αn }∞ n=0 and ∞ zeros at the poles of the Carathéodory function for {αn }n=0 (i.e., once stripped). If these have a pole in common, the situation is slightly different. In addition, zδ0 D(z) has a pole at ∞− and a zero at 0+ , so in place of the A(∞− ) − A(∞+ ) of (5.13.37), we have A(∞− ) − A(0+ ). This describes the major differences. Remarks and Historical Notes. See [391] and its notes for the theory and history of periodic OPUC. [391] uses the function it calls M(z) related to δ0 D(z) by M(z) = 2ρ0 z(δ0 D)(z) While [390] introduced δ0 D, the connection of M(z) and δ0 D was not realized in [390, 391].

CHAPTER 6

Toda Flows and Symplectic Structures Having discussed periodic Jacobi matrices, we would be remiss if we did not discuss the closely related Toda lattice dynamical system. So even though it is definitely an aside, we provide the high points in this chapter.

6.1. Overview The structure that the spectra of periodic Jacobi matrices induce on Jacobi parameters is striking. [(0, ∞) × R]p , consisting of points (an , bn )pn=1 , is decomposed into its isospectral tori, generically of dimension p − 1 with some degenerate tori of lower dimension. The fibration into tori is reminiscent of another structure which we will discuss in Section 6.2. A completely integrable system is a manifold of dimension 2ℓ with ℓ Poisson commuting “independent” functions. If the sets where these functions have constant values are compact, then phase space is fibered into tori of dimension ℓ with some degenerate lower-dimensional tori. Of course, there is a dimension counting issue: [(0, ∞) × R]p has dimension 2p but the tori here are not of dimension p, but p − 1. We will see shortly why that is not a problem. Our main goal in this chapter will be to explore the completely integrable system on Jacobi matrices that helps “explain” the fibration into tori. Along the way, we will prove a technical fact about derivatives of coefficients of ∆ with respect to Jacobi parameters that will be an important ingredient in the proof of the Killip–Simon theorem for periodic OPRL; see Section 8.5. The Toda lattice was originally formulated in terms of the Hamiltonian H(p1 , . . . , pN , q1 , . . . , qN ) =

N X j=1

1 2

p2j

+γ

N −1 X

eqj −qj+1

(6.1.1)

j=1

Here γ is a fixed positive coupling constant which is usually set to 1, but which we will want to include. We will also consider the periodic 425

426

6. TODA FLOWS AND SYMPLECTIC STRUCTURES

Toda lattice where HP (p1 , . . . , pN , q1 , . . . , qN ) =

N X

1 2

p2j

j=1

+γ

N −1 X

qj −qj+1

e

j=1

qN −q1

+e

(6.1.2) To distinguish it from the periodic case, we will sometimes use the phrase “free Toda flow” for the solutions of the equation of motion associated to the Hamiltonian (6.1.1). We consider the equations of motion in Poisson bracket (PB) form (discussed in Section 6.2)

{pj , pk } = 0 so

df = {H, f } dt {qj , qk } = 0 {pj , qk } = δjk

N X ∂f ∂g ∂f ∂g {f, g} = − ∂pj ∂qj ∂qj ∂pj j=1

(6.1.3) (6.1.4)

(6.1.5)

Thus, (6.1.3) says dqj = pj dt

dpj = −γ(eqj −qj+1 − eqj−1 −qj ) dt

(6.1.6)

with special formulae for j = 1, N. Note that the physical setup is a bit unphysical. Particle j only interacts with particles j −1 and j +1 (as one might expect in a lattice), but the particle positions are not forced to be ordered. Moreover, the potential is highly nonsymmetric, minimum potential energies occur as qj − qj+1 → −∞, and there is a kind of hard core: if the energy is E, then qj − qj+1 ≤ log(E/γ). Flaschka [129] and Manakov [287] found a remarkable change of variables √ γ 1 (qj −qj+1 ) 1 bj = − 2 pj aj = e2 (6.1.7) 2 In the free case, we only have a1 , . . . , aN −1 . In the periodic case, we have √ γ 1 (qN −q1 ) aN = e2 (6.1.8) 2 not independent since a1 . . . aN =

γ N/2 2N

(6.1.9)

6.1. OVERVIEW

427

The Hamiltonians are H=2

N X

b2j

+4

j=1

HP = 2

N X

N −1 X

a2j

(6.1.10)

a2j

(6.1.11)

j=1

b2j

+4

j=1

N X j=1

and the fundamental PB becomes

{bj , aj } = − 14 aj 1 4

{bj , aj−1} = aj−1

(6.1.12) (6.1.13)

γ drops out in the free case and only enters the periodic case through (6.1.9). The equations of motion (6.1.3) (equivalent to (6.1.6)) become daj = aj (bj+1 − bj ) (6.1.14) dt dbj = 2(a2j − a2j−1 ) (6.1.15) dt with the proviso in the free case for (6.1.15): we interpret a0 = aN = 0, and in the periodic case, a0 = aN in (6.1.15), and bN +1 = b1 in (6.1.14) for j = N. One can now understand the reason the tori are only dimension N − 1, not N. The a, b variables PN have Poisson brackets that are degenerate. In the free case, j=1 bj Poisson commutes (i.e., has zero Q Poisson bracket) with all a and b. In the periodic case, N j=1 aj also PN Poisson commutes (as does j=1 bj ). Thus, in both cases, we need to P QN restrict to N j=1 pj = β, and in the periodic case to j=1 aj = α (which P is no restriction to the q’s; in p, q language, we are fixing N j=1 pj and PN j=1 qj ). In either case, we get 2N − 2-dimensional manifolds with nondegenerate Poisson brackets. The natural completely integrable systems then have invariant tori of dimension 12 (2N − 2) = N − 1. A hint of the connection to Jacobi matrices and invariance of the spectrum is seen in the fact that the free Hamiltonian (6.1.10) is given by H = 2 Tr(JN2 ;F ) (6.1.16) with JN ;F given by (1.2.30) and the periodic Hamiltonian (6.1.11) by HP = 2 Tr(JN2 ;P ) where JN ;P is the J(θ) of (5.3.8) with θ = 0.

(6.1.17)

428


For complete integrability on our 2N − 2 phase space, one needs N − 1 Poisson commuting functions and they will be Tr(JNℓ ;F ) and Tr(JNℓ ;P ), ℓ = 2, 3, . . . , N, respectively. ℓ = 1 is not included since P it is N j=1 bj and constant on the manifold. We stop at ℓ = N since ℓ Tr(A ), ℓ = 1, . . . , N for any N × N matrix determine the eigenvalues λ1 , . . . , λN , and so Tr(AN +1 ), Tr(AN +2 ), . . . . Section 6.2 is a tutorial on symplectic manifolds and completely integrable systems, while Section 6.3 provides background on a piece of linear algebra (the QR factorization) needed later. Section 6.4 provides a first proof that in the free case, the PBs of the traces are zero: it goes from PBs of a, b to PBs of the orthogonal polynomials, and from there to PBs of eigenvalues and their spectral weights. Section 6.5 then solves the free Toda lattice in the eigenvalue-weight coordinates. Section 6.6 provides a second proof the PBs of traces are zero, using Lax pairs, and Section 6.7 completes that analysis using the QR algorithm to link the two approaches. Section 6.7 also completes the proof that {Tr(JNℓ ;F ), Tr(JNk ;F )} = 0 since Section 6.6 only does the calculation for ℓ = 2 and general k. Section 6.8 turns to PBs for the periodic case and Section 6.9 proves an important independence result when all gaps are open. Finally, Section 6.10 has some remarks on the OPUC analog. 6.2. Symplectic Dynamics and Completely Integrable Systems In this section, we describe Hamiltonian dynamics on general manifolds and prove a key theorem about completely integrable systems. We suppose the reader is familiar with the basics of manifold theory, including the definition of tangent space, Tp (M), cotangent space, Tp∗ (M), vector fields, forms, and flows; see [53, 89, 261, 278, 406, 407]. M will be a C ∞ manifold. We only sketch the proofs; for details, see [4, 26, 278, 291, 297, 426]. Two-forms can be viewed as functions on M with values at p ∈ M in the antisymmetric bilinear maps from Tp (M) × Tp (M), that is, given p ∈ M, a two-form, Ω, and X, Y ∈ Tp (M), Ωp (X, Y ) is a number linear in each of X and Y with the other fixed, and Ωp (X, Y ) = −Ωp (Y, X)

(6.2.1)

If {xj }nj=1 is a local coordinate system near p, then ( ∂x∂ j )nj=1 is a basis for Tp (M) and {dxj }nj=1 for Tp∗ (M). We normalize dxj ∧ dxℓ (j 6= ℓ) by ∂ ∂ (dxj ∧ dxℓ ) , = 12 (δjm δℓq − δℓm δjq ) (6.2.2) ∂xm ∂xq

6.2. SYMPLECTIC DYNAMICS

429

The half is there so we can write n X Ωp = Ωkℓ (p) dxk ∧ dxℓ

(6.2.3)

k,ℓ=1

with Ωkℓ (p) = −Ωℓk (p)

and have

∂ ∂ Ωp , = Ωmℓ (p) ∂xm ∂xℓ Every form defines a map Ω∗p : Tp (M) → Tp∗ (M) by

Ω∗p (X)(Y ) = Ωp (X, Y )

(6.2.4) (6.2.5)

(6.2.6)

equivalently, Ω∗p

X n

∂ aj ∂xj j=1

=

n X

aj Ωjk (p) dxk

(6.2.7)

j,k=1

A form is called nondegenerate at a point p if Ω∗p is a bijection; equivalently, if det(Ωmℓ (p)) 6= 0. Notice, by (6.2.4), that det(Ωmℓ ) = (−1)n det(Ωmℓ ) is 0 if n is odd, so only even-dimensional manifolds can have nondegenerate forms. The key definition is Definition. A symplectic manifold is a manifold, M, with distinguished two-form, Ω, nondegenerate at every point and closed, that is, dΩ = 0 (6.2.8) In (6.2.8), d is the canonical differential from ℓ-forms to (ℓ + 1)forms. Recall that a vector field is a smooth function on M taking values in Tp (M) which, given that tangent vectors are equivalence classes of curves, are the same as first-order differential operators. P Thus, vector fields map C ∞ (M) to itself. In local coordinates, X = nj=1 aj ∂x∂ j P ∂f and Xf = nj=1 aj ∂x . Vector fields define flows, and conversely. In j good cases (always if M is compact), flows can be globally defined: There is ϕt : M → M, C ∞ maps for all t ∈ R and ϕt ◦ ϕs = ϕt+s

(6.2.9)

The relation to X is that d f (ϕt (x)) = (Xf )(ϕt (x)) dt We will often write exp(tX) for ϕt .

(6.2.10)

ϕt=0 = id

430


In general, functions, f , define one-forms df but not vector fields. On a symplectic manifold, one can use (Ω∗ )−1 to map one-forms to vector fields, and so associate functions to vector fields. The Hamiltonian vector field, Xf , associated to an arbitrary function f on a symplectic manifold is defined by df = Ω∗ (Xf ) (6.2.11) In local coordinates n n X X ∂ ∂f Xf = aj aj = − (Ω−1 )jk (6.2.12) ∂x ∂x j k j=1 k=1

(the minus sign comes from antisymmetry of Ω−1 and the flip of order from (6.2.7)). The Poisson bracket (aka PB), {f, g}, is defined by {f, g} = Xf g

(6.2.13)

By (6.2.10), if ϕt is the flow defined by Xf (the Hamiltonian flow), then d g = {f, g} (6.2.14) dt t=0

In particular, by (6.2.10), g is invariant under the Hamiltonian flow generated by f if and only if {f, g} = 0. By (6.2.12), n X ∂f ∂g {f, g} = (Ω−1 )jk (6.2.15) ∂xj ∂xk j,k=1

which implies, by the antisymmetry of Ω, that {f, g} = −{g, f }

(6.2.16)

An intrinsic way of seeing this is to note {f, g} = Xf g = dg(Xf ) = Ω∗ (Xg )(Xf ) = Ω(Xg , Xf )

(6.2.17)

from which the antisymmetry is obvious. Note, in particular, (6.2.16) implies that Xf f = 0

(6.2.18)

So, by (6.2.14), df =0 dt under the flow generated by f —this is energy conservation. One advantage of the PB formalism is that it makes it easy to compute changes in the form of Hamiltonian equations under a change

6.2. SYMPLECTIC DYNAMICS

431

of variables. In this regard, two versions of the chain rule are invaluable. First, ℓ X ∂G {f, G(g1 , . . . , gℓ )} = {f, gj } (6.2.19) ∂g j j=1 which follows from the chain rule for differential operators and {f, · } = Xf · . Using (6.2.16), one can iterate this to obtain a general formula for change of variables {F (f1 , . . . , fm ), G(g1 , . . . , gℓ )} =

m X ℓ X k=1 j=1

{fk , gj }

∂F ∂G ∂fk ∂gj

(6.2.20)

In particular, if M has coordinates (pj , qj )m j=1 with {pj , pk } = {qj , qk } = 0, then m X ∂F ∂G ∂G ∂F {F (p, q), G(p, q)} = {pj , qk } − (6.2.21) ∂p ∂p j ∂qk j ∂qk k,j=1 Recall that the Lie bracket of two vector fields is defined by [X, Y ] = XY − YX

(6.2.22)

as a composition of differential operators. It is also a vector field. The fact that dΩ = 0 has the following important consequences: Theorem 6.2.1. On a symplectic manifold, (i) [Xf , Xg ] = X{f,g}

(6.2.23)

(ii) (Jacobi identity)

{f, {g, h}} = {{f, g}, h} + {g, {f, h}}

(6.2.24)

(iii) Hamiltonian flows preserve the symplectic form. Remarks. 1. Maps preserving the symplectic form are called canonical transformations or, in more modern discussions, symplectomorphisms. 2. (6.2.24) is often written in the more symmetric form {f, {g, h}} + {g, {h, f }} + {h, {f, g}} = 0

(6.2.25)

3. An invariant way to see (6.2.25) is to prove for general two-forms Ω, LHS of (6.2.25) = c dΩ(Xf , Xg , Xh ) for suitable constant c so that (6.2.25) is equivalent to dΩ = 0.

432


Sketch. (i) Using (6.2.12), one easily computes [Xf , Xg ] and sees that it is X{f,g} , plus some terms involving derivatives of (Ω−1 )jk . Since dΩ = 0, we have ∂ ∂ ∂ (Ω)jℓ + (Ω)ℓk + (Ω)kj = 0 (6.2.26) ∂xk ∂xj ∂xℓ and this plus ∂x∂ k Ω−1 = −Ω−1 ( ∂x∂ k Ω)Ω−1 (matrix multiplication) implies the terms involving derivatives of Ω cancel. This proves (6.2.23). (ii) By (6.2.23), Xg Xh f − Xh Xg f = X{g,h} f

(6.2.27)

which, given (6.2.13), implies (6.2.24).

(iii) If ϕft is the flow generated by Xf and (ϕft )∗ g = g ◦ ϕft

(6.2.28)

then, because { · , · } determines (Ω−1 )jk and so Ω, invariance of Ω is equivalent to ϕft ({g, h}) = {ϕft (g), ϕft (h)} (6.2.29) The derivative with respect to t of (6.2.29) is exactly (6.2.24). Since (6.2.29) holds at t = 0 and derivatives are equal, we have (6.2.29) in general. This has a number of important consequences. Suppose M is a symplectic manifold, so of even dimension 2m. The m-fold wedge product Ω ∧ · · · ∧ Ω = det(Ω)dx1 ∧ · · · ∧ dx2m , called the canonical volume form, is, by nondegeneracy, an everywhere nonzero 2m-form. Since Ω is invariant, so is this volume form. Thus, Corollary 6.2.2 (Liouville’s Theorem). Any Hamiltonian flow preserves the canonical volume form. Secondly, we can set up several equivalences: Theorem 6.2.3. Let f1 , . . . , fℓ be ℓ functions on a symplectic manifold M. Then the following are equivalent: (i) For all 1 ≤ j, k ≤ ℓ, {fj , fk } = 0 (6.2.30) (ii) For all j, the flows exp(tXfj ) leave {fk }ℓk=1 invariant. If these hold, then (iii) For 1 ≤ j, k ≤ ℓ, [Xfj , Xfk ] = 0 (iv) The flows exp(tXfj ) and exp(sXfk ) commute.

(6.2.31)

6.2. SYMPLECTIC DYNAMICS ~

(v) The map ϕf : (t1 , . . . , tℓ ) 7→ exp( flows are global) obeys ~

Pℓ

j=1 tj Xfj )

~

433

on Rℓ (assuming all

~

ϕft+s = ϕft ϕfs

(6.2.32)

for all t, s ∈ Rℓ . Remarks. 1. Notice that if {f, g} is a constant, [Xf , Xg ] = 0, so (6.2.31) does not imply (6.2.30). 2. One can also show (iii), (iv), (v) are equivalent. Sketch. (i) ⇔ (ii) by (6.2.14). (ii) ⇒ (iii) by (6.2.27). This in turn implies (iv) by standard results on Lie derivatives, and similarly, they imply (v). Poisson commuting functions are said to be in involution. ℓ functions, f1 , . . . , fℓ are said to be independent at p0 ∈ M if and only if (df1)(p0 ), . . . , (dfℓ )(p0 ) are linearly independent. The implicit function theorem then implies ~

Mpf0 = {p | fj (p0 ) = fj (p), j = 1, . . . , ℓ}

(6.2.33)

intersected with a small neighborhood of p0 is a submanifold of dimension dim(M) − ℓ. Put down a Riemann metric near p0 . grad(f ) is the vector field associated to df under this metric. {grad(fj )}ℓj=1 ~ are all orthogonal to tangent vectors to Mpf . On the other hand, if ~ {fj , fk } = 0, exp(tXfj ) leaves Mpf0 invariant, and so the Xfj are all ~ tangent to Mpf0 , and so orthogonal to all grad(fk ). If f1 , . . . , fℓ are independent, the grad(fk ) are independent, as are the Xfj since Ω and the Riemann metric are nondegenerate. Given the orthogonality, we get 2ℓ independent vectors at p0 . We have thus proven: Proposition 6.2.4. If M is a symplectic manifold of dimension 2m and f1 , . . . , fℓ are in involution and independent at some point p0 ∈ M, then ℓ ≤ m. ~ If ℓ = m, then Mpf0 is of dimension m, {Xfj }m j=1 span its tangent n space and {grad(fj )}j=1 span the normal subspace to this tangent space. Definition. A completely integrable system on a symplectic manifold, M, of dimension 2m is a set f1 , . . . , fm of functions in involution which are linearly independent at almost all points in M. Finally, here are the tori: Theorem 6.2.5 (Arnold–Jost–Liouville Theorem). Let {f1 , . . . , fm } be a completely integrable system on a symplectic

434


manifold, M, of dimension 2m. Let T be a connected compact set on which all fj are constant and so that at each point of T, the fj are independent. Then: P m (i) For any p0 ∈ T, {exp( m j=1 tj Xfj )p0 | t ∈ R } = T . (ii) T is diffeomorphic to an m-dimensional torus. Sketch. (i) Fix p0 . Let ϕ : Rm → T by ~

ϕ(t) = ϕ~ft (p0 )

~

Clearly, ϕ maps to Mpf0 since fj are constant and so must lie in its ~ connected component. Since T has a neighborhood N with N ∩ Mpf0 = T (by independence and the implicit function theorem), ϕ maps to T . By compactness, the flow is complete, that is, defined for all t. f~ Since {Xfj }m j=1 are independent at any point, p1 , in T , t → ϕ~t (p1 ) ~

~

~

is a C ∞ bijection for |t| small and so, by ϕ~ft+~s = ϕ~ft ϕ~sf , the range of ϕ is open. Suppose pn → p∞ and pn ∈ Ran(ϕ). Then, for n large, pn lies ~ ~ in the image of ϕft on p∞ . So there is t with ϕf−t (p∞ ) = pn . Since pn is in Ran(ϕ), there is s with ϕ(s) = pn . Thus, ϕ(t + s) = p∞ , that is, Ran(ϕ) is closed. By connectedness, (i) holds. (ii) Let G ⊂ Rm be {t | ϕ(t) = p0 }. Clearly, G is a closed subgroup. Since ϕ is a diffeomorphism of a small neighborhood of 0 ∈ Rm to p0 (by the independence and implicit function theorem), G is discrete, then ϕ˜ : Rm /G → M is a bijection, so compactness implies G is an m-dimensional lattice and Rm /G is an m-dimensional torus.

Remarks and Historical Notes. In some ways, our symplectic manifolds will come via the ad hoc introduction of the PBs (6.1.12)/(6.1.13), but there are two natural classes of symplectic manifolds both related to the Toda lattices. First, the cotangent bundle of any manifold has a natural one-form, ω, and Ω = dω defines a closed nondegenerate two-form; see [4, 25, 26, 278]. Second, any Lie group defines an action on its Lie algebra and so on the dual of the Lie algebra. Orbits under this action are called coadjoint orbits. Using the Lie bracket, they have a natural symplectic form; see [197, 221, 347, 379, 451] for further discussion. As we will explain in the Notes to Section 6.7, this is also connected to Toda lattices. Completely integrable systems were heavily studied in the nineteenth century with important contributions, via striking examples, by Jacobi, Neumann, and Kovalevskaya. They fell into the background after Poincaré’s proof that celestial dynamics was not integrable and

6.3. QR FACTORIZATION

435

the focus on ergodicity to explain statistical mechanics. With the discovery by Gardner, Greene, Kruskal, and Miura [141] that KdV has an infinity of conserved quantities, there was an explosion of interest in the subject that has continued for the past forty years. The Lax formalism [268], which we discuss in Section 6.6, has been a central element of most of the examples found since then. Missing from our discussion is the existence of angle variables. Under the hypotheses of Theorem 6.2.5, one can prove that there is a neighborhood, N, of T so that N ∼ = M × Tm with M ⊂ Rm a neighborm hood of 0 and T the m torus, so that if (y1 , . . . , ym ) are coordinates on M and θ1 , . . . , θm (θ ∈ [0, 2π)) coordinates in Tm , then {yj , yk } = 0, {θj , θk } = 0, {yj , θk } = δjk and the f ’s are functions only of y’s. The y’s are called action variables and the θ’s angle variables. Angle variables are important in the study of perturbations, including KAM theory. For angle variables in free Toda, see [301], and in periodic Toda, see [33, 34, 193]. 6.3. QR Factorization In this section, we discuss an elementary piece of linear algebra that we will need later. While we will be mainly interested in finite matrices, the semi-infinite case presents no difficulty. The decomposition we will discuss is not about linear transformations but about matrices, that is, bases matter, and we will be talking about explicit n × n matrices and semi-infinite matrices, that is, operators on Cn and on ℓ2 (Z+ ). As we have done for Jacobi matrices, we label such vectors vj , j = 1, . . . , n or j = 1, . . . . Matrices have the form (aij ), i, j = 1, . . . , n or i, j ∈ Z+ ≡ {1, 2, . . . }. Definition. An upper triangular matrix is one with ajk = 0 if j > k, that is, it consists of diagonal elements and potentially nonzero elements above the diagonal. R will denote the set of upper triangular matrices that are strictly positive on diagonal, that is, ajj > 0

(6.3.1)

Notice that R is closed under products and, at least in the finite matrix case, if A ∈ R, it is invertible and A−1 ∈ R, that is, in the case of n × n matrices, R is a subgroup of GL(n, R) or GL(n, C). We will let U stand for the group of unitary matrices. Theorem 6.3.1 (QR Decomposition). Let A be a bounded matrix with bounded inverse. Then there exist unique Q ∈ U and R ∈ R so that

436


A = QR

(6.3.2)

Moreover, Qδ1 =

Aδ1 kAδ1 k

(6.3.3)

Proof. We begin by noting that U ∩ R = {1}

(6.3.4)

for if A ∈ U ∩ R, then unitarity of A and the fact that the first column is of the form (a11 , 0, 0, . . . )t implies |a11 | = 1 and then, since a11 > 0, we have a11 = 1. Unitarity of A means column j (j ≥ 2) is orthogonal to (1, 0, 0, . . . )t , so a1j = 0, that is, A has the form   1 0 0 ... ... 0   A= (6.3.5) 0  A˜ .. .

where A˜ ∈ U ∩ R so, by an obvious induction, A = 1, proving (6.3.4). (6.3.4) implies uniqueness, for if Q1 R1 = Q2 R2

(6.3.6)

−1 Q−1 2 Q1 = R2 R1

(6.3.7)

and R1 is invertible, then lies in U ∩ R, showing Q1 = Q2 and R1 = R2 . For existence, {Aδj }N j=1 (N finite or infinite) are linearly independent since A is invertible, so we can use Gram–Schmidt to define {ej }N j=1 inductively so the e’s are orthonormal and Aδj =

j X

rkj ek

(6.3.8)

k=1

with

rjj > 0

(6.3.9)

and so

Aδ1 kAδ1 k Now define a unitary matrix, Q, by e1 =

Qδj = ej

(6.3.10)

(6.3.11)

6.3. QR FACTORIZATION

437

N Since A is invertible, the {ek }N k=1 are a basis, for if ψ ⊥ {ek }k=1 , by ∗ (6.3.8), hψ, Aδj i = 0, A ψ = 0, so ψ = 0. Define a matrix, R, by ( rkj k ≤ j (R)kj = (6.3.12) 0 R>j

Clearly, R ∈ R and −1

−1

Q (Aδj ) = Q =

X j

X j k=1

−1

rkj ek

k=1

rkj δk

= Rδj

(6.3.13)

that is, Q A = R, proving (6.3.2). (6.3.3) is (6.3.10) plus (6.3.11). Theorem 6.3.2. If the matrix A of Theorem 6.3.1 is real, then Q is orthogonal (i.e., real and unitary) and R is real. Proof. Clearly, the vectors {ej }N j=1 are real, so R is real and Q is orthogonal. One reason the QR factorization is important in numerical analysis is the QR algorithm. Given A invertible, write A by (6.3.2) and let A1 = RQ = Q−1 AQ

(6.3.14)

Since A and A1 are unitarily equivalent, they have the same eigenvalues. The map A → A1 is called one step in the QR algorithm. It can be iterated, that is, one writes A1 = Q1 R1 and then A2 = R1 Q1 . The remarkable fact is that in very many cases, one can prove that An converges to an upper triangular, sometimes even diagonal, matrix. Indeed, we will prove in Section 6.7: Theorem 6.3.3. Let J be an n × n Jacobi matrix that is strictly positive. Let J (1) , J (2) , . . . be the results of repeatedly applying the QR algorithm to J. Then each J (n) is a Jacobi matrix and J (n) converges exponentially fast to a diagonal matrix whose eigenvalues are those of J. Thus, a practical method for effective numerical approximation of eigenvalues of a positive symmetric matrix is to first use Gram–Schmidt to find a basis in which the matrix is triangular and then to use this iterated QR algorithm. Remarks and Historical Notes. The QR algorithm as a numerically convergent method goes back at least to Francis [136]. See the Notes to Section 6.7 for a discussion of works connecting it to the Toda

438


lattice. Typical of generalizations of Theorem 6.3.3 are the following: If A is a finite symmetric matrix which is strictly positive with distinct eigenvalues, then the QR algorithm converges to a diagonal matrix; see, for example, Olver [323]. The QR algorithm is connected to the Iwasawa decomposition of the semisimple Lie group GL(n, C); see, for example, Helgason [190]. 6.4. Poisson Brackets of OPs, Eigenvalues, and Weights As noted in Section 6.1, the free Toda lattice can be interpreted as the Hamiltonian equations of motion associated to the Poisson brackets (aka PBs) {bk , ak } = − 14 ak

k = 1, . . . , N − 1

{bk , ak−1 } = 14 ak−1

k = 2, . . . , N

(6.4.1) (6.4.2)

(all other brackets are zero) and H=2

N X

b2j

+4

j=1

N −1 X

a2j

(6.4.3)

j=1

Our goal here is to show, first of all, that there is a suitable symplectic structure in which these are the PBs and then to compute the −1 PBs for the orthogonal polynomials generated by {aj , bj }N j=1 ∪ {bN } and for the eigenvalues and spectral weights of the associated Jacobi matrix, JN ;F . In particular, we will prove the Toda flow leaves the spectrum of JN ;F invariant. −1 N Of course, {(ak )N k=1 , (bk )k=1 } is an odd-dimensional space, so it cannot be a symplectic manifold. Related to this is that (6.4.1)/(6.4.2) imply that X N bj , ak = 0 k = 1, . . . , N − 1 (6.4.4) j=1

so that { · , · } is not nondegenerate. In fact, we need to fix β and look at the submanifold where N X

bj = β

j=1

+1 −1 N We will use R2N for the set of (aj )N + j=1 , (bj )j=1 with aj > 0.

(6.4.5)

6.4. POISSON BRACKETS OF OPS, EIGENVALUES, AND WEIGHTS

439

+1 Proposition 6.4.1. Fix β real. Let Xβ ⊂ R2N be the set of + 2N +1 (a, b) ∈ R+ obeying (6.4.5). Use a1 , . . . , aN −1 , b1 , . . . , bN −1 for coordinates on Xβ , and define X Ω=4 (a−1 (6.4.6) ℓ daℓ ) ∧ dbk 1≤ℓ≤k≤N −1

Then Ω is a closed nondegenerate two-form P which induces the PBs −1 (6.4.1)/ (6.4.2) (where bN is the function β − N j=1 bj ). Proof. If {xj }2L j=1 are local coordinates and Ω=

2L X

j,k=1

Ωjk (x) dxj ∧ dxk

(6.4.7)

where Ωjk = −Ωkj is a symplectic form on a 2L-dimensional manifold, then the Hamiltonian vector field Hxj is given by Hx j =

2L X k=1

where

So implies

(j)

αk

∂ ∂xk

2L X ∂ (j) Ωmk αm = δkj Ω Hx j , = δjk ⇒ ∂xk m=1 (j) αm = (Ω−1 )jm = −(Ω−1 )mj (j)

{xj , xk } = Hxj (xk ) = αk = −(Ω−1 )kj

(6.4.8)

(6.4.9)

(6.4.10) (6.4.11)

Thus, the coefficients of Ω are given by the negative of the inverse of the matrix of PBs. Next, we suppose the 2L coordinates are written in two blocks of L, say, p1 , . . . , pL and q1 , . . . , qL . If (in our case, U = 0, but later we will want U 6= 0)   U W  Ω= (6.4.12) −W t 0 then Ω is invertible if and only if W is, and   t −1 0 −(W )  Ω−1 =  W −1 W −1 U(W t )−1

(6.4.13)

440


The PBs, (6.4.1)/(6.4.2), we are interested in have this form if (p1 , . . . , pN −1 , q1 , . . . , qN −1 ) = (b1 , . . . , bN −1 , a1 , . . . , aN −1 ) with U ≡ 0 and 1  a − 14 a1 0 ... 4 1 1 a − 14 a2 . . . (W t )−1 =  0 (6.4.14) 4 2 ... ... ... ... = D(1 − M)

with D thediagonal matrix Dkj = 0 1 0 ... 0 0 1 ... . Thus,

δkj 14 ak

(6.4.15)

and M the standard nilpotent

... ... ... ...

W = ([D(1 − M)]t )−1 = D −1 (1 + M t + (M t )2 + · · · + (M t )N −1 )  −1  4a1 0 0 = 4a−1 (6.4.16) 4a−1 0 1 2 ... ... ...

Thus, Ω, given by (6.4.6), is nondegenerate and leads to the required 2 PB. That Ω is closed follows from a−1 ℓ daℓ = (d log(aℓ )) and d = 0.

We are mainly interested in the PBs of the functions of JN ;F given by the eigenvalues and weights. These are complicated functions of the a’s and b’s, so the key will be to use some intermediate functions, namely, the coefficients of the monic OPs. −1 Theorem 6.4.2. Given Jacobi parameters in R2N , let + N (Pn , Qn )n=1 be the monic OPs and second kind monic (i.e., given by (3.2.12) with pn replaced by Pn ) polynomials. Then, for n = 1, . . . , N,

{Pn (x), Pn (y)} = {Pn−1 (x), Pn−1 (y)} = 0 (6.4.17) Pn (x)Pn−1 (y) − Pn (y)Pn−1(x) − Pn−1 (x)Pn−1 (y) 2{Pn (x), Pn−1 (y)} = x−y (6.4.18) {Pn (x), Pn (y)} = {Qn (x), Qn (y)} = 0 (6.4.19) Pn (x)Qn (y) − Pn (y)Qn (x) 2{Pn (x), Qn (y)} = − + Qn (x)Qn (y) x−y (6.4.20) Remarks. 1. While one tends to think of Pn (x) as a function of a n−1 single variable, x, in fact, it is a function of x and also (aj )j=1 ∪ (bj )nj=1. In (6.4.17), x and y are fixed and we mean PBs in the a’s and b’s! In essence, these PBs encode information on the PBs of the coefficients of Pn and Pn−1 . 2. (6.4.18)/(6.4.20) hold for x 6= y and then in a limit.


441

3. If S, T are polynomials, [S(x)T (y) − S(y)T (x)]/(x − y) is called their Bezoutian. Proof. We begin by proving (6.4.17)/(6.4.18) by induction. With P0 (x) = 1, P−1 (x) = 0, we see they hold when n = 0. So suppose they hold for n and let us check {Pn+1 (x), Pj (y)} for j = n + 1, n. As preliminary, we claim {a2n , Pn (x)} = − 12 a2n Pn−1 (x)

for

(6.4.21)

Pn (x) = (x − bn )Pn−1 (x) − an−1 Pn−2 (x) (6.4.22) n−1 n−2 Pn−1 , Pn−2 are only functions of {bj }j=1 and {aj }j=1 , and an only fails to Poisson commute with bn and bn+1 , so {a2n , Pn (x)} = {a2n , −bn }Pn−1 (x)

(6.4.23)

Pn+1 (x) = (x − bn+1 )Pn (x) − a2n Pn−1 (x)

(6.4.25)

{(x − bn+1 )Pn (x), (y − bn+1 )Pn (y)} = 0

(6.4.26)

{a2n Pn−1 (x), a2n Pn−1 (y)} = 0

(6.4.27)

=

by (6.4.1). Now use

− 12

a2n Pn−1 (x)

{aj }n−1 j=1 , bn+1

Since Pn (x) is a function of {bj }nj=1 and with Pn (x), so by the induction hypothesis,

(6.4.24)

Poisson commutes

n−2 Similarly, Pn−1 is a function of {bj }n−1 j=1 and {aj }j=1 , so an Poisson commutes with Pn−1 (x), and by induction,

Thus, by (6.4.22),

−{Pn+1 (x), Pn+1 (y)} = {(x − bn+1 )Pn (x), a2n Pn−1 (y)} − (x ↔ y) (6.4.28) Now {XY, WZ} = XW {Y, Z} + XZ{Y, W } + Y W {X, Z} + YZ{X, W }, so (6.4.28) is a sum of four terms with one zero since {bn+1 , Pn−1 (y)} = 0. The other terms are 2t1 = 2(x − bn+1 )a2n {Pn (x), Pn−1 (y)} − (x ↔ y)

= (x − y)a2n {Pn (x), Pn−1 (y)} (6.4.29) 2 Pn (x)Pn−1 (y) − Pn (y)Pn−1 (x) = (x − y)an − Pn−1 (x)Pn−1 (y) x−y (6.4.30)

where (6.4.29) comes from the symmetry of {Pn (x), Pn−1 (y)} under x ↔ y.

442


Next, 2t2 = 2(x − bn+1 ){Pn (x), a2n }Pn−1 (y) − (x ↔ y) = a2n (x − y)Pn−1(x)Pn−1 (y)

(6.4.31)

by (6.4.21). Finally, 2t3 = −2{bn+1 , a2n }(Pn (x)Pn−1 (y) − (x ↔ y)) = −a2n (Pn (x)Pn−1 (y) − Pn (y)Pn−1(x))

(6.4.32)

which shows that t1 + t2 + t3 = 0, proving (6.4.18) for n + 1. Similarly, by (6.4.25), using (6.4.26), {Pn+1 (x), Pn (y)} = {−a2n Pn−1 (x), Pn (y)}

= −a2n {Pn−1 (x), Pn (y)} − Pn−1 (x){a2n , Pn (y)} (6.4.33)

The first term is evaluated by induction and the second by (6.4.22)—it cancels one part of the first term, giving 2 Pn (y)Pn−1 (x) − Pn (x)Pn−1 (y) 2{Pn+1(x), Pn (y)} = an y−x Pn (y)(Pn+1(x) − (x − bn+1 )Pn (x)) − (x ↔ y) =− y−x (6.4.34) =

Pn+1 (x)Pn (y) − Pn+1 (y)Pn (x) − Pn (x)Pn (y) x−y

proving (6.4.18) for n + 1. We get (6.4.34) using (6.4.25). This proves (6.4.17)/(6.4.18) inductively. To get (6.4.19)/(6.4.20), we note that Pn (x) is det(x − Jn;F ), while Pn−1 (x) is the minor of (n, n) while Qn (x) is the minor of (1, 1). There is an obvious symmetry that says Pn (x; a1 , . . . , an−1 , b1 , . . . , bn ) = Pn (x; an−1 , . . . , a1 , bn , . . . , b1 ) (6.4.35) Qn (x; a1 , . . . , an−1 , b1 , . . . , bn ) = Pn−1 (x; an−1 , . . . , a1 , bn , . . . , b1 ) (6.4.36) (Pn−1 is not dependent on the last a and b nor Qn on the first a and b). Notice that under this reordering of variables, the signs of {aj , bk } flip, so all Poisson brackets change signs. Thus, (6.4.19)/(6.4.20) follow by the change of variables.


443

Finally, we turn to the spectral representation of Jn;F . We can write hδ1 , (JN ;F − z)−1 δ1 i = where the ρ’s are not independent since N X

ρj = 1

N X j=1

ρj λj − z

ρj > 0

(6.4.37)

(6.4.38)

j=1

The λ’s are ordered by

λ 1 < λ2 < · · · < λN

(6.4.39)

and if (6.4.5) holds, then N X

λj = Tr(JN ;F ) = β

(6.4.40)

j=1

Our analysis in Section 1.3 shows {(a, b) ∈ R2N +1 } (6.4.5) holds is mapped bijectively to the set of (λ, ρ) obeying (6.4.38)/(6.4.39). The +1 λ’s and ρ’s are functions on R2N so we can ask about their PBs. But + also (a1 , . . . , aN −1 , b1 , . . . , bN −1 ) → (ρ1 , . . . , ρN −1 , λ1 , . . . , λN −1 ) is a coordinate change and we can ask about its Jacobian. We will be able to answer both! Theorem 6.4.3. We have {λj , λk } = 0

1 ≤ j, k ≤ N

1 2

{λj , ρk } = [δjk ρj − ρj ρk ]

1 ≤ j, k ≤ N

(6.4.41) (6.4.42)

Remarks. 1. We will discuss {ρj , ρk } in the Notes—we don’t use it, so we don’t make the calculation explicitly. 2. Notice the right side P of (6.4.42) P sums to zero, summed over either j or k, consistent with ρk and xj being constant. Proof. We have that

N Y

(x − λj )

(6.4.43)

QN (x) X ρj = PN (x) x − λj j=1

(6.4.44)

PN (x) =

j=1

and, by Cramer’s rule, that

N

444


so that QN (x) =

N X

ρj

j=1

Y (x − λk )

(6.4.45)

k6=j

It follows that {PN (x), PN (y)}|x=λj , y=λk = {λj , λk }

Y Y (λj − λℓ ) (λk − λm ) ℓ6=j

m6=k

(6.4.46) where one makes the substitution only after evaluating all the PBs. Thus, {PN (x), PN (y)} = 0 implies (6.4.41). Once one knows that (6.4.41) holds, we get from (6.4.45) that Y Y {PN (x), QN (y)}|x=λj , y=λk = −{λj , ρk } (λj − λℓ ) (λk − λℓ ) ℓ6=j

If j 6= k, since PN (x)|x=λj for the second,

m6=k

(6.4.47) = 0, the first term in (6.4.20) vanishes. As

QN (x)QN (y)|x=λj , y=λk = ρj ρk

Y Y (λj − λℓ ) (λk − λm ) ℓ6=j

(6.4.48)

m6=k

which leads to (6.4.42) for j 6= k. If j = k, the first term in (6.4.20) is 00 if one sets x = λj , y = λj directly. One needs to set y = λj and take the limit as x → λj . We get Y Y −ρj (λj − λℓ ) (λk − λm ) (6.4.49) ℓ6=j

m6=k

yielding the extra term in (6.4.42) when j = k.

That the λj ’s all Poisson commute and there are N −1 independent ones gives the promised complete integrability. However, as we will see, this is not on a compact set (we will need to pass to the periodic case to get compactness). As a final result from these calculations, we note: −1 Theorem 6.4.4. (a) In terms of the coordinates {λj , ρj }N j=1 , the symplectic form Ω has the form

2

N X j=1

for suitable Uij .

dλj ∧ ρ−1 j dρj +

X i,j

Uij dλi ∧ dλj

(6.4.50)


445

−1 (b) The Jacobian of the change of variables from {aj , bj }N j=1 to −1 {λj , ρj }N j=1 is Q −1 ∂(a, b) 2−(N −1) N j=1 aj = (6.4.51) QN ∂(x, p) ρ j j=1 P −1 Remark. In (6.4.50), dλN is shorthand for − N j=1 dλj and dρN PN −1 for − j=1 dρj . Alternatively, while not coordinate one-forms, dλN and dρN are legitimate one-forms.

Proof. (a) We make a change of variables that helps “explain” the form of (6.4.42). Let ρk k = 1, . . . , N − 1 (6.4.52) yk = log ρN P Since N 1 ρk = 1, one can invert this via eyj ρj = j = 1, . . . , N − 1 (6.4.53) PN −1 y [1 + ℓ=1 e ℓ ] 1 ρN = (6.4.54) PN −1 y [1 + ℓ=1 e ℓ ] P N −1 mapping {(ρ1 , . . . , ρn ) | N . j=1 ρj = 1; ρj > 0} to R Moreover, {λj , yk } = {λj , log(ρk )} − {λj , log(ρN )} = ( 12 δjk − ρj ) − (−ρj ) =

1 2

δjk

(6.4.55)

It follows by (6.4.13) (with W −1 = 12 1) that Ω=2

N −1 X j=1

dλj ∧ dyj +

X i,j

Uij dλj ∧ dλj

(6.4.56)

PN −1 −1 Since dyj = ρ−1 k dρk − ρN dρN and j=1 dλj = −dλN , this implies (6.4.50). (b) By (6.4.50), the (N − 1)-fold wedge product Ω ∧ · · · ∧ Ω = 2N −1 (N − 1)! =2

N −1

(N − 1)!

N ^ X

j=1 k6=j

ρ−1 k (dλk ∧ dρk )

N Y X j=1

k6=j

ρ−1 k

(6.4.57)

dλ1 ∧ dρ1 ∧ · · · ∧ dλN −1 ∧ dρN −1 (6.4.58)

446


=2

N −1

Y N −1 (N − 1)! ρk dλ1 ∧ dρ2 ∧ · · · ∧ dλN −1 ∧ dρN −1 k=1

(6.4.59)

since XY j=1 k6=j

ρ−1 k

=

N Y

k=1

ρ−1 k

X N j=1

ρj

=

N Y

ρ−1 k

(6.4.60)

k=1

On the other hand, by (6.4.6), NY −1 N −1 −1 Ω∧···∧Ω = 4 (N − 1)! aj da1 ∧ db1 ∧ · · · ∧ daN −1 ∧ dbN −1 j=1

(6.4.61) Since is the absolute value of the coefficients in da1 ∧ · · · ∧ dbN −1 = C dλ1 ∧ · · · ∧ dρN −1 , this proves (6.4.51). ∂(a,b) | ∂(λ,ρ) |

Remarks and Historical Notes. The analog of {λj , λk } = 0 for periodic boundary conditions under the symplectic form given by (6.4.1)/(6.4.2) is due to Flaschka [129], and (6.4.41)/(6.4.42) for Jn;F is implicit in Moser [301]. The proof via the OP brackets in Theorem 6.4.2 is due to Cantero–Simon [67] with closely related calculations (brackets of m-functions) in Faybusovich–Gekhtman [126] and Gekhtman–Nenciu [143]. The Jacobian relation (6.4.51) appeared first in Dumitriu–Edelman [111] via an indirect calculation. A more direct proof using forms is in Forrester–Rains [135]. The idea we follow of getting it, via Poisson brackets, is due to Deift (unpublished). For analogs of the results of this section for OPUC, see [67, 143] and Killip–Nenciu [218, 219]. The ρj are not angle variables conjugate to the λ’s because we do not have {ρj , ρk } = 0. Instead, via {QN (x), QN (y)} = 0 and the computed {ρj , λk }, one obtains (see, e.g., [67]) X ρj ρk ρm X ρj ρk ρm ρj ρk {ρj , ρk } = − + (6.4.62) λj − λk m6=j λj − λm m6=k λk − λm 6.5. Spectral Solution and Asymptotics of the Toda Flow In this section, we will begin by noting that the PBs of Theorem 6.4.3 allow an immediate solution of the Toda equations of motion in (λj , ρj ) coordinates, and then we will use OPs to deduce the asymptotics of the original Jacobi parameters. Here the main result will be

6.5. SPECTRAL SOLUTION

447

Theorem 6.5.1. Let an (t), bn (t) solve the Toda equations in Flaschka form (6.1.14)/ (6.1.15) and let λ1 < λ2 < · · · < λN be the eigenvalues of the Jacobi matrix J(0) with parameters an (0), bn (0). Then lim aj (t) = 0

(6.5.1)

lim bj (t) = λN +1−j

(6.5.2)

lim bj (t) = λj

(6.5.3)

|t|→∞

t→∞ t→−∞

Indeed, if c=

min

j=1,...,N −1

λj+1 − λj

(6.5.4)

then aj (t) is O(e−c|t|) and |bj (t) − bj (±∞)| is O(e−2c|t| ). We begin with the equations for λj and ρj . Theorem 6.5.2. If J(t) solves (6.1.14)/ (6.1.15) for the Jacobi paN rameters, then its eigenvalues {λj (t)}N j=1 and weights {ρj (t)}j=1 obey λj (t) = λj (0)

(6.5.5)

e2tλj ρj (0) ρj (t) = PN 2tλk ρ (0) k k=1 e

(6.5.6)

Proof. By (6.2.14), Hamilton’s equation of motion for the values of an arbitrary smooth function on the manifold takes the form df = {H, f } (6.5.7) dt Shift to the coordinates λ1 , . . . , λN −1 , y1 , . . . , yN −1 of (6.4.52). Since, by (6.4.3), H = 2Tr(J 2 ) =2

N X

λ2j

(6.5.8) (6.5.9)

j=1

PN −1

where λN = β − j=1 λj and {λj , λk } = 0, {λj , yk } = 12 δjk (by (6.4.55)), we first have {λN , yj } = − 12 (6.5.10) and thus, d d λj = 0 yj = 4(λj − λN ) 21 (6.5.11) dt dt so λj (t) = λj (0)

yj (t) = yj (0) + 2t(λj (0) − λN (0))

(6.5.12)

448


Plugging this into (6.4.53)/(6.4.54) leads to (6.5.6).

Henceforth, since λj (0) is constant, we write it as λj . We focus now on t → +∞; the analysis as t → −∞ is similar except for the ordering of λ’s: the largest, −λj , that is, −λ1 , is relevant in place of λN . dρ has the form PN 2tλj ρj (0)δλj j=1 e (6.5.13) dρt = PN 2tλj ρ (0) j j=1 e

For t very large, the overwhelming largest weight is at λN since etλN ≫ etλN−1 ≫ · · · , next largest at λN −1 R , . . . . Since Pj (y, t), the orthogonal polynomial for dρt , minimizes |P (y)|2 dρt (y) among all monic polynomials of degree j, the best strategy is to put its zeros very near the j largest weights. Proving this is the key to going from Theorem 6.5.2 to Theorem 6.5.1: Proposition 6.5.3. Let Pj ( · , t) be the OPs for dρt and let be the zeros ordered by

(j) {xk (t)}jk=1

(j)

(j)

(j)

x1 > x2 > · · · > xj

(6.5.14)

Let c be given by (6.5.4). Then (i) kPj ( · , t)k2L2(dρt ) ≤ e−2t(λN −λN−j ) (λN −λ1 )2j ρN (0)−1

j = 1, 2, . . . , N−1 (6.5.15)

(ii) For t large, we have (j)

|xk (t) − λN +1−k | ≤

c 2

(6.5.16)

(iii) For large t, kPj ( · , t)k2L2(dρt ) ≥

( 2c )2j e−2t(λN −λN−j ) ρN −j (0) ρN (0)

(6.5.17)

(iv) −(j−1) c − λN +1−k | ≤ (λN − λ1 )j ρk (0)−1/2 e−t(λN+1−k −λN−j ) 2 (6.5.18) Q Proof. (i) Pj minimizes the norm, so picking Q(x) = jk=1 (x − λN +1−k ), we see for q = N − j, N − j − 1, . . . , 1, (j) |xk (t)

|Q(λq )| ≤ (λN − λ1 )j


449

and thus (since Q vanishes at λN , . . . , λN −j+1), kPj ( · , t)k2L2(dρt )

≤

kQk2L2 (dρt )

≤ (λN − λ1 ) 2j −2tλN

≤ (λN − λ1 ) e

since

N X j=1

and

N −j

X q=1

2j

N −j

X

ρq (t)

q=1

−1 2tλN−j

ρN (0) e

e2tλj ρj (0) ≥ ρN (0)e2tλN

e2tλq ρq (0) ≤ e2tλN−j

N X q=1

ρq (0) ≤ e2tλN−j

(6.5.19)

(6.5.20)

(6.5.21)

(ii) Let mq be given by (j)

mq = min |xk − λq |

(6.5.22)

k=1,...,j

If we show for q = N, N − 1, . . . , N + 1 − j that mq → 0, we see that there is at least one zero in each (λN +1−k − 2c , λN +1−k + 2c ). This gives j disjoint intervals, so there has to be exactly one of j zeros in each interval, proving (6.5.16). At λq , Pj (λq ) ≥ (mq )j (6.5.23) so kPj k2 ≥ (mq )2j e−2t(λN −λq ) ρq (0)ρN (0)−1 (6.5.24) By (6.5.15), (mq )2j ≤ (λN − λ1 )2j e−2t(λq −λN−j ) ρq (0)−1

goes to zero since λq > λN −j . (iii) By (6.5.16), we have no zero within mN −j ≥

c 2

(6.5.25)

of λN −j , so

c 2

(6.5.26)

and thus (6.5.24) implies (6.5.17). (iv) Since only one zero is within 2c of any λq , we can improve (6.5.23) to Pj (λq ) ≥ mq ( 2c )j−1 (6.5.27) and so improve (6.5.25) to m2q ( 2c )2j−2 ≤ (λN − λ1 )2j e−2t(λq −λN−j ) ρq (0)−1 which implies (6.5.18).

(6.5.28)

450


Proof of Theorem 6.5.1. Since kPj k aj = kPj−1k

(6.5.29)

the upper and lower bounds in (6.5.15)/(6.5.17) imply Cj e−t(λN+1−j −λN−j ) ≤ aj ≤ Dj e−t(λN+1−j −λN−j )

(6.5.30)

for nonzero constants Cj , Dj , which shows aj → 0 and is O(e−ct) as t → ∞. Next, note that the xj−1 term in the recursion relation j Y

(x −

(j) xk )

k=1

= (x − bj )

implies that bj =

j−1 Y

so (6.5.32) implies

) + O(xj−2)

(6.5.31)

k=1

j X k=1

By (6.5.18), as t → ∞,

(j−1)

(x − xk

(j) xk

−

j−1 X

(j−1)

xk

(6.5.32)

k=1

xk → λN +1−k

(j)

(6.5.33)

bj → λN +1−j

(6.5.34)

proving (6.5.3). Once we have an = O(e−c|t| ), the differential equation (6.1.15) implies that dbn /dt = O(e−2c|t| ), which implies |bn (t) − bn (±∞)| = O(e−2c|t| ). In addition to the Toda flow, one can define generalized Toda flows. Pick a C 1 function, G, on R. We want a Hamiltonian H=

N X

G(λj ) = Tr(G(J))

(6.5.35)

j=1

−1 N which is a polynomial of degree ℓ in {aj }N j=1 and {bj }j=1 if G is a polynomial of degree ℓ. Then Theorem 6.5.2 immediately extends:

Theorem 6.5.4. If J(t) solves the Hamiltonian equations of the generalized Toda flow associated to G, then the measure dρt associated to J(t) is PN 1 tG′ (λj ) 2 ρj (0)δλj j=1 e dρt = PN 1 tG′ (λ ) (6.5.36) j 2 e ρ (0) j j=1


451

The proof of Theorem 6.5.1 also extends so long as there is a nondegeneracy condition G′ (λj ) 6= G′ (λk ) (6.5.37) for all j 6= k (see the Notes for a discussion of the degenerate case).

Theorem 6.5.5. If J(t) solves the Hamiltonian equations of the generalized Toda flow associated to G and if the eigenvalues of J(0) obey the nondegeneracy condition (6.5.37), define g1 , . . . , gN to be the reordering of {G′ (λj )}N j=1 obeying g1 < g2 < · · · < gN

(6.5.38)

˜ j be defined by gj = G′ (λ ˜ j ). Then with Let λ c=

1 min 4 j=1,...,N −1

gj+1 − gj

(6.5.39)

we have aj (t) = O(e−c|t| ) ˜ N +1−j bj (t) → λ

(6.5.40) as t → ∞

(6.5.41)

|bj (t) − bj (±∞)| = O(e−2c|t| )

(6.5.43)

˜j bj (t) → λ

as t → −∞

(6.5.42)

and

Remarks and Historical Notes. Theorem 6.5.1 is due to Moser [301]. Our proof using zeros of OPs is taken from Simon [397]. The analysis of generalized Toda flows using critical points of the spectrum goes back to work of Deift–Li–Tomei [100] and Deift–Nanda–Tomei [101]. If one transfers the asymptotics back to p, q variables, one sees as t → ±∞, qj+1 − qj → ∞, so as t → +∞, p1 < p2 < · · · < pN , and as t → −∞, pN < · · · < p2 < p1 . The remarkable fact is that if + − limt→±∞ pj (t) = p± j , then pj = pN +1−j . Even though there are multiple particles, there is no momentum transfer, or put more precisely, the transfer preserves the set of values. This is, of course, a sign of the multiple conservation laws. If G′ has a degeneracy, the analysis is a little more complicated. One needs to break up λj ’s into groups with equal G′ (λj ). If λj1 , . . . , λjℓ is a group with equal G′ (λj ) and there are m λj ’s with larger G′ (λ), then as t → ∞, the block {bj }m+ℓ ∪ {aj }m+ℓ−1 approaches the Jacobi m+1 Pm+1 ℓ parameters for the measure k=1 ρt=0 ({λjk })δλk . This is discussed in Simon [397].

452


Asymptotics of semi-infinite Toda and generalized Toda flows can be found in Deift–Li–Tomei [100], Deift–Nanda–Tomei [101], Golinskii [172], and Simon [397]. 6.6. Lax Pairs One of the striking elements of our analysis so far is that the flow is isospectral—that is, the eigenvalues of JN ;F are preserved. Lax [268] found a general class of isospectral flows, and we will note here that the Toda flow fits into this framework. Let B be a smooth function from the set of all selfadjoint matrices to the set of skew adjoint (skew adjoint means B ∗ = −B) matrices or B might be defined on some manifold of selfadjoint matrices. Suppose C(t) is a curve of selfadjoint matrices (lying in the domain of definition of B) solving the differential equation dC(t) = [B(C(t)), C(t)] (6.6.1) dt B and C are called a Lax pair and (6.6.1) is called a Lax differential equation. Note that there is no assumption that there is an underlying symplectic form or Hamiltonian. We will not address when (6.6.1) has a solution but suppose a solution is given. The point is Theorem 6.6.1. If C(t) obeys (6.6.1), then there is a unitary family, W (t), smooth in t, so W (0) = 1 and C(t) = W (t)−1 C(0)W (t)

(6.6.2)

m

In particular, for any m, Tr(C(t) ) is constant (equivalently, C(0) → C(t) is isospectral). Remark. W (t) are called the Lax unitaries. Proof. Consider the differential equation dW (t) = −W (t)B(C(t)) (6.6.3) dt Here B(C(t)) is given, so this is a linear differential equation with smooth coefficients which has unique global solutions by standard techniques. We consider the solution with W (0) = 1. Since B ∗ = −B, we see so

dW ∗ (t) = B(C(t))W ∗ (t) dt

(6.6.4)

d WW∗ = 0 dt

(6.6.5)

6.7. THE SYMES–DEIFT–LI–TOMEI INTEGRATION

453

Since W W ∗ |t=0 = 1, we see W (t) is unitary. Let D(t) = W (t)C(t)W ∗(t) (6.6.6) Then, by (6.6.3)/(6.6.4), d dC ∗ D(t) = W (t) W (t) − W (t)[B(C(t)), C(t)]W ∗ (t) dt dt =0 (6.6.7) by (6.6.1). Thus, D(t) = C(0), proving (6.6.2). Since W is unitary, (6.6.2) says the flow preserves the spectrum. Given any Jacobi matrix, J, define B(J) by 

b1 a1 0 a1 b2 a2 J =  0 a2 b3 .. .. .. . . .

 ··· · · ·  · · · .. .



0 a1 0 −a1 0 a2 B(J) =   0 −a2 0 .. .. .. . . .

 ··· · · ·  · · · (6.6.8) .. .

Then, by a simple calculation (a special case of a more elaborate calculation in the next section!),   2a21 a1 (b2 − b1 ) 0 ··· a1 (b2 − b1 ) 2a22 − 2a21 a2 (b3 − b2 ) · · ·  [B(J), J] =  (6.6.9) 0 a2 (b3 − b2 ) · · ·  .. .. .. .. . . . . Thus, (6.6.1), the Lax differential equation, is just the Toda differential equation (6.1.14)/(6.1.15)! This provides a second proof of the isospectral nature of the Toda flow. In the next section, we will recover the action on the weights, ρj .

Remarks and Historical Notes. Lax pairs were introduced by Lax [268] in a seminal paper in the context of the KdV equation “explaining” the isospectral solution found by Gardner, Greene, Kruskal, and Miura [141]. Its relevance to Toda flows was discovered by Flaschka [129] and Moser [302]. 6.7. The Symes–Deift–Li–Tomei Integration: Calculation of the Lax Unitaries In this section, we will answer several hanging questions: (a) From the Lax pair point of view, how can one get the dynamics of the weights? (b) Can one find the Lax unitaries?

454


(c) What are the Lax pairs for the generalized Toda flow? The key will be the QR factorization, and as a bonus, we will have a new understanding of the QR algorithm and its convergence to diagonal form under iteration. Lemma 6.7.1. If J is a Jacobi matrix and R1 , R2 are in R, then (R1 JR2 )jk = 0

if j > k + 1

(6.7.1)

Remark. A matrix H with Hjk = 0 if j > k + 1 is called upper Hessenberg; it has only one possible nonzero diagonal below the main, precisely one diagonal below. Proof. In (R1 JR2 )jk =

X

(R1 )jℓ Jℓm (R2 )mk

(6.7.2)

ℓ,m

if the summand is nonzero, then ℓ ≥ j, m ≥ ℓ − 1, k ≥ m, so k ≥ j − 1; that is, if k < j − 1, then all terms in the sum are zero.

Theorem 6.7.2 ([415, 416, 100]). Let J ≡ J(0) be a given Jacobi matrix. Let G be a C 1 function on R. Let exp( 41 tG′ (J)) = Qt Rt

(6.7.3)

be the QR factorization, and define J(t) = Q−1 t JQt

(6.7.4)

Then (i) J(t) is a Jacobi matrix. (ii) If λj (t), ρj (t) are eigenvalues and weights of dρt , the spectral measure of J(t), then λj (t) = λj (0) ≡ λj e ρj (t) = PN

1 2

tG′ (λj )

k=1 e

1 2

(6.7.5) ρj (0)

tG′ (λk )

ρk (0)

(6.7.6)

Remarks. 1. Of course, given Theorem 6.5.4, this implies J(t) solves the generalized Toda flow—we will say more about this later. 2. Qt is, of course, the Lax unitary. For the Toda case, where 1 ′ G (J) = J, this calculation of the Lax unitary is due to Symes [415, 4 416]; for generalized Toda, it was discovered by Deift–Li–Tomei [100]. Proof. (i) Let At ≡ exp( 41 tG′ (J)) which is obviously invertible with At J = JAt . Thus, −1 J(t) = Rt Rt−1 Q−1 t JQt Rt Rt

(6.7.7)

6.7. THE SYMES–DEIFT–LI–TOMEI INTEGRATION

455

−1 = Rt A−1 t JAt Rt

= Rt JRt−1

(6.7.8)

Rt and Rt−1 lie in R, so by the lemma, J(t)jk = 0

(6.7.9)

if j−k > 1. But J(t) is symmetric, so (6.7.8) always holds if j−k < −1. Thus, J(t) is tridiagonal and symmetric. Since (6.7.8) and (6.7.2) imply that J(t)j j−1 = (Rt )jj Jj j−1 (Rt−1 )j−1 j−1 (6.7.10) we see J(t) is positive off-diagonal, that is, J(t) is a Jacobi matrix, as claimed. (ii) Let e1 , . . . , eN be normalized eigenvectors of J, that is, Jej = λj ej

(6.7.11)

Then (6.7.5) is immediate from (6.7.4) and unitarity of Qt . Moreover, ρj (t) = |hej , Qt δ0 i|2 = 1

′

|hej , At δ0 i|2 kAt δ0 k2

(6.7.12) (6.7.13)

by (6.3.3). But At ej = e 4 tG (λj ) ej and |hej , δ0 i|2 = ρj (0), which leads to (6.7.6). This allows us to compute the other element of the Lax pair for general G and redo the calculation (6.6.9). Given any real symmetric matrix, A, define π(A) antisymmetric by   j k (6.7.14)  0 j=k

Theorem 6.7.3. Let G be a C 1 function and let J, Qt , Rt , J(t) be given by Theorem 6.7.2. Then dQt = −Qt Bt dt

(6.7.15)

Bt = π( 41 G′ (J(t)))

(6.7.16)

d J(t) = [Bt , J(t)] dt

(6.7.17)

where and

456


Remarks. 1. Since Tr(G(J)) is the Hamiltonian that generates the generalized flow in question, (6.7.17) can be rewritten {Tr(G(J)), J} = [ 41 G′ (J), J]

(6.7.18)

{Tr(G(J)), Jkℓ } = ([ 14 G′ (J), J])kℓ

(6.7.19)

Bt = π(J(t))

(6.7.20)

intended in a matrix element equality, that is, for all k, ℓ, 2. For the Toda lattice, G′ (λ) = 4λ and is given by (6.6.8). Proof. By (6.7.3),

d G (J) = (Qt Rt ) (Qt Rt )−1 (6.7.21) dt dQt −1 dRt −1 −1 = Qt + Qt R Q (6.7.22) dt dt t t Multiply by Q−1 on the left and Qt on the right, using (6.7.4) to get t dQt dRt −1 1 G′ (J(t)) = Q−1 + R (6.7.23) t 4 dt dt t ˙ Since Qt is orthogonal, Q−1 t Qt is skew symmetric and so 0 on diagt t onal. Since Rt is upper triangular, dR is also, and thus, so is dR Rt−1 . dt dt Thus, below diagonal, that is, for j > k, 1 ′ −1 dQt G (J(t)) = Qt 4 dt jk jk 1 4

′

so if Bt is given by (6.7.15), we have (6.7.16) for j > k and then, by antisymmetry, for all j, k. (6.7.17) is immediate from (6.7.15) and (6.7.4). We have thus found a Lax pair representation for the dynamics of generalized Toda flows. Suppose now J > 0 and G is a C 1 function with G′ (x) = 4 log(x) for x ≥ inf(σ(J)). Then for n = 1, 2, . . . , At=n = J n

In particular, At=1 = J = Qt=1 Rt=1 and Jt=1 = Q−1 t (Qt Rt )Qt = Rt Qt The time one flow is just the QR algorithm! The time n flow can be seen to be the n times iterated flow. We thus have

6.8. COMPLETE INTEGRABILITY OF PERIODIC TODA FLOW

457

Proof of Theorem 6.3.3. As just noted, the n times iterated QR algorithm is a generalized Toda flow at time n. Now use Theorem 6.5.5. Remarks and Historical Notes. Symes [415, 416] discovered the approach of this section for Toda lattices. Deift–Li–Tomei [100] developed this for generalized flows, emphasizing that it could be applied to convergence of the QR algorithm. See Simon [397] for the OPUC analog. We already mentioned that the QR algorithm had a group theoretic interpretation in terms of the Iwasawa decomposition. There is a group theoretic version of the Toda chain found by Kostant [236] that, in part, motivated Symes [415]. The symplectic manifolds in this point of view are coadjoint orbits. This has spawned a huge literature, of which we mention [125, 144, 219, 237, 274, 284, 320, 322, 358, 374]. Some of them extend this approach to CMV matrices and Schur flows (see Section 6.9). 6.8. Complete Integrability of Periodic Toda Flow and Isospectral Tori We turn now to the periodic case of greatest interest to us in this book. The main result in this case is Theorem 6.8.1. Let ∆(x, {an , bn }N n=1 ) be the discriminant for period N Jacobi matrices. Then under the Poisson brackets (6.1.12) and (6.1.13), one has {∆(x), ∆(y)} = 0 (6.8.1) If λj (θ) are the eigenvalues of J(θ) (given by (5.3.8)), with 0 < θ, θ′ < π, {λj (θ), λk (θ′ )} = 0 (6.8.2) and this remains true for θ = 0 or π at points of nondegeneracy. If ℓ, ℓ′ are arbitrary, ′

{Tr(J(θ)ℓ ), Tr(J(θ′ )ℓ ) = 0

(6.8.3)

Partial Proof. We discuss various proofs of one of these below— here we want to discuss their relation. For θ 6= 0, π, λj (θ) are the roots of ∆(x) − 2 cos θ = 0 and are simple roots. Thus, (6.8.2) follows from {∆(x) − 2 cos θ, ∆(y) − 2 cos θ′ } = 0 by setting x to λj (θ) and y to λk (θ′ ), as in the proof of Theorem 6.4.3 In this calculation, one uses (a1 . . . aN )−1 is in the Poisson center since, unlike P, ∆ is not monic. One can go backwards using the fact that ∆(x) − 2 cos θ = Q (a1 . . . aN )−1 N j=1 (x − λj (θ)).

458


Clearly, (6.8.2) implies (6.8.3). To get from (6.8.3), one uses a piece of combinatorics (see the Notes) that shows if λ1 , . . . , λN are P P ℓ distinct, then sq = ij ℓ, then we can only shift index by one and so not reach the rows (1 and N) where JN ;F and JN ;P differ. The optimal r is N2 (or N 2−1 if N is odd) which limits ℓ to a lot less than the ℓ ≤ N − 1 that we need. The clever way that van Moerbeke [438] (following a similar KdV analysis in McKean– van Moerbeke [298]) is to look at the matrix with periodic boundary conditions but period 2N! So fix {aj , bj }N j=1 and let J2N ;2P be the 2N × 2N matrix which is J2N ;P for the Jacobi parameters of period 2N obtained by repeating {aj , bj }N j=1 two times (and is really of period N). Here is the key fact: Theorem 6.8.3. For ℓ = 2, 3, . . . , N, we have ℓ−1 ℓ 1 [ 4ℓ π(J2N ;P ), J2N ;P ] = 2 {Tr(J2N ;P ), J2N ;P }

(6.8.28)

In particular, for ℓ, ℓ′ = 1, 2, . . . , N, we have ′

ℓ ℓ {Tr(J2N ;P ), Tr(J2N ;P )} = 0

(6.8.29)

Proof. (6.7.18) for G(x) = xℓ says that for 2N ×2N free boundary conditions with 2N − 1 distinct a’s and 2N distinct b’s, we have ℓ−1 ℓ [ 4ℓ π(J2N ;F ), J2N ;F ] = {Tr(J2N ;F ), J2N ;F }

(6.8.30)

If we look at row N, since ℓ − 1 ≤ N − 1, we have

ℓ−1 ℓ−1 ℓ [ 4ℓ π(J2N ;P ), J2N ;P ]N q = [ 4 π(J2N ;F ), J2N ;F ]N q

since the differing matrix elements at 2N, 1 cannot be linked in ℓ − 1 ℓ ℓ steps to site N. Also, the difference of Tr(J2N ;F ) and Tr(J2N ;P ) Poisson commutes with (J2N ;P )N q . However, in (6.8.30), we first compute { , } and then set aN +j = aN , bN +j = bN (j = 1, . . . , N), while in (6.8.28), we have this equality and then compute { , }. By the periodicity, this makes the PB in (6.8.28) twice as large, explaining the 12 . For ℓ = 2, . . . , N, the argument that led to (6.8.25) goes from PN ℓ (6.8.28) to (6.8.29). For ℓ = 1, Tr(J2N ;P ) = 2 j=1 bj lies in the Poisson center, and so has zero PBs.

462


At first sight, we have not helped the situation much because, while we now have N functions of the eigenvalues, we also have 2N rather than N eigenvalues! However, the new eigenvalues, which are the N periodic and the N antiperiodic eigenvalues, are not independent. This can be seen by the fact that a1 . . . aN and the roots of ∆(x) − 2 determine the roots of ∆(x) + 2! So let JN ;A refer to the antiperiodic boundary condition operator, that is, J(θ) of (5.3.8) with θ = π. Notice, since J2N ;2P has eigenvalues which are the union of those of JN ;P and JN ;A , that for any ℓ, ℓ ℓ ℓ Tr(J2N ;2P ) = Tr(JN ;P ) + Tr(JN ;A )

Proposition 6.8.4. For ℓ = 1, 2, . . . , N − 1, we have

(6.8.31)

Tr(JNℓ ;A ) = Tr(JNℓ ;P )

(6.8.32)

ℓ ℓ Tr(J2N ;2P ) = 2 Tr(JN ;P )

(6.8.33)

Tr(JNN;A ) = Tr(JNN;P ) − 4Na1 . . . aN

(6.8.34)

Moreover, N Tr(J2N ;2P )

=

2 Tr(JNN;P )

− 4Na1 . . . aN

(6.8.35)

Proof. In computing (JNℓ ;P )jj , we have products of ℓ matrix elements that change the index by either 0, ±1 or ±N −1. Since ℓ ≤ N −1, the number that changes by N −1 must equal the number that changes by −(N − 1), if we are to return to j. Thus, aN appears an even number of times and so is unchanged by the replacement aN → −aN . This proves (6.8.32). This plus (6.8.31) implies (6.8.33). Since ∆(JN ;P ) = 2 and ∆(JN ;A ) = −2, we have that Since

Tr(∆(JN ;P ) − ∆(JN ;A )) = 4N

(6.8.36)

∆(x) = (a1 . . . aN )−1 xN + lower order by (6.8.32) in the left side of (6.8.36), all terms cancel but (a1 . . . an )Tr(JNN;P − JNN;A ), so (6.8.36) implies (6.8.34). Then (6.8.34) implies (6.8.35). Second Proof of Theorem 6.8.1. For ℓ, ℓ′ ≤ N − 1, (6.8.3) follows from (6.8.33) and (6.8.29). Since a1 . . . aN is in the Poisson center, (6.8.33) yields (6.8.3) for ℓ = N. Thus, λ1 , . . . , λN Poisson commute, and so (6.8.2) holds for θ = 0. Remarks and Historical Notes. Theorem 6.8.1 is due to Flaschka [129, 130]; the first proof we give is from [130] and the second from [129] and van Moerbeke [438].

6.9. INDEPENDENCE OF TODA FLOWS

463

There is a third proof in Cantero–Simon [67] who obtain it as a limit of free cases—the Tr(JNℓ ;F ) converge to moments of the density ℓ of states, not Tr(Jm;P (θ)), so the argument is subtle. Basically, one can compute the θ-dependence using the fact that these are roots of ∆(λ) − 2 cos θ and then relate moments of the density of states to integrals of Tr(JNℓ ;P (θ)) over dθ/2π. Our discussion of Lax pairs follows van Moerbeke [438]. P For the fact that, given λ1 , . . . , λN , sq = i1 ε} is finite and there is a finite-dimensional projection, P , so PA = AP and σ(A(1 − P )) ↾ Ran(1 − P )) = σ(A) \ S. w Since ϕn −→ 0 and P is finite-dimensional, kP ϕn k → 0, so dµϕn (S) → 0 and, by the argument that led to (7.2.12), lim infk(A − λ)ϕn k ≥ dist(λ, σ(A) \ S)

≥ dist(λ, σess (A)) − ε

Since ε is arbitrary, we get (7.2.10). If α ∈ σess (A), the spectral projection for (α − ε, α + ε) is infinitedimensional, so it contains an infinite orthonormal set {ϕn }. The set w has ϕn −→ 0, kϕn k = 1, and lim infk(A − λ)ϕn k ≤ |λ − α| + ε. By

472

7. RIGHT LIMITS

picking from the sequences for dist(αn + λ) → dist(σess (A), λ), and w εn → 0, we get ϕn −→ 0, kϕn k = 1 with limk(A − λ)ϕn k ≤ dist(λ, σess (A))

so there is equality by (7.2.9).

We can prove (7.2.5): Proof of (7.2.5). Given λ ∈ σ(Jr ) and ε, first find (using (7.2.9)) ϕ with kϕk = 1 and k(Jr − λ)ϕk ≤ 2ε . Then find ϕ˜ with kϕk ˜ = 1, ϕ˜ ε 1 having finite support, and kϕ − ϕk ˜ ≤ 2 kJr k+|λ| , so k(Jr − λ)ϕk ˜ ≤ε

(7.2.13)

Pick N so ϕ˜j = 0 if |j| > N. Now let mj be chosen so (7.1.1) holds. For j such that mj > N, define ϕ(j) by ϕ(j) (n) = ϕ(n ˜ − mj ) (7.2.14) w

Since mj > N, kϕ(j) k = 1, and since mj → ∞, ϕ(j) −→ 0. By (7.1.1), lim sup k(J − λ)ϕ(j) k = k(Jr − λ)ϕk ˜ ≤ε j→∞

By Proposition 7.2.2, dist(λ, σess (J)) ≤ ε

(7.2.15)

Since ε is arbitrary, λ ∈ σess (J).

The following is stronger than (7.2.6):

Theorem 7.2.3. Let J be a half- or whole-line Jacobi matrix. (a) If (7.2.2) holds for a u obeying |un | ≤ C(1 + |n|)k

(7.2.16)

for some k, then λ ∈ σ(J). (b) For almost every λ in σ(J), there is a solution of (7.2.2) obeying (7.2.16) with k = 1. Remarks. 1. k = 1 in (b) can be replaced by any k > 12 . 2. (7.2.16) can be replaced by lim|n|→∞ |n|−1 log(1 + |un |) = 0. 3. In (b), “almost every” means with respect to a spectral measure. Proof. We will discuss the half-line results; the whole-line case is similar. (a) We claim lim inf n→∞

|un+1 |2 + |un |2 =0 |u1|2 + · · · + |un−1|2

(7.2.17)

7.2. THE ESSENTIAL SPECTRUM

473

for if not, for large n and some ε > 0, (|u1 |2 + · · · + |un+1 |2 ) ≥ (1 + ε)(|u1 |2 + · · · + |un−1 |2 )

(7.2.18)

which implies n X j=1

|uj |2 ≥ C(1 + |ε|)n/2

violating (7.2.16). Let u˜(N ) be defined by

) u˜(N n

Then

( un = 0

  0 (N ) ((J − λ)˜ u )j = an−1 un  −a u n n+1

n≥N n>N j= 6 N, N + 1 j=N j =N +1

(7.2.19)

(7.2.20)

(7.2.21)

from which we get

|uN |2 + |uN +1 |2 k(J − λ)˜ u(N ) k2 ≤ 2 sup |aj | k˜ u(N ) k2 |u1 |2 + · · · + |uN −1 |2 j

(7.2.22)

So for a subsequence Nj → ∞,

k(J − λ)˜ u(Nj ) k →0 ku(Nj ) k

which, by (7.2.8), implies λ ∈ σ(J). R (b) Since p2n (x) dµ(x) = 1, we have Z X ∞ (n + 1)−2 p2n (x) dµ(x) < ∞

(7.2.23)

(7.2.24)

n=0

so for a.e. x,

∞ X

(n + 1)−2 p2n (x) < ∞

(7.2.25)

(n + 1)−1 |pn (x)| ≤ C(x)

(7.2.26)

n=1

which implies Use pn−1 for the un .

As a preliminary for the final step, we need Proposition 7.2.4. If Jr ∈ R(J), then L(Jr ) ⊂ R(J).

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7. RIGHT LIMITS

Proof. Let J˜ ∈ L(Jr ) be such that with |mj | > ∞ and for j → ∞, (r)

(r)

amj +n → a ñ

bmj +n → ˜bn

(7.2.27)

aℓp +n → a(r) n

bℓp +n → b(r) n

(7.2.28)

Pick ℓp → ∞ so that as p → ∞, For each j, pick p(j) so that (a) (b)

ℓp(j) > 2|mj |

(7.2.29)

(r) −j |aℓp(j) +n − a(r) n | + |bℓp(j) +n − bn | ≤ 2

(7.2.30)

for |n| ≤ |mj | + |j|. Then for all n fixed, as j → ∞, aℓ +m +n → a ñ bℓ +m +n → ˜bn p(j)

j

p(j)

j

and mj + ℓp(j) ≥ |mj | → ∞ so J˜ ∈ R(J).

Example 7.2.5. Let an ≡ 1, bn = 1 if n = 1, 4, 9, 16, . . . and (r) 0 otherwise. Then R(J) consists of Jr ’s with an ≡ 1 and either (r) (r) (r) bn ≡ 0 or bn has one n0 for which is bn0 = 1 and all others are 0 (all n0 occur). Each L(Jr ) has only the element a ñ ≡ 1, ˜bn ≡ 0. Thus, L(R(J)) ⊂ R(J) but strictly smaller. It can also happen that L(R(J)) = R(J). Proposition 7.2.6. If there exist Jk ∈ R(J) and nk and solutions ϕ of Jk ϕ(k) = λ(k) ϕ(k) so that λ(k) → λ∞ , and for some C < ∞, (k)

(k)

max |ϕj | ≤ C|ϕ(k) nk |

|j−nk |≤k

(7.2.31)

then there exists Jr ∈ R(J) with λ∞ ∈ σ∞,pp (Jr ). In particular, ∪Jr ∈R(J) σ∞,pp (Jr ) is closed. Proof. Let

(k)

(k) uj

ϕnk +j

(7.2.32) (k) ϕnk and J˜k be Jk -translated by nk units (which also lies in Jr ). Then =

(k)

sup |uj | ≤ C

(7.2.33)

|j|≤k

(k) and u0 = 1. By compactness, we can pass to a subsequence so J˜k(ℓ) → Jr and u(k) → u∞ . Clearly, Jr u∞ = λ∞ u∞ and ku∞ k∞ ≤ C so λ∞ ∈ σ∞,pp (Jr ).

We can now describe our strategy for completing the proof. By Theorem 7.2.3(b), if λ ∈ σess (J), there exist distinct λm → λ and u(m) obeying (7.2.16) with k = 1. We will proceed as follows:


475

(1) If u(m) is unbounded, we will find Jr ∈ R(J) so λ(m) ∈ σ∞,pp (Jr ). Thus, by the last proposition, λ is in some σ∞,pp (Jr ) or, for all m large, u(m) is bounded. (2) If u(m) is not exponentially decaying, we find Jr ∈ R(J) so λ(m) ∈ σ∞,pp (Jr ). (3) Thus, the only way to avoid λ ∈ ∪ σ∞,pp (Jr ) is for each λm to be an eigenvalue with exponentially decaying eigenvector, and we will prove they have to move to infinity in a way to get λ ∈ σ∞,pp (Jr ) for some Jr . Proposition 7.2.7. If u obeys (7.2.2) and (7.2.16) for some k but u∈ / ℓ∞ , then there is Jr ∈ R(J) and u˜ 6= 0 in ℓ∞ so Jr u˜ = λ˜ u

(7.2.34)

Proof. By the last proposition, it suffices, for each k, to find Jr and u˜ obeying (7.2.34) with u˜0 = 1 and max |˜ un | ≤ 2

(7.2.35)

|n|≤k

Fix k and for m = 1, 2, . . . , let qm =

max

m(k−1)≤n<mk

|un |

(7.2.36)

By hypothesis, qm → ∞, so there are infinitely many m’s, say m ∈ M, with qm ≥ max qℓ (7.2.37) ℓ≤m

If there are infinitely many m ∈ M with

qm+1 ≤ 2qm

(7.2.38)

then, by using compactness, we get a Jr and u˜ obeying (7.2.35). If qm+1 ≥ 2qm with m ∈ M, then clearly, m + 1 ∈ M. So if (7.2.38) fails to hold infinitely often, then there is m0 with m ∈ M for m ≥ m0 and qm0 +ℓ ≥ 2ℓ qm0 for all ℓ, violating (7.2.16). Thus, (7.2.38) holds infinitely often and we get the required u˜. Proposition 7.2.8. If (7.2.2) has a solution, u ∈ ℓ∞ , then either there is a Jr ∈ R(J) and a bounded nonzero u˜ obeying (7.2.34) or else, for some A, B > 0, we have |un | ≤ A exp(−Bn)

(7.2.39)

Proof. If un fails to go to zero, find mj → ∞ and |umj | ≥ ε > 0, and using a subsequence so un+mj , an+mj , and bn+mj all converge for Jr and u˜ obeying (7.2.34) with |˜ u0 | > ε and k˜ uk∞ ≤ kuk∞ < ∞. Thus, either there is a solution of (7.2.34) or un → 0.

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7. RIGHT LIMITS

As above, we need only find for each k, Jr and u˜ obeying (7.2.35). Fix j and define qm by (7.2.36). Let M be the set of m’s with qm ≥ max qℓ ℓ≥m

Since un → 0, M is infinite. If there are infinitely many m’s with qm−1 ≤ 2qm , we can find a solution obeying (7.2.35). On the other hand, if qm−1 ≥ 2qm for all large m, then u decays exponentially. Remark. This proposition proves that if λ ∈ σ(J) \ σess (J), then the corresponding eigenvectors decay exponentially; see the Notes for discussion. Proof of Theorem 7.2.1. We need only prove (7.2.7). Since λ is not isolated, there exist λ(m) → λ and solutions u(m) of Ju(m) = λ(m) u(m) obeying (7.2.16). By Proposition 7.2.6, if there are bounded solutions of some Jr u˜(m) = λm u˜(m) , then there is a Jr with Jr u˜ = λ˜ u ∞ for u˜ ∈ ℓ . By Propositions 7.2.7 and 7.2.8, this happens, unless for all large m, u(m) obeys (7.2.39) so that each λ(m) is an eigenvalue. So we consider only that case. Pick u(m) so ku(m) k2 = 1. (m) Since un → 0 as n → ∞, |u(m) | takes its maximum value at some (m) point—let nm be picked so |unm | = ku(m) k∞ and so that nm is the largest such point. If nm → ∞ as m → ∞, we can use a limit point of (m) u. Thus, we need u(m) /unm to get a bounded solution of some Jr u˜ = λ˜ only consider the case supm |nm | ≡ N < ∞. Since u(m) → 0 weakly, (m) sup|n|≤n |un | = Cm → ∞ and thus, ku(m) k∞ = Cm → 0

(7.2.40)

As in the proof of the last proposition, we need only find, for each k, solutions of (7.2.34) obeying (7.2.35). By following the proof of that proposition, we see one cannot do this unless one has a B1 , an N1 , and an M1 so that for all m > M1 and n > N1 , we have |un(m) | ≤ ku(m) k∞ e−B(n−N1 ) But then, by (7.2.40), as m → ∞, X 2 |u(m) n | → 0 n≥N1

By (7.2.40) again, N1 X n=1

2 (m) 2 |u(m) k∞ → 0 n | ≤ N1 ku

(7.2.41)


477

so ku(m) k2 → 0, violating the choice ku(m) k2 = 1. This contradiction shows that one can always construct the required bounded solutions. This completes the proof of Theorem 7.2.1. Q Example 7.2.9. Let S ≡ {x1 , . . . , xℓ } ⊂ R. Let P (λ) = ℓj=1(λ − xj ). It is easy to see that σess (J) ⊂ S if and only if P (J) is compact. For ℓ = 1, this happens if and only if an → 0, bn → x1 . But for ℓ ≥ 2, the conditions on the a’s and b’s are murky. Theorem 7.2.1 clarifies this. For example, we must have an an+1 . . . an+ℓ → 0 as n → ∞ so the J’s have to be direct sums of finite matrices of size at most ℓ. The possible limits are those finite Jacobi matrices with spectrum in S. For example, if ℓ = 2, the limits are the 1 × 1 matrices b1 = x1 or b = x2 and the 2 × 2 matrices ab x1 +xa 2 −b where b and a are related by b(x1 + x2 − b) − a2 = x1 x2 . Example 7.2.10. If Te is the isospectral torus associated to a finite gap set e and if R ⊂ Te, then by Theorem 7.2.1, σess (J) = e. Remarks and Historical Notes. Last–Simon [265] looked at onesided right limits J˜r and proved σess (J˜r ) ⊂ σess (J), but their arguments prove (7.2.5). (7.2.3) and its proof are from Last–Simon [266]. In the spectral theory community, these ideas go back to geometric approaches to the HVZ theorem; see [266] for references. There are three other independent threads that consider limits of differential or difference operators or some subclass, especially the almost periodic case. One thread using Fredholm operators goes back to Favard [122] with later developments by Muhamadiev [303, 304], Shubin [377, 378], Rabinovich [348], and Chandler-Wilde–Linder [74, 75]. In particular, Rabinovich has results close to (7.2.3) and ChandlerWilde–Linder have the result (7.2.4) on σ∞,pp (Jr ) Two other threads involve C ∗ -algebras (see Georgescu–Iftimovici [148] and M˘antoiu [288]) and what has been called “collectively compact operator approximation theory” (see Anselone [20] and references therein). The result that σess (J) ⊂ S if and only if P (J) is compact is due to Krein in Akhiezer–Krein [16]. The other parts of Example 7.2.9 are from [266]. Example 7.2.10, also from [266], answered a conjecture of Simon [391]. The history of this and related conjectures is discussed in the Notes to Section 8.1. There is a huge literature on exponential decay of discrete eigenfunctions of Schrödinger operators, of which the seminal works are

478

7. RIGHT LIMITS

Combes–Thomas [88] and Agmon [6]. For CMV matrices, this is discussed in [391, Sect. 10.14], and the same analysis works for Jacobi matrices. This is superior to Proposition 7.2.8 since the Combes– Thomas method gives explicit positive lower bounds on the constant B in (7.2.39), while the proof of Proposition 7.2.8 does not. Still it is interesting to see this new approach to exponential decay. 7.3. The Last–Simon Theorem on A.C. Spectrum In this section, we will prove that Theorem 7.3.1 (Last–Simon [265]). For any Jr ∈ R, Σac (Jr ) ⊃ Σac (J)

(7.3.1)

Indeed, on Σac (J), Jr has a.c. spectrum of multiplicity two. We will use the following characterization of Σac (J): Theorem 7.3.2 (Last–Simon [265]). Let 1 ♯ Kn (x0 , x0 ) < ∞ N = x0 lim inf n+1

(7.3.2)

Then up to sets of measure zero, Proof. Since

R

Z

Σac (J) = N ♯

1 K (x , x0 ) dµ n+1 n 0

(7.3.3)

= 1, by Fatou’s lemma,

1 Kn (x0 , x0 ) dµ ≤ 1 (7.3.4) n+1 so supp(dµ) ⊂ N, so all the more so, up to sets of Lebesgue measure zero, Σac (J) ⊂ N ♯ (7.3.5) On the other hand, by Theorem 3.11.7 (scaling (−2, 2) up to some interval I containing supp µ), up to sets of measure zero, 1 {x | w(x) = 0} ⊂ x lim inf Kn (x, x) = ∞ = R \ N ♯ (7.3.6) n+1 lim inf

which implies

N ♯ ⊂ Σac (J)

(7.3.7)

Next, we need the invariance of Σac under rank one perturbations. Suppose H = L2 (R, dµ) for µ a probability measure, ϕ ≡ 1 ∈ H, (A0 f )(x) = xf (x), and for λ ∈ R, Aλ f = A0 f + λhϕ, f if

(7.3.8)

7.3. THE LAST–SIMON THEOREM ON A.C. SPECTRUM

479

Theorem 7.3.3. ϕ is cyclic for Aλ and the spectral measure, dµλ , for Aλ and ϕ, that is, Z dµλ (x) 1 Fλ (z) = = ϕ, ϕ (7.3.9) x−z Aλ − z z∈ / R, obeys

Fλ (z) =

F0 (z) 1 + λF0 (z)

(7.3.10)

In particular, if dµλ(x) = wλ (x) dx + dµλ,s

(7.3.11)

w0 (x) |1 + λF0 (x + i0)|2

(7.3.12)

then for a.e. x, wλ (x) = and

Σac (Aλ ) = Σac (A)

(7.3.13)

Proof. We have that (Aλ − z)−1 = (A0 − z)−1 − (Aλ − z)−1 (Aλ − A0 )(A0 − z)−1 (7.3.14) which leads to Fλ (z) = F0 (z) − λF0 (λ)Fλ (z)

(7.3.15)

which implies (7.3.10). Also, by (7.3.14), if ψ is orthogonal to (Aλ − z)−1 ϕ for all z, then ψ is orthogonal to (A0 − z)−1 ϕ for all z, so ψ = 0. Thus, ϕ is cyclic for Aλ . By (2.3.55), 1 wλ (x) = Im Fλ (x + i0) (7.3.16) π Since (7.3.10) implies Im Fλ (z) =

Im F0 (z) |1 + λF0 (x)|2

(7.3.17)

we get (7.3.12), which implies (7.3.13) given that {x | F0 (x+i0) = −1/λ or F0 (x + i0) = ∞} has Lebesgue measure zero. Corollary 7.3.4. Let A be a bounded selfadjoint operator and F a finite rank selfadjoint operator. Then Σac (A + F ) = Σac (A) Indeed, the multiplicities are equal.

(7.3.18)

480

7. RIGHT LIMITS

Remarks. 1. Traditionally, this is done via scattering theory; see the Notes. 2. By using cyclic sets of vectors when there is not a single cyclic vector (or by taking direct sums), one form of the spectral theorem is that any A is unitarily equivalent to multiplication by x on L2 (R, dµ) where now dµ is a matrix- (or operator-) valued measure. One can still write dµ(x) = W (x) dx + dµs (7.3.19) but now W is an operator. One shows Σ(k) ac (A) = {x | rank(W ) = k}

(7.3.20)

(k) Σ(k) ac (A + F ) = Σac (A)

(7.3.21)

is independent (Lebesgue a.e.) of the representation and the equal multiplicities statement means Proof. By diagonalizing F , we see any F is a sum of selfadjoint rank one operators, so it is sufficient to prove it for the case F = λ(ϕ, ·)ϕ with kϕk = 1. Let H1 = span of {(A − z)−1 ϕ | z ∈ / R}. Then ⊥ H1 and so H1 are invariant for A and A + F and A ↾ H1⊥ = (A + F ) ↾ H1⊥

(7.3.22)

Σac ((A + F ) ↾ H1 )) = Σac (A ↾ H1 )

(7.3.23)

By the theorem,

and are multiplicity one. (7.3.22)/(7.3.23) imply (7.3.18) and (7.3.21). Closely related to the last theorem is Theorem 7.3.5. Let J be a Jacobi matrix and J1 the once-stripped Jacobi matrix (see Theorem 3.2.4). Then Σac (J) = Σac (J1 )

(7.3.24)

Remarks. 1. In a sense, J1 is the “rank one perturbation” with b1 = ∞, so this result is a special or, at least, limiting case of Corollary 7.3.4. See the Notes for a discussion of this infinite coupling theory. 2. This is essentially an OPRL analog of (2.6.15). Proof. By (3.2.28), m1 (z) = (−z + b1 − a21 m1 (z))−1

(7.3.25)

w(z) = w1 (x)|−x + b1 − a1 m1 (x + i0)|−2

(7.3.26)

so, by (2.3.55),


481

so as in the proof of Theorem 7.3.3, up to sets of Lebesgue measure zero, {x | w(x) 6= 0} = {x | w1 (x) 6= 0} (7.3.27)

Now let µ be the measure for J and µ1 for J1 . Just as there is a sup of measures discussed in Lemma 2.16.9, there is an inf η = µ ∧ µ1

(7.3.28)

dη = n(x) dx + dηs

(7.3.29)

with and one has n(x) = min(w(x), w1(x)) and, in particular, by (7.3.27), up to sets of Lebesgue measure zero, {x | n(x) 6= 0} = Σac (J) By (3.2.16), the second kind polynomials obey Z a21 qn (x)2 dµ1 (x) = 1

Thus, by (3.2.19), Z Define

kTn (x)k2 dη(x) ≤ (1 + a2n )(1 + a−2 1 )

n X 1 2 N = x lim inf kTj (x)k < ∞ n j=1

(7.3.30)

(7.3.31)

(7.3.32)

(7.3.33)

Then we have the following variant of Theorem 7.3.2:

Theorem 7.3.6 (Last–Simon [265]). Up to sets of measure zero, N = Σac (J)

(7.3.34)

N ⊂ N ♯ = Σac (J)

(7.3.35)

supp(dη) ⊂ N

(7.3.36)

Σac (J) ⊂ N

(7.3.37)

Proof. Clearly, On the other hand, since sup(an ) < ∞, by (7.3.32) and Fatou’s lemma, so, by (7.3.30),

482

7. RIGHT LIMITS

Recall that the transfer matrix, Tkj (z; J), can be defined as mapping to aukk+1 for solutions of (3.2.6) and for J, a half-line Jacobi uk matrix (k, j ≥ 1) or whole-line matrices. In terms of the transfer matrix Tn we discussed above, uj+1 aj uj

Tn = Tn1

(7.3.38)

Tkj Tjℓ = Tkℓ

(7.3.39)

Tkj = Tk Tj−1

(7.3.40)

and since we have Moreover, since det(Tj ) = 1, kTj−1 k = kTj k

(7.3.41)

kTkj k ≤ kTk k kTj k

(7.3.42)

so, by (7.3.40), Thus, by (7.3.32) and the Schwarz inequality, we get for half-line Jacobi matrices, J, Z kTkj (x; J)k dηJ (x) ≤ sup(1 + a2n ) (1 + a−2 (7.3.43) 1 ) n

Let us denote the right side of (7.3.43) by K(J). Suppose now that (7.3.2) holds. Then as ℓ → ∞, Tk+mℓ j+mℓ (x; J) → Tkj (x; Jr ) Theorem 7.3.7. For any Jr ∈ R, we have Z (i) kTkj (x; Jr )k dηJ (x) ≤ K(J)

1/2 Z X n 1 2 kT±k ±1 (x; Jr )k dηJ (x) ≤ K(J) (ii) n k=1 X 1/2 Z n 1 2 (iii) lim inf kJ±k ±1 (x; Jr )k dηJ (x) ≤ K(J) n k=1

(7.3.44)

(7.3.45) (7.3.46) (7.3.47)

Proof. (i) This follows from (7.3.43), (7.3.44), and Fatou’s lemma. (ii) By (7.3.42), if S is a set with n elements, X 1/2 X 1/2 1 1 2 2 kTkj k ≤ kTj k kTk k (7.3.48) n n k∈S

k∈S


so, by the Schwarz inequality and (7.3.32), 1/2 Z X 1 2 kTkj k dηJ ≤ K(J) n k∈S

483

(7.3.49)

By Fatou’s lemma and (7.3.44), this leads to (7.3.46). (iii) This follows from (7.3.46) and Fatou’s lemma.

Proof of Theorem 7.3.1. By (7.3.46), we see that if (Jr )± are the half-line Jacobi matrices obtained from Jr (i.e., (Jr )± have Jacobi (n) (n) (n) (n) ∞ parameters (an , bn )∞ n=1 and (a−n , b1−n )n=1 ), then 1/2 Z n 1X ± 2 kTk (x; Jr )k dηJ (x) < ∞ (7.3.50) lim inf n k=1 By (7.3.30), we see a.e. on Σac (J), we have that the lim inf is finite so, by Theorem 7.3.6, Σac (Jr± ) ⊂ Σac (J) (7.3.51)

Since Jr and Jr+ ⊕ Jr− differ by a rank two operator (by replacing a0 by 0), by Corollary 7.3.4, (7.3.51) implies (7.3.1) with the multiplicity two statement.

Remarks and Historical Notes. As indicated, Theorems 7.3.1 and 7.3.6 are from [265] and the use of Fatou’s lemma to get (7.3.5) is there. But the other direction, (7.3.7), is obtained there through the use of subordinacy theory; the idea we use here to exploit the Máté– Nevai variational principle seems to be new. The subordinacy theory yields more, namely, µs (N) = 0 [265]. (Note: µs (N ♯ ) 6= 0; indeed, µs (R \ N ♯ ) = 0.) The spectral theory of rank one perturbations goes back to Aronszajn and Donoghue [27, 108]. Implicit in their work is the invariance of a.c. spectrum. For further discussion, see Simon [381, Ch. 11 and 12]. Apparently without realizing the relevance of this work (even Kato’s 1976 book [210] makes no mention of this work of Aronszajn and Donoghue!), invariance of the a.c. spectrum under finite rank perturbations was obtained by Kato [209] using scattering theory methods at about the same time as their work. The scattering approach also works for trace class perturbations; see Reed–Simon [355]. As mentioned, Theorem 7.3.5 can also be obtained using rank one perturbations at infinite coupling; see Gesztesy–Simon [160].

484

7. RIGHT LIMITS

7.4. Remling’s Theorem on A.C. Spectrum We saw in the last section that Jr ’s in R(J) have a.c. spectrum of multiplicity two on Σac (J). In this section, we will prove they are actually reflectionless there, so we will have to begin with a definition of reflectionless. In Section 5.4, we proved a property called reflectionless for whole-line periodic Jacobi matrices. We proved Theorem 5.4.17 which had a number of conditions that turn out to hold for a.e. λ0 . We explored this further in Section 5.13. The natural notion is reflectionless on a set—indeed, on a Lebesgue measure class, that is, an equivalence class of Borel sets under the equivalence relation defined by A ≡ B if and only if |A△B| = 0. Theorem 7.4.1. Let Σ be a measure class and J a two-sided Jacobi matrix with bounded Jacobi parameters. The following are equivalent: (i) For a.e. λ ∈ Σ and all n ∈ R, Re Gnn (λ + i0) = 0

(ii) For a.e. λ ∈ Σ and three successive n’s, (7.4.1) holds. (iii) For a.e. λ ∈ Σ and all n, a2n m(λ + i0, Jn+ ) = m(λ + i0, Jn− )−1

(7.4.1)

(7.4.2)

(iv) For a.e. λ ∈ Σ and one n, (7.4.2) holds,

a2n m(λ + i0, Jn+ ) = m(λ + i0, Jn− )−1

(v) If u± n (λ, J) are the Weyl solutions for λ ∈ C+ normalized by u± 0 (λ, J) = 1

then for a.e. λ ∈ Σ and all n,

+ u− n (λ + i0, J) = un (λ + i0, J)

(7.4.3)

(7.4.4)

Proof. We recall some basic formulae we will need (see Theorem 5.4.12 and its proof): Jℓ+ has Jacobi parameters {an+ℓ , bn+ℓ }∞ n=0 Jℓ−

has Jacobi parameters

{aℓ−n , bℓ+1−n }∞ n=1

u+ ℓ+1 (λ) aℓ u+ ℓ (λ) − u (λ) m(λ, Jℓ− ) = − ℓ− aℓ uℓ+1(λ)

m(λ, Jℓ+ ) = −

Gℓℓ (λ) =

− u+ ℓ (λ)uℓ (λ) − − + aℓ (u+ ℓ+1 (λ)uℓ (λ) − uℓ+1 (λ)uℓ (λ))

(7.4.5) (7.4.6) (7.4.7) (7.4.8) (7.4.9)

7.4. REMLING’S THEOREM ON A.C. SPECTRUM

485

1 (7.4.10) − m(λ, Jℓ− )−1 initially for λ ∈ C+ , then by taking boundary values for a.e. λ ∈ R + i0. We will prove that (v) ⇒ (iii) ⇒ (iv) ⇒ (v) and (iii) ⇒ (i) ⇒ (ii) ⇒ (v). (v) ⇒ (iii). (7.4.3), (7.4.7), and (7.4.8) imply (7.4.2). =−

a2ℓ m(λ, Jℓ+ )

(iii) ⇒ (iv) is trivial.

(iv) ⇒ (v). By translation invariance, we can suppose n = 0, in which case by (7.4.1), we see (7.4.3), (7.4.7), and (7.4.8) imply (7.4.4). Here we use the fact that the difference equation is second-order, so it suffices for (7.4.4) to hold at n = 0, 1. (iii) ⇒ (i) is immediate from (7.4.10). (i) ⇒ (ii) is trivial.

(ii) ⇒ (v). By translation invariance, we can suppose that the three successive values are n = 0, ±1. Since boundary values of mfunctions cannot vanish on sets of positive Lebesgue measure (see Theorem 2.3.21), for a.e. λ ∈ Σ, (a) and (d) of Theorem 5.4.17 hold. Similarly, since w(λ) 6= 0 for λ ∈ C+ , for a.e. λ ∈ Σ, w(λ + i0) 6= 0, so (b) of that theorem holds for a.e. λ. By hypothesis, (c) holds, so by that theorem, (7.4.4) holds. Definition. Let J be a whole-line Jacobi matrix. If any and hence all of (i)–(v) hold on Σ, we say J is reflectionless on Σ. If Σ ⊂ e ⊂ R with |Σ| > 0 and e compact, we let R(Σ, e) denote those whole-line Jacobi matrices which are reflectionless on Σ and have σ(J) ⊂ e. Theorem 7.4.2. If J ∈ R(Σ, e), then Σ ⊂ Σac (J). Indeed, J has multiplicity 2 on Σ. Proof. Let m± (z) = m(z, J0± ) and let G(z) = G00 (z). By hypothesis, Re G00 (λ+i0) for λ ∈ Σ. Since |{λ | G00 (λ+i0) = 0}| = 0, for a.e. λ ∈ Σ, we have Im G00 (λ + i0))neq0. Thus, for a.e. λ ∈ Σ, by (7.4.10), Im a20 m+ (λ + i0) − m− (λ + i0)−1 ) > 0

(7.4.11)

Im a20 m+ (λ + i0) = Im −m− (λ + i0)−1

(7.4.12)

But, by (7.4.2) on Σ, so a.e. on Σ,

J0+

Thus, on Σ, ⊕ lary 7.3.4, J has also.

J0−

Im m± (λ + i0) > 0 (7.4.13) has a.c. spectral multiplicity 2. By Corol

486

7. RIGHT LIMITS

In many cases, µsing (Σ) = 0, but not for all possible Σ; see Theorem 7.4.8 below and the Notes. We can now state Remling’s theorem whose proof we postpone to the end of the section: Theorem 7.4.3 (Remling’s Theorem [357]). Let Jr ∈ R(J) be a right limit of a half-line Jacobi matrix, J. Then Jr is reflectionless on Σac (J). To apply this theorem, we need to know about reflectionless operators. The following is critical: Theorem 7.4.4 (Kotani [238]). Let Σ ⊂ e ⊂ R with e compact and Σ a Borel set with |Σ| > 0. Put the product topology on {{an , bn }0n=−∞ }. Then the set, L(Σ, e), of such Jacobi parameters obtained by restriction from R(Σ, e) is compact, and there is a continuous function F : L(Σ, e) → (0, ∞) × R so that for any {an , bn }∞ n=−∞ ∈ R(Σ, e), we have (a1 , b1 ) = F ({an , bn }0n=−∞ ) (7.4.14) Remark. By {an , bn }∞ n=1 .

iteration,

{an , bn }0n=−∞

then

determine

all

Proof. By (3.2.28), − (−m(z, J0− ))−1 = z − b0 − a2−1 m(z, J−1 )

(7.4.15)

so by the continuity of the map from half-line J’s to m ↾ Σ (in the weak topology discussed in (2.3.73)), we see that − −1 2 + {an , bn }∞ n=−∞ 7→ a0 m(λ + i0, J0 ) − m(λ + i0, J0 )

is continuous. Thus, the set on which it is 0 is compact. On this set, −m(z, J0− )−1 is a continuous function of {an , bn }−1 n=−∞ ∪{b0 } by (7.4.15). + 2 So by (7.4.2) on R(Σ, e), a0 m(λ + i0, J0 ) ↾ Σ is a continuous function of {an , bn }0n=−∞ . Thus, by (3.2.31), F is continuous as claimed. Theorem 7.4.5 (Kotani [239]). Let F be a finite subset of (0, ∞)× R. Let Σ, e be given with |Σ| > 0 and Σ ⊂ e. Then there exists a p so that every {an , bn }∞ n=−∞ ∈ R(Σ, e) with (an , bn ) ∈ F for all n has period p. Remark. We are only claiming p is a period for (an , bn ), that is, an+p = an not that p is the minimal period.

bn+p = bn

(7.4.16)


487

Proof. Let R(Σ, e, F ) be the set of {an , bn }∞ n=−∞ ∈ R(Σ, e) with every (an , bn ) ∈ F . Pick ε > 0 so that for all (α, β) 6= (α′ , β ′ ) both in F , |α − α′ | + |β − β ′ | ≥ ε

(7.4.17)

Then, with F given by (7.4.14), for each (α0 , β0 ) ∈ F ,

F −1 ({(a′ , b′ )} | |a′ − α0 | + |b′ − β0 | < 2ε )

is open and so depends only on finitely many {aj , bj }j≤0. Since F is finite, we see that there are k and Fe defined on {{an , bn }0n=−k+1} to (0, ∞) × R, so on R(Σ, e, F ), Fe gives the value of (a1 , b1 ). Thus, by iteration, the k block (−k + 1, 0) determines (1, k), that is, there is a map H from allowed values in F k to itself. In the same way, there is a map from (1, k ′ ) to (−k ′ + 1, 0) for some ′ k . So by increasing k (if k ′ > k), we can suppose that H is invertible. Since F k is finite, for any allowed value α ∈ F k , there must be a repeated entry among α, H(α), H 2(α), . . . . But if H k (α) = H k+q (α), then by invertibility, H q (α) = α, that is, the corresponding J has period kq. Since F k is finite, there is maximal period, r, for all the J’s in R(Σ, e, F ). Then p = r! is a common period of all such J’s. Theorem 7.4.6 (Remling [357]). Let F be a finite subset of (0, ∞) × R. Let J be a half-line Jacobi matrix with each (an , bn ) ∈ F . Suppose J has some a.c. spectrum, that is, |Σac (J)| > 0. Then J is eventually periodic, that is, for some p and N, we have for n ≥ N, an+p = an

bn+p = bn

Remarks. 1. Eventually periodic matrices are finite rank perturbations of strictly periodic J’s, so by Theorem 5.3.7 and Corollary 7.3.4, they have some a.c. spectrum. 2. This shows, for example, that if an ≡ 1 and bn = 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, . . . , then J has purely singular continuous spectrum. Proof. Put some metric on F Z , say ∞ X ′ ′ d({a, b}, {a , b }) = 2−n (|an − a′n | + |bn − b′n |)

(7.4.18)

n=−∞

Extend J to Z by setting (an , bn ) to some fixed point in F for n ≤ 0. Let J (m) have parameters {an+m , bn+m }∞ n=−∞ . I claim that min d({a(m) , b(m) }, {a(r) , b(r) }) → 0

Jr ∈R(J)

(7.4.19)

488

7. RIGHT LIMITS

as m → ∞. For if not, there is mj → ∞ and ε so for all Jr , d({a(mj ) , b(mj ) }, {a(r) , b(r) }) ≥ ε

(7.4.20)

But, by compactness, there is a subsequence so J (mjk ) converges to some Jr . Thus, (7.4.20) fails and (7.4.19) must hold. By Remling’s theorem, each Jr is reflectionless on Σac (J). So, by Theorem 7.4.1, each Jr is periodic with period p. Pick ε so small that d((a, b), (a′ , b′ )) < ε ⇒ sup (|ak − a′k | + |bk − b′k |) 0≤k≤2p (7.4.21) ′ ′ < min |α − α | + |β − β | ′ ′ (α,β)6=(α ,β )∈F

Pick M so m > M ⇒ minJr ∈R(J) d(J (m) , Jr ) < ε. Thus, if m > M, J from m → m + 2p must be equal a unique element of Jr . But from m+ p to m+ 3p, the same is true. On the overlap from m+ p to m+ 2p, they must agree. So, by periodicity of Jr , J must agree with a single Jr from M onward. As a second application, here is a very strong sparse potential theorem: Theorem 7.4.7 (Remling [357]). Let J be a bounded half-line Jacobi matrix so there exist M fixed and mj → ∞ with (a) lim inf j→∞ (|amj − 1| + |bmj |) > 0 (b) M ≤ |ℓ| ≤ j ⇒ |amj +ℓ − 1| + |bmj +ℓ | = 0 Then J has no a.c. spectrum. Proof. By passing to a subsequence of mj , we find Jr ∈ R(J) with (r) (r) (i) |a0 − 1| + |b0 | = 6 0 (r) (r) (ii) aℓ = 1 and bℓ = 0 for |ℓ| ≥ M By (ii), Jr takes only finitely many values. By Theorem 7.4.5, if there is |Σ| > 0 on which Jr is reflectionless, then Jr is periodic. So by (ii), (r) (r) aℓ = 1, bℓ = 0 for all ℓ, but that violates (i). We conclude that Jr is not reflectionless on any Σ with |Σ| > 0. By Remling’s theorem, |Σac (J)| = 0, that is, J has no a.c. spectrum. Clearly, reflectionless Jacobi matrices are important, so one defines a reflectionless measure, Σ ⊂ R, as a measure, µ, on R so that Z dµ(x) (7.4.22) Fµ (z) ≡ x−z has Re Fµ (x + i0) = 0 for a.e. x ∈ Σ (7.4.23)


489

As we have seen, if (here dµs is dx-singular) dµ = w(x) dx + dµs

(7.4.24)

then w(x) > 0 for a.e. x ∈ Σ. We want to explore when dµs (Σ) = 0 which often holds. For finite gap sets, we will see this in Section 7.5, but there is a very general result: Theorem 7.4.8 (Poltoratski–Remling [343]). Let µ be a reflectionless measure. Let Σ0 = x ∈ Σ | lim sup(2δ)−1 |Σ ∩ (x − δ, x + δ)| > 0 (7.4.25) δ↓0

Then

µs (Σ0 ) = 0

(7.4.26)

Note that in many cases Σ0 = Σ. For example, if Σ is a finite gap set, the lim sup in (7.4.25) is 12 at endpoints and 1 at interior points. There is a class of sets called homogeneous sets for which Σ0 = Σ. See the Notes for further discussion. The proof of the result depends on the following (whose proof is in the references in the Notes): Theorem 7.4.9 (Poltoratski’s Theorem [342]). For any measure of compact support µ and any f ≥ 0 with f ∈ L1 (R, dµ), lim ε↓0

Ff µ (x + iε) = f (x) Fµ (x)

(7.4.27)

for µs a.e. x. Corollary 7.4.10. (i) If ν is mutually singular to µ, then for µs a.e. x, Fν (x + iε) lim =0 (7.4.28) ε↓0 Fµ (x + iε) (ii) For any measure ν and µs a.e. x, lim ε↓0

Fν (x + iε) 0 and (7.4.46) proves X(z) =

A(z) ≥

Similarly, A(u + i0) ≤ 1 implies A(z) +

1 2

1 2

Im X(z)

Im X(z) ≤ 1

(7.4.37) (7.4.38)


491

Thus, π Im X(z) ≤ π 2 π ± F (z) = Fµ (z) exp ± X(z) 2 0 ≤ πA(z) ±

so

(7.4.39)

are both Herglotz functions. Since X(z) → 0 at ∞,

1 1 F (z) = − + o z z ±

(7.4.40)

at ∞ and F is real outside supp(dµ) since X is analytic and Im X = 0 ¯ Thus, F is a discrete m-function. outside Σ. By Corollary 7.4.10, for µs a.e. x, we have lim ε↓0

F ± (x + iε) ≡ f ± (x) Fµ (x + iε)

(7.4.41)

exists and is finite. By (7.4.39), F + (z)F − (z) ≡1 Fµ (z)2

(7.4.42)

Thus, for µs a.e. x, f + (x)f −1 (x) = 1, so for µs a.e. x, f ± (x) is strictly positive. It follows that for µs a.e. x, lim ε↓0

F + (x + iε) = (0, ∞) F − (x + iε)

(7.4.43)

so for µs a.e. x, lim Im X(x + i0) = 0 ε↓0

Since Im X(x + iε) ≥ ε ≥

(7.4.44)

Z

χΣ dµ 2 2 |x−u|≤ε (x − u) + ε

1 |(x − ε, x + ε) ∩ Σ| 2ε

(for |x − u| ≤ ε ⇒ [(x − u)2 + ε2 ]−1 ≥ (2ε2 )−1 ), we see for µs a.e. x, lim δ −1 |(x − δ, x + δ) ∩ Σ| = 0 so µs (Σ0 ) = 0.

492

7. RIGHT LIMITS

Finally, we turn to the proof of Remling’s theorem. TK

x-ref?

Remarks and Historical Notes. The main results of this section are from Remling [357]. The proof of his main theorem (Theorem 7.4.3) relies on ideas of Pearson [327, 328], especially two papers of Breimesser–Pearson [55, 56]). However, these earlier papers did not realize the link to reflectionless potentials nor the deep applications on finite-valued potentials and on the extensions of the Denisov– Rakhmanov theorem. An important precursor which provided guideposts for the applications is the work of Kotani [238], who considered Schrödinger operators. Simon extended this work to Jacobi matrices with an ≡ 1 [382] and to OPUC [391, Sect. 10.11]. Kotani (or its Jacobi analog) considered stochastic Jacobi matrices which involve certain families depending on a parameter, ω, in a probability measure space where many properties are a.e. constant. In this case, Kotani proved that a.e. Jω was reflectionless on its a.c. spectrum. In [239], he proved Theorems 7.4.4 and 7.4.5. Sparse potentials go back to a basic paper of Pearson [326]; see the notes to [391, Sect. 12.5] for further references. Theorem 7.4.5 depended on two ideas: (a) Because of the form of open sets in product spaces, the values of the function F of (7.4.14) are determined up to ε by {an , bn }0−k+1 for some k. (b) By a compactness argument, for any ε > 0, for n large, J (n) is within ε of R(J) for any fixed metric on the product space of Jacobi parameters. Finiteness entered in Theorem 7.4.6 only because then ε bounds imply equality so long as ε obeys (7.4.20). Thus, without finiteness, one gets the following result which Remling calls the Oracle theorem: Theorem 7.4.11 (Remling [357]). Let Σ ⊂ e ⊂ R with |Σ| > 0 and e compact. Then for any ε > 0, there are k and function F : [(0, ∞) × R]k → (0, ∞) × R, so that for any half-line Jacobi matrix, J, with Σ ⊂ Σac (J) ⊂ σ(J) ⊂ e

(7.4.45)

we have for n ≥ N, which can be J-dependent, that |(an , bn ) − F ({aj , bj }n−1 j=n−k )| < ε

(7.4.46)

Theorem 7.4.9 is due to Poltoratski [342]. For a simple proof using rank one perturbation theory (starting with Theorem 7.3.3), see Jakˇsić– Last [203]. Theorem 7.4.9 does not require compact support but only

7.5. PURELY REFLECTIONLESS JACOBI MATRICES

493

R that Fµ exists, for example, if (x2 + 1)−1 dµ(x) < ∞. It also holds for signed measures and general f ∈ L1 (R, dµ). Theorem 7.4.8 is due to Poltoratski–Remling [343]. For cases where Σ is a homogeneous set, (i.e. ∃ε, δ0 > 0, so for all δ < δ0 and all x ∈ Σ, |(x − δ, x + δ) ∩ Σ| ≥ 2εδ) and supp(dµ) = Σ, Sodin–Yuditskii [404] found all dµ’s reflectionless on Σ (as isospectral torus) and found that µs = 0. Earlier extensions where supp(dµ) are bigger than Σ are found in Gesztesy–Zinchenko [?]. It is easy to construct reflectionless measures on {x0 } ∪ [−2, 2] for any x0 ∈ / [−2, 2], so if Σ has isolated points, µs can be nonzero. Much more subtle are examples where µ is reflectionless on Σ but µ has pure points or even singular continuous components on Σ (obviously, such examples must have Σ0 6= Σ). These are constructed in Nazarov– Volberg–Yuditskii [309]. For consideration of Jacobi matrices, one definition is adequate, but for continuum Schrödinger operators, one wants to allow measures which are not of compact support and not finite weight, but only obey (2.3.86). One defines p to be reflectionless on Σ if for some choice of A and Re G(i), Re G(x + i0) = 0 on Σ for G given by (2.3.87). [343] prove Theorem 7.4.8 in this context. 7.5. Purely Reflectionless Jacobi Matrices on Finite Gap Sets Let e ⊂ R be a finite gap set. In Section 5.13, we defined the isospectral torus, Te, of whole-line Jacobi matrices and proved that if J ∈ Te, then J is reflectionless on e and σ(J) = e

(7.5.1)

We also showed that J had purely a.c. spectrum of multiplicity two. Here we will prove a converse: Theorem 7.5.1 (Sodin–Yuditskii [404]). Let e ⊂ R be a finite gap set. Let J be a whole-line operator which obeys (7.5.1) and which is reflectionless on e. Then J ∈ Te.

Proof. Let e be given by (5.5.127) and (5.5.128). By Craig’s theorem (Theorem 5.4.18), there exist x1 , . . . , xℓ with xj ∈ [βj , αj+1 ] so Qℓ j=1 (z − xj ) (7.5.2) G00 (z) = − Qℓ+1 [ j=1 (z − αj )(z − βj )]1/2

By this explicit formula, (−G00 (z))−1 has real boundary values on R \ e∪{xj }ℓj=1 , is bounded on eint , and has poles exactly at those xj ’s which are in the interior of gaps.

494

7. RIGHT LIMITS

By (5.4.50), (−G00 (z))

−1

= z − bn +

where

Z

d(µ+ + µ− )(x) x−z

Z

dµ+ (x) x−z Z dµ− (x) − a2−1 m(z, J−1 )= x−z a20 m(z, J0+ )

=

(7.5.3)

(7.5.4) (7.5.5)

By the properties of (−G00 (z0 ))−1 above, we see (µ+ + µ− ) has pure points at xj ’s in the gaps and otherwise is purely a.c. and supported on e. If xj is a pole of m(z, J0+ ), there is a solution of Ju = xj u which is − ℓ2 at +∞ with u(0) = 0. Similarly, if xj is a pole of m(z, J−1 ), there 2 is a solution ℓ at −∞ with u(0) = 0. If xj were a pole of both, then Ju = xj u would have an ℓ2 solution. But then σ(J) would not be just e. Thus, each xj in the interior of the gap is a pole of either m(z, J0+ ) or of m(z, J0− )−1 , but not both. By the reflectionless hypothesis, we can take m(z, J0+ ) on S+ ∩ C+ − −1 and a−2 on S− ∩ C+ and extend to S+ and S− and sew 0 m(z, J0 ) together to a single meromorphic function on S. By the above, there is exactly one pole in each two-sheeted gap (if xj is at βj or αj+1 , there is a square root divergence which is a pole the way poles at branch points on S are counted). Thus, m has ℓ + 1 poles and so is a minimal Herglotz function on S. The analysis of Theorem 5.13.11 shows that J is the corresponding point in the isospectral torus. Remark. While we used the theory of minimal Herglotz functions in the above, the fact that J is determined by the xj ’s plus a left/right choice can be seen directly from reflectionless J’s. For the reflectionless condition implies Im a20 m(x + i0, J0+ ) =

1 2

Im −G00 (x + i0)−1

(7.5.6)

This plus the choice of poles and residues (which come from residues of −G00 (z)−1 ) determine µ+ , and so a20 m(x + i0, J0+ ), and thus J0+ and a0 . Similarly, m(z, J0− )−1 determine J0−1 . Remarks and Historical Notes. That reflectionless J’s in the finite gap case are the isospectral torus (i.e., Theorem 7.5.1) goes back to Sodin–Yuditskii [404]; see also [159, 164, 425]. Our proof follows Remling [357].

7.6. THE DENISOV–RAKHMANOV–REMLING THEOREM

495

7.6. The Denisov–Rakhmanov–Remling Theorem Given the last two sections, we immediately have the following beautiful result: Theorem 7.6.1 (Denisov–Rakhmanov–Remling Theorem; Remling [357]). Let e be a finite gap set. Let J be a half-line Jacobi matrix with σess (J) = Σac (J) = e (7.6.1) Then, with Te the isospectral torus, R(J) ⊂ Te

(7.6.2)

Remarks. 1. This is even interesting in case e = [−2, 2] (due to Denisov [103]), in which case the conclusion is an → 1, bn → 0 since Te is a single point. 2. In colloquial language, J approaches Te at infinity. 3. The periodic case shows that it can happen that R(J) is only a subset of Te. Proof. Let Jr ∈ R(J). By Theorem 7.2.1, σess (J) = e implies σ(Jr ) ⊂ e

(7.6.3)

Σac (Jr ) ⊃ e

(7.6.4)

σ(Jr ) = Σac (Jr ) = e

(7.6.5)

By Theorem 7.3.1 and Σac (J) = e,

and, by Theorem 7.4.3, Jr is reflectionless. Thus, and Jr is reflectionless. By Theorem 7.5.1, Jr ∈ Te

(7.6.6)

Remarks and Historical Notes. Rakhmanov [350, 351] proved that, for OPUC, if Σac (C) = ∂D, then αn → 0. For alternate proofs and the involved history ([350] had an error!), see [391, Ch. 9]. Denisov [103] then proved that, for OPRL, if σess (J) = Σac (J) = [−2, 2], then an → 1, bn → 0. Earlier, Bello–López [36], using ideas from López [280], had shown that if Σa is the σess for the CMV matrix associated to αn ≡ a > 0 and if σ(C) = Σac (C) = Σa , then {αn } is in the L´ opez class αn+1 |αn | → a →1 (7.6.7) αn

496

7. RIGHT LIMITS

In [391], Simon realized that (7.6.7) was equivalent to saying that αn approached the isospectral torus for Σa and he conjectured the result for the general periodic case. Damanik–Killip–Simon [93] proved this periodic conjecture using the magic formula machinery we discuss in Chapter 8, and they conjectured the result for general finite gap e. Their conjecture was then proven by Remling [357]. For more on the history of results on approach to the isospectral torus, see the Notes to Section 8.1.

CHAPTER 8

Szeg˝ o and Killip–Simon Theorems for Periodic OPRL In this chapter, we turn to a synthesis of the theory of periodic Jacobi matrices studied in Chapters 5 and 6 with the perturbation theory of Chapters 3 and 4. 8.1. Overview We have looked at four results on perturbations of the Jacobi matrix, J0 , with an ≡ 1, bn ≡ 0: (i) Weyl-type results that an → 1, bn → 0 ⇒ σess (J) = [−2, 2] (ii) Denisov–Rakhmanov-type results that σess (J) = Σac (J) = [−2, 2] implies an → 1, bn → 0 (iii) Szeg˝o–Shohat–Nevai-type results relating a Szeg˝o condition plus P 1 eigenvalue bounds to boundedness of N n=1 log(an ) 2 (iv) Killip–Simon-type results relating a pseudo-Szeg˝o condition plus 3 eigenvalue bounds to ℓ2 conditions of the form 2 X (an − 1)2 + b2n < ∞ (8.1.1) n

In this chapter, we want to focus on perturbation results of this type for general periodic J0 . A key initial question is what replaces the limit point an ≡ 1, bn ≡ 0, for which it is not hard to see that J0 alone is not enough. The answer, given what we have seen, especially since we addressed (i) and (ii) in Chapter 7, should be obvious: The single point J0 for the case an ≡ 1, bn ≡ 0 needs to be replaced instead by the isospectral torus. For a history that led to this realization, see the Notes. We addressed (i) in Section 7.4 and (ii) in Section 7.6. Question (iii) will be addressed in Section 8.4 and question (iv) in Section 8.6. Remarkably, we will be able to do this by reducing things to an MOPRL involving perturbations of An ≡ 1, Bn ≡ 0. The key will be to form the discriminant, ∆J0 , of J0 , and given a Jacobi matrix, J, to look at ∆J0 (J). The key will be to show that ∆J0 (J), which is a p × p block Jacobi matrix, has An → 1, Bn → 0 if and only if J approaches the 497

498

˝ AND KILLIP–SIMON THEOREMS FOR PERIODIC OPRL 8. SZEGO

isospectral torus. Indeed, we will prove in Section 8.2 that for a wholeline bounded Jacobi matrix (with (Su)n = un−1 and e = σess (J0 )) that ∆J0 (J) = S p + S −p ⇔ J ∈ Te

(8.1.2)

something that has been dubbed the magic formula. Section 8.3 discusses a technical issue relating the spectral measure for ∆J0 (J) and for J. Section 8.5 relates a Hilbert–Schmidt condition on ∆J0 (J)−(S p +S −p ) to ℓ2 convergence of J to Te. Section 8.7 discusses the OPUC case, which turns out to also be related to MOPRL, not MOPUC. Remarks and Historical Notes. The first perturbations of periodic problems where the proper set of limits was found was the OPUC case where αn ≡ a, a constant with a 6= 0. López, in a series of papers, some with collaborators [31, 36, 37, 279], with important followup by Khrushchev [214, 216] focused on the López class, {αn }∞ n=0 , so that for some a > 0, αn+1 |αn | → a →1 (8.1.3) αn Simon [391] realized that (8.1.3) is equivalent to αn approaching (0) the isospectral torus of αn ≡ a (which is αn ’s with αn = aeiθ for a fixed θ) and conjectured OPUC analogs of all four results. These were then proven (for (i)) by Last–Simon [266] and by Damanik–Killip–Simon [93]. Most of this chapter will follow [93]. 8.2. The Magic Formula We define S : ℓ2 (Z) → ℓ2 (Z) by

(Su)k = uk−1

(8.2.1)

As we have discussed, the key to understanding perturbations of periodic Jacobi matrices is a characterization of the isospectral torus: Theorem 8.2.1 (The Magic Formula [93]). Let J be a bounded twosided Jacobi matrix. Let ∆ be the discriminant of a period-p periodic Jacobi matrix J0 with e = σess (J0 ). Then J ∈ Te ⇔ ∆(J) = S p + S −p

(8.2.2)

Remarks. 1. ∆ is a polynomial so, by ∆(J), we mean the operator obtained by replacing the variable in ∆(z) by the operator J. 2. We emphasize that J is not assumed a priori to be periodic. 3. As we have discussed (see the end of Section 5.13), Te can be viewed either as a class of one-sided matrices (using minimal Herglotz

8.2. THE MAGIC FORMULA

499

functions) or as two-sided reflectionless operators. Here, obviously, we have two-sided in mind. We will first prove that ∆J0 (J0 ) = S p + S −p

(8.2.3)

which is a large part of the ⇒ half of (8.2.2). Proposition 8.2.2. (8.2.3) holds. Proof. Both sides are periodic of period p, that is, commute with S p so, as in the discussion in Section 5.3, we can use the Fourier transform, F , of (5.3.18)/(5.3.19) to “diagonalize” them as matrices on Cp . One sees directly that (F (S p + S −p )F −1 f )n (θ) = 2 cos(θ)fn (θ)

(8.2.4)

On the other hand, by (5.3.24), (F ∆J0 (J0 )F −1 f )n (θ) = [∆J0 (J0 (θ))f ]n (θ) which, by (5.4.9), is the right side of (8.2.4).

(8.2.5)

Lemma 8.2.3. Let J0 , J be two period p Jacobi matrices. The following are equivalent: (i)

σess (J) = σess (J0 )

(8.2.6)

(ii)

∆J = ∆J0

(8.2.7)

(iii)

J ∈ Te where e = σess (J0 )

(8.2.8)

Remark. The essential spectra are the same for half- and wholeline periodic J’s, so it does not matter in (8.2.6) which we mean! Proof. (i) ⇒ (ii). Let e = σess (J0 ). Then ∆J0 (z) = κz p + . . . where κ = C(e)−1 , the inverse of the logarithmic capacity of e. Moreover, the potential theorist’s Green’s function, Ge(z), is related to ∆ by (5.4.26) on account of Theorems 5.4.9 and 5.5.17. Put differently, ∆J0 (z) = exp(Ge(z)) + exp(−Ge(z))

(8.2.9)

Thus, e determines ∆J0 and thus, (8.2.6) implies (8.2.7). (ii) ⇒ (i). Immediate from (see Section 5.4) σess (J) = ∆−1 ([−2, 2])

(8.2.10)

(i), (ii) ⇔ (iii). This depends on the definition of isospectral torus used, but all wind up being those periodic J’s with σess (J) = e, so (i) ⇔ (iii).


500

Let Q be an operator on ℓ2 (Z). We say Q has finite width if there is a k so that supp(u) ⊂ [n, m] ⇒ supp(Qu) ⊂ [n − k, m + k]

(8.2.11)

equivalently, if the matrix, Qmn , of Q has

Qmn = 0 if |m − n| ≥ k + 1

(8.2.12)

Dmn = dm δmn

(8.2.13)

A diagonal matrix has the form

Q has finite width with k if and only if there are diagonal matrices {D (j) }kj=−k so that k X Q= D (j) S j (8.2.14) j=−k

Lemma 8.2.4 (Na˘ıman’s Lemma). Let Q be a bounded operator on ℓ of finite width so that for some p, 2

[Q, S p + S −p ] = 0

(8.2.15)

[Q, S p ] = 0

(8.2.16)

Then Remark. For Q of the form (8.2.14), (8.2.16) is equivalent to each (j) (j) (j) D having the form (8.2.13) with dm periodic, that is, dm+p = dm . (j)

Proof. Write Q=

k X

D (j) S j

(8.2.17)

j=−ℓ

with D (k) 6= 0 6= D (−ℓ) . (Note: −ℓ ≤ k, but k or ℓ is allowed to be negative.) Looking at matrix elements of [Q, S p + S −p ]mn with n = m + k + p shows that (k)

(k) Dm m = Dm+p m+p

(8.2.18)

so [D (k) , S p ] = 0. Thus, [D (k) , S −p ] = 0 also and [Q − D (k) S k , S p + S −p ] = 0

(8.2.19)

[D (j) , S p ] = 0

(8.2.20)

By induction, one sees that each D (j) obeys which implies (8.2.16).

Lemma 8.2.5. Let P be a polynomial and J a Jacobi matrix. Suppose P (J) = 0. Then P is the zero polynomial.

8.2. THE MAGIC FORMULA

501

Proof. If P is not the zero polynomial, then P (z) = b0 z ℓ + b1 z ℓ−1 + · · · + bℓ

(8.2.21)

for some b0 6= 0. But then P (J)1 ℓ+1 = b0 a1 a2 . . . aℓ 6= 0 a contradiction to P (J) = 0.

(8.2.22)

Proof of Theorem 8.2.1. If J ∈ Te, then by (8.2.7), ∆J0 (J) = ∆J (J) = S p + S −p by (8.2.3). Conversely, suppose J is any two-sided Jacobi matrix and ∆J0 (J) = S p + S −p Since [J, ∆J0 (J)] = 0

(8.2.23)

[J, S p + S −p ] = 0

(8.2.24)

[J, S p ] = 0

(8.2.25)

we have By Na˘ıman’s lemma, that is, J is periodic with period p. By (8.2.3), ∆J (J) = S p + S −p

(8.2.26)

P (z) = ∆J (z) − ∆J0 (z)

(8.2.27)

P (J) = 0

(8.2.28)

so if we have By Lemma 8.2.5, P ≡ 0, that is, ∆J = ∆J0 . By Lemma 8.2.3, J ∈ TJ0 . Remarks and Historical Notes. Theorem 8.2.1 is due to Damanik– Killip–Simon [93]. Na˘ıman’s lemma is from Na˘ıman [306], who had other ideas approaching the magic formula. The proof we give here using Na˘ıman’s lemma follows a suggestion of L. Golinskii.

502


8.3. The Determinant of the Matrix Weight The strategy is now clear. Take a half-line Jacobi matrix, J, which is “near” the isospectral torus at infinity. Then we expect ∆(J) to be near S p + S −p at infinity. If we use sum rules for ∆(J), we can hope to get effective sum rules for J. Specifically, let dρJ be the spectral measure for J so dρJ (x) = ω(x) dx + dρJ,s (x)

(8.3.1)

On the other hand, ∆(J) is a p × p block Jacobi matrix and the corresponding measure dρ∆(J) is a p × p matrix-valued measure which we can write dρ∆(J) (E) = W (E) dE + dρ∆(J),s (E) (8.3.2) The sum rules that we invoke from Chapter 4 involve det(W (E)). So the question is how this is related to ω(x) for those p values of x solving ∆(x) = E. This is what we want to compute in this section. Here is the main result: Theorem 8.3.1. Let J0 be a period p Jacobi matrix with σess (J0 ) = e (0) (0) and discriminant ∆, so ∆−1 ([−2, 2]) = e. Let {an , bn }∞ n=1 be the Jacobi parameters of J0 . Let J be a half-line Jacobi matrix with σess (J) ⊂ e

(8.3.3)

and dρJ (x) its spectral measure. Then dρJ has the form (8.3.1) with supp(ω) ⊂ e. Let {an , bn }∞ n=1 be the Jacobi parameters for J. Then ∆(J) is a p × p matrix-valued Jacobi matrix with spectral measure dρ∆(J) (E), where σess (∆(J)) ⊂ [−2, 2] (8.3.4) and dρ∆(J) of the form (8.3.2). Given E ∈ (−2, 2), let x1 < · · · < xp be the p solutions of ∆(x) = E (8.3.5) Then Y Y p p 2p−2j −1 (0) p ω(xj ) (8.3.6) det(W (E)) = [aj ] [aj ] j=1

j=1

Remark. e and ∆ only depend on the isospectral torus. The J0 (0) (0) dependence of (8.3.6) is (a1 . . . ap )p = C(e)p is also only e-dependent. Proof. We put ∆(J) into block form by placing δ1 , . . . , δp into block 1, δp+1 , . . . , δ2p into block 2, etc. Thus, Z F (E)(dρ∆ (E))jk = hδj , F (∆(J))δk i (8.3.7)

8.3. THE DETERMINANT OF THE MATRIX WEIGHT

503

The orthogonality of pj (x) in dρJ (x) implies orthogonality of pj (J)δ1 in ℓ2 (Z), so δj = pj−1(J)δ1 (8.3.8) int Taking into account that ∆ is a p to 1 map of e to (−2, 2), we see that p X Wjk (E) = ω(xℓ )(|∆′ (xℓ )|)−1 pk−1(xℓ )pj−1(xℓ ) (8.3.9) ℓ=1

where we use

dE dx = dE dx Thus, if Mkℓ = pk−1 (xℓ )

−1

= |∆(x)|−1 dE

k = 1, . . . , p; ℓ = 1, . . . , p

(8.3.10)

(8.3.11)

and Aℓm = δℓm ω(xℓ )(|∆′ (xℓ )|)−1

(8.3.12)

W = MAM t

(8.3.13)

then and det(W ) = det(M)2 det(A) Y Y p p 2 ′ −1 = det(M) ω(xk ) |∆ (xk )| k=1

(8.3.14) (8.3.15)

k=1

To compute det(M), we note that k−1 Y −1 aj xk−1 + lower order pk−1 (xℓ ) = ℓ

(8.3.16)

j=1

and the lower-order terms can be removed by subtracting rows. Thus, Y p k−1 Y −1 ) (8.3.17) det(M) = aj det(xk−1 ℓ k=1

=

Y p

j=1

ap−j j

j=1

−1 Y (xj − xk )

(8.3.18)

j>k

recognizing det(xk−1 ) as a Vandermonde determinant. ℓ On the other hand, xj solve ∆(x) − E = 0, so −1 Y Y p p (0) aj (x − xk ) ∆(x) − E = j=1

k=1

(8.3.19)


504

Thus, ′

∆ (xj ) =

Y p j=1

and

(0) aj

−1 Y (xj − xk )

2 p p Y Y Y (0) p (xj − xk ) = [aj ] |∆′ (xj )| j=1

j>k

j=1

(8.3.21)

j=1

We conclude that Y −2 Y p p p Y (0) p p−j 2 det(M) = aj [aj ] |∆′ (x′j )| j=1

(8.3.20)

k6=j

(8.3.22)

j=1

(8.3.14) and (8.3.22) imply (8.3.6).

Corollary 8.3.2. Under the hypotheses of Theorem 8.3.1, Z 2 (4 − E 2 )−1/2 log(det(W (E))) dE > −∞ (8.3.23) −2

if and only if Z

σess (J)

dist(x, R \ σess (J))−1/2 log(ω(x)) dx > −∞

(8.3.24)

Proof. By (8.3.6) and a change of variables, (8.3.23) is equivalent to

Z

σess (J)

(4 − ∆(x)2 )−1/2 |∆′ (x)| log(ω(x)) > −∞

(8.3.25)

Near the edges of σess (J), including edges of open gaps, (4 − ∆(x)2 ) ∼ dist(x, R\σess (J))−1/2 and ∆′ (x) is bounded above and away from zero. At interior points other than closed gaps, (4−∆(x)2 )−1/2 and |∆′ (x)| are bounded above and away from from zero. At a closed gap, 4−∆(x)2 has a double zero, so |∆′ (x)| has a simple zero cancelled by the first-order infinity in (4 − ∆(x)2 )−1/2 . Thus, dist(x, R \ σess (J))1/2 (4 − ∆(x)2 )−1/2 |∆′ (x)| is bounded above and away from zero globally on σess (J), and so (8.3.25) is equivalent to (8.3.24). If one looks at (4 − ∆(x)2 )α |∆′ (x)|, one only has the cancellation at closed gaps if α = − 21 , but if all gaps are open, the above argument works on all of σess (J), and we obtain (we care mainly about α = 12 and α = − 12 ):

8.4. A SHOHAT–NEVAI THEOREM FOR PERIODIC JACOBI MATRICES 505

Corollary 8.3.3. Let α > −1 and let J0 have all gaps open. Under the hypotheses of Theorem 8.3.1, Z 2 (4 − E 2 )α log(det(W (E))) dE > −∞ (8.3.26) −2

if and only if Z

σess (J)

dist(x, R \ σess (J))α log(ω(x)) dx > −∞

Remarks and Historical Notes. Damanik–Killip–Simon [93].

(8.3.27)

These calculations are from

8.4. A Shohat–Nevai Theorem for Periodic Jacobi Matrices Given the calculation of the last section, the magic formula and Theorem 4.5.1, it is easy to obtain a Szeg˝o-type theorem, specifically an analog of the Shohat–Nevai theorem for perturbations of periodic Jacobi matrices. Our goal is to prove: Theorem 8.4.1. Let J0 be a period-p periodic Jacobi matrix with (0) (0) Jacobi parameters {an , bn }∞ n=1 . Let e = σess (J0 ). Let J be a half-line Jacobi matrix with Jacobi parameters {an , bn }∞ n=1 and Suppose that

X

E∈σ(J)\σess (J)

σess (J) = e

(8.4.1)

dist(E, σess (J))1/2 < ∞

(8.4.2)

and the spectral measure dρ of J has the form Then

Z

σess (J)

dρ(x) = ω(x) dx + dρs (x)

(8.4.3)

dist(x, R \ σess (J))−1/2 log(ω(x)) > −∞

(8.4.4)

if and only if lim sup

m Y aj (0)

j=1

(0)

aj

(0)

>0

(8.4.5)

Remarks. 1. Because a1 . . . ap = C(e)p , (8.4.5) is equivalent to m Y aj lim sup >0 (8.4.6) C(e) j=1

506


2. There is no assertion about aj or bj having a limit. Using very different methods, we will prove in Section 9.13 that there is (0) (0) (0) (0) {an , bn } ∈ Te so that |an − an | + |bn − bn | → 0. 3. The hypotheses (8.4.1), (8.4.2), and (8.4.4) imply the hypotheses of Theorem 8.6.1, so if all gaps are open, the equivalent hypotheses (8.4.2)/(8.4.4) or (8.4.2)/(8.4.5) imply ℓ2 approach to the isospectral torus as in that theorem. Lemma 8.4.2. Under the hypotheses of Theorem 8.4.1, suppose ∆J0 (J) has Jacobi parameters {An , Bn }∞ n=1 . Then p Y p Y akp+j−1+ℓ det(Ak ) = (8.4.7) (0) a j=1 ℓ=1 kp+j−1+ℓ Proof. Since Ak is lower triangular, det(Ak ) = (Ak )11 (Ak )22 . . . (Ak )pp

(8.4.8)

= (∆(J))(k−1)p+1 (k−1)p+1+p (∆(J))(k−1)p+2 (k−1)p+2+p (8.4.9) because of where Ak sits in ∆(J). (0)

−1 p ∆(J) = (a1 . . . a(0) p ) J + lower order

(8.4.10)

so, for any m, (0)

p −1 ∆(J)m m+p = (a1 . . . a(0) p ) )Jm m+1 (0)

−1 = (a1 . . . a(0) p ) am am+1 . . . am+p−1

(8.4.11)

(0)

Given that am is periodic, (8.4.9) and (8.4.11) imply (8.4.7). (0)

Proof of Theorem 8.4.1. By the fact that aj /aj above and away from 0, and (8.4.7), we see that (8.4.5) ⇔ lim sup[det(|A1 |) . . . det(|An |)] > 0

is bounded (8.4.12)

where we use the fact that since det(Aj ) > 0,

det(|Aj |) = |det(Aj )| = det(Aj )

By Corollary 8.3.2, (8.3.23) ⇔ (8.4.4). If we prove that X (8.4.2) ⇒ (|E| − 2)1/2 < ∞

(8.4.13)

(8.4.14)

E ∈[−2,2] / E∈σ(∆(J))

then Theorem 8.4.1 follows from Theorem 4.5.1. By the spectral mapping theorem, E ∈ σ(∆(J)) ⇔ E = ∆(E ′ ) with E ′ ∈ σ(J)

(8.4.15)

8.5. CONTROLLING THE APPROACH TO THE ISOSPECTRAL TORUS 507

Moreover, since all gaps in which eigenvalues occur are open, there are c > 0, d so that for all E ′ ∈ σ(J) \ σess (J), c(|∆(E)| − 2) ≤ dist(E, σess (J)) ≤ d(|∆(E)| − 2)

(8.4.16)

which verifies (8.4.14).

Remarks and Historical Notes. The results in this section are from Damanik–Killip–Simon [93]. However, the periodic case is a special case of the finite gap case, where there is earlier work by Widom and Peherstorfer–Sodin–Yuditskii that overlaps Theorem 8.4.1. See the Notes to Section 9.13 for further discussion. 8.5. Controlling the ℓ2 Approach to the Isospectral Torus In this section, we will control the relation of Hilbert–Schmidt estimates on ∆J0 (J) − (S p + S −p ) to ℓ2 approach of the Jacobi parameters of J to the isospectral torus, Te, with e = σess (J). This is preliminary to proving a Killip–Simon-type theorem for periodic perturbations. We need to begin by considering the definition of the distance of the tail of J to Te. Definition. Given two bounded sequences {an , bn }∞ n=1 and ′ ′ ∞ {an , bn }n=1 of Jacobi parameters, we define ′

′

dm ((a, b), (a , b )) =

∞ X k=0

e−k (|am+k − a′m+k | + |bm+k − b′m+k |) (8.5.1)

a metric defining the infinite product topology on {an+m , bn+m }∞ n=1 . We also define d˜m ((a, b), (a′ , b′ )) =

p−1 X k=0

(|am+k − a′m+k | + |bm+k − b′m+k |)

(8.5.2)

Given a set, T , of Jacobi parameters, we set

dm ((a, b), T ) = inf{dm ((a, b), (a′ , b′ )) | (a′ , b′ ) ∈ T }

(8.5.3)

{right limits of (a, b)} ⊂ Te ⇔ lim dm ((a, b), Te) = 0

(8.5.4)

and similarly for d˜m . Notice that because Te is a translation invariant set and the translates of a bounded (a, b) lie in a compact set, we have that m→∞

The main result of this section is the following:


508

Theorem 8.5.1. Let J0 be a period-p periodic Jacobi matrix with all gaps open and let ∆J0 be its discriminant. Let J be a bounded Jacobi matrix with Jacobi parameters {an , bn }∞ n=1 . Let An , Bn be the block Jacobi parameters of ∆J0 (J). Then the following are equivalent: (i) ∆J0 (J) − S p − S −p is a Hilbert–Schmidt operator on ℓ2 ({0, 1, 2, . . . }). (ii) X Tr(Bn2 + (|An | − 1)2 ) < ∞ (8.5.5) n

(iii)

X m

(iv)

X m

dm ((a, b), TJ0 )2 < ∞

(8.5.6)

d˜m ((a, b), TJ0 )2 < ∞

(8.5.7)

We begin by proving equality of sums of dm ’s and d˜m under great generality. This will require the following technical-looking result: (0)

(0)

Lemma 8.5.2. Fix ε > 0. Let {an , bn }∞ n=1 be the Jacobi parameters of some period-p periodic Jacobi matrix in an isospectral torus, Te. Let (an , bn )∞ n=1 be a set of bounded Jacobi parameters with ε < an < ε−1

(8.5.8)

There exists C depending only on ε and Te so that for all m and all n ≥ m, n−p+1

|an −

a(0) n |

+ |bn −

b(0) n |

≤ d˜m ((a, b), (a(0) , b(0) )) + C

X

r=m

d˜r ((a, b), Te) (8.5.9)

Proof. Decrease ε if necessary, so the an ’s of every element of Te obeys (8.5.8). Define for (b1 , . . . , bp ) ∈ Rp , (a1 , . . . , ap ) ∈ (ε, ε−1)p , p X (0) f (a1 , . . . , ap ) = [log(aj ) − log(aj )] (8.5.10) j=1 p

g(b1 , . . . , bp ) =

X j=1

By (5.5.122), (1) (1) {aj , bj }pj=1

Pp

j=1

(1)

log(aj ) =

(0)

[bj − bj ] Pp

(0) j=1 log(aj ) = −1

log(C(e)) for any

∈ Te. Since log is Lipschitz on (ε, ε ), we conclude that f (am , . . . , an+p−1) ≤ C1 d˜m ((a, b), Te)

(8.5.11)


Pp

By (5.4.13), ∆(x) determines (1)

j=1 bj

(1)

also, so

Pp

(1) j=1 bj

for any {aj , bj }pj=1 ∈ Te. As with (8.5.11), we obtain g(bm , . . . , bm+p−1 ) ≤ d˜m ((a, b), Te)

=

Pp

(0) j=1 bj

(8.5.12)

Thus, by (8.5.12), |bn − bn−p | = |g(bn−p+1, . . . , bn ) − g(bn−p , . . . , bn−1 )| ≤ dñ−p+1((a, b), Te) + dñ−p ((a, b), TJ ) 0

(8.5.13)

Similarly, using (8.5.11), |log(an ) − log(an−p )| ≤ C1 [dñ−p+1((a, b), Te) + dñ−p ((a, b), Te)] (8.5.14)

Since exp is Lipschitz on (log ε, log ε−1 ),

|an − an−p | ≤ C2 [dñ−p+1 ((a, b), Te) + dñ−p ((a, b), Te)]

(8.5.15)

Thus, by periodicity of a(0) , b(0) , we see (0)

(0)

(0) |an − a(0) n | + |bn − bn | ≤ |an−p − an−p | + |bn−p − bn−p | + (1 + C2 )[dñ−p+1((a, b), Te) + dñ−p ((a, b), Te)] (8.5.16)

We can now prove (8.5.9) by induction. For m ≤ n ≤ m + p − 1, the sum disappears on the right of (8.5.9) and the result is immediate from the definition of d˜m . For m + p ≤ n ≤ m + 2p − 1, we obtain the result from the original case using (8.5.11). The general result follows by induction. Proposition 8.5.3. Let J0 be a period-p periodic Jacobi matrix with isospectral torus Te. For all ε > 0, there is a constant C so that for all Jacobi parameters (an , bn )∞ n=1 with (8.5.8), we have X X e2(1−p) d˜m ((a, b), Te)2 ≤ dm ((a, b), Te)2 (8.5.17) m

m

≤C

X m

d˜m ((a, b), Te)2

(8.5.18)

Proof. (8.5.17) is trivial since, except for a weight bounded below by e−(p−1) , the sum in dm includes all terms in d˜m . For the other direction, the lemma implies dm ((a, b), Te) ≤ C1

∞ X j=0

e−j d˜m+j ((a, b), Te)

(8.5.19)

510


Thus, by a Schwarz inequality and 2

dm ((a, b), Te) ≤ C2 so X m

∞ X j=0

≤ C2 ≤C

−j j=0 e

< ∞,

e−j d˜m+j ((a, b), Te)2 ∞ X

2

dm ((a, b), Te) ≤ C2

P∞

m,j=0 ∞ X n=0

∞ X n=0

(8.5.20)

e−j d˜m+j ((a, b), Te)2

dñ ((a, b), Te)2

X n

−j

e

j=0

dñ ((a, b), Te)2

Remark. While the technicalities may obscure this, the key fact (1) (1) (1) (1) that lets us use p-fold sums is that in Te, (a1 , . . . , ap−1 ; b1 , . . . , bp−1 ) P Q (1) (1) determine ap and bp by the constancy of pj=1 bj and pj=1 aj over Te. Having seen (iii) ⇔ (iv) in Theorem 8.5.1, we turn to the other easy equivalence (i) ⇔ (ii).

Proposition 8.5.4. For any J0 , J we have (i) ⇔ (ii) in Theorem 8.5.1. Remark. Bn and An as block Jacobi parameters for ∆J0 (J) depend on J and J0 . ˜ since Hilbert– Proof. For two block Jacobi matrices, J and J, Schmidt norms are squares of matrix elements, X X ˜ 2 = ˜ n k2 + 2 kJ − Jk kBn − B kAn − Añ k22 (8.5.21) I2

I2

n

I

n

Thus,

k∆J0 (J) − S p − S −p k2I2 =

X n

Tr(Bn2 ) + 2Tr((An − 1)2 )

This plus Theorem 4.6.7 implies (i) ⇔ (ii).

(8.5.22)

We turn now to the most subtle part of Theorem 8.5.1, namely that (i) ⇔ (iii), which will depend on the all-gaps-open hypothesis. Lemma 8.5.5. Let F be a C ∞ map of an open set U ⊂ Rn to Rℓ with ℓ < n. Suppose for some y0 ∈ Rℓ , T = F −1 ({y0})

(8.5.23)


is a smooth compact manifold of dimension n − ℓ, and for all x0 ∈ T, rank((∇F )(x0 )) = ℓ Then for any compact neighborhood, K, of T, there are constants cK , dK in (0, ∞) so that for all x ∈ K, cK |F (x) − y0 | ≤ dist(x, T ) ≤ dK |F (x) − y0 |

(8.5.24)

Proof. That this holds locally near any x1 ∈ K follows from the implicit function theorem. Compactness then implies the global result on K. For any {aj , bj }pj=1 in ((0, ∞) × R)p , we can define p + 1 functions c0 , . . . , cp by p X ∆J0 (a,b) (λ) = ck λ k (8.5.25) k=0

where J0 (a, b) is the periodic Jacobi matrix with parameters {aj , bj }pj=1. Let F : (0, ∞) × Rp → Rp+1 by Fk (a, b) = ck (a, b). Then

Proposition 8.5.6. At any set of periodic Jacobi parameters for which J0 has all gaps open, rank[(∇F )(a, b)] = p + 1

(8.5.26)

Proof. Since ∇F maps R2p to Rp+1 , this is equivalent to saying {∇ck }pk=0 as vectors in R2p are linearly independent. That is the content of Theorem 6.9.1. Lemma 8.5.7. Let χk be the projection in ℓ2 onto {δj }kj=1 . For any compact subset, K, in ((0, ∞) × R)p of period-p Jacobi matrices, there are constants cK , dK so for all J ∈ K and (y0 , . . . , yp ) ∈ Cp+1 ,

2

p

p

2 p X X

X j 2 j

≤

|y | ≤ d y J χ (8.5.27) y J χ cK j p+1 j K j p+1

j=1

I2

j=1

j=0

I2

Proof. {J ℓ χp+1 }pℓ=0 are linearly independent since J ℓ χp+1 has nonzero elements in position 1 ℓ + 1 and zero elements in positions 1 j + 1 for j = ℓ + 1, . . . , p + 1. Thus, the matrix Mℓk = Tr(χp+1J ℓ J k χp+1 )

ℓ, k = 0, . . . , p

(8.5.28)

is strictly positive definite. By continuity, 0 < inf kMℓk k ≤ sup kMℓk k < ∞ J∈K

which leads directly to (8.5.27).

(8.5.29)

J∈K

512


Now view J, J0 , two period-p periodic Jacobi matrices, as two-sided. Then ∆J0 (J) is a two-sided block Jacobi matrix with constant A’s and B’s we will denote by AJ0 (J), BJ0 (J). Proposition 8.5.8. Let J0 be a periodic Jacobi matrix with all gaps open and isospectral torus Te. Then for any compact neighborhood, K, of Te in ((0, ∞)×R)p, there are cK , dK in (0, ∞) so that all for period-p J with Jacobi parameters in K, cK (kAJ0 (J) − 1k2 + kBJ0 (J)k2 ) ≤ dist(J, Te)2

≤ dK (kAJ0 (J) − 1k2 + kBJ0 (J)k2 ) (8.5.30)

Proof. Use ∼ to indicate two sides have a ratio bounded above and away from zero on compacts. Clearly, for any operator on ℓ2 , ℓ → kMχℓ kI2

is monotone in ℓ (since it is the sum of the squares of rows 1, . . . , ℓ). Thus, k[∆J0 (J) − (S p + S −p )]χp kI2 ≤ k[∆J0 (J) − (S p + S −p )]χp+1 kI2

≤ k[∆J0 (J) − (S p + S −p )]χ2p kI2 (8.5.31)

while for n = 1, 2, k[∆J0 (J) − (S p + S −p )]χnp k2I2 = nkBJ0 (J)k2I2 + 2nk(AJ0 (J) − 1)k2I2 (8.5.32) so kAJ0 (J) − 1k2 + kBJ0 (J)k2 ∼ k[∆J0 (J) − S p + S −p ]χp+1 k2 (8.5.33) By the magic formula, p

[∆J0 (J) − (S + S

−p

)]χp+1 =

p X

cℓ J ℓ χp+1

(8.5.34)

ℓ=0

where cℓ are the difference of the coefficients of the polynomials ∆J0 and ∆J . By Lemma 8.5.5, p X p −p 2 |cℓ |2 (8.5.35) k[∆J0 (J) − (S + S )]χp+1 k ∼ ℓ=0

By Lemma 8.5.5 and Proposition 8.5.6, p X |cℓ |2 ∼ dist(J, Te)2

(8.5.36)

ℓ=0

(8.5.33), (8.5.35), and (8.5.36) imply (8.5.30).


Proposition 8.5.9. Let k ≤ ℓ. Then ∆J0 (J)kℓ for any bounded ℓ+α Jacobi matrix depends only on {bj }j=k−α and {aj }ℓ+α−1 j=k−α where α is the 1 greatest integer less than or equal to 2 [p − (ℓ − k)].

Proof. Each J changes index by at most 1, so J m , m = 0, 1, . . . , p, can change index by at most p steps of 0, ±1, ℓ − k steps are needed to get from k to ℓ. The remaining steps have to go both up and back, so they cannot go higher than ℓ + α or below k − α.

Corollary 8.5.10. Fix J0 . Let k ≤ ℓ and α given by Proposition 8.5.9. Then for any K and all J, J˜ whose Jacobi parameters obey sup [|bj | + |˜bj | + |aj | + |˜ aj |] ≤ K (8.5.37) j

there is a CK so that ˜ kℓ | ≤ CK |∆J (J)kℓ − ∆J (J) 0

0

sup k−α≤j≤ℓ+α

[|bj − ˜bj | + |aj − a ˜j |] (8.5.38)

Proof. Immediate from Proposition 8.5.9, given that for J0 fixed, ∆J0 (J)kℓ is a polynomial in a fixed number of variables with fixed coefficients. Lemma 8.5.11. (a) For any Jacobi matrix, J, and ℓ = 1, 2, . . . , m = 1, 2, . . . , (J ℓ )m m+ℓ = am am+1 . . . am+ℓ−1

(8.5.39)

and for ℓ = 2, 3, . . . , m = 1, 2, . . . , ℓ

(J )m m+ℓ−1 = am . . . am+ℓ−2

X ℓ−1 j=0

bm+j

(8.5.40)

(b) For J0 is periodic of period p ≥ 2 and m = 1, 2, . . . , (am . . . am+p−1 ) ∆J0 (J)m m+p = (0) (8.5.41) (0) [am . . . am+p−1 ] X p−1 (0) (0) (0) −1 ∆J0 (J)m m+p−1 = [am . . . am+p−1 ] (am . . . am+p−2 ) (bm+j − bm+j ) j=0

(8.5.42)

Proof. (a) Since J can increase index by at most one, (J ℓ )m m+ℓ = (Jm m+1 ) . . . (Jm+ℓ−1 m+ℓ )

(8.5.43)

proving (8.5.39), while ℓ

(J )m m+ℓ−1 =

ℓ−1 X j=0

(J j )m m+j Jm+j m+j (J ℓ+j−1)m+j m+ℓ−1

(8.5.44)

514


which, given (8.5.39), proves (8.5.40). (b) By (5.4.13), ∆J0 (J) =

(0) (aj

−1 . . . a(0) p )

p

J −

p−1 X

(0) bj+1 J p−1

+ O(J

j=0

p−2

)

(8.5.45)

which, given (a), (J p−k )m m+p = (J p−k )m m+p−1 = 0 if k = 2, 3, . . . , and the periodicity of a(0) and b(0) , implies (8.5.41) and (8.5.42). Lemma 8.5.12. If ∆J0 (J) − (S p + S −p ) ∈ I2 , then X (0) 2 (i) (an an+1 . . . an+p−1 − a(0) n . . . an+p−1 ) < ∞

(8.5.46)

n

2 p−1 X X (0) (bn+j − bn+j ) < ∞

(ii)

n

(8.5.47)

j=0

Proof. For a Hilbert–Schmidt operator, any subset of matrix elements lies in ℓ2 , so by (8.5.41), X (0) −1 |an . . . an+p−1[a(0) − 1|2 < ∞ (8.5.48) n . . . an+p−1 ] n

(0)

(0)

which, given that an . . . an+p−1 is n-independent, implies (8.5.46). Similarly, (8.5.42) implies (8.5.47) if we note that aj bounded and (0) (0) an . . . an+p−1 → a1 . . . ap > 0 implies inf(aj ) > 0, so (0)

−1 inf (a(0) m . . . am+p−1 ) (am . . . am+p−2 ) > 0 m

(8.5.49)

Lemma 8.5.13. If ∆J0 (J) − (S p + S −p ) ∈ I2 , then X (an+p − an )2 < ∞

(8.5.50)

n

X n

(bn+p − bn )2 < ∞

(8.5.51)

Proof. Since a difference of ℓ2 sequences is ℓ2 , (8.5.46) implies (0) (since an is periodic) X (an+p − an )2 (an+1 . . . an+p−1)2 < ∞ (8.5.52) n

which, given that inf(aj ) > 0, implies (8.5.50).


Similarly, since p−1 X j=0

(bn+1+j − bn+j ) = bn+p − bn

(8.5.53)

(8.5.47) implies (8.5.51).

Proof of Theorem 8.5.1. Given what we have proven already, we only need (iv) ⇒ (i) and (ii) ⇒ (iii).

(iv) ⇒ (i). In Proposition 8.5.8, α ≤ 12 (p − (ℓ − k)), so (ℓ + α) − (k − α) = 2α + ℓ − k ≤ p

(8.5.54)

|∆J0 (J)kℓ − (S p + S −p )kℓ | ≤ C d˜m ((a, b), TJ0 )

(8.5.55)

Since J is bounded and ∆J0 (J)kℓ − (S p + S −p )kℓ is a polynomial in at most p consecutive a, b pairs which vanishes on TJ0 , for some m (dependent on k, ℓ) and some C (independent of k, ℓ), we have that Since a fixed m occurs for most p2 (kℓ) pairs (m = k − α, so m ≤ k ≤ m + p and k ≤ ℓ ≤ k + p), with k ≤ ℓ, we see (the 2 comes from k ≤ ℓ and ℓ ≤ k pairs) X k∆J0 (J) − (S p + S −p )k2I2 ≤ 2Cp2 d˜m ((a, b), TJ0 )2 (8.5.56) m

(i) ⇒ (iii). Let J (k) be the period-p Jacobi matrix that equals J on block k, that is, for ℓ = 1, . . . , p, (k)

(k)

bℓ = bkp+ℓ By Lemma 8.5.13, X sup n

aℓ = akp+ℓ (k)

(k)

|bj − bj | + |aj − aj |

(k+1)p≤j≤(k+2)j−1

(8.5.57) 2

−∞

(8.6.4)

e

Proof. By Theorem 8.5.1,

(8.6.1) ⇔ ∆J0 (J) − (S p + S −p ) ∈ I2

(8.6.5)

(8.6.2) ⇒ σess (∆J0 (J)) = [−2, 2]

(8.6.6)

σess (∆J0 (J)) = [−2, 2] + (8.6.3) ⇒ (8.6.2)

(8.6.7)

By the spectral mapping theorem, while

8.6. A KILLIP–SIMON THEOREM FOR PERIODIC JACOBI MATRICES 517

By Corollary 8.3.3, (8.6.4) ⇔ (8.3.26)

(8.6.8)

As in the proof in Section 8.4, all gaps open means dist(E, σess (J0 )) ≈ dist(∆J0 (E), [−2, 2]) so (8.6.3) ⇔

X

E∈σ(∆J0 (J))\[−2,2]

(|E| − 2)3/2 < ∞

(8.6.9)

(8.6.10)

We have thus proven equivalence of all conditions in the theorem and conditions on ∆J0 (J), so Theorem 4.6.1 completes the proof of this theorem. Example 8.6.2. Let J(t) be a curve in Te (thought of as half-line Jacobi matrices) so kJ ′ (t)k = O(t−2/3 ). Thus, Z kJ ′ (t)k2 dt < ∞ (8.6.11) and J(t) may not have a limit. For example, if we think of the torus Te as Rℓ /Zℓ , picking any unit vector η ∈ Rℓ , we can take J(t) = [t1/3 η], the equivalence class of t1/3 η, in which case J(t) does not have a limit. Let {an (t), bn (t)}∞ n=1 be the Jacobi parameters of J(t) and let J be the matrix with Jacobi parameters aJn = an (n), bJn = bn (n). Then J is not asymptotic to any fixed J0 ∈ Te, although all right limits lie in Te. Moreover, by (8.6.11), it is easy to see that X d˜m (J, J(n))2 < ∞ (8.6.12) n

so J obeys (8.6.1). Thus, in particular, Σac (J) = e. The point of this example is that one might have thought while we can only prove in theorem 7.6.1 that the right limits lie in Te, it might be that there is a single orbit as the limit points (as we will see (in Section 9.13) happens if a Szeg˝o condition holds). This example shows that, in fact, the limit points can be the entire isospectral torus even though Σac = σess = e. Remarks and Historical Notes. from Damanik–Killip–Simon [93].

This theorem and its proof are

518


8.7. Sum Rules for Periodic OPUC We want to summarize here the main differences between the OPRL and OPUC results of the type discussed in Sections 8.4 and 8.6 and state, without proof, the OPUC results. (1) For OPUC, the discriminant obeys ∆(z) = ∆(1/¯ z)

(8.7.1)

so ∆(z) is real on ∂D. Thus, if C is a unitary CMV matrix, then ∆(C)∗ = ∆(C)

(8.7.2)

Moreover, if C0 has period p = 2k, then ∆(z) has the form ∆C0 (z) =

k X

yj z j

(8.7.3)

j=−k

for suitable y. Since C is five-diagonal, ∆C0 (C) has 2k = p diagonals above and below the main, so it is a block Jacobi matrix with p × p blocks. There is still a magic formula. Namely, for any period p = 2k CMV matrix, C0 , and any two-sided CMV matrix, C, ∆C0 (C) = S p + S −p

(8.7.4)

if and only if C ∈ Te with e = σess (C0 ). The only difference from OPRL is that An and Bn can have complex elements. However, B is still selfadjoint, that is, Bn† = Bn , and An is still lower triangular and positive on diagonal. The moral is that even for OPUC, it is MOPRL not MOPUC that is relevant! (2) For OPRL, we have that if e has all gaps open, the flows generated by the coefficients of ∆ (other than the constants (a1 . . . ap )−1 and Pp −1 (a1 . . . ap ) ( j=1 bj )) are linearly independent. This was used critically in Section 8.5. For OPUC, it remains an open question to prove that the analog always holds. What is known (proven in Simon [391]) is that for a generic e, the normal bundle is spanned by the derivatives of the coefficients of ∆. (0) (0) (3) In changing from dm to d˜m , we used the fact that in Te, (aj , bj )p−1 j=1 (0) (0) determine ap , bp . For OPUC, the analog is not true. But, of course, (0) (0) {αj }pj=1 determine αp+1 , so if one defines d˜m as a sum over p + 1, things work, but that change is needed. Here are the two theorems:

8.7. SUM RULES FOR PERIODIC OPUC

519

Theorem 8.7.1. Let C0 be a period-p (p = 2k) periodic CMV matrix (0) with Verblunsky coefficients {αn }∞ n=0 and let e = σess (C0 ). Let C0 be a CMV matrix with Verblunsky coefficients {αn }∞ n=0 and σess (C) = e

(8.7.5)

and spectral measure dµ = w(θ) Suppose that

X

E∈σ(C)\e

Then

Z

e

if and only if

dθ + dµs 2π

dist(E, σess (C))1/2 < ∞

(8.7.6)

(8.7.7)

dist(eiθ , ∂D \ e)−1/2 log(w(θ)) > −∞

(8.7.8)

m Y ρj

(8.7.9)

lim sup

(0)

j=1

ρj

>0

Theorem 8.7.2. Fix p = 2k. There is a dense open set U ⊂ Dp so that if C0 is periodic with period p and with Verblunsky coefficients {αn }p−1 n=0 ∈ U, then with e = σess (C0 ) and Te the isospectral torus, we have for any C with Verblunsky coefficients {αn }∞ n=1 that ∞ X dm ((α), Te)2 < ∞ (8.7.10) m=1

if and only if (i) (ii)

X

σess (C) = e

(8.7.11)

dist(E, e)3/2 < ∞

(8.7.12)

E∈σ(C)\e

(iii) dµ has the form (8.7.6) with Z dist(eiθ , ∂D \ e)1/2 log(w(x)) dx > −∞

(8.7.13)

e

Remarks and Historical Notes. The result on linear independence of flows is in Simon [391, Sect. 11.10]. The other results are from Damanik–Killip–Simon [93].

CHAPTER 9

Szeg˝ o’s Theorem for Finite Gap OPRL 9.1. Overview In this chapter, we consider a general finite gap set, e, of the form e=

ℓ+1 [

j=1

[αj , βj ]

α1 < β1 < α2 < · · · < βℓ+1

(9.1.1)

and prove a Szeg˝o–Shohat–Nevai theorem and Szeg˝o asymptotics for suitable measures, µ, with σess (µ) = e. The key is to find an analog of the map z → z + z −1 of D to C ∪ {∞} \ [−2, 2], which was central to Chapter 3. Thus, we seek an analytic map x : D → C ∪ {∞} \ e

(9.1.2)

x(z) = x(w) ⇔ ∃γ ∈ Γ so that w = γ(z)

(9.1.3)

Since the right side of (9.1.2) is not simply connected, we cannot hope that x is a bijection. Instead, we will want a many-to-one map. The inverse image, x−1 (w), of a single point will be a countable discrete set and we will deal with this set by finding a group, Γ, of analytic bijections of D so that Groups of analytic bijections of D with {γ(z)}γ∈Γ discrete are called Fuchsian groups. This approach to finite gap spectral theory was pioneered by Sodin–Yuditskii [404] and Peherstorfer–Yuditskii [336] and developed from a sum rule point of view by Christiansen–Simon– Zinchenko [82, 83, 84]. In Sections 9.2–9.4, we discuss general analytic bijections on D: individual maps in the first two sections and groups of such maps in the third. Section 9.5 constructs the map x and Section 9.6 studies the detailed structure for a finite gap set and its associated group and fundamental region. Section 9.7 finds functions vanishing at {γ(z0 )}γ∈Γ and relates these functions to a potential theory on e. Section 9.8 completes the general theory by proving a technical continuity result (that x and Γ are continuous in {αj , βj }ℓ+1 j=1 ) that is important in Section 9.12. An important role throughout is played by character automorphic functions, that is, analytic (or meromorphic or harmonic) functions, f , 521

522

˝ THEOREM FOR FINITE GAP OPRL 9. SZEGO’S

on D (or larger sets) which obey f (γ(z)) = c(γ)f (z)

(9.1.4)

where c : Γ → ∂D and obeys c(γγ ′ ) = c(γ)c(γ ′ ). c is a character of Γ and the set of all such characters, Γ∗ , is isomorphic to the ℓ-dimensional torus. It is no coincidence that Γ∗ is isomorphic to the isospectral torus and a natural map of Te to Γ∗ will play a central role in our proof of Szeg˝o asymptotics. In Section 9.9, we turn to applying the machinery developed earlier to spectral theory, proving a step-by-step sum rule which, in Section 9.10, proves the following version of the Szeg˝o–Shohat–Nevai theorem: Theorem 9.1.1. Let µ be a nontrivial probability measure on R with σess (µ) = e. Let {an , bn }∞ n=1 be its Jacobi parameters and J its Jacobi matrix. Suppose µ has the form dµ = w(x) dx + dµs and that

X

E∈σ(J)\σess (J)

Then

Z

e

if and only if

dist(E, e)1/2 < ∞

w(x)dist(x, R \ e)−1/2 dx > −∞ lim sup

a1 . . . an >0 C(e)n

(9.1.5) (9.1.6)

(9.1.7) (9.1.8)

If (9.1.6) and (9.1.7) (equivalently, (9.1.8)) hold, we say µ ∈ Sz(e), the Szeg˝o class for e. In Section 9.13, we will prove that if µ ∈ Sz(e), (0) (0) there is {an , bn } ∈ Te, an element of the isospectral torus, so that (0) lim |an − a(0) n | + |bn − bn | = 0

n→∞

(9.1.9)

a result that has not been proven using the methods of Chapter 8 even for the periodic case. To obtain this result (and an associated Szeg˝o asymptotics on the polynomials), we rely on machinery developed in Sections 9.11 and 9.12. In Section 9.11, we define Θ-functions, natural character automorphic functions on C ∪ {∞} \ Λ(Γ) (Λ(Γ) is the set of limit points of Γ, a closed nowhere dense subset of ∂D discussed in Section 9.4) with given zeros and poles. As a bonus of this theory, we will prove the case of Abel’s theorem we need in Section 5.12. In Section 9.12, we associate a Jost function, yet another character automor(0) (0) phic function, to any µ ∈ Sz(e). It will turn out that the {an , bn }∞ n=1

9.2. FRACTIONAL LINEAR TRANSFORMATIONS

523

of (9.1.9) is determined by the fact that the Jost functions for µ and for µ(0) have the same character. 9.2. Fractional Linear Transformations Since we need more about fractional linear transformations (FLTs) than what is in the elementary books, our discussion begins with a rapid minicourse on the subject. We will not describe the Riemann sphere in terms of stereographic projection but as P, the complex projective line. In C2 \ {0}, we say . u, v are equivalent, written u = v, if and only if there is λ ∈ C \ {0} so u = λv. It is easy to see this defines an equivalence relation. The equivalence classes, associated with (complex) lines in C2 , are elements of P. P contains a distinguished element ∞ = [ 10 ], that is, the line with second coordinate 0. (Here [ · ] means equivalence class of · .) P \ {∞} can be put in one-one correspondence with C by associating z with π∞ C by defining π ˜∞ on [ z1 ]. Put equivalently, we can map P \ {∞} −→ 2 2 C \ {u ∈ C | u2 = 0} by u1 π ˜∞ (u1 , u2 ) = (9.2.1) u2 π ˜∞ is constant on equivalence classes, and so induces π∞ on P \ {∞} by π∞ ([u]) = π ˜∞ (u). −1 Similarly, if 0 ∈ P is defined by 0 = π∞ (0), that is, 0 = [ 01 ], we can define π ˜0 on C2 \ {u | u1 = 0} by u2 π ˜0 (u1, u2 ) = (9.2.2) u1 and induce π0 . The domains of π0 and π∞ overlap in P \ {0, ∞} which each maps to C \ {0}, and −1 π0 π∞ : π∞ [P \ {0, ∞}] → π0 [P \ {0, ∞}]

is, according to (9.2.1)/(9.2.2), given by 1 (9.2.3) z Thus, we have a local coordinate system with transition maps given by analytic functions which defines a complex variables analog of manifolds called Riemann surfaces. In practical terms, we associate P with C∪{∞}, that is, we normally use z = π∞ (u) as our coordinates, shifting to 1/z as a coordinate near infinity. −1 π0 π∞ (z) =

524


If T is an invertible linear map on C2 , clearly T maps C2 \ {∞} to . . itself and u = v implies T u = T v, so T induces an invertible map fT from P to itself. If T = ( ac db ), we have az+b z a b z cz+d fT = = 1 c d 1 1

which in local coordinate z (from π∞ ) is  az+b d   cz+d z 6= ∞, − c fT (z) = ac if z = ∞  ∞ if z = − dc

(9.2.4)

so FLTs are just the maps on the Riemann sphere induced by linear transformations. We already used this notion in Section 2.5.

Example 9.2.1. If c = 0, then fT (∞) = ∞ and fT is the affine map a b fT (z) = z + (9.2.5) d d where det(T ) 6= 0 implies d 6= 0 6= a. If det(T ) = 1, then ad = 1, so d−1 = a and fT (z) = a2 z + ba (9.2.6) We will summarize this below. As a second example, 1 f (z) = or f (reiθ ) = r −1 e−iθ z which inverses the radius and complex conjugates. To get pure inversion, one can define 1 r(z) = (9.2.7) z¯ which is the inversion in the unit circle, ∂D. More generally, inversion in the circle |z − z0 | = r has the form r2 z¯ − z¯0 Note that (9.2.7)/(9.2.8) are not analytic and not FLTs! T (z) = z0 +

(9.2.8)

We summarize the first example in Proposition 9.2.2. If fT (∞) = ∞ and det(T ) = 1, then fT (z) = a2 z + ab

(9.2.9)

A big point of defining FLTs as linear maps on P is that clearly, fT fS = fT S The way to compose FLTs is matrix multiplication.

(9.2.10)


525

Proposition 9.2.3. If T, S ∈ GL(2, C) (2 × 2 invertible matrices), then fT = fS if and only if T = λS for some λ ∈ C \ {0}.

Proof. By using (9.2.10), it suffices to consider the case S = id, since fT = fS ⇔ fT S −1 = fid . But then fT (w) = w means T w1 = λw w1 for w = 0, z0 , 2z0 . Since z10 = 12 01 + 12 2z10 , we see (by looking at the two components) that 2λ2z0 = 2λz0 λ0 + λ2z0 = 2λz0

which implies λ0 = λz0 = λ2z0 . Since z0 is arbitrary, T = λ0 1.

Given S, we can always pick λ so det(T ) = 1. We will henceforth do so unless noted explicitly. This determines T up to ± sign. Thus, if SL(2, C) is the set of 2 × 2 matrices of determinant 1, the map T → fT is two-to-one with kernel = {±1}, that is, if F is the group of all FLT, then F = SL(2, C)/{±1} (9.2.11) Thus F is often called PSL(2, C). One immediate advantage of the matrix connection is: Lemma 9.2.4. fT ([u]) = [u] if and only if u is an eigenvector of T . Proof. fT ([u]) = [u] ⇔ T (u) = λu.

Proposition 9.2.5. Let f ∈ F and suppose f leaves three points fixed. Then f = id. Proof. Any 2 × 2 matrix with three distinct eigenvectors (not counting multiples) is a multiple of the identity. Theorem 9.2.6. Fix w0 , w1 , w2 distinct. Then, for each distinct z0 , z1 , z2 , there is exactly one f ∈ F with f (wj ) = zj

(9.2.12)

for j = 0, 1, 2. Proof. Uniqueness is immediate from Proposition 9.2.5. For if f1 and f2 solve (9.2.12), then g = f1 f2−1 has z0 , z1 , z2 as fixed points and so, by the proposition, f1 f2−1 = id, proving uniqueness. For existence, we note first that it suffices to handle the case z0 = 0, z1 = 1, z2 = ∞. For if f takes (w0 , w1 , w2 ) to (0, 1, ∞) and g takes (z0 , z1 , z2 ) to (0, 1, ∞), then g −1 f solves (9.2.12). Given distinct w0 , w1 , w2 , the FLT w1 − w2 w − w0 f (w) = (9.2.13) w1 − w0 w − w2

526


solves (9.2.12) with (z0 , z1 , z2 ) = (0, 1, ∞).

As an immediate consequence, we see that F is exactly the set of bianalytic homeomorphisms (aka conformal maps) of P to itself: Corollary 9.2.7. If f is a bijection of P to itself which is analytic (in a local coordinate sense), then f ∈ F. Proof. Without loss, we can suppose f leaves 0, 1, ∞ fixed since we can replace f by f g −1, where g ∈ F has g(0) = f (0), g(1) = f (1), g(∞) = f (∞). 1 Since f (∞) = ∞, h(w) = f (1/w) is analytic near w = 0, has h(0) = 0 ′ and h (0) 6= 0 since it is single-valued near w = 0. It follows that |h(w)| > C|w| near w = 0, so |f (z)| ≤ C −1 |z| near infinity. By Louiville’s theorem, f is a degree one polynomial, hence f (z) = z since there is a unique affine function with f (0) = 0, f (1) = 1. Remark. A useful way of thinking of this is that any analytic map of P to P is given by a rational function and it has to be a degree one polynomial if it is a bijection. F is a group and, as is with any group, its conjugacy classes are of interest. Definition. f, g ∈ F are called conjugate if and only if there is h ∈ F so hf h−1 = g. Theorem 9.2.8. For T, S ∈ SL(2, C) neither equal to ±1, fT is conjugate to fS if and only if Tr(S) = ±Tr(T )

(9.2.14)

and then we are in one of the following family of classes: (i) Parabolic: Tr(S) = ±2; one conjugacy class; f has one fixed point. An element in the class is f (z) = z + 1

(9.2.15)

(ii) Elliptic: Tr(S) ∈ (−2, 2), so Tr(S) = 2 cos θ. Classes labelled by θ ∈ (0, π2 ]. f has two fixed points. An element in this class is f (z) = e2iθ z

θ ∈ (0, π/2]

(9.2.16)

(iii) Hyperbolic: Tr(S) ∈ ±(2, ∞), so Tr(S) = ±2 cosh ϕ, ϕ ∈ (0, ∞). f has two fixed points. An element in this class is f (z) = e−2ϕ z

ϕ ∈ (0, ∞)

(9.2.17)


527

(iv) Loxodromic: Tr(S) ∈ {z | Im(z) 6= 0}, so Tr(S) = ±(eα+iθ + e−α−iθ ) for some α ∈ (0, ∞) and θ ∈ (0, π). f has two fixed points. An element in this class is f (z) = e−2α−2iθ z

α ∈ (0, ∞), θ ∈ (0, π)

(9.2.18)

−1 fW fS fW ,

Proof. If fT = then T = ±WSW −1 , so of course, (9.2.14) holds. For the converse, we note that if we prove that one of models (9.2.15)–(9.2.18) is in each conjugacy class, we see there is only one conjugacy class with a given ±Tr(T ), proving that (9.2.14) implies conjugacy. Every 2 × 2 matrix T has one or two (generically, two) eigenvectors, and so one or two fixed points, which we denote z1 , z2 (if there is only one, we do not define z2 ). If g maps z1 to ∞ and z2 to 0, then gT g −1 has z1 = ∞, and if there is a z2 , it is 0. Thus we can be sure there is a conjugate f to fT with f (∞) = ∞ and, unless Tr(T ) = ±2 (since det(T ) = 1 and two equal algebraic eigenvalues implies the eigenvalues are both +1 or both −1), f (0) = 0. By Proposition 9.2.2, f (∞) = ∞ implies f has the form (9.2.9). If f (0) = 0, b = 0 and thus a 0 T = fT (z) = a2 z (9.2.19) 0 a−1 and Tr(T ) = a + a−1 . Without loss, we can suppose |a| ≤ 1, since interchanging 0 and ∞ interchanges a and a−1 . (ii), (iii), (iv) correspond precisely to |a| = 1 with a 6= ±1, a real with |a| < 1, and |a| < 1, Im(a) 6= 0, respectively. That leaves the case where T has a single eigenvector, which means there is a W with 1 1 −1 ±W T W = ≡S 0 1 which has TS (z) = z + 1.

One advantage of these models is that they immediately make clear the asymptotics of f (n) (z) ≡ f ◦ · · · ◦ f (z) repeated n times:

Theorem 9.2.9. Let f be an element of F with fixed points, z1 and z2 (if there is a second). Then (a) If f is hyperbolic or loxodromic, for one of the fixed points, say z1 , we have f (n) (w) → z1 as n → +∞ for any w 6= z2 , and for each fixed w, the approach is exponentially fast. As n → −∞, f (n) (w) → z2 for any w 6= z1 . (b) If f is parabolic, for any w, f (n) (w) → z1 as n → ±∞ and the approach is O(1/n).

528


(c) If f is elliptic, either f is periodic, that is, f (p) = 1 for some p, or else f (n) (w) is dense in an orbit which is a closed curve, and f (n) (w) is almost periodic. Remarks. 1. Near any given point, we can measure distances in a local coordinate system, and ideas like “approach exponentially fast” are independent of coordinate system. There is a natural metric on P, namely, ρ([u], [v]) = min(kx − yk | x ∈ [u], y ∈ [v], kxk = kyk = 1), and one could use that. This metric can be described in terms of stereographic projection. 2. The curves in (c) are circles. Our proof will show they are circles in a special case, and once one has Theorem 9.2.13 below, it will follow they are always circles. Proof. For the last three models, where z2 = ∞ and z1 = 0 (in the terminology of this theorem), the claims are obvious, but the claims are preserved by conjugation. For the parabolic case, for |w| near ∞, ρ(w, ∞) ∼ 1/w and 1/(w0 + n) goes to ∞ as O(n−1 ).

x-ref?

Example 9.2.10. It pays to look at the parabolic case in more detail. Consider the case z f (z) = z+1 where z = 0 is the parabolic fixed point. f = fT with T = ( 11 01 ), so T n = ( n1 01 ) and i n+i f (n) (i) = = 2 ni + 1 n +1 The asymptotic approach is tangent to the real axis, but unlike the hyperbolic case where the approach to the asymptotic tangent is exponential, Im f (n) (i) = O(1/n2). It is easy to see that for any nonreal z0 , f (n) (z0 ) has Re f (n) (z0 ) = O(1/n) and |Im f (n) (z0 )| = O(1/n2). The flow lines for hyperbolic and parabolic examples are shown in Figure TK. Note the name parabolic is not connected with the asymptotic parabolic relation of Re f (n) and Im f (n) ! Next, we turn to the role of circles and lines under FLTs. C2 has its natural Euclidean inner product. Given a selfadjoint matrix B, hu, Bui is not a function of [u], but since hλu, Bλui = |λ|2 hu, Bui whether it is positive, zero, or negative, is constant on equivalence classes. Theorem 9.2.11. Let

α β J= ¯ β γ

(9.2.20)


be a selfadjoint matrix (i.e., α, γ real). Let z z CJ = z ∈ C ∪ {∞} ,J =0 1 1 z z ± CJ = z ∈ C ∪ {∞} ± ,J >0 1 1

529

(9.2.21)

(9.2.22) z z with the convention that, for z = ∞, we replace h , J i by 1 1 1 1 h 0 , J 0 i. Let det(J) < 0. Then (a) If α = 0, CJ is the straight line (real line, not complex line) (z = x + iy) γ (Re β)x + (Im β)y = − (9.2.23) 2 (b) If α 6= 0, then CJ is the circle 2 z + β = − det(J) (9.2.24) α |α|2

Every circle or line has this form. CJ± are the two connected components of P \ CJ . Remarks. 1. α = 0 is equivalent to ∞ ∈ CJ . 2. If α > 0 (resp. α < 0), CJ+ is the outside (resp. inside) of CJ . Proof.

z z ¯ +γ ,J = α¯ z z + 2 Re(βz) 1 1

(9.2.25)

If α = 0, we get (9.2.23). If α 6= 0,

2 β |β|2 RHS of (9.2.25) = α z + + γ − α α ¯

which leads to (9.2.24). To get the line ax + by = c, take 0 a + ib J= a − ib −2c

To get the circle |z − z0 |2 = r 2 , take 1 −z0 J= −¯ z0 −r 2 + |z0 |2 Example 9.2.12. If J= then h

z 1

1 0 0 −1

, J z1 i = |z|2 − 1. CJ is ∂D; CJ− is D.

(9.2.26)


530

If J= then h plane.

z 1

0 i −i 0

(9.2.27)

, J z1 i = −2 Im z, so CJ = R and CJ− is C+ , the upper half

Theorem 9.2.13. f ∈ F takes circles and lines into circles and lines. For any two circles (or circle and line, or two lines), there is an f ∈ F taking the first to the second. Proof. Since z z z z ∗ T , JT = , T JT 1 1 1 1

(9.2.28)

we have that

fT [CJ ] = CT ∗JT

(9.2.29)

proving circles/lines go to circles and lines. Clearly, {z | |z − z0 | = r0 } goes to {z | |z − z1 | = r1 } under f (z) = rr01 (z − z0 + z1 ), so any circle can go to any other circle, and by a translation and rotation, any line goes to any line. Since f (z) = 1/z takes {z | |z − 1| = 1} to Re z = 12 , we see that we can get from one particular circle to one particular line, so by the beginning of this paragraph, from any circle to any line. Example 9.2.14. Among the most famous FLTs are 1 z−1 f (z) = i z+1

(9.2.30)

and its inverse

1 − iw 1 + iw f maps D (resp. ∂D) to C+ (resp. R) with θ iθ f (e ) = tan 2 f −1 (w) =

(9.2.31)

and, of course, f −1 takes C+ to D, and

f −1 (tan(ψ)) = e2iψ

(9.2.32)

Next we want to look at setwise invariances of CJ and CJ± under some fT . It will help to consider first the special case C+ and R considered in Example 9.2.12.


531

Proposition 9.2.15. Let A ∈ GL(2, C). Then fA maps R ∪ {∞} to itself if and only if there is a real matrix B and eiψ ∈ ∂D so that A = eiψ B

(9.2.33)

fA maps C+ to C+ if det(B) > 0 and C+ to C− if det(B) < 0. In particular, A ∈ SL(2, C) maps C+ onto C+ if and only if A ∈ SL(2, R), that is, A has all real entries. Proof. Clearly, if B is real, fB maps R to R and so, by projective equivalence, so does A of the form (9.2.33). Conversely, if f ∈ F maps R ∪ {∞} to itself, let f −1 (0) = w0 , −1 f (1) = w1 , f −1 (∞) = w2 . Then f is given by (9.2.13) and so is fC for C ∈ GL(2, R). Thus, A = λC = eiψ B with B = |λ|C ∈ GL(2, R). If det(A) = 1, the B in (9.2.33) has det(B) real, so eiψ is ±1 or ±i. Thus, det(B) = 1 or det(B) = −1. If B = ( ac db ) ∈ GL(2, R), ai + b fB (i) = ci + d so ad − bc det(B) Im fB (i) = = (9.2.34) 2 |ci + d| |ci + d|2 so taking C+ to C+ (resp. C− ) corresponds to A ∈ SL(2, R) (resp. iA ∈ SL(2, R)). We are now ready for the main theorem on invariance of circles and disks:

Theorem 9.2.16. Let CJ+ be the disk or half-plane described by (9.2.22). Let T ∈ SL(2, C). Then fT is a bijection of CJ+ to itself if and only if T ∗JT = J (9.2.35) + If this happens for CJ and CJ , then T cannot be loxodromic. If T ∗JT ≥ J (9.2.36) + then fT maps CJ into itself. Remarks. 1. (9.2.35) can be rewritten T −1 = J −1 T ∗J

(9.2.37)

T ∗ = JT −1 J −1

(9.2.38)

or 2. The parabolic model (9.2.15) and hyperbolic model (9.2.17) take C+ onto C+ and the elliptic model (9.2.16) takes D onto itself. By conjugacy, we see that any nonloxodromic f ∈ F fixes some disk or half-plane.

532


Proof. By (9.2.29), we see (9.2.35) (resp. (9.2.36)) implies fT maps CJ+ bijectively to CJ+ (resp. maps CJ+ into CJ+ ). For the converse, let 0 i Jr = (9.2.39) −i 0 so h z1 , Jr z1 i = 2 Im z and CJ+ is C+ . For any Hermitean J with det(J) < 0, we can find S ∈ SL(2, C) so J = S ∗Jr S

α 0 0 β

(9.2.40)

For we can find U ∈ SU(2),so U −1 JU = for α > 0, β < 0, and α−1/2 0 ∗ 1 0 then if V = U , then V JV = ( 0 −1 ). Finally, there is a 0 |β|−1/2 0 unitary, W, in SU(2), so W ∗ ( 10 −1 ) W = Jr . Thus, S = W −1 V −1 yields (9.2.40). Given S and T ,

T ∗JT = J ⇔ (STS −1)∗Jr STS −1 = Jr

(9.2.41)

fT maps C+ onto C+ ⇒ T ∗Jr T = Jr

(9.2.42)

Jr T −1 Jr−1 = T t

(9.2.43)

Thus it suffices to show that for T ∈ SL(2, C),

A little calculation proves that for any T ∈ SL(2, C), Thus T ∗Jr T = Jr ⇔ T t = T ∗ ⇔ T¯ = T

⇔ T ∈ SL(2, R) ⇔ fT maps C+ to C+

by Proposition 9.2.15. Finally, if fT has a fixed disk, there is a conjugate of fT fixing C+ , so a conjugate of T in SL(2, R). But the trace of A ∈ SL(2, R) is real, so T cannot be loxodromic. Remark. One reason we will discuss loxodromic maps so rarely is that we will be interested in FLTs which map D to itself. Example 9.2.12, continued. For J of the form (9.2.26), the T ’s that leave D invariant obey 1 0 1 0 ∗ T T = (9.2.44) 0 −1 0 −1 This group is called SU(1, 1).


Proposition 9.2.17. T ∈ SU(1, 1) if and only if a c T = c¯ a¯

533

(9.2.45)

with |a|2 − |c|2 = 1

(9.2.46)

Proof. That a T of the form (9.2.45)/(9.2.46) has det(T ) = 1 and obeys (9.2.38) are straightforward computation. Conversely, if T = ( ab dc ) has determinant 1, then JT −1 J −1 = ( db ac ), so by (9.2.38), we see (9.2.44) implies d = a ¯, c = ¯b. We have not discussed uniqueness of fixed circles because they are not; there is always an infinite family. Theorem 9.2.18. (a) If T is hyperbolic, T fixes all circles (or lines) through the two fixed points and no others. (b) If T is elliptic, T fixes all circles orthogonal to all the circles (or lines) through its two fixed points and no others. (c) If T is parabolic and its fixed point, z0 , is finite, T leaves fixed exactly one straight line which goes through z0 . It leaves fixed all circles through z0 tangent to this line and no other circles. Proof. Without loss, we can assume ∞ is a fixed point and if there is a second, it is zero; essentially we can take the models Theorem 9.2.8. For the statements to be proved are conjugacy invariant. (a) The model is T (z) = az with a real and less than 1. This clearly leaves each straight line through 0 invariant—precisely all “circles” through 0 and ∞. It is not hard to see that no other circle or line is invariant. (b) The model is T (z) = e2iθ z. This leaves every circle centered at 0 invariant and no other circle or line. These circles are precisely the curves orthogonal to all lines through 0 which are the “circles” through 0 and ∞. (c) The model is z → z + 1. Its only invariant lines or circles are Im(z) = 21 a for a real. To see what happens when the fixed point is finite, move it to 0 and note Im( z1 ) = 12 a ⇔ |z + ai |2 = a12 ⇔ 1 |z + ai | = |a| , the circles through 0 tangent to R. There is another way to understand invariance of circles and disks involving the cross-ratio:

534


Definition. If z2 , z3 , z4 are distinct, one defines the cross-ratio by z1 − z3 z2 − z4 (z1 , z2 , z3 , z4 ) = (9.2.47) z1 − z4 z2 − z3

Remarks. 1. One takes the obvious limit if some zj is ∞, for example, (z1 , ∞, z3 , z4 ) = (z1 − z3 )/(z1 − z4 ). 2. There are some obvious covariances, for example, (z2 , z1 , z3 , z4 ) = (z1 , z2 , z3 , z4 ) (z3 , z4 , z1 , z2 ) = (z1 , z2 , z3 , z4 ) 3. Notice that (9.2.13) can be rewritten f (w) = (w, w1, w0 , w2 ) allowing a reinterpretation of cross-ratios. In C2 , we can define the two-form v ∧ w given v, w ∈ C2 . Once one picks ω 6= 0 in ∧2 (C2 ), we can define v × w to be the number v ∧ w = (v × w)ω

For example, with the right choice of ω, (v × w) = v1 w2 − w2 v1

(9.2.48)

(v1 × v3 )(v2 × v4 ) (v1 × v4 )(v2 × v3 )

(9.2.49)

Proposition 9.2.19. Given v1 , v2 , v3 , v4 in C2 , the quantity

is a function only of [vj ]. Moreover, if vj = ratio.

zj 1

, its value is the cross-

Proof. Since v×w is bilinear, (9.2.49) is invariant under vj → λj vj z1 z2 with λj ∈ C. Since, for the choice (9.2.48), 1 × 1 = z1 − z2 , we have the cross-ratio formula. Theorem 9.2.20. For any FLT, f , we have (f (z1 ), f (z2 ), f (z3 ), f (z4 )) = (z1 , z2 , z3 , z4 )

(9.2.50)

Proof. Let f = fT . Since (T v ∧ T w) = det T (v ∧ w), we see T v × T w = det(T )(v × w) and (9.2.46) is immediate from (9.2.49). Theorem 9.2.21. Fix z2 , z3 , z4 distinct. Then {z | (z, z2 , z3 , z4 ) ∈ R} is the unique circle or line containing z2 , z3 , z4 .

(9.2.51)


535

Proof. (z, 1, 0, ∞) ≡ z, so the set in (9.2.51) is precisely the real axis which is the unique circle or line containing 1, 0, ∞. Now use Theorems 9.2.13 and 9.2.20. Remark. One can prove this theorem directly and use it for an alternate proof of Theorem 9.2.13. The last topic in our presentation of FLTs is the study of reflections and the closely related issue of isometric circles and behavior of Euclidean lengths. One of our intermediate goals will be to generalize to the geometry associated with the group, F, the well-known fact about Euclidean geometry that any proper Euclidean motion is a product of two Euclidean reflections. (For two dimensions, the proof goes as follows: If f (z) = z + z0 , f is the product of reflection in the lines Re(z¯ z0 ) = 0 and Re(z¯ z0 ) = 12 |z0 |2 , while a rotation by angle θ is a product of reflections in two lines through the center of rotation with angle θ/2.) We will see an element in F is a product of two FLT reflections if and only if it is not loxodromic. But first we need to define FLT reflections. Definition. An antilinear map on C2 is a map, T , that obeys ¯ for λ ∈ C. T (u + v) = u + v, T (λu) = λu The union of the sets of linear and antilinear invertible maps is a group. An antilinear map preserves lines and so also induces a map ˜ the group of extended FLTs is the set of all maps induced by on P. F, linear and antilinear transformations. c defined on C2 by c uu12 = uu¯¯12 is antilinear; its induced map, which we will also call c, obeys c(z) = z¯ (9.2.52) z z¯ since c 1 = 1 . If T is antilinear on C, A = T c is linear so A has the form ( ac db ), so T = Ac and a¯ z+b (9.2.53) c¯ z+d where we can suppose ad − bc = 1. We will call such maps anti-FLTs. fT (z) =

Proposition 9.2.22. For any three points in C ∪ {∞}, there is a unique anti-FLT that fixes them. It pointwise fixes the circle or line they determine. Its square is the identity. Any circle or line has an anti-FLT that fixes it pointwise. If the circle is |z − z0 | = r, then the map is r2 f (z) = z0 + (9.2.54) z¯ − z¯0

536


Remarks. 1. Especially since we talked about circles left setwise fixed by some f ∈ F, we emphasize in that earlier discussion we meant setwise fixed, that is, z ∈ C ⇒ f (z) ∈ C. Here we mean pointwise fixed, that is, z ∈ C ⇒ z = f (z). 2. The map in (9.2.54) is called the reflection (or inversion) in the circle |z − z0 | = r.

Proof. By conjugacy, we can suppose the three points are 0, 1, ∞. In that case, c leaves them fixed and c2 = 1 and c leaves R pointwise fixed. If L, an anti-FLT, also leaves 0, 1, ∞ fixed, Lc is an FLT leaving 0, 1, ∞ fixed, hence the identity, so L = Lc2 = (Lc)c = c

One could check (9.2.54) by a suitable mapping of the circle to R, but it is easier to note that f is antilinear and r2 f (z0 + reiθ ) = z0 + = z0 + reiθ r eiθ There is a geometric connection between reflections and circles left setwise fixed by the reflections: Theorem 9.2.23. Let R be a reflection in the circle C1 and suppose C2 is another distinct circle. Then R[C2 ] = C2 (as sets) if and only if C1 and C2 intersect in two points and intersect orthogonally. Proof. By a conjugacy, we can suppose C1 = R and R = c. If C1 and C2 intersect not at all or in a single point (including only at ¯ + or −C ¯ + ) and so it ∞), then C2 lies on one side of C1 (i.e., all in C cannot possibly be left invariant by c. Thus C1 and C2 must intersect in two points which, by conjugacy, we can take as 0 and ∞. Thus C2 is a straight line through 0. Such a line is invariant under c if and only if C2 = R or iR, so C2 6= C1 ⇒ C2 is orthogonal to C1 . The following pieces of geometry will be very important in our analysis of the Fuchsian group associated to C \ e:

Proposition 9.2.24. Let f be the reflection in the circle |z − z0 | = r. Let z, w lie outside the disk {u | |u − z0 | ≤ r}. Then |f (z) − f (w)| =

r 2 |z − w| |z − z0 | |w − z0 |

(9.2.55)

Remark. (9.2.55) is always true! We state the result this way to emphasize the size contraction that takes place for distances outside the disk.


537

Proof. By scaling and translation, it suffices to consider the disk where |u| = 1, that is, f (z) = 1/¯ z . Then |z − w| 1 1 |f (z) − f (w)| = − = (9.2.56) z¯ w¯ |z| |w|

We are heading towards proving a map is the product of two reflections if and only if it is nonloxodromic. Let us first look at products of reflections:

Theorem 9.2.25. Let f = R1 R2 be a product of two reflections in circles or lines C1 , C2 . Then (a) If C1 and C2 intersect in two points, f is elliptic. (b) If C1 and C2 intersect in one point, f is elliptic. (c) If C1 and C2 do not intersect, f is hyperbolic. Proof. Again, we have conjugacy conditions so we can move the intersection or other points where it is convenient. (a) Move the intersection points to zero and infinity. C1 and C2 are now straight lines through 0. If they met in angle θ, the product of reflections is rotation by angle 2θ, hence elliptic. (b) Move the intersection point to infinity. Then C1 and C2 are parallel lines. The product of R1 and R2 is translation in the direction perpendicular to these lines by a distance twice the distance between them and is parabolic. (c) By an FLT, we can move C2 to R and C1 to a circle about i of radius r < 1. Let C (n) be the image of |z − i| ≤ r under f n . Two points, z, w, in C (0) go to z¯, w ¯ ∈ C− , so |¯ z − i| ≥ 1 and similarly for w. By (9.2.55), |f (z) − f (w)| ≤ r 2 |z − w|. Thus, if z, w ∈ C (0) , then |f n (z) − f (n) (w)| ≤ r 2n |z − w|. In particular, |f (n+1) (z) − f (n) (z)| ≤ r 2n |f (1) (z) − z| ≤ r 2n+1 . f (n) (z) converges to a point exponentially fast so F is hyperbolic or loxodromic. The ray {z | z = ia, a < 1} is taken into itself under f , so the images of points in the intersection of C (0) and that ray approach the fixed point on the ray from a fixed direction. Hence, f is hyperbolic, not loxodromic. Remark. One can use the calculation in Proposition 9.2.31 below instead of these arguments, but we prefer the geometry. Corollary 9.2.26. A loxodromic map is not the product of two reflections.

538


Theorem 9.2.27. Any nonloxodromic map, f , is the product of two reflections. One of the reflections can be required to be in a circle or line containing any z0 not a fixed point of f . First Proof. By conjugacy, we need only consider the basic models. As we have already seen, rotation about 0 (basic model for elliptic) is a product of a reflection in two lines through 0, one of which can be arbitrary (but it fixes the second). Similarly, z0 → z0 + 1 is the product of reflections in the lines Re z = a and Re z = a + 21 with a arbitrary. r2 Finally, if fj (z) = rj2 /¯ z for j = 1, 2, then (f1 ◦ f2 )(z) = r12 z, showing 2 our hyperbolic model is a product of reflections which can be arranged to contain any point different from 0 and ∞.

This proof is simple; we will give a second proof not for the sake of a second proof, but because it introduces important notions. Euclidean distances, which are not invariant under most FLTs, will be critical here and will be important in Section 9.6. While we have not mentioned it explicitly, we have used in passing ˜ preserve orthogonality of curves; more generally, that elements in F they all locally preserve angles (the f ∈ F are conformal in that they ˜ \ F are anticonformal in that they also preserve orientation; the f ∈ F reverse orientation). Infinitesimal Euclidean lengths scale under f near z0 by a factor |f ′ (z0 )| where f ′ = ∂f /∂z if f is analytic and ∂f /∂ z¯ is f anti-analytic. Here is one important consequence of (anti)conformality: ˜ Let A = {z0 + reiθ | θ0 < θ < θ1 } Proposition 9.2.28. Let f ∈ F. be an arc of the circle, C, of z with |z − z0 | = r. Suppose f −1 (∞) ∈ /C so f [C] is also a circle. Then the angular fraction of (2π) subtended by f [A] in f [C] is Z 2π Z θ1 dθ dθ ′ iθ |f (z0 + re )| |f ′ (z0 + reiθ )| (9.2.57) 2π 2π θ0 0 Proof. Immediate from the fact that angular fractions are ratios of arc length and the fact that f locally scales by f ′ in all directions.

Let

Here is a simple but basic calculation: ˜ not have infinity as a fixed point. Proposition 9.2.29. Let f ∈ F z0 = f −1 (∞)

Then for some r, |f ′ (z)| =

r2 |z − z0 |2

(9.2.58) (9.2.59)


539

Proof. If f is given by (9.2.4) and det ( ac db ) = 1, a straightforward calculation shows 1 (9.2.60) f ′ (z) = (cz + d)2 so (9.2.59) holds with r = c−1 and z0 = − dc , the point that goes into ∞ under f . The proof is identical in the anti-analytic case. The circle {z | |z − z0 | = r} = {z | |cz + d| = 1}

(9.2.61)

is called the isometric circle. Distances inside this circle C expand under f , and outside C they compress. C is precisely the set of points where |f ′ (z)| = 1. Theorem 9.2.30 (Ford’s Theorem, Part 1). Let f ∈ F and let C be its isometric circle. Then f [C] is a circle with the same radius but with center f (∞) and is the isometric circle for f −1 . Let R be the reflection in the circle C, and Q the reflection in the line, which is the perpendicular bisector of the line between f −1 (∞) and f (∞). For any θ, let Aθ be rotation by angle θ about z0 = f −1 (∞), that is, Aθ (z) = z0 + eiθ (z − z0 ). Then for some θ, f = QRAθ

(9.2.62)

Remarks. 1. We will see shortly that if f is nonloxodromic, θ = 0, that is, f = QR. 2. Since Aθ is a product of reflections, we see that any loxodromic f is a product of four reflections. ˜ is the reflection in f [C], then QRQ−1 = R ˜ and f = RQA ˜ 3. If R θ. 4. If f (∞) = f −1 (∞) (but ∞ is not fixed), we have a subtle situation since there is no line to bisect. Here is what is going on. If f = fT with det(T ) = 1 and T = ( ac db ), f (∞) = ac , and f −1 (∞) = − dc , so f (∞) = f −1 (∞) means a = −d or Tr(T ) = 0. Thus, T has eigenvalues ±i and f 2 = id. Let z0 , z1 be the two fixed points of f . Let C be a circle of radius 12 |z1 − z0 | centered at 12 (z0 + z1 ). Let R be the reflection through C and Q through the line through z0 and z1 . Let g = QR which also equals RQ in this case. g 2 is also the identity and g has the same fixed points as f since Q and R leave both points fixed. Thus f = g and C is the isometric circle. 5. Of course, f (∞) = ac and the isometric circle for f −1 is |cz −a| = 1 as can also be seen by inverting the matrix.

540


Proof. Since f is isometric on C, f [C], which is a circle, has the same size circumference, so same radius. Since f −1 maps f [C] isometrically to C, f [C] is the isometric circle for f −1 , so its center is (f −1 )−1 (∞) = f (∞). R maps C isometrically to itself and Q maps C isometrically to f [C]. Thus, QR and f are both isometries of C to f [C], so (QR)−1 f is an isometry of C to itself, hence on C a rotation. −1 Thus for some Aθ , A−1 θ (QR) f leaves C pointwise fixed. Since C has −1 more than two points, Aθ (QR)−1 f = id, so (9.2.62) holds. Proposition 9.2.31 (Ford’s Theorem, Part 2). If z1 ≡ f (∞) 6= f (∞) ≡ z2 and f is given by (9.2.62), then f = fT where T ∈ SL(2, C) and |z1 − z2 | −iθ/2 Tr(T ) = 2 e (9.2.63) 2r In particular, if f is nonloxodromic, then θ = 0 and −1

f = QR

(9.2.64)

Remarks. 1. This provides another proof that any nonloxodromic map is a product of two reflections. Since a preliminary conjugation can take any point to infinity and Q leaves ∞ fixed, we can arrange for any nonfixed point to be on one of the reflection circles. 2. We see once again that if two circles do not intersect, then f is hyperbolic (since then 12 |z1 − z2 | > r and Tr(T ) > 2 by (9.2.63)), tangency means parabolic and intersection means elliptic. Proof. Euclidean transformations preserve length and scalings do not change ratios, so without loss, we can make a preliminary conjugacy so that f −1 (∞) = i, f (∞) = −i, and thus Q is c, complex conjugation. Let r be the radius of the isometric circle which is thus |z − i| = r. With these changes Aθ (z) = i + eiθ (z − i) R(z) = i +

r2 z¯ + i

(9.2.65) (9.2.66)

so f (z) = QRAθ (z) = −i + = fS (z) where

r 2 e−iθ z−i

−i r 2 eiθ − 1 S= 1 −i

(9.2.67) (9.2.68) (9.2.69)


541

det(S) = −r 2 eiθ , so to get T ∈ SL(2, C), we take T = S/(−ireiθ/2 ), which has a trace given by (9.2.63) (since |z1 − z2 | = 2). We emphasize that isometric circles are not preserved by FLTs that are not Euclidean motions, but their geometry can be very useful. If C is the isometric circle of f , its (open) inside, Di , will be called the initial disk and C ≡ Ci the initial circle. Cf ≡ f [C] will be called the final circle and its inside, Df , the final disk. Here is the basic geometry: Theorem 9.2.32. (a) The initial circle, Ci , is mapped by f into the final circle, Cf . The exterior of the initial disk (C\Di ) maps to the final disk, Df , and the initial disk maps to the exterior, C \ Df , of the final disk. For f −1 , just interchange “initial” and “final” in these statements. ¯i ∪ D ¯f. (b) All fixed points of f lie in D (c) In the elliptic case, the two fixed points are the two points in which Ci and Cf intersect. (d) In the parabolic case, the unique fixed point is the point in which Ci and Cf intersect. (e) In the hyperbolic case, the fixed points are symmetric under Q, one lies in Di and one in Df , and they lie on the line segment strictly between the centers of the disks. The attracting fixed point lies in Df and the other in Di . ¯i Remark. We will actually show that fixed points lie in Df ∪ D −1 ¯ f (since f (z) = z ⇒ f (z) = z). (Df ∪ D ¯ i) ∩ and similarly in Di ∪ D (Df ∪ Di ) = Di ∪ Df ∪ (∂Df ∩ ∂Di ). We stated the simpler form in (b) since we analyze in more detail in the other parts. Proof. (a) is immediate from the QR representation (9.2.64) since R maps Di to its exterior and Q maps Di to Df . ¯ i , Rx lies in Di and QRx in Df , so if f (x) = x, (b) If x lies in C \ D ¯ i. x ∈ Df . Thus x ∈ Df ∪ D

(c),(d) If Cf and Ci intersect (but are distinct), the intersection points are also on the line defining Q, so they are left invariant by both Q and R, and so by f . In the parabolic case, where the circles only touch at a single point, there is one fixed point; in the elliptic case, there are two of each. In both cases, the intersections account for all fixed points. (e) In this case, the disks are disjoint. R maps points in Df into Di and then Q back into Df . Thus, knowing all points in Df —except


542

possibly one gets mapped under iteration to the attracting fixed point— guarantees that this fixed point, call it z0 , lies in Df . Since RQ = f −1 , f (Qz0 ) = QRQz0 = Qf −1 (z0 ) = Qz0 , we see Qz0 is also a fixed point, so the other fixed point lies in Q[Df ] = Di . Let w be the point on Ci on the segment from the center of Ci to the center of Cf . Let L be a half-line from w through Cf and off to ∞. R maps this to the segment from w to the center of Ci and Q maps that to the segment from Qw to the center of Df . Thus, this segment is mapped into itself and so, as above, the attracting fixed point must lie in this segment. The argument behind the proof of (b) which says fixed points must ¯f ∪ D ¯ i also shows that if |f (z) − z| is small, z must be close to lie in D ¯f ∪ D ¯ i: D Theorem 9.2.33. Let f ∈ F with f (∞) 6= ∞. Let Di and Df be the initial and final disks. Then either z ∈ Di or dist(z, Df ) ≤ |z − f (z)|

(9.2.70)

Remarks. 1. Since we will talk about another metric in the next section, we emphasize that dist( · , Df ) is here in the Euclidean metric. 2. This implies dist(z, Df ∪ Di ) ≤ |z − f (z)|

¯ f , so (9.2.70) holds. Proof. If z ∈ / Di , then f (z) ∈ D

(9.2.71)

By (9.2.60), we have and

x-ref?

Di = {z | |f ′ (z)| > 1}

(9.2.72)

¯ i = {z | |f ′ (z)| < 1} C\D

(9.2.73)

Remarks and Historical Notes. Given how fundamental FLTs are to so many parts of mathematics, it is unfortunate how little they are discussed in basic courses (which, e.g., don’t discuss the hyperbolic, parabolic, elliptic splitting), and that this discussion doesn’t talk about projective space. The course description of the Riemann sphere is via stereographic projection—admittedly useful—but not as basic as the P point of view. Most of the material in this section is classical (from the nineteenth century), although our discussion has some more modern elements. Key figures in these classical developments are Möbius, Schwarz, Klein, and especially Poincaré. For further discussion of PSL(2, C), see TK.

¨ 9.3. MOBIUS TRANSFORMATIONS

543

The use of isometric circles and the representation f = QR for nonloxodromic transformations was emphasized especially by Ford; see, for example, [134]. ˜ = QRQ, the reflection in the isometric circle for f −1 , then If R 2 ˜ something that can easily be proven directly. It f = QRQR = RR, is simple in various ways to use geometric structures defined by f to get f 2 as a product of reflections. The neat thing about Ford’s idea of using a perpendicular bijector is that is “takes the square root.” 9.3. M¨ obius Transformations In this section, we will discuss FLTs that take D onto D (equivalently, take D into D and ∂D to ∂D). Of course, by Theorem 9.2.13, the FLTs which are bijections of any disk or half-plane are conjugate to bijections of the disk, so this section could also describe analytic bijections of, say, C+ . That said, there are often good reasons to study C+ ( as we will explain in the Notes). But we will need D later, so we x-ref? study these maps in this guise. An FLT which is a bijection of D we will call a M¨ obius transformation. We use M for the family of Möbius transformations. This is nonstandard terminology since “Möbius transformation” is typically used as a synonym for FLT, but it is useful to have a standard term. It will be very useful to have Möbius transformations that map any point in D to any other point. As usual, if we do it for a fixed endpoint, we can do it for any other, for if fz0 takes z0 to 0, then fw−1 f maps z0 0 z0 to w0 . Proposition 9.3.1. Let z0 ∈ D. Then fz0 (z) =

z − z0 1 − z¯0 z

(9.3.1)

maps D onto D and has fz0 (z0 ) = 0.

Proof. f is analytic in {z | |z| < |z0 |−1 } and so in a neighborhood of D. Moreover, |fz0 (eiθ )| = |eiθ −z0 |/|e−iθ −¯ z0 | = 1, so by the maximum principle, f maps by calculating, f−z0 · fz0 = 1 since D into D.2 But 1 −z0 1 z0 1 0 ), so f is an analytic bijection of D. = 1 − |z | ( 0 −¯ z0 1 z¯0 1 0 1 Clearly, fz0 (z0 ) = 0. The second main result that we will need to analyze all Möbius transformations is a general one about analytic bijections, which we don’t know a priori are FLTs restricted to D:

544


Theorem 9.3.2. If f : D → D is an analytic bijection and f (0) = 0, then for some θ ∈ [0, 2π), f (z) = eiθ z

(9.3.2)

Proof. We begin with the Schwarz lemma (Proposition 2.3.4) which implies that |f (z)| ≤ |z|. But since f −1 also maps D to D and f −1 (0) = 0, we have that |f −1 (z)| ≤ |z|. Setting w = f −1 (z), we see |w| ≤ |f (w)|, so |f (z)/z| = 1 on D. By the maximum principle, f (z)/z is constant. Theorem 9.3.3. If f : D → D is an analytic bijection, then f is a Möbius transformation. In fact, if f (z0 ) = 0, then for some θ ∈ [0, 2π), f (z) = eiθ fz0 (z)

(9.3.3)

where fz0 is given by (9.3.1). Proof. f fz−1 maps D onto D and takes 0 to 0, so this follows from 0 Proposition 9.3.1 and Theorem 9.3.2. The remarkable fact about this is that analytic bijections of D automatically have meromorphic continuations to all of P. This is not quite as surprising as it might seem at first. If |zn | → 1, f (zn ) cannot converge to a point, w0 , in D because f (z) near w0 means z must be near f −1 (w0 ), and so must have |z| near |f −1 (w0 )|. Thus, |f (z)| → 1 as |z| → 1. If we knew f had a continuous extension of D to D, then we could extend f to C ∪ {∞} by f (z) = f (1/¯ z)

−1

(9.3.4)

which is trivially meromorphic in D ∪ C \ D and analytic across ∂D by the reflection principle and the fact that |f (eiθ )| = 1. There is a version of the Schwarz reflection principle that only requires that Im g vanishes. That can be applied to i log|f |. In any event, we have (9.3.4) for any Möbius transformation. In the last section, we saw that FLTs could be labelled by three complex variables, f (0), f (1), f (∞), so F has real dimension 6. Here we saw that Möbius transformations are parametrized by one complex variable z0 = f −1 (0) and one real variable, so M is three-dimensional. Moreover, we see M topologically is D × ∂D. iθ By Theorem 9.2.16, any f ∈ M is nonloxodromic. f (z) = e z is iθ/2 0 elliptic (it is fT for T = e 0 e−iθ/2 has det(T ) = 1 and Tr(T ) ∈ 1 −z0 [−2, 2]). fz0 is hyperbolic since it is fT for T = (1 − |z0 |2 )−1/2 −¯ z0 1


545

has det(T ) = 1 and Tr(T ) = 2/(1−|z|2)1/2 > 2. The parabolic example is (1 + i)z − i f (z) = iz + 1 − i 1+i −i (T = i 1−i has determinant 1 and trace 2, and a little calculation shows |f (eiθ )| = 1.) Thus, all nonloxodromic possibilities occur. Here is what one can say about fixed points: Theorem 9.3.4. Let f ∈ M not be the identity. Then (a) If f is elliptic, it has one fixed point at z0 in D and one fixed point in C \ D at 1/¯ z0 . (b) If f is hyperbolic or parabolic, all the fixed points of f lie in ∂D. Proof. By (9.3.4), if f ∈ M has a fixed point z0 , then 1/¯ z0 is also a fixed point, so if there is a fixed point not in ∂D, there is one, call it z0 in D. −1 If f (z0 ) = z0 , then h ≡ g−z f g−z0 maps zero to zero, and so is 0 iθ h(z) = e z which is elliptic, and thus f is elliptic. This proves (b). All that remains is the proof that elliptic elements of M cannot have their fixed points on ∂D. As we have seen, if f has a fixed point off ∂D, it has a second at the reflected point. Thus, if f has a fixed point on ∂D, it must have two. Let g be a map in F that takes these two fixed points to zero and infinity and some other point, z2 , on ∂D to ±1. g thus maps ∂D to R and so if we pick the ±1 for g(z2 ) properly, D maps to C+ . Since h ≡ gf g −1 fixes zero and infinity and is elliptic, it has the form h(z) = eiθ z. No such map takes C+ to C+ , which proves (a). Remark. We will see later (see the discussion after Proposition 9.3.8) a geometric way to understand why parabolic and hyperbolic maps have their fixed points on ∂D. Obviously, if f, g ∈ M are conjugate in M, they are conjugate in F but, in principle (and in practice!), they could be conjugate in F but not in M. Put differently, if C ⊂ F is a class in F and C ∩ M = 6 ∅, C ∩ M is one or more classes in M. Here is the breakdown: Theorem 9.3.5. (a) Each hyperbolic conjugacy class in F intersects M. Two hyperbolic elements in M are conjugate in M if and only if they are conjugate in F. Hyperbolic conjugacy classes in M are labelled by a ∈ (0, 1) with z−a fa (z) = (9.3.5) 1 − az The associated T in SU(1, 1) has Tr(T ) = 2/(1 − |a|2 )1/2 .

546


(b) Each elliptic conjugacy class in F intersects M, and for θ ∈ (0, π/2), its intersection is two classes in M labelled by ±θ. The F-class with θ = π/2 (Tr(T ) = 0) intersects M in a single class of M. All elliptic classes are labelled by θ ∈ ±(0, π/2). An element in the class is fθ (z) = e2iθ z (9.3.6) The associated trace is 2 cos θ. (c) The single parabolic class in F intersects M and the intersection is two classes of M of which representative elements are f± (z) = These have Tr(T ) = 2.

(1 ± i)z ∓ i iz + 1 ∓ i

(9.3.7)

Remark. The f± in (9.3.7) has

n2 in ± 2 1+n 1 + n2 and iterates approach 1 asymptotically tangent to ∂D but from the top (resp. bottom) for f+ (resp. f− ). In F, they are conjugate via g(z) = z −1 , but that maps D to C \ D and is not in M. (n)

f± (0) =

Proof. (a), (c) It is easier to look at the conjugate of M that maps C+ to C+ , that is, SL(2, R). In the hyperbolic case, we can find a conjugate in SL(2, R) that takes any hyperbolic map to one whose fixed points are 0 and ∞ and with 0 the attracting fixed point. The classes in SL(2, R) are thus z 7→ az with a ∈ (0, 1), as they are in SL(2, C). In the parabolic case, we can take the fixed point to infinity. The map is then Tb (z) = z + b with b ∈ R \ 0. By a scaling map in SL(2, R), we can conjugate that to T±1 but T+1 and T−1 are not conjugate in SL(2, R). The conjugacy in SL(2, C) is by z → −z which maps C+ to C− . (b) By conjugating with fz0 , we can suppose the elliptic map has zero as a fixed point, so of the form (9.3.6). For distinct θ’s, these are not conjugate in M, although conjugation with z → 1/z takes fθ to f−θ . Next, we want to discuss the Ford representation when f ∈ M. Note that f ∈ M has f (∞) = ∞ if and only if f (0) = 0, so the condition that f not leave ∞ fixed is f (z) 6≡ eiθ z. Theorem 9.3.6. Let f ∈ M not be a rotation about 0. Then the isometric circle of f has a center outside D and is orthogonal to ∂D. z = 0 lies outside both the initial and final disks for f and on the


547

(Euclidean) perpendicular bisection of the line between the center of Di and Df . f (0) lies in Df . Proof. We know f maps C \ D to itself, so f −1 (∞) ∈ C \ D, which says the center of the circle lies outside D. We know f = fT for T = ( ac¯ a¯c ). Then f −1 (∞) = − a¯c¯ and f (∞) = ac¯ . Since |f −1 (∞)| = |f (∞)|, they are equidistant from 0, which means that 0 lies on the perpendicular bisector of the line between f −1 (∞) and f (∞). Thus, in the Ford factorization of f = QR, Q maps D to D, so R = QF maps D to D. By Theorem 9.2.23, the isometric circle is orthogonal to ∂D. With f = fT and T = ( ac¯ ac ) and |a|2 − |c|2 = 1, we have that |¯ cz + a ¯| = 1 is the isometric circle. Since |¯ c·0+a ¯| = |a| > 1, (if f is not a rotation), 0 is outside Di . Since Df is the initial circle for f −1 , 0 is also outside Df . f (0) ∈ Df since C \ Di is mapped to Df by f (see Theorem 9.2.32). Remarks. 1. There is a quantitative way of seeing that f (0) lies inside Df , namely, since |f (0)| = | ac |, and rf , the radius of Di , is | 1c |, we have rf2 = |f (0)−2 | − 1 (9.3.8)

since |a|2 − |c|2 = 1. 2. This theorem illustrates Theorem 9.2.32. If f is parabolic, Ci and Cf intersect on ∂D (since Ci and Cf are orthocircles). If T is elliptic, Ci and Cf intersect in points inside and outside. If T is hyperbolic, the line from center to center intersects ∂D, giving the fixed point on that line segment. Definition. An orthocircle is a circle or line in C that intersects ∂D in two points with orthogonal intersections. The extended Möbius transformations are those extended FLTs that ˜ Since c is such map D onto D. The set of such maps we denote by M. a map, one easily sees: ˜ is of the form g or gc for some Proposition 9.3.7. Every f ∈ M g ∈ M. A reflection is an extended M¨ obius transformation if and only if the line or circle in which one reflects is an orthocircle. Proof. The first statement is immediate and the second follows from Theorem 9.2.23. One big difference between M and F is that there is a Riemannian metric (on D) that is left fixed by all elements of M, while there cannot be such a metric on P left invariant by all elements of F since:

548


Proposition 9.3.8. If (X, ρ) is a metric space, f : X → X an isometry (i.e., ρ(f (x), f (y)) = ρ(x, y) for all x, y), then there cannot be an x0 and x∞ 6= x0 so that f (n) (x0 ) → x∞ . Proof. Since f is continuous, f (f (n) (x0 )) → f (x∞ ) but f (f (n) (x0 )) = f (n+1) (x0 ), so x∞ is a fixed point. But then ρ(f (n+1) (x0 ), x∞ ) = ρ(f (n+1) (x0 ), f (x∞ )) = ρ(f (n) (x0 ), x∞ ) = · · · = ρ(x0 , x∞ ) 6= 0. Thus f (n) (x0 ) does not converge to x∞ . This contradiction proves the result. Thus, isometries cannot have attracting fixed points, so there is no metric (let alone Riemann metric) on P in which hyperbolic or parabolic maps are isometries. The reason we can define a metric on D in which hyperbolic or parabolic maps are isometries is that the attracting fixed points are not in D (but in ∂D). This will not be a problem because the metric will diverge as we approach ∂D. The following calculation is the key to the invariant metric: Theorem 9.3.9. Let f be an extended M¨ obius transformation. Then 1 − |f (z)|2 (9.3.9) |f ′ (z)| = 1 − |z|2 Proof. If g is an antilinear extended Möbius transformation, then f = cg is in M and |f ′ (z)| = |g ′(z)| and |f (z)| = |g(z)|, so (9.3.9) for f implies it for g, that is, we can suppose f ∈ M, that is, f = fT with a c T = (9.3.10) c¯ a¯ where det(T ) = |a|2 − |c|2 . As we computed in (9.2.60),

1 |¯ cz + a ¯ |2 On the other hand, since (the cross terms cancel) |f ′ (z)| =

we see that

(9.3.11)

|az + c|2 − |¯ cz + a ¯|2 = (|a|2 − |c|2 )(|z|2 − 1) |f (z)|2 − 1 =

(9.3.12) and (9.3.11) imply (9.3.9).

|z|2 − 1 |¯ cz + a ¯ |2

(9.3.12) 2

The standard Euclidean Riemannian structure will be called d z. The Poincaré metric on D is defined to be the one associated to the Riemann structure (1 − |z|2 )−2 d2 z (9.3.13)


549

Put differently, the length of a smooth curve γ : [0, 1] → D is Z 1 L(γ) = |γ ′ (s)|(1 − |γ(s)|2)−1 ds (9.3.14) 0

and

ρ(x, y) = inf{L(γ) | γ(0) = x, γ(1) = y}

(9.3.15)

˜ Then g preserves the Poincaré RieTheorem 9.3.10. Let g ∈ M. mann structure (9.3.13), the length (9.3.14), and the metric (9.3.15). Proof. It suffices to prove preservation of the Riemann structure. Since g is conformal or anticonformal, it preserves angles, so we need only show infinitesimal lengths get mapped properly. The mapping is, of course, by |f ′ (z)|. (9.3.9) is precisely this statement, that is, |df | |dz| = 2 1 − |f | 1 − |z|2

(9.3.16)

The metric has a 12 (1 − |z|)−1 divergence as |z| → 1 whose integral diverges logarithmically, so we expect ρ(0, z) to look like 12 log(1−|z|)−1 as |z| ↑ 1. That is part of the following: The set D with the Poincaré metric is called the D-model of the hyperbolic plane. Theorem 9.3.11. (i) The geodesic from 0 to z ∈ D is the straight line segment between them. (ii) We have that ρ(z, 0) is given by so that as |z| ↑ 1, ρ(z, 0) =

1 2

tanh(ρ(z, 0)) = |z|

log((1 − |z|)−1 ) +

1 2

(iii) For any z, w ∈ D,

tanh(ρ(z, w)) =

log 2 + O(1 − |z|) |z − w| |1 − z¯w|

(9.3.17) (9.3.18)

(9.3.19)

(iv) The geodesics in the D-model of the hyperbolic plane are precisely segments of the orthocircles. Proof. (i) Because the Poincaré metric is conformal, for any curve from 0 to z, if zˆ = z/|z|, then |γ ′ (s)|2 = [Re(γ ′ (s)ˆ z )]2 + [Im(γ ′ (s)ˆ z )]2 ≥ Re(γ ′ (s)ˆ z )2

(9.3.20)


550

that is, the infinitesimal length is larger than its radial component. Since the metric is invariant under rotations, d|γ(s)| 1 ′ |γ (s)| ≥ (9.3.21) 1 − |γ(s)|2 ds with equality only if arg(γ(s)) is constant. This shows the minimal length path has arg(γ(s)) constant, and so it is the straight line. (ii) By (i), γ(s) = sz, so |γ ′ (s)| =

|z| 1 − |γ(s)|2

and thus Z

1

|z| ds ρ(0, z) = = 2 0 1 − |zs| = arctanh(|z|)

Z

|z|

0

dy 1 − y2

d since dy arctanh(y) = (1 − y 2)−1 . This proves (9.3.17). To get (9.3.18), we note (9.3.17) with |z| = r, we have

1 − e−2ρ =r 1 + e−2ρ so (1 − r)

−1

2e−2ρ = 1 + e−2ρ

−1

(9.3.22)

=

1 2

e2ρ +

1 2

(9.3.23)

which implies (9.3.18). (iii) By the invariance of ρ under f ∈ M, ρ(z, w) = ρ(fz (z), fz (w)) w−z = ρ 0, 1 − z¯w so (9.3.17) implies (9.3.19). (iv) The geodesic from z to w is taken into the geodesic from 0 to gz (w) by gz . Thus this geodesic is the image under gz−1 of a diameter, so a segment of an orthocircle. Remark. A convenient way of rewriting (9.3.22) is e−2ρ(0,z) =

1 − |z| 1 + |z|

(9.3.24)


551

Notice that given an orthocircle and a point not on that circle, we can find multiple orthocircles which contain the point but do not intersect the original circle, for by a Möbius transformation, we can suppose the point is 0 and it is obvious that multiple diameters avoid a given orthocircle. That is, if parallel lines mean infinite geodesics which are nonintersecting, Euclid’s fifth postulate fails. This is a homogeneous geometry that is a realization of Lobachevsky’s plane. Analogous to the fact that M is the set of holomorphic bijections of D, we can describe all isometries. Theorem 9.3.12. Let f : D → D be any continuous function which ˜ is an isometry in the Poincaré metric. Then f ∈ M. ˜ are isometries, we see M ˜ Remark. Since we have seen all f ∈ M is the set of all isometries.

Proof. Let f (0) = z0 , f ( 12 ) = w0 . Then (gz0 ◦ f )(0) = 0. Since gz0 ◦ f is an isometry, ρ((gz0 ◦ f )( 21 ), 0) = ρ((gz ◦ f )( 12 ), (gz ◦ f )(0)) = ρ( 12 , 0). Since ρ(w, 0) is a monotone function of |w|, |(gz0 ◦ f )( 12 )| = 12 . Thus, by following gz0 by a rotation about zero, we find h ∈ M, so h ◦ f take 0 to 0 and 12 to 12 . It thus takes the geodesic from 0 to 21 and its continuation setwise to itself, that is, h ◦ f maps (−1, 1) to itself. Since h ◦ f is one-one and continuous, either h ◦ f [C+ ∩ D] ⊂ C+ ∩ D or in C− ∩ D. By replacing h by ch, we can be sure the image is in C+ ∩ D, that is, we can find ˜ so that h∈M (h ◦ f )(0) = 0

(h ◦ f )( 21 ) =

1 2

(h ◦ f )(C+ ∩ D) ⊂ C+ ∩ D

˜ If we prove h ◦ f is the identity, then f = h−1 ∈ M. Let w lie in C+ ∩ D. The two sets S0 = {w1 | ρ(w1 , 0) = ρ(w, 0)} and S1 = {w1 | ρ(w1 , 12 ) = ρ(w, 12 )} are circles (S0 is a circle by (9.3.17) and S1 is an image under a Möbius transformation of a circle about 0, and so a circle). These circles are distinct (look at their real points) and contain w and w. ¯ Since circles can intersect in at most two points, S1 ∩ S0 = {w, w}. ¯ But (h ◦ f )(w) ∈ S1 ∩ S0 and is in C+ so (h ◦ f )(w) = w. Thus, h ◦ f = id on C+ ∩ D and similarly on C− ∩ D and so, by continuity, on D. Next, we want to look at which points in D are closer to z than w. For Euclidean geometry, this is answered by the perpendicular bisector. The same is true here but the bisector is an orthocircle:


552

Theorem 9.3.13. Fix z0 6= z1 both in D. Then {w | ρ(w, z0 ) = ρ(w, z1 )}

is an orthocircle. Removing this orthocircle from D yields two open connected components with z0 and z1 in the two components. In the component with z0 , we have ρ(w, z0 ) < ρ(w, z1 ), and vice-versa within the other. Proof. Suppose first z0 = ia, z1 = −ia with 0 < a < 1 and Im w > 0, w ∈ D. We claim ρ(w, z0 ) < ρ(w, z1 ). By (9.3.19), this is equivalent to

where

|(w − ia)(1 + iaw)| ¯ < |(w + ia)(1 − iaw)| ¯ LHS = A + B RHS = A − B

A = −ia + ia|w|2 A is pure imaginary, so

(9.3.25)

B = w + a2 w¯

|Re(A + B)| = |Re B| = |Re(A − B)|

Since |w| < 1 and |a| < 1, Im A < 0, and since Im w > 0, Im B > 0. Thus |Im(A + B)| < |Im(A − B)|, proving (9.3.25). This proves the result in the special case z0 = ia, z1 = −ia. In general, let w be the geodesic midpoint of the geodesic from z0 to z1 . Let g ∈ M take w to 0. Since it preserves geodesics and hyperbolic lengths, it must map z0 and z1 to equidistant points from 0 on the same line through zero. By a further rotation, we see any pair is equivalent to the special case under a hyperbolic isometry. Corollary 9.3.14. Let r be a reflection in an orthocircle, C. Let w, z be on the same side of C (and not on C). Then ρ(w, z) < ρ(w, r(z))

(9.3.26)

Proof. Since ρ is preserved by γ ∈ M, we can suppose the orthocircle is (−1, 1). Then C is the perpendicular bisector of points equidistant from z, r(z) = z¯, and (9.3.26) is the final assertion of the theorem. Theorem 9.3.15. For any f ∈ M, the hyperbolic perpendicular bisection of the hyperbolic line from 0 to f (0) is the part of the boundary, ∂Df , of the final circle, Df , inside D. Proof. f −1 is the reflection in ∂Df followed by reflection in the line, L, which is the Euclidean bisector of the line between the centers of Df and Di . By Theorem 9.3.6, 0 ∈ L, so for w ∈ D ∩ ∂Df , |f −1 (w)| = |w|

(9.3.27)

9.4. FUCHSIAN GROUPS

553

Since ρ(0, z) is a function of |z| only, we have

ρ(0, f −1 (w)) = ρ(0, w)

(9.3.28)

But since f is a ρ-isometry, ρ(f (0), w) = ρ(0, f −1 (w)) Thus w lies on the hyperbolic perpendicular bisector.

(9.3.29)

Remarks and Historical Notes. Mainly TK. SL(2, Z) TK x-ref? Katok [211] proves Theorem 9.3.13 in the UHP model where the x-ref? calculation is less messy. 9.4. Fuchsian Groups In this section, we will say something about general Fuchsian groups as a preliminary to the study in the two next sections of the ones of interest for finite gap Jacobi matrices. This will hardly be a comprehensive look at the subject—our example, as we will explain in the next two sections, will be infinitely nicer than more typical cases, so we we can avoid discussions of all sorts of subtleties. Our main theme here will be equivalences of various measures of discreteness and of a critical number called the Poincaré index. Given f ∈ M, there are various measures of how “large” f is, that is, how far it is from the identity. We can write f = fT with det(T ) = 1 and use kT k; we can look at (1 − |f (0)|)−1, e2ρ(f (0),0) , or |f ′(0)|−1 , or replace f (0) by f (z) for some other z ∈ D. Our initial goal will be to prove an equivalence in the quantitative sense of upper and lower bounds on ratios. We begin with what happens at a fixed z for a single f: Theorem 9.4.1. Let f = fT lie in M. Then:

2 (1 − |f (z)|) 1 − |z|2

(a)

1 − |f (z)| ≤ |f ′ (z)| ≤

(b)

1 2

(c)

(kT k22 + 2)−1 = 14 (1 − |f (0)|2 )

(1 − |f (z)|) ≤ e−2ρ(f (z),0) ≤ (1 − |f (z)|)

(9.4.1) (9.4.2) (9.4.3)

where det(T ) = 1, k·k2 is Hilbert–Schmidt norm, and ρ is the Poincaré metric. Remark. All norms on 2 × 2 matrices are equivalent, so for any norm, (9.4.3) says 1 − |f (0)| ∼ kT k−2 in the sense that the ratio in either direction is bounded by some constant.


554

Proof. (a) By (9.3.9), 1 − |f (z)| ≤ (1 − |f (z)|)(1 + |f (z)|) = (1 − |z|2 )|f ′(z)| ≤ |f ′ (z)|

and

|f ′ (z)| =

2 1 + |f (z)| (1 − |f (z)|) ≤ (1 − |f (z)|) 2 1 − |z| 1 − |z|2

(b) (9.3.24) implies (9.4.2) if we note that 1 1 ≤ ≤1 2 1 + |f (z)| α γ

(c) If T = ( γ¯ α¯ ), then |f (0)| = | αγ |, so 2 γ 1 2 1 − |f (0)| = 1 − = α |α|2

while

kT k22 = 2|α|2 + 2|γ|2 = 4|α|2 − 2

Corollary 9.4.2. Fix z0 ∈ D and ε > 0. Then {fT | |fT (z0 )| ≤ 1 − ε} is compact in M. Proof. The set is clearly closed. By (9.4.2) and |ρ(f (z0 ), 0) − ρ(f (0), 0)| ≤ ρ(f (z0 ), f (0)) = ρ(z0 , 0), we see 1 − |f (0)| is bounded away from 0 on the set in question. So, by (9.4.3), kT k is bounded above, implying compactness. The following shows all quantities are comparable as z, w run through fixed compact subsets of D: Theorem 9.4.3. For any f ∈ M and z, w ∈ D,

e−2ρ(0,f (z)) ≤ e2ρ(z,w) (9.4.4) e−2ρ(0,f (w)) Proof. By the triangle inequality and the fact that f is a ρisometry, e−2ρ(z,w) ≤

|ρ(0, f (z)) − ρ(0, f (w))| ≤ ρ(f (z), f (w)) = ρ(z, w)

We will soon use Γ for certain countable subgroups of M. But for a while, we will use Γ to denote a countable family in M which need not (yet) be a group. Theorem 9.4.4. Let Γ be a countable subset of M. Then the following are equivalent: (i) For every z0 ∈ D and every r < 1, {f ∈ Γ | |f (z0 )| < r} is finite. (ii) For one z0 ∈ D and every r < 1, {f ∈ Γ | |f (z0 )| < r} is finite.


555

(iii) For every compact subset K ⊂ D and every r < 1, {f ∈ Γ | inf z0 ∈K |f (z0 )| < r} is finite. (iv) For every z0 ∈ D and every η > 0, {f ∈ Γ | |f ′(z0 )| > η} is finite. (v) For one z0 ∈ D and every η > 0, {f ∈ Γ | |f ′ (z0 )| > η} is finite. (vi) For every compact K ⊂ D and every η > 0, {f ∈ Γ | supz0 ∈K |f ′(z0 )| > η} is finite. (vii) For every C, {f ∈ Γ | f = fT , T ∈ SL(2, C), kT k < C} is finite.

Remarks. 1. We don’t include e−2ρ(f (z),0) results since they are trivially equivalent to (i), (ii). 2. For families of FLTs, discreteness of orbits (a condition like (i)) implies an analog of (vii) but not vice-versa; see the Notes. x-ref? 3. If these conditions hold, we say Γ is a discrete family. Proof. Immediate from Theorems 9.4.1 and 9.4.3.

Definition. A Fuchsian group, Γ, is a discrete subgroup of M. Theorem 9.4.4 does not use the quantitative equivalence of Theorems 9.4.1 and 9.4.3. The following does: Theorem 9.4.5. Let Γ be a discrete family in M. Fix s > 0. Then the following are equivalent: X (i) (1 − |f (z)|)s < ∞ for one z ∈ D (9.4.5) f ∈Γ

(ii)

X f ∈Γ

(iii)

X f ∈Γ

(iv)

X f ∈Γ

(v)

X f ∈Γ

(vi)

X f ∈Γ

(vii)

(1 − |f (z)|)s < ∞ for all z ∈ D |f ′ (z)|s < ∞ for one z ∈ D

(9.4.6)

|f ′ (z)|s < ∞ for all z ∈ D e−2sρ(0,f (z)) < ∞ for some z ∈ D

(9.4.7)

e−2sρ(0,f (z)) < ∞ for all z ∈ D

X

T |fT ∈Γ

kT k−2s < ∞

Proof. Again immediate from Theorems 9.4.1 and 9.4.3.

For Fuchsian groups, the series in (9.4.5)–(9.4.7) for z = 0 are, depending on the author, called Poincaré series. The inf over s for

556


which these sums converge is called the critical exponent. If it converges for some s, we will say that s is a Poincaré exponent. Convergence for s = 1 implies Blaschke products Y BΓ (z, w) ≡ bf (w) (z) (9.4.8) f ∈Γ

x-ref? x-ref?

converge where bw (z) is given by (2.3.67). We will also see later that it is important for the groups we consider that the critical exponent is strictly less than 1. Poincaré [TK] used his series to construct automorphic functions; see the Notes TK. Example 9.4.6. Let Γ be a Fuchsian group with a single generator, f . If f is elliptic, it must be periodic to assure discreteness, and all series are finite. If f is hyperbolic, f (n) (z) approaches a limit in ∂D exponentially fast as n → ±∞ (different limits for +∞ and −∞), so 1 − |f (n) (z)| ≤ e−C|n| and the critical index is 0. If f is parabolic, 1 − |f (n) (z)| is O(n−2) and the critical index is 12 . Since this example is a little subtle, let us give the details. Parabolic elements of SU(1, 1) have the form 1 + ia aeiψ T =± ae−iψ 1 − ia

for some a ∈ R and ψ ∈ [0, 2π). The unique eigenvector in this case is (1 −ie−iψ )t . For this T we have (with ± taken to be +) 1 + ina naeiψ n T = nae−iψ 1 − ina

Picking a = 1, e−iψ = i, we have (1 − in)z − in f (n) (z) = (9.4.9) inz + (1 − in) Thus n2 − in f (n) (0) = (9.4.10) 1 + n2 The fixed point is 1 = w∞ and f (n) (0) → w∞ . We have 1 |w∞ − f (n) (0)| = (9.4.11) (1 + n2 )1/2 1 1 − |f (n) (0)|2 = (9.4.12) 1 + n2 As claimed, 1 − |f (n) (0)|2 = O(1/n2) even though the distance to the fixed point is O(1/n). The asymptotic direction is ∂D. This is the phenomenon explained in Example 9.2.10.


557

We are heading towards a proof that if fn is a sequence in M with |fn (0)| → 1, then |fn (0) − fn (z)| → 0 for any fixed z (so orbits in Fuchsian groups will have the same limit points). The idea will be that near ∂D, the Euclidean distance is much smaller than the hyperbolic distance, at least if the hyperbolic distance is not too large. The following expresses this idea quantitatively: Proposition 9.4.7. Let z, w ∈ D. Then ρ(z, w) ≤ while

|z − w| 1 − max(|z|, |w|)2

|z − w| ≤ (1 − max(|z|, |w|)2)ρ(z, w) e4ρ(z,w)

(9.4.13)

(9.4.14)

Proof. The (Euclidean) straight line from z to w is a possible trial geodesic for the hyperbolic metric, so its hyperbolic length bounds ρ(z, w), that is, with γ(t) = tz + (1 − t)w, Z 1 |dγ(t)| 2 −1 ρ(z, w) ≤ ≤ sup [1 − |γ(t)| ] |z − w| 2 t 0 1 − |γ(t)|

which is (9.4.13). Similarly, suppose |z| ≥ |w|. With fz0 given by (9.3.1), the hyperbolic geodesic from z to w is γ(t) = f−z (tfz (w)) z + tζ = 1 + z¯tζ

(9.4.15)

where ζ = fz (w), Using this as a trial for the Euclidean distance, Z 1 |z − w| ≤ |dγ(t)| 0

≤ max(1 − |γ(t)|2 )ρ(z, w)

R1

(9.4.16)

since 0 |dγ(t)|/(1 − |γ(t)|2 = ρ(z, w) since it is the length of the geodesic. By (9.4.15), 1 − |γ(z)|2 = But, by (9.3.24),

(1 − |z|2 )(1 − |tζ|2) 1 − |z|2 ≤ |1 + z¯tζ|2 (1 − |ζ|)2

1 − |ζ| ≥ e−2ρ(ζ,0) = e−2ρ(z,w)

(9.4.16)–(9.4.18) imply (9.4.14).

(9.4.17)

(9.4.18)

558


Remark. The occurrence of e4ρ needed! For if w = 0, then |z − w| (9.3.18)). Thus for ρ(z, w) large, we requires e4ρ ! For the application we harmless.

might be surprising, but it is = |z|, ρ(z, 0) ∼ 12 (1 − |z|) (by must cancel the 1 − |z|2 which want, ρ is bounded and e4ρ is

Theorem 9.4.8. Let {fn }∞ n=1 be a family in M with lim fn (z0 ) = w0 ∈ ∂D for some z0 ∈ D. Then fn (z) → w0

as n → ∞ for each z ∈ D, uniformly in compact subsets of D. First Proof. Since fn is a hyperbolic isometry, ρ(fn (z), fn (ζ)) = ρ(z, ζ), so by (9.4.14), |fn (z) − fn (ζ)| ≤ (1 − |fn (z)|2 )ρ(z, ζ)e4ρ(z,ζ)

(9.4.19)

so |fn (z)| → 1 ⇒ |fn (z)−fn (ζ)| → 0 uniformly on compact subsets.

Second Proof. By Theorems 9.4.1 and 9.4.3, |fn (z)| → 1 implies for any compact K ⊂ D, sup |fn′ (ζ)| → 0

(9.4.20)

ζ∈K

Thus if z, ζ ∈ {η | |η| ≤ r < 1},

|fn (z) − fn (ζ)| ≤ |z − ζ| sup |fn′ (η)| → 0

|η|≤r

Henceforth, we will use Γ to denote only Fuchsian groups and we will generally use the symbol γ for a generic element in Γ. Definition. A point, w0 ∈ ∂D, is called a limit point for Γ if and only if there exists {γn }∞ n=1 ⊂ Γ so γn (0) → w0 . The set of all limit points is denoted by Λ(Γ). A point in ∂D \ Λ(Γ) is called an ordinary point. By Theorem 9.4.8, the limit points are the same if we take limit points of any γn (z) with z ∈ D. By compactness, there are always limit points so long as Γ is not finite—we will discuss this below. Notice that (9.4.21) Λ(Γ) = {γ(0) | γ ∈ Γ} ∩ ∂D and so Λ(Γ) is always closed in C and in ∂D. Also, notice that, as we have seen, γ(0) lies in the final disk for γ and so in the initial disk for γ −1 . Moreover, the radii of these disks go to zero as γ(0) → ∂D. Thus, we have


559

Proposition 9.4.9. For any Fuchsian group, Λ(Γ) = {center of isometric circles for γ ∈ Γ} ∩ ∂D = {center of final disks for γ ∈ Γ} ∩ ∂D

(9.4.22) (9.4.23)

One other immediate result about Γ is Proposition 9.4.10. For any Fuchsian group, the fixed points of all hyperbolic and parabolic elements are limit points. Proof. For parabolic γ, γ n (0) converges to the unique fixed point as |n| → ∞. For hyperbolic γ, γ ±n (0) converges to the two fixed points as n → +∞. Limit points help us understand when we can extend comparison results for γ ′ (z) and γ ′ (0) to z’s in ∂D. Theorem 9.4.11. (a) For all z ∈ ∂D and f ∈ M, we have that |f ′(0)| ≤ 4|f ′(z)|

(9.4.24)

(b) Let K be a compact subset of the ordinary points for Γ. Then there is a constant C < ∞ so that for all γ ∈ Γ, sup{|γ ′ (z)| | z ∈ D, arg z ∈ K} ≤ C|γ ′ (0)|

(9.4.25)

Proof. By (9.2.59), for any f ∈ M with wf ≡ f −1 (∞) the center of the isometric circle, |f ′(0)| |z − wf |2 = |f ′ (z)| |wf |2

(9.4.26)

To get (9.4.23), we note that since |z| = 1 and |wf | > 1 (see Theorem 9.3.6), |z − wf | |z| ≤1+ ≤2 |wf | |wf | To get (9.4.25), let S = {z ∈ D | arg z ∈ K} ∪ {0} By Proposition 9.4.9 and Theorem 9.3.6, {wγ | γ ∈ Γ, γ 6= 1} ∩ S = ∅, so since both sets are closed, d ≡ min(|z − wγ | | z ∈ S, γ ∈ Γ \ {1}) > 0

Since |wγ | = |γ(0)|−1 (see (9.3.4)), we get, by (9.4.26), |γ ′ (z)| 1 = ′ |γ (0)| |z − wγ |2 |γ(0)|2

(9.4.27)

560


so (9.4.25) holds with C = d−2 [inf(|γ(0)|2 | γ ∈ Γ \ {1})]

(9.4.28)

Remark. If there are γ ∈ Γ with γ(0) = 0, γ ′ is constant and we can drop them from consideration in (9.4.28) and earlier in the proof. If | · | is Lebesgue measure of a set in ∂D, we have Z 1 |γ[K]| = |γ ′ (eiθ )| dθ |K| |K| K so, by (9.4.25), if K is disjoint from the limit points, |γ[K]| 1 ≤ C|γ ′ (0)| |γ ′ (0)| ≤ 4 |K| Thus:

(9.4.29)

(9.4.30)

Theorem 9.4.12. Let K ⊂ ∂D be a compact subset of the regular points for a Fuchsian group Γ. Then for each s > 0, X X |γ ′ (0)|s < ∞ ⇔ |γ[K]|s < ∞ (9.4.31) γ∈Γ

γ∈Γ

Next, we want to study possible sets that can be Λ(Γ) for some Γ. For this, it will be useful to note that since Γ is a set of maps each analytic in some neighborhood of D, they define maps of ∂D to ∂D. Clearly, if γn (0) → w0 , then (γ ◦ γn )(0) → γ(w0 ), so this action maps Λ(Γ) onto itself, and since γ is invertible on ∂D, of ∂D \ Λ(Γ) to itself. Here is a key fact:

Lemma 9.4.13 (Three-Point Lemma). Let w0 ∈ Λ(Γ). Let w1 , w2 be points in ∂D so w0 , w1 , w2 are distinct. Then there exists γn ∈ Γ so that either γn (w1 ) → w0 or γn (w2 ) → w0 . Remark. If γ0 is a hyperbolic map with fixed points w0 and w1 and Γ = {γ0n | n ∈ Z}, then there is no γn ∈ Γ so γn (w1 ) → w0 . This shows we need two extra points in general.

Proof. By passing to a subsequence, we can find γn ∈ Γ so γn (0) → w0 and γn−1 (∞) → w3 for some w3 ∈ ∂D. Clearly, since w1 6= w2 , one is distinct from w3 , say, w1 is. Since |γn (0)| → 1, the radius, rn , of the isometric circle of γn goes to zero. Thus for n large, w1 is not in the initial disk. By Theorem 9.2.32, for such n, both 0 and w1 map into the final disk, so |γn (0) − γn (w1 )| ≤ 2rn → 0. We need a final technical result before we can get to a more thorough analysis of Λ(Γ):


561

Lemma 9.4.14. (a) If f ∈ M is elliptic and g ∈ M does not leave fixed the fixed points of f , then gf g −1f −1 is hyperbolic. (b) If f and g are FLTs with g parabolic and f not fixing the point left invariant by g, then f g n is hyperbolic for large n. Proof. (a) By conjugation, we can suppose f (0) = 0 so f = fT iθ 0 with T = e0 e−iθ . Let g = fS where S = ( ac¯ a¯c ). Then T S −1T −1 = a ¯ −e2iθ c and −e−2iθ c¯ a Tr(ST S −1 T −1 ) = 2[|a|2 − sin 2θ|c|2 ] > 2

since θ 6= 0, π (since f 6= 1), |a|2 −|c|2 = 1, and c 6= 0 since g(0) = Thus, gf g −1f −1 is hyperbolic.

c a ¯

6= 0.

(b) By a conjugation, g leaves ∞ fixed, so g = fS with S = ( 10 11 ) and f = fT with T = ( ac db ) where c 6= 0 since f (∞) 6= ∞. We have |Tr(T S n )| = |a + d + cn| > 2 for n large.

Theorem 9.4.15. Let Γ be a Fuchsian group. (a) Λ(Γ) is empty if and only if Γ is a finite cyclic group with an elliptic generator. (b) Λ(Γ) is a single point if and only if Γ is infinite cyclic with a parabolic generator. (c) If Λ(Γ) has at least two points, Γ has hyperbolic elements and the fixed points of these elements are dense in Λ(Γ). Proof. (a) By Proposition 9.4.10, if Λ(Γ) is empty, Γ can only contain elliptic elements. By (a) of the last lemma, those have to have common fixed points, so Γ is a subgroup of a group of two-dimensional rotations. The only such groups that are discrete are the finite cyclic groups. (b) The group cannot contain any hyperbolic elements since they have two fixed points in Λ(Γ). It cannot contain only elliptic elements since it if does, they either have common fixed points, in which case Λ(Γ) is empty, or some distinct fixed points, in which case, there are hyperbolic elements by (a) of the last lemma. Thus Γ has parabolic elements. Those elements must fix the unique point of Λ(Γ). By (b) of the last lemma, it cannot also have elliptic elements because elliptic elements in M do not have fixed points in ∂D, so Γ would have hyperbolic elements.

562


Thus, by the analysis in Example 9.4.6, Γ is a subgroup of {fT | 1+ia aeiψ T = ae−iψ 1−ia with ψ fixed. This group is isomorphic to R under the variable a. All discrete subgroups of R are cyclic. (c) By the analysis above, if Γ has a parabolic element and Λ(Γ) has more than one point, it will have hyperbolic elements guaranteed by (b) of the last lemma. If it has an elliptic element and Λ(Γ) is nonempty, it will have hyperbolic elements generated by (a) of the last lemma. Thus Γ has hyperbolic elements as claimed. If Λ(Γ) has exactly two points, they must be the fixed points of this hyperbolic element—proving the second assertion in this part. If Λ(Γ) has a point, w0 , which is not a hyperbolic fixed point, there must be two hyperbolic fixed points w1 , w2 associated with a hyperbolic element, γ0 . By the three-point lemma, there is γn so γn (w1 ) or γn (w2 ) converges to w0 . But γn (wj ) are the fixed points of the hyperbolic element γn γ0 γn−1 . Theorem 9.4.16. The set Λ(Γ) is one of the following possibilities: (a) The empty set, in which case Γ is a finite cyclic group with an elliptic generator. (b) A single point, in which case Γ is an infinite cyclic case with a parabolic generator. (c) Two points. (d) A nowhere dense, perfect set (aka a Cantor set). (e) The whole circle. Remarks. 1. If case (e), Γ is called a type 1 Fuchsian group. In cases (a)–(d), a type 2 Fuchsian group. 2. We will analyze the specific cases in (c) after the theorem. 3. Recall that a perfect set, S, is a closed set where any x ∈ S is a limit point of points in S \ {x}. Such sets are always uncountable.

Proof. It suffices to show that if Λ(Γ) has three or more points but is not all of ∂D, then it is nowhere dense and perfect. Since it is not all of ∂D, its complement, which is open, has two distinct points, w1 and w2 . If w0 ∈ Λ(Γ), by the three-point lemma, there exist γn so γn (w1 ) → w0 or γn (w2 ) → γ0 . Since γn (∂D \ Λ(Γ)) = ∂D \ Λ(Γ), each γn (wj ) ∈ ∂D \ Λ(Γ) so ∂D \ Λ(Γ) is dense and thus Λ(Γ) is nowhere dense. To see that Λ(Γ) is perfect, let w0 ∈ Λ(Γ). If w0 is not a hyperbolic fixed point, it is a limit of such points by Theorem 9.4.15 and so a limit of other points in Λ(Γ). If w0 is a hyperbolic fixed point for γ ∈ Γ, since Λ(Γ) has at least three points, there is a point, w1 , in Λ(Γ) neither w0 nor the other fixed


563

point of γ, so either γ0n (w1 ) → w0 or γ0−n (w1 ) → w0 . In either case, the points are not w0 (since it is fixed by γ0 ). Thus w0 is a limit of other points of Λ(Γ), and so the set is perfect. Example 9.4.17. We will analyze the case of two limit points. Before beginning, it pays to note that if we classify up to conjugacy, the classes with Λ(Γ) empty are one for each n ∈ Z, the order of the group, since all elliptic elements in M of exact order n are conjugate. For #(Λ(Γ)) = 1, there is a single class since all parabolic elements are conjugate. For calculations, it is easier to use conjugacy from SU(1, 1) to SL(2, R), in which case we can assume the fixed points are 0 and ∞. The only possible T ’s where fT map the set of two fixed points to themselves are a 0 0 b T (a) = S(b) = −1 0 a−1 b 0 with a, b ∈ (0, ∞). One class of discrete examples are infinite cyclic which have a single hyperbolic generator. Since {a, a−1 } is a conjugacy invariant, these are classified by a ∈ (1, ∞) with the group being γn (z) = a2n z, n ∈ Z. Since T (c)S(b)T (c)−1 = S(c2 b) while T (c)T (a)T (c)−1 = T (a), up to conjugacy, we can suppose, if the group is not infinite cyclic, that it contains S(1). We then get a class of groups isomorphic to Z s Z2 with a2n γn+ (z) = a2n z γn− (z) = − z Again, a ∈ (1, ∞) is a conjugacy invariant. Next, we prove two general results about Poincaré indices: Theorem 9.4.18 (Poincaré [340]). For any Fuchsian group, the Poincaré series X |γ ′ (0)|s (9.4.32) γ∈Γ

converges for s = 2.

Proof. By Theorem 9.4.4 (iii), for each r, {γ ∈ Γ | γ(z0 ) = z0 for some |z0 | < r} is finite, so the set of points in D left fixed by some nonidentity γ ∈ Γ is discrete. Pick z0 so γ(z0 ) 6= z0 for all γ ∈ Γ, γ 6= 1. Since {γ(z0 ) | γ ∈ Γ} is discrete, we have δ = min ρ(z0 , γ(z0 )) > 0 γ6=1

(9.4.33)


564

Let

δ Q = w ∈ D ρ(z0 , w) < 2

We claim

(9.4.34)

γ 6= γ ′ ⇒ γ[Q] ∩ γ ′ [Q] = ∅ (9.4.35) ′ ′ For if w ∈ γ[Q] ∩ γ [Q], then ρ(w, γ(z0 )) < δ/2 and ρ(w, γ (z0 )) < δ/2, so ρ(γ(z0 ), γ ′ (z0 )) = ρ(z0 , γ −1 γ ′ (z0 )) < δ violating the definition (9.4.33). Thus (9.4.35) holds. Since the {γ[Q]} are disjoint and lie in D, with vol( · ) the Euclidean volume, X vol(γ[Q]) < ∞ (9.4.36) γ∈Γ

Since γ is conformal and this is two-dimensional volume, Z vol(γ[Q]) = |γ ′ (z)|2 d2 z Q ≥ min |γ ′ (z)|2 vol(Q) z∈Q

≥ C|γ ′ (0)|2

where we use

min |γ ′ (z)|2 ≥ min (1 − |γ(z)|)2 z∈Q

(by (9.4.1))

z∈Q

≥ min e−4ρ(γ(z),0)

(by (9.4.2))

≥ e−4ρ(γ(0),0) A

(by the triangle inequality)

z∈Q

where

(9.4.37)

A = exp −4 max ρ(z, 0) z∈Q

Thus

min |γ ′ (z0 )|2 ≥ Ae−4ρ(γ(0),0) z∈Q

A (1 − |γ(0)|)2 4 A ′ ≥ |γ (0)|2 16

≥

(by (9.4.2)) (by (9.4.3))

verifying (9.4.37). Clearly, (9.4.37) plus (9.4.36) imply the Poincaré series converges for s = 2. Theorem 9.4.19 (Burnside [63, 64]). For any type 2 Fuchsian group, the Poincaré series (9.4.32) converges for s = 1.


565

This will depend on Lemma 9.4.20. Let z0 ∈ ∂D be an ordinary point for a Fuchsian group, Γ. Then there exists δ > 0 so that if then for all γ 6= 1 in Γ.

I = {z0 eiθ | |θ| ≤ δ}

(9.4.38)

γ[I] ∩ I = ∅

(9.4.39)

Proof. If not, then there exist γn ∈ Γ different from 1 and wn ∈ ∂D so that |wn − z0 | < 1/n and |γn (wn ) − z0 | < 1/n. We first claim that 1 − |γn (0)| → 0 (9.4.40) for if not, since Γ is discrete, there is a subsequence γn(j) = γ0 for some γ0 ∈ Γ and then taking n → ∞, γ0 (z0 ) = z0 implying z0 is a limit point (by Proposition 9.4.10). But z0 is, by hypothesis, not a limit point. Thus (9.4.40) holds. By Theorem 9.2.30 and |γn (wn ) − wn | < 2/n, we see either wn ∈ Di (γn ), the initial circle of γn , or wn is within 2/n of the final circle, Df (γn ), of γn . Thus 3 (9.4.41) dist(z0 , Di (γn ) ∪ Df (γn )) ≤ n By Theorem 9.2.32, γn (0) lies in Df (γn ) and γn−1 (0) lies in Di (γn ), so if rn is the radius of Di (γn ), 3 3 dist(z0 , γn (0)) ≤ 2rn + or dist(z0 , γn−1 (0)) ≤ 2rn + n n By (9.4.40), rn → 0 so z0 is a limit point of Γ. This contradiction proves that (9.4.39) holds for some δ. Proof of Theorem 9.4.19. Find I of the form (9.4.14) so that (9.4.39) holds and so that I˜ = {z0 eiθ | |θ| ≤ δ/2} is in the ordinary ′ ˜ ˜ points. As in the proof of Theorem 9.4.19, (9.4.39) implies γ[I]∩γ [I] = −1 ∅ for all γ 6= γ in Γ, so if | · | is now a one-dimensional Lebesgue measure on ∂D, X ˜ ≤ 2π |γ(I)| (9.4.42) γ∈Γ

so (9.4.31) implies (9.4.32) converges for s = 1.

As a final topic in this section, we want to discuss fundamental domains for Γ.


566

Definition. Let Γ be a Fuchsian group. A fundamental domain for Γ is a closed set Ω ⊂ D so that (i)

Ω = Ωint

(9.4.43)

(ii)

γ[Ωint ] ∩ Ωint = ∅ for all γ ∈ Γ, γ 6= 1 [ γ[Ω] = D

(9.4.44)

(iii)

(9.4.45)

γ

Remarks. 1. The term “closed” here means in the relative topology on D, not necessarily closed in C. 2. Thus Ω contains one point from “most” orbits but can contain multiple points from orbits that intersect ∂Ω. 3. In Section 9.6, we will consider “fundamental domains” which are not closed but rather picked so each orbit contains exactly one point in the domain. Definition. Given a point, z0 ∈ D, and Fuchsian group, Γ, the Dirichlet domain Dz0 (Γ) is defined by Dz0 (Γ) = {w ∈ D | ρ(w, z0 ) = inf ρ(w, γ(z0 ))} γ∈Γ

(9.4.46)

We will also let ◦

D z0 (Γ) = {w ∈ D | ρ(w, z0 ) < ρ(w, γ(z0 )) for all γ ∈ Γ, γ 6= 1} (9.4.47) We define I = {z0 ∈ D | ∃ γ 6= 1, γ ∈ Γ, γ(z0 ) = z0 }, which we have proven earlier, is always a discrete set. Most Dirichlet domains are fundamental: Theorem 9.4.21. For any z0 ∈ D \ I and r < 1, there is a finite set, S, of γ ∈ Γ so that \ Dz0 (Γ) ∩ {z | |z| ≤ r} = {w | ρ(w, z0 ) ≤ ρ(w, γ(z0 ))} (9.4.48) γ∈S

◦

Dz0 is the interior of Dz0 (Γ) and is dense in Dz0 (Γ). Dz0 (Γ) is a fundamental domain. Proof. Fix z0 and r. By discreteness, the set S with γ min ρ(γ(z0 ), w) ≤ max ρ(z0 , w) |w|≤r

is finite, so if γ ∈ / S,

|w|≤r

{z | |z| < r} ⊂ {w | ρ(w, z0 ) ≤ ρ(w, γ(z0 ))}

and therefore (9.4.48) holds.

9.5. COVERING MAPS FOR MULTICONNECTED REGIONS

567

RHS of (9.4.48) is a subset of D bounded by a finite number of arcs from {z | |z| = r} and arcs from the orthocircles which, by Theorem 9.3.13, are the set of points equidistant from z0 and γ(z0 ) for some γ ∈ S, γ 6= 1, and the interior of Dz0 (Γ) ∩ {z | |z| ≤ r} is the “inside” ◦

of this boundary curve. It follows that D z0 (Γ) given by (9.4.47) is the ◦

interior of Dz0 (Γ) and is dense in it. Clearly, (9.4.44) holds for D z0 (Γ) and (9.4.45) for Dz0 (Γ). Finally, we can describe D0 (Γ) in terms of isometric circles: Theorem 9.4.22. For γ ∈ Γ, a Fuchsian group so 0 ∈ / I, let Di (γ) be the (open) initial disk and Df (γ) be the (open) final disk. Then \ D0 (Γ) = (D \ [Df (γ)]) (9.4.49) γ∈Γ γ6≡1

=

\

γ∈Γ γ6≡1

(D \ [Di (γ)])

(9.4.50)

Remark. Because of (9.4.50), D0 (Γ) is sometimes called the Ford fundamental domain. Proof. Since Di (γ) = Df (γ −1 ), (9.4.49) is equivalent to (9.4.50). By Theorem 9.3.15, D \ Df (γ) = {z | ρ(z, 0) ≤ ρ(z0 , γ(0))}

Thus, by (9.2.73), D0 (Γ) =

\

γ6≡1

{z ∈ D | |γ ′ (z)| < 1}

(9.4.51)

Remarks and Historical Notes. TK

x-ref?

9.5. Covering Maps for Multiconnected Regions Our main goal in this section is to discuss the following result: Theorem 9.5.1. Let e be a closed subset of the Riemann sphere C ∪ {∞} so that S+ ≡ (C ∪ {∞}) \ e (9.5.1) is connected. Suppose that e contains at least three points. Then there exists a Fuchsian group, Γ, and function x : D → S+

(9.5.2)

568


which is locally one-one, that is, x′ is everywhere nonvanishing, and so that x(z) = x(w) ⇔ ∃ γ ∈ Γ with γ(z) = w (9.5.3)

Remark. x′ everywhere nonvanishing at (9.5.3) implies each γ ∈ Γ is parabolic or hyperbolic. We will provide a proof of the special case when e has a component with more than one point

(9.5.4)

Of course, since components are connected, such a component is uncountable if not a single point. In our applications, e is a finite union of nontrivial closed intervals in R, so (9.5.4) holds. In the Notes, we will discuss the general case. As we also explain there, the conclusion of the theorem fails if e has only one or two points. The proof and interpretation of the theorem depend on the theory of covering spaces, which in turn relies on the theory of fundamental groups. We assume familiarity with the necessary homotopy theory; see the Notes. We will provide a synopsis of the main parts of the covering space theory that we need. Definition. Let X be an arcwise connected space. A covering space is an arcwise connected space, Y, and a map, f (the covering map), f : Y → X so that Ran(f ) = X, and for every x ∈ X, there is an open arcwise connected neighborhood, U, of x so that f −1 [U] is a union of disjoint arcwise connected sets {Uα }α∈A with f a homeomorphism of Uα and U. (i) It is not hard to see that any continuous curve in X, γ : [0, 1] → X can be lifted to Y, that is, for any y0 ∈ f −1 (γ(0)), there is γ˜ : [0, 1] → Y so f ◦ γ˜ = γ and γ˜ (0) = y0 . This lift is unique. Similarly, any homotopy in X can be lifted to Y. (ii) Pick base points, x0 and y0 , in X and Y with f (y0) = x0 . Any closed loop, γ, in Y with γ(0) = γ(1) = y0 is mapped into one, f ◦ γ in X. The fact that homotopies lift shows that on the level of equivalent classes, this map is injective, that is, f∗ : π1 (Y, y0 ) → π1 (X, x0 ) maps one-one to an image subgroup GY ≡ f∗ [π1 (Y, y0)]. Let y1 ∈ f −1 [{x0 }] ≡ FY and let γ be a curve with γ(0) = y0 , γ(1) = y1 . Then f ◦ γ is a closed loop in X and the lifting of homotopies shows it is nontrivial in π1 (X, x0 ) if y1 6= y0 . Indeed, one shows this association of points in FY into classes of elements of π1 (X, x0 ) is a bijection of FY and left cosets of GY , that is, to π1 (X, x0 )/GY . f1

f2

(iii) Two covers Y1 −→ X, Y2 −→ X are called isomorphic if there is a homeomorphism, Q : Y1 → Y2 so f2 ◦Q = f1 . The analysis in (iv) shows


569

this happens if and only if GY1 = GY2 and that, then, Q is uniquely determined (to make this precise, one needs to speak of spaces with distinguished points, so y1 ∈ Y1 , y2 ∈ Y2 , fj (yj ) = x0 , and Q(y1 ) = y2 ). Moreover, if any x ∈ X has a simply connected neighborhood, then every subgroup, G, of π1 (X, x0 ) enters as some GY . Thus, in that case, there is a one-one correspondence between subgroups of π1 (X, x0 ) and equivalence classes of covering maps. (iv) In particular, if we demand GY = {1}, so π1 (Y, y0 ) = {1}, we get a distinguished cover called the universal covering space, which is equivalent to any simply connected cover, that is, cover with π1 (Y, y0 ) = {1}. (v) Each element [γ] of π1 (X, x0 ) induces a map τ[γ] on Y, called the deck transformation that obeys f ◦ τ[γ] = f

(9.5.5)

determined by also requiring if γ is a loop in X, with γ(0) = γ(1) = x0 , its lift γ˜ with γ˜ (0) = y0 has γ(1) = τγ (y0 ). τγ is the identity map if and only if [γ] ∈ GY and any other τ[γ] leaves no points fixed. Thus, π1 (X, x0 )/GY acts simply on Y and orbits {τ[γ] (y) | γ ∈ π1 (X, x0 )} are all of f −1 [{f (y)}]. In particular, if Y is the universal cover, π1 (X, x0 ) acts transitively on each f −1 [{f (y)}] and f (y1 ) = f (y0 ) ⇔ ∃ [γ] ∈ π1 (X, x0 ) s.t. τ[γ] (y1 ) = y0

(9.5.6)

(vi) If X is a connected Riemann surface, that is, a one-dimensional complex manifold with a distinguished set of charts whose transition functions are all analytic, then the fact that covering maps are local f homeomorphisms allows one to make any cover Y −→ X into a Riemann surface in such a way that f is analytic. It is then easy to see that the deck transformations are bianalytic homeomorphisms of Y. (vii) By combining the uniqueness of universal covering spaces with the local analytic structures, one sees that if f, g : D → X are both analytic covering maps, then there is a Möbius transformation, h, with f ◦h=g

(9.5.7)

for uniqueness implies there is a homeomorphism h, and the fact that f locally has an analytic inverse implies h is analytic locally, and so analytic globally. The relevance of this to Theorem 9.5.1 is now clear. The fact that Γ is a Fuchsian group, where each γ ∈ Γ (γ 6= e) has no fixed points, lets one find for any z0 ∈ D, a disk, D, about z0 so {γ(D)}γ∈Γ are disjoint, which implies that x is a covering map. Since D is simply

570


connected, it is the universal cover. On the other hand, if the universal cover is D, the covering map can be taken as x and the family of deck transformations as Γ. Thus, Theorem 9.5.1 is equivalent to the statement that as Riemann surfaces, the cover of S+ is D. One proof of the theorem (discussed in the Notes) relies on the fact (due to Poincaré) that the only simply connected Riemann surfaces are D, C, and the Riemann sphere. Instead, we use a proof going back to Rado [349] which is based on the usual proof of the Riemann mapping theorem. We begin by describing that proof not merely because we will use the Riemann mapping theorem in our proof but because most of the steps in the proof of the Riemann mapping theorem are identical to steps in the proof we will give of Theorem 9.5.1. One downside is that, because it relies on a compactness argument, our proof is not constructive. Consider three properties of a connected Riemann surface: (i) The surface is topologically simply connected in the sense that any closed curve (with base point) is homotopic to the trivial curve. (ii) Contour integrals around closed contours of functions analytic on the surface are zero; we call such surfaces holomorphically simply connected. (iii) When the surface is a subset of C ∪ {∞}, its complement is connected. It is fairly easy to see that (ii) ⇔ (iii) (see the reference in the Notes) and that (i) ⇒ (ii). Theorem 9.5.2 (Riemann Mapping Theorem). Let Ω ⊂ C∪{∞} be a connected open region so that C ∪ {∞} \ Ω has at least two points and so that Ω is analytically simply connected. Then there is an analytic bijection h : D → Ω. This theorem then implies (ii) ⇒ (i). We can suppose without loss, by a preliminary fractional linear transformation, that ∞ ∈ / Ω and thus, that Ω & C. Instead of constructing h, we construct its inverse, so pick z0 ∈ Ω finite and define

R = {f : Ω → D | f (z0 ) = 0, f ′ (z0 ) > 0, f (z) = f (w) ⇒ z = w} (9.5.8) We will prove: (a) R is nonempty. (b) R ∪ {f ≡ 0} is compact. (c) Given f ∈ R, if Ran(f ) 6= D, then there exist g ∈ R and ϕ : D → D, ϕ(z) 6≡ z so that f =ϕ◦g (9.5.9)


571

(d) These imply Theorem 9.5.2 Simple connectedness comes in via: Lemma 9.5.3. Let f be an analytic function on a holomorphically simply connected region or simply connected and connected Riemann surface, Ω, which is everywhere nonvanishing. Then there is an analytic function, g, on Ω with g 2 = f . √Remark. There are exactly two such g’s. We will write them as ± f. Proof. Pick z0 ∈ Ω and α0 so α02 = f (z0 ). Define Z z ′ f (w) 1 dw g(z) = α0 exp 2 z0 f (w)

(9.5.10)

Since f is nonvanishing, f ′ /f is analytic on Ω and, by the holomorphic simple connectivity, the integral is a single-valued analytic function equal to (a branch of) log[f (z)/f (z0 )]. We can now do step (a). Lemma 9.5.4. R is nonempty if Ω & C. Proof. Pick w0 ∈ / Ω. Thus, f (z) = z − w0 is nonvanishing and, by Lemma 9.5.3, we can find an analytic function, g(z), so g(z)2 = f (z). If g(z1 ) = ±g(z2 ), then z1 − w0 = z2 − w0 , so z1 = z2 . Thus, if w1 = g(z1 ), then −w1 ∈ / Ran(g). Since g is analytic and nonconstant, Ran(g) is open, so for some δ, {w | |w − w1 | < δ} ⊂ Ran(g). By the above, {−w | |w − w1 | < δ} ∩ Ran(g) = ∅, that is, on Ω, 1 1 g(z) + w1 ≤ δ

so h(z) ≡ δ/(g(z) + w1 ) maps Ω to D. By composing h with a suitable Möbius transformation, we get F taking Ω to D with F (z0 ) = 0 and F ′ (z0 ) > 0. Since h is one-one, so is F . Next we do step (c). Lemma 9.5.5. Let f ∈ R and suppose there exists w0 ∈ D \ Ran(f ). Then there exists g ∈ R and ϕ : D → D so ϕ(z) 6≡ z, and so (9.5.9) holds. In particular (with strict inequality), g ′ (z0 ) > f ′ (z0 ) Proof. Let T1 (z) =

z − w0 1−w ¯0 z

(9.5.11) (9.5.12)


572

so T1 ◦ f is nonvanishing, and thus we can pick a branch of p H(z) ≡ T1 (f (z)) √ which maps to D since w ∈ D ⇒ ± w ∈ D. Let T2 (z) =

|H ′(z0 )| z − H(z0 ) H ′(z0 ) 1 − H(z0 )z

(9.5.13)

(9.5.14)

(note that H is one-one, so H ′ (z0 ) 6= 0) and let g(z) = (T2 ◦ H)(z)

(9.5.15)

ϕ(z) = T1−1 ((T2−1 (z))2 )

(9.5.16)

Define so ϕ : D → D and ϕ′ (T2 (0)) = 0 so ϕ(z) 6≡ z. By construction, (9.5.9) holds, so by the chain rule, f ′ (z0 ) = ϕ′ (0)g ′(z0 )

(9.5.17)

By (9.5.15), g ′(z0 ) = T2′ (H(z0 ))H ′ (z0 ) = so

|H ′(z0 )| 1 − |H(z0 )|2

ϕ′ (0) > 0

(9.5.18) (9.5.19)

Since ϕ is analytic in a neighborhood of D and maps D to D, Z 2π dθ ′ (9.5.20) ϕ (0) = e−iθ ϕ(eiθ ) 2π 0

has |ϕ′ (0)| ≤ 1 with equality only if ϕ(z0 ) = cz for |c| = 1, inconsistent with ϕ′ (T2 (0)) = 0. Thus, ϕ′ (0) < 1 (9.5.21) so, by (9.5.17), we have (9.5.11).

Remarks. 1. The square root is used in constructing g to be sure that the inverse is two-to-one so ϕ′ (0) < 1. 2. Rather than rely on a theoretical proof that (9.5.11) holds, one can do a direct calculation to find that 1 + |w0 | ′ g ′ (z0 ) = p f (z0 ) (9.5.22) 2 |w0 | We prefer the indirect argument rather than rely on a calculation that “happens to work.”


573

Proof of Theorem 9.5.2. Let fn be a sequence of functions in R that converge to some function, f , uniformly on compact subsets of Ω. Then for each fixed w ∈ Ω, fn (z) − fn (w) has a single zero at z = w, so by Hurwitz’s theorem, either f has the same property or else f (z) ≡ f (w), so f ≡ 0 (since f (z0 ) = 0). Thus, either f ≡ 0 or f ∈ R. It follows that R ∪ {0} is closed in this topology. By Montel’s theorem, R is compact in this topology of uniform convergence on compact subsets since R ⊂ {f | kf k∞ ≤ 1}

By compactness and by continuity of f 7→ f ′ (z0 ), we can find f0 ∈ R with f0′ (z0 ) = sup {f ′(z0 )} (9.5.23) f ∈R∪{0}

Since R is not empty, > 0 and f0 ∈ R. If f0 is not onto D, we can, by Lemma 9.5.5, find g ∈ R so g ′ (z0 ) > f0 (z0 ), violating (9.5.23). Thus, f0 is a bijection and h = f0−1 provides the required map of D to Ω. f0′ (z0 )

Proof of Theorem 9.5.1 when (9.5.4) holds. Let π : U → S+ be the universal covering space of S+ . As before, suppose ∞ ∈ / S+ and pick some z0 ∈ U. We will define R = {f : U → D | f (z0 ) = 0, f ′ (z0 ) > 0, f (z) = f (w) ⇒ π(z) = π(w)}

Step (c) in the earlier strategy holds without any change—the argument that proves Lemma 9.5.5 only needed Ω simply connected and U is simply connected. Step (b) is also essentially unchanged: Montel’s and Hurwitz’s theorems remain true on U, and if fn ∈ R and w is fixed, the zeros of fn (z) − fn (w) are contained in π −1 [{π(w)}]. That leaves step (a). In the general case, this requires an argument exploiting the elliptic modular function; see the Notes. Given our assumption (9.5.4), it is easy. Let e1 be the assumed component with more than one point. Then C ∪ {∞} \ e1 ⊃ S+ and is simply connected. So, by the Riemann mapping theorem (indeed, by part (a) of its proof!), there is a one-one f0 : C ∪ {∞} \ e1 → D with f0 (z0 ) = 0, f0′ (z0 ) > 0. Let f = f0 ◦ π, so f ∈ R. Following the proof of Theorem 9.5.2, we see that there exists f : U → D which is onto and in R. For all z ∈ D, π is constant on f −1 [{z0 }], so we can define x to be this common value. By construction, f is locally one-one and π is locally one-one, so x is locally one-one. For given w ∈ D, pick z in U with g(z) = w and a neighborhood U of z and which g and π are one-one. So on g[U], f = g −1 ◦ π is one-one.

574


Given z0 in S+ , let U be a connected open neighborhood of z0 so π −1 (U) is a collection of connected open sets, {Uα }α∈A , which are disjoint in U so that π is a homeomorphism on each Uα to U. Thus, if α, β ∈ A, there is a unique homeomorphism παβ : Uα → Uβ to π ◦ παβ = π. We claim {z ∈ Uα | f (παβ (z)) = f (z)} is both open and closed. By continuity of f and π, it is obviously closed. On the other hand, suppose that f (παβ (z)) = f (z) and that zn → z has f (παβ (zn )) 6= f (zn ). Then since x is locally one-one, x(f (παβ (zn ))) 6= x(f (zn )) for n large. But x(f (zn )) = π(zn ) and π(παβ (zn )) = π(zn ), so we have a contraction. Thus, for each α, β, either f ◦ παβ = f on Uβ or f [Uα ] and f [Uβ ] are disjoint. This shows that x is a covering map. Thus, the fundamental group of S+ acts as a Fuchsian group and (9.5.3) holds. Remarks and Historical Notes. The Riemann mapping theorem appeared in 1851 in Riemann’s inaugural dissertation [359], but its proof depended on ideas (which he called Dirichlet’s principle) that at the time were not rigorous and even now rely on regularity of the boundary. The first general proof was found by Osgood [324] in 1900 (see Walsh [444] for Osgood’s proof in modern language). Osgood was isolated in the U.S. and his proof not widely noted—the now standard proof which we give here is based on ideas of Carathéodory [68] and Koebe [232, 233] in 1912–1915. The uniformation theorem, sometimes called the Poincaré or Klein– Poincaré theorem, states that every simply connected Riemann surface is analytically equivalent to one of three standard models: the Riemann sphere, C, or D. It is due to Poincaré [341] based in part on results of Klein [225], with important clarifications by Koebe [228, 229, 230, 231]. As we have discussed, the fundamental group acts on the universal cover of a Riemann surface as a group of analytic isomorphisms with no fixed points. The Riemann sphere has no analytic isomorphisms with no fixed points, so it is not the universal cover of any surface but itself. The only analytic isomorphisms of C with no fixed points are for the form z → z + a for some a in C. The only discrete subgroups are isomorphic to Z or to Z2 . The quotient by Z2 is a torus, and by Z a cylinder which is the same as the once punctured plane. All other Riemann surfaces have D as universal cover, providing one proof of Theorem 9.5.1. This also shows that Theorem 9.5.1 fails if e has only one or two points. The idea of using the standard proof of the Riemann mapping theorem that we use to prove Theorem 9.5.1 is from Rado [349] (see also [175]) who says he used in part ideas of Fejér and F. Riesz.

9.6. THE FUCHSIAN GROUP OF A FINITE GAP SET

575

To get the full version of Theorem 9.5.1, one needs to use the elliptic modular function, λ(τ ), defined on the upper half-plane, C+ (see, e.g., Ahlfors [7] for the definition and proof of properties). Let Γ be the group of fractional linear transformations induced by the elements of SL(2, C), ( ac db ) where a, d are odd integers and b, c are even integers. Then λ(τ ) = λ(τ ′ ) ⇔ ∃ γ ∈ Γ s.t. γ(τ ) = τ ′ (9.5.24) and Ran(λ) = C\{0, 1}. Since C+ is simply connected (and analytically isomorphic to D), this provides an explicit model where the set e in (9.5.1) is {0, 1, ∞} = e0 . For general e with at least three points, by a fractional linear transformation, we can suppose e0 ⊂ e. For any z0 ∈ C \ e, find w0 ∈ C+ with λ(w0 ) = z0 and let f be a local inverse of λ, defined originally near z0 (by (9.5.24) and the fact that nonidentity elements in Γ have no fixed points, λ′ is everywhere nonvanishing). Using (9.5.24), it is not hard to see that f can be continued along any curve in S+ , although it will be a multivalued function on S+ . On U, it defines a single-valued function by the monodromy theorem (see Ahlfors [7]). By construction, λ ◦ f (z) = π(z) so f on U obeys f (w) = f (z) ⇒ π(z) = π(w)

(9.5.25)

By composing f with a suitable fractional linear transformation mapping C+ to D, we get an element in R. The rest of the proof is then unchanged. For books on basic topology—fundamental group and covering spaces—see [139, 24, 305, 439]. For background in complex analysis (such as Montel’s and Hurwitz’s theorems), see Ahlfors [7], Stein– Shakarchi [410], and Lang [260]. It is interesting to see how starting with f : S+ → D (but not onto) constant on the fibers π −1 [{z0 }], we get an f which is a bijection of U and D. In step (c), when we take the square roots, we essentially halve the set of points of points where f has a given value. 9.6. The Fuchsian Group of a Finite Gap Set We specialize to e, a finite gap set of the form (5.12.1). We normalize the covering map x : D → C ∪ {∞} \ e ≡ S+ by requiring x(0) = ∞

lim zx(z) > 0

z→0 z6=0

(9.6.1)

By Theorem 9.5.1, there is a unique such map and an associated Fuchsian group, Γ, which is isomorphic to π1 (S+ ), and so a free nonabelian group on ℓ generators. Since any γ ∈ Γ acts freely on D, 0 ∈ / I = ∅,

576


so there is an associated Ford fundamental domain, D0 (Γ). Our goal in this section is to study the group, Γ, and the fundamental domain, D0 (Γ). In particular, we will prove a theorem critical to step-by-step sum rules that the Poincaré critical exponent is strictly smaller than 1. We will begin by analyzing a fundamental domain, F , which will ◦

turn out to be essentially D0 (Γ) (more precisely, F int will be D 0 (Γ)). Consider in S+ , P ≡ C ∪ {∞} \ [α1 , βℓ+1 ], that is, we remove the ℓ gaps, ∪ℓj=1 (βj , αj+1), from S+ . P is connected and simply connected. For any z0 ∈ P, all curves γ : [0, 1] → P with γ(0) = ∞ and γ(1) = z0 are homotopic, so the lift to the universal cover, γ˜ , with γ˜ (0) = 0 ∈ D has γ˜ (1), the same for all such γ’s. This allows us to define a unique branch of x−1 on P whose range is connected and contains 0. The image of this branch we will call F int (for now, int is a symbol; later it will be the interior of F ). Thus, F int is a connected open subset of D for which x is a bijection of F int and P. Consider first what x does to (−1, 1). Since S+ is invariant under complex conjugation, x(¯ z ) is also a locally bijective map of D to S+ , which clearly obeys (9.6.1). Thus, by uniqueness, x must obey x(¯ z ) = x(z)

(9.6.2)

Thus, x, and so x′ , is real on (−1, 1) \ {∞}. By (9.6.1), x′ (w) < 0 for w real and near zero, so since x′ is never zero or ∞, we see that x′ (w) < 0

if w ∈ (0, 1) ∪ (−1, 0)

(9.6.3)

Thus, x maps (0, 1) to a part of (βℓ+1 , ∞) in a monotone decreasing way, so lim x(w) = ∞ (9.6.4) w↓0

We claim that lim x(w) = βℓ+1 w↑1

(9.6.5)

for if the limit (which exists by monotonicity) were some y > βℓ+1 , we would be unable to lift the curve in P that runs from ∞ to y. Thus, one inverse image of the curve in S+ that runs from βℓ+1 up to ∞ and then from ∞ up to α1 is exactly (−1, 1) (run from 1 to −1). By the action of the Fuchsian group, the other inverse images are images of (−1, 1) under Möbius transformations, and so a set of orthocircles. Pick some point z0 in the gap (β1 , α2 ). There is a covering map ˜ : D → S+ with x ˜ (0) = z0 and x ˜ ′ (0) < 0. As above, this map must take x (−1, 1) onto (β1 , α2 ) and all inverse images of (β1 , α2 ) are orthocircles. ˜ are related by a Möbius transformation by Remark (vii) But x and x


577

in the last section. Thus, under x−1 also, all images of (β1 , α2 ) are orthocircles. We have thus proven: Proposition 9.6.1. The inverse images under x of any gap (βj , αj+1 ), j = 1, . . . , ℓ, or of (βℓ+1 , ∞) ∪ {∞} ∪ (∞, α1 ) are a family of orthocircles. Note that since x′ (z) < 0 for w ∈ (−1, 1), near (−1, 1), x(z) reverses the sign of Im z. By continuity, one sees that x−1 maps P ∩ C+ onto D ∩ C− and P ∩ C− onto D ∩ C+ . Consider now what happens as z ∈ P ∩ C− approaches a gap. Since x is a covering map, x−1 has a limit which lies in an inverse image in a gap—thus, in an orthocircle that lies entirely in D ∩ C+ . By (9.6.2), as we approach the gap from the other side, x−1 goes to the conjugate orthocircle. The boundary of F int is thus 2ℓ orthocircles. Since there are bands between gaps, these orthocircles are a finite distance apart. We thus have shown that Proposition 9.6.2. In D, the topological boundary of F int consists of ℓ orthocircles in C+ and their complex conjugates. There is a finite distance in D between the ends of distinct orthocircles. We will use C1+ , . . . , Cℓ+ to denote the orthocircles in C+ ∩D, labelled going clockwise. We let Cj− = Cj+ be their conjugates. Cj± are arcs e± and call complete orthocircles. of full circles, which we denote by C j ± ± ± ^ e ∩ D. Notice also that γ(C e± ) = γ(C Thus, C = C ). j

j

j

j

Thus, there are 2ℓ orthocircles and their interiors removed to get int F . Figure 9.6.1 shows the way this looks for a case with ℓ = 2. The shaded region is the inverse image of P ∩ C− .

Figure 9.6.1. The fundamental region

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Consider now a curve in S+ as shown in the lower half of Figure 9.6.2 starting at ∞, going in C− to a gap, crossing the gap, and returning to ∞ in C+ . The lift leaves P when the gap is crossed. The lift is thus shown in the upper half of Figure 9.6.2.

Figure 9.6.2. Fuchsian group generators ˜ , a cover map taking (−1, 1) to the crossed map, If we had used x the two halves would be complex conjugate, so in Figure 9.6.2, the two pieces of lift curve are images under inversion in the orthocircle corresponding to the gap. In particular, the other endpoint is just the image of 0 under this inversion. The same argument shows that for any point on (−1, 1), the image under the deck transformation associated to this curve is inversion in the circle. Let γ be the deck transformation and r + reflection in the circle. Then γ −1 r + is an conjugate linear extended FLT which leaves (−1, 1) fixed. It must be complex conjugation c(z) = z¯ −1 +

+

(9.6.6) +

Thus, γ r = c, so r γ = c or γ = r c (we used here (r + )2 = (c)2 = 1). We have thus proven: Theorem 9.6.3. Let rj+ be the inversions in Cj+ for j = 1, . . . , ℓ. Let c be given by (9.6.6). Let γj = rj+ c

(9.6.7)

Then Γ is the free nonabelian group generated by {γ1, . . . , γℓ }. If rj− is reflection in Cj− , then crj+ c = rj− , so by (rj+ )2 = c2 = 1, we see γj−1 = rj− c (9.6.8)


We can now define F by F =F

int

∪

ℓ [

Cj+

579

(9.6.9)

j=1

Thus, F is a strict fundamental domain in the sense that it contains one point from each orbit {γ(z)}γ∈Γ . Its interior is indeed F int . We will use F¯ in two different ways: sometimes the closure in D, that is, F¯ = F int ∪

ℓ [

(Cj+ ∪ Cj− )

(9.6.10)

j=1

and sometimes the closure in D, including some boundary points in ∂D. We will return to F and Γ shortly, but first we want to use F to extend x beyond D. Let zn lie in F with |zn | → 1. x(zn ) lies in the Riemann sphere which is compact, so without loss, we can pass to a subsequence so that x(zn ) has a limit, x∞ . Suppose x∞ ∈ S+ . There is then z∞ ∈ F so that x(z∞ ) = x∞ . But x is one-one on F so all nearby points for x(z) have z near z∞ , that is, |zn | → |z∞ | < 1. It follows that x∞ ∈ e and, in particular, is real. In particular, since all limit points are real, we see that Im x(z) → 0 as |z| → 1 with z ∈ F . It follows from the strong form of the reflection principle (see Ahlfors [7, Thm. 4.24]) that if we define x on C \ D with values in C ∪ {∞} by x(z) = x(1/¯ z) (9.6.11) ¯ ¯ then x can be continued across ∂ F ∩ ∂D (where here F means in D). Combining (9.6.2) and (9.6.11), we get x(1/z) = x(z)

(9.6.12)

Given that we have continued outside D, it will be useful to define extended versions of F int and F . By Feint , we mean the union of F int , e we mean, {z | z¯−1 ∈ F int }, and the interior in ∂D of ∂D ∩ F¯ . By F, following (9.6.9), ℓ [ e+ Fe = Feint ∪ C (9.6.13) j

j=1

e Moreover, for any distinct γ, γ ′ ∈ Γ, Feint is, indeed, the interior of F. γ[F ] ∩ γ ′ [F ] = ∅ and [ γ[F ] = C ∪ {∞} \ [Λ(Γ) (9.6.14) γ∈Γ

We claim that

580


Theorem 9.6.4. Let Λ(Γ) be the set of limit points for Γ. Then x, defined by (9.6.11), can be defined on ∂D \ Λ(Γ) so that x is analytic from C ∪ {∞} \ Λ(Γ) to C ∪ {∞}. Moreover, (i) If γ ∈ Γ, defined from D to D, is extended (analytically) to a map of C ∪ {∞} to C ∪ {∞}, then for all z ∈ C ∪ {∞} \ Λ and all γ ∈ Γ, x(γ(z)) = x(z) (9.6.15) (ii) x′ (z) 6= 0 so long as x(z) ∈ / {αj , βj }ℓ+1 j=1 . ℓ+1 (iii) At points with x(z) ∈ {αj , βj }j=1 (necessarily z ∈ ∂D), we have x′ (z) = 0

x′′ (z) 6= 0

(9.6.16)

Proof. As we explained, analyticity across ∂F ∩ ∂D follows from the reflection principle. (i) follows from the fact that it holds for z ∈ D by analytic continuation. (9.6.15) then implies analyticity across ∪γ γ[∂F ∩ ∂D] = ∂D \ Λ. Now let Cj± denote the full orthocircle, not just the part in D. Then x is real exactly on ℓ [ [ + − γ R ∪ (Cj ∪ Cj ) ∪ ∂D \ Λ(Γ) (9.6.17) j=1

γ∈Γ

The first union is over disjoint sets and the last set intersects all the others orthogonally. The special set in (9.6.17) is displayed in Figure 9.6.3.

x is locally one-one on D, so x′ (z) 6= 0 for z ∈ D and then, by (9.6.11), on C \ D. As an analytic function, if x(z) − x(z0 ) has a kth order zero at z0 , there are 2k asymptotic rays at relative angle 2π/2k near z0 on which x is real. Thus, x′ (z) 6= 0 on all points in (9.6.17), except the points in [ ℓ [ + − γ R ∪ (Cj ∪ Cj ) ∩ (∂D \ Λ(Γ)) (9.6.18) γ∈Γ

j=1

where four real rays come in at 90◦ angles. At these points, the zero of x(z) − x(z0 ) is double, so x′′ (z) 6= 0. If we note that the set in (9.6.18) is exactly x−1 ({αj , βj }ℓ+1 j=1 ), we have (ii) and (iii). Remark. This says x−1 has square root behavior at points in {αj , βj }ℓ+1 j=1 .


581

Figure 9.6.3. Three generations of γ[Cj± ] Thus, x is locally one-one on the complement of the set in (9.6.18) and it is locally two-one at those points. But the image points, {αj , βj }ℓ+1 j=1 , are precisely the branch points of S, so we introduce a modified map, x♯ , to be a map from C \ Λ(Γ) to S and define it so that (9.6.11) is replaced by x♯ (1/¯ z ) = τ (x(z))

(9.6.19)

where τ (z+ ) = z− is the reflection on S discussed in Section 5.12. (9.6.19) is for z ∈ D. For z ∈ D, we also have x♯ (z) = x(z)

(9.6.20)

interpreting C ∪ {∞} \ e as S+ . Then we have proven that

Theorem 9.6.5. x♯ : C ∪ {∞} \ Λ(Γ) → S is a covering map.

Of course, C ∪ {∞} \ Λ(Γ) is not simply connected, so this is not the universal cover. Example 9.6.6 (One gap set). Let ℓ = 1. Then π1 (S + ) is Z while S is a torus so π1 (S) is Z2 . Γ has a single hyperbolic generator, γ1 , and Γ = {(γ1)n | n ∈ Z} ∼ = Z. Unlike the case ℓ ≥ 2 where Λ(Γ) is infinite, in this case there are only two limit points: the two fixed points of γ1 . Notice that C ∪ {∞} with two points removed is homeomorphic to the punctured plane, C \ {0}, so its π1 is Z. As a covering map, x♯ induces a map of π1 (C ∪ {∞} \ Λ(Γ)) = Z to π1 (S) = Z2 . This image is the


582

group generated by a loop around one band. The loops around the gap on both sheets generate the quotient and label γ ∈ Γ. We return to the group, Γ, and its action on F and on D.

Proposition 9.6.7. Let {γj }ℓj=1 be the generators given by (9.6.7). Then every element γ ∈ Γ can be written uniquely as γ = αw(γ) . . . α2 α1

(9.6.21)

where each αk is a γj or γj−1 with the convention that for no j = 1, . . . , w(γ) − 1 is αj+1 αj = 1. If, for k = 0, 1, 2, . . . , then for k ≥ 1,

Γk = {γ | w(γ) = k}

#Γk = (2ℓ)(2ℓ − 1)k−1 In addition, any γ ∈ Γ2m has a unique representation, γ = s1 . . . s2m

(9.6.22) (9.6.23) (9.6.24)

where each sk is an rj± , and for all j = 1, . . . , 2m − 1, sj+1 6= sj . Similarly, any γ ∈ Γ2m+1 has the form γ = s1 . . . s2m+1 c

(9.6.25)

Remarks. 1. w(γ) is called the word length or length of γ. 2. Among all representations of γ as a product of γj ’s and γj−1 ’s, (9.6.21) is the one of minimal length. Proof. Γ is the free nonabelian group generated by {γj }ℓj=1 , so any γ has a product representation of the form (9.6.21). If some αj+1 αj = 1, remove them and so end up with a shorter representation of that form. Since Γ is free, all such products in Γk are distinct (with the no αj+1αj = 1 condition). α1 can be chosen in 2ℓ ways. Since α2 6= α1−1 , it can only be chosen in (2ℓ − 1) ways. This leads to (9.6.23). Given (9.6.8) and (rj+ )2 = c2 = 1, we get γj = crj−

γj−1 = crj+

(9.6.26)

In addition, crj± c = rj∓ (9.6.27) Thus, any representation of the form (9.6.21) leads to one of the form (9.6.24)/(9.6.25). Later, we will need the fact that w(γ n ) grows linearly in n. We are heading towards a proof that w(γ n ) ≥ |n| − 1 + w(γ)

(9.6.28)

Call γ solid if the representation (9.6.21) has α1 αw(γ) 6= 1 or if w(γ) = 1.


583

Lemma 9.6.8. Any γ has the form γ = γ0 γ1 γ0−1

(9.6.29)

where γ1 6= 1 and is solid. Proof. There is a first k0 with −1 αk0 6= αw(γ)+1−k 0

(9.6.30)

γ0 = (αk0 −1 . . . α1 )−1

(9.6.31)

for if w(γ) is odd, k0 = 21 (w(γ)−1) works. If w(γ) is even, then (9.6.30) holds for k0 = w(γ)/2 since αj+1αj 6= 1. Let if k0 6= 1 and γ0 = 1 if k0 is 1. Let

γ1 = αw(γ)+1−k0 . . . αk0

(9.6.32)

By (9.6.30), γ1 is solid and not 1. By construction, (9.6.29) holds. Proposition 9.6.9. Given γ, find a representation (9.6.29) with γ1 solid and let s(γ) = w(γ1 ) (9.6.33) Then γ n = γ0 γ1n γ0−1 (9.6.34) n is the (9.6.21) representation of γ so w(γ n ) = 2w(γ0) + |n|w(γ1)

= w(γ) + (|n| − 1)s(γ)

(9.6.35)

In particular, (9.6.28) holds.

Proof. Since γ1 is solid, the (9.6.21) representation of γ1n is just n times that of γ1 repeated. We next want to define some subsets of D that keep track of how many γj ’s or γj−1 ’s we need to get to these sets, starting in F . Since F is a fundamental domain, [ D= γ[F ] (9.6.36) γ∈Γ

and the union is over disjoint sets. We define [ Dk = γ[F¯ ]

(9.6.37)

γ : w(γ)≤k

and

Rk = D \ Dk (9.6.38) Returning to Figure 9.6.3, D0 = F¯ is the intersection of D and the exterior of the four big circles and R0 is the part of D inside those

584


circles. D1 is the exterior of the 12 = 4 × 3 next biggest circles and R1 the interior of the 12 circles. D1 \ D0 are the four images of F under γ1 , γ2, γ1−1 , γ2−1 (up to some edges). The interior of the 36 = 4 × 32 smallest circles is R2 and their complement is D2 . ¯ k be the closure of Rk in D and Finally, let R ¯ k ∩ ∂D ∂Rk = R (9.6.39) We are heading towards a proof of a major geometric theorem which will be critical in our proof of step-by-step sum rules.

Theorem 9.6.10 (Beardon’s Theorem). For some positive constants C0 , C1 , we have |∂Rk | ≤ C0 e−C1 k

(9.6.40)

Remark. | · | means dθ/2π measure. As we will see below (see Theorem 9.6.13 and the Notes), this is equivalent to the fact that there is a Poincaré index, s, for Γ with s < 1—and it is in this form that Beardon stated his theorem (for more general Fuchsian groups). As noted, ∂Rk contains 2ℓ(2ℓ − 1)k−1 arcs. It is not hard to see that the maximum radius of disks in Rk decays exponentially (see Lemma 9.6.16 below), while the number of arcs grows exponentially. (9.6.40) says the size decrease wins out by a bit. We note that \ ¯k Λ(Γ) = R (9.6.41) k

=

\

∂Rk

(9.6.42)

k

Before turning to a proof of Theorem 9.6.10, we want to note a number of consequences. The first can be proven without using anything as powerful as (9.6.40), but since we have it, we will use it. Corollary 9.6.11. Every γ ∈ Γ, γ 6= 1 is hyperbolic.

Remarks. 1. Indeed, our proof shows that if Tγ ∈ SU(1, 1) is defined by γ = fTγ , then inf γ6=1 |Tr(Tγ )| > 2. 2. In the lead-up to the proof of Theorem 9.6.10, we will prove (9.6.43). Proof. The length of the arcs in ∂Rk is comparable to the radii of the circles in Rk . Thus, (9.6.40) implies

¯ k} ≤ C e0 e−C1 k (9.6.43) sup{|w − z| | w, z inside the same circle of R


585

By construction, γ n (0) lies in one of these circles for k = w(γ n ) and, as we will show below (see (9.6.48)), γ n (0)/|γ n (0)| lies in the same circle. Thus, e0 e−C1 w(γ n ) 1 − |γ n (0)| ≤ C (9.6.44) By (9.6.28), e0 e−C1 (n−1) 1 − |γ n (0)| ≤ C (9.6.45) which implies that approach to the limit is exponential, so γ is hyperbolic. The main use that we will have for Theorem 9.6.10 is Theorem 9.6.12. Let f be an analytic function so that for some C > 0, {z | |Im f (z)| > Cn} ⊂ Rn (9.6.46) Then \ H p (D) (9.6.47) f∈ p Cn} ⊂ ∂Rn

(9.6.49)

|{eiθ | |Im f (reiθ )| > Cn}| ≤ C0 e−C1 n

(9.6.50)

Thus, for any p < ∞, sup r

Z

|Im f (reiθ )|p

dθ 0. This completes k=1 (1 − e the sketch of what follows. We will need the following, which is essentially a restatement of Proposition 9.2.28: Proposition 9.6.14. Let η be a conjugate analytic function of z in a neighborhood of ∂D, η ′ = ∂η/∂ z¯ its derivative. Suppose η maps ∂D to itself and let Q ⊂ ∂D. Then Z dθ |η[Q]| = |η ′ (eiθ )| (9.6.65) 2π Q Remark. | · | is in dθ/2π measure. Proof. η is anticonformal, so it infinitesimally stretches or contracts distances by |η ′ |. Since dθ is arclength in Euclidean metric, (9.6.65) is immediate. Corollary 9.6.15. Under the hypotheses of Proposition 9.6.14, if Q1 , Q2 are any two subsets of ∂D, then |η[Q1 ]| inf Q1 |η ′ (eiθ )| |Q1 | ≥ |η[Q2 ]| supQ2 |η ′ (eiθ )| |Q2 |

(9.6.66)


588

Proof. Immediate from (9.6.65), which implies sup |η ′ (eiθ )| |Q| ≥ |η[Q]| ≥ inf |η ′(eiθ )| |Q| Q

Q

(9.6.67)

Let Cγ be the outer circle of γ[F ], as discussed in the proof of Theorem 9.6.13 and let Aγ be the arc of ∂D inside Cγ . We need to prove |Aγ | decreases exponentially in w(γ). Let rj± be the reflections in Cj± . (rj± )′ has Cj± as isometric circle. Outside Cj± , |(rj± )′ | < 1. Let ± ′ iθ b = max max |(r ) (e )| (9.6.68) j ± j,±

eiθ inside some other Ck

Since the Cj± are a strictly positive distance from each other, b 0

(9.6.77)

Qγ = ∂γ[F ] ∩ ∂D (9.6.78) which is 2ℓ arcs between the 2ℓ − 1 orthocircles inside Cγ . For each γ, |Qγ | > 0, so |Qγ | m = min >0 (9.6.79) w(γ)≤n0 |Aγ | as a finite min of positive numbers. By (9.6.77) and (9.6.21), for any γ, we have |Qγ | ≥ mf (9.6.80) |Aγ | Write γ ′ ⊲ γ if γ ′ = γ˜ γ with w(γ ′) = w(˜ γ ) + w(γ), that is, γ ′ = αw(γ ′ ) . . . α1 and γ = αw(γ) . . . α1 (same α’s). Given γ, the γ ′ ’s with γ ′ ⊲ γ and w(γ ′) = w(γ) + 1 number 2ℓ − 1 and the corresponding A′γ ’s are the arcs between the area making up Qγ , that is, X |A′γ | ≤ (1 − mf )|Aγ | (9.6.81) γ ′ ⊲γ w(γ ′ )=w(γ)+1

This implies, by summing (9.6.81) over all words of length k, X X |Aγ | ≤ (1 − mf ) |Aγ | (9.6.82) γ : w(γ)=k+1

so, by induction,

X

γ : w(γ)=k

γ : w(γ)=k

|Aγ | ≤ (1 − mf )k

(9.6.83)

590


But ∂Rk =

[

Aγ

(9.6.84)

w(γ)=k

so (9.6.83) implies which proves (9.6.40).

|∂Rk | ≤ (1 − mf )k

(9.6.85)

This completes what we want to prove abut the Γ and F associated to a finite gap set. The reader may have noticed that we did not use any isometric circles. We end this section with some alternate proofs that use that technology. Proposition 9.6.17. Let γ1 be solid. Then Df (γ1 ) and Di (γ1 ) lie inside distinct Cj± and so are disjoint. Remark. By Theorem 9.2.32, this implies γ1 is hyperbolic. Thus, by Lemma 9.6.8, any γ 6= 1 in Γ is hyperbolic, that is, we have a second proof of Corollary 9.6.11. Proof. Suppose first w(γ1 ) is even. Then γ1 = s1 . . . s2m with each sk one of rj± and no sj+1 = sj . Thus, as above, if s1 = rj± , then γ1 (0) lies inside Cj±1 . But, by Theorem 9.4.22, all Df (γ1 )’s lie inside some Ck± and, by Theorem 9.3.6, γ1 (0) lies in Df (γ1 ). We conclude Df (γ1 ) lies inside Cj±1 . Similarly, since Di (γ1 ) = Df (γf−1 ), we see Di (γ1 ) is inside Cj±2m since γ1−1 = s2m . . . s1 . Since s2m 6= s1 , the initial and final circles lie inside distinct Cj± as claimed. The analysis in the odd case is similar. Finally, we want to provide a different proof of the key Theorem 9.6.10: Lemma 9.6.18. Let γ ∈ M have γ(0) 6= 0 and let θ(z) be the angle (in (−π, π]) between z ∈ ∂D and the ray from 0 through the center of Di (γ). Then |γ ′ (z)| is a function of |θ(z)| only and monotone decreasing as |θ(z)| increases. Proof. By covariance, we can suppose the center of Di (γ) is on (1, ∞) at β. Then, by (9.2.61), |γ ′(z)| = c−2 |eiθ(z) − β|−1

= c−2 (1 + β 2 − 2β cos(θ(z)))−1

is clearly monotone decreasing in θ(z).

(9.6.86)


591

Sketch of Proof of Theorem 9.6.10. For j = 1, . . . , ℓ, let Cj+ be the jth orthocircle in C+ and let A+ j be the arc in ∂D it cuts ℓ,r off. Let Qj be the two arcs in ∂F ∩ ∂D adjacent to A+ j on the left and + + + − ℓ right (so Qj is between Cj−1 and Cj with C0 ≡ C1 and Qrj between + + Cj+ and Cj+1 with Cℓ+1 ≡ Cℓ− ). Let qj =

− |Q+ j ∪ Qj | |∂D \ Aj |

− be the fraction of the remainder of ∂D taken by Q+ j ∪ Qj . Let q = min qj > 0 j

(9.6.87)

(9.6.88)

We will prove that for any γ,

from which

|Qγ | ≥q |Aγ |

(9.6.89)

|∂Rk | ≤ (1 − q)k

(9.6.90)

|γ(Q± |Q± j )| j | ≥ |γ(∂D \ Aj )| |Aj |

(9.6.91)

as in the other proof. As noted above, Di (γ) lies inside some Cj± , say Cj+ for simplicity of notation. Thus, ∂Cj+ goes under γ into Cγ , ∂D \ Aj into all of Aγ , and ± ′ Q± j into parts of Qγ . Since |γ | is decreasing by the lemma and Aj is closest to Di (γ), we have that

which implies

− − |γ(Q+ |Q+ |Qγ | j ) ∪ γ(Qj )| j ∪ Qj | ≥ ≥ ≥ qj ≥ q |Aγ | |γ(Aj )| |Aj |

proving (9.6.88).

Remarks and Historical Notes. The use of explicit covering maps in spectral theory and the structure of Fuchsian groups goes back to Sodin–Yuditskii [404] and has been developed by Peherstorfer– Yuditskii [336, 337] and Christiansen–Simon–Zinchenko [82, 83, 84, 85]. The basic picture with orthocircles in complex symmetric positions, one in C+ for each gap is from [404]. The importance of Beardon’s theorem in the finite gap case is due to Christiansen–Simon–Zinchenko [83, 84].


592

x-ref?

What we call Beardon’s theorem is a special case of a much more general theorem of Beardon [35]: he proved that any finitely generated Fuchsian group, Γ, for which Λ(Γ) is not dense in ∂D, has a Poincaré index of convergence s < 1. He also proved that this implies the set of limit points has Hausdorff dimension less than one. For more on Hausdorff dimensions of limit sets of Fuchsian groups, see TK. Beardon’s general result is much more difficult to prove because of the need to accommodate parabolic and elliptic elements. Our proof of Theorem 9.6.10 here is new and was arrived at in discussion with Jacob Christiansen and Maxim Zinchenko. We proved that (9.6.40) implies (9.6.52) for some s < 1. One can go backwards and show (9.6.52) for s < 1 implies (9.6.40). For |Aγ | is comparable to 1 − |γ(0)| so (9.6.40) is equivalent to X e0 e−kC1 1 − |γ(0)| ≤ C (9.6.92) γ : w(γ)=k

On the other hand, one proves

1 − |γ(0)| ≤ D0 e−w(γ)D1

so X

γ : w(γ)=k

1 − |γ(0)| ≤

X

γ : w(γ)=k

(1 − |γ(0)|) (D0 e−kD1 )1−s

≤ D01−s e−k(1−s)D1

so (9.6.52) implies (9.6.92).

(9.6.93)

s

X γ

(1 − |γ(0)|)− s

(9.6.94)

9.7. Blaschke Products and Green’s Functions The analog of what we did for a single interval is that, given a measure, dµ, with σess (dµ) = e, we form its m-function, m(z), on C \ σ(dµ), meromorphic on C \ e and define on D, M(z) = −m(x(z))

(9.7.1)

M(γ(z)) = M(z)

(9.7.2)

This function is automorphic in that for all γ, That is, automorphic functions, f , are defined on D and obey f (γ(z)) = f (z)

(9.7.3)

for all γ ∈ Γ and z ∈ D. We will mainly want to consider meromorphic functions obeying (9.7.3), but occasionally we will also want to allow f to be a real harmonic or subharmonic function.

9.7. BLASCHKE PRODUCTS AND GREEN’S FUNCTIONS

593

One of the first things we want to do is remove zeros and poles. For example, even if there were no bound states, we needed to consider M(z)/z in case e = [−2, 2]. As in that case, m has a zero at ∞, so M has a zero at z = 0. But then, by (9.7.2), it has zeros at all points in {γ(0)}γ∈Γ. So we have to divide out by an infinity of zeros even in the simplest cases. That will lead us to Blaschke products and, as a bonus, we will find a remarkably simple connection to the logarithmic potential for e. Recall that s = 1 is a Poincaré index if for one, and hence all, z0 ∈ D, we have X (1 − |γ(z0 )|) < ∞ (9.7.4) γ∈Γ

and, in particular, if Γ is of the second kind, (9.7.4) holds (see Theorem 9.4.19). This is, of course, exactly a Blaschke condition, (2.3.69). Thus, by Proposition 2.3.16, Theorem 9.7.1. If Γ is a Fuchsian group for which (9.7.4) holds for one, and hence all, z0 ∈ D, the function (b defined by (2.3.67)) Y B(z, z0 ) = b(z, γ(z0 )) (9.7.5) γ∈Γ

is an absolutely convergent product which defines a function of z on D analytic there, vanishing exactly at the points {γ(z0 )}γ∈Γ with simple zeros there. Moreover, if Λ(Γ) 6= ∂D, then B has an analytic continuation to a neighborhood of ∂D \ Λ(Γ). On ∂D \ Λ(Γ), |B(eiθ , z0 )| = 1

(9.7.6)

B( · , z0) then also has a meromorphic continuation to (C ∪{∞}) \ Λ(Γ) with poles exactly at {1/ γ(z0 )}γ∈Γ where all poles are simple. Remark. B( · , z0) is called a Fuchsian Blaschke product, or sometimes just a Blaschke product. The case z0 = 0 is special, so we will write B(z) ≡ B(z, z0 = 0)

(9.7.7)

Proof. By (3.3.3) and (3.3.4), one has for any z0 ∈ D and z ∈ C \ {¯ z0−1 } that |bz0 (z) − 1| ≤

1 + |z| (1 − |z0 |) |1 − z¯ z0 |

(9.7.8)


594

−1

from which one concludes that for z ∈ / {γ(z0 ) }γ∈Γ = Λ(Γ) ∪ −1

{γ(z0 ) }γ∈Γ ≡ P(z0 ), we have X |bγ(z0 ) − 1| < ∞

(9.7.9)

γ∈Γ

with a bound uniform on compact subsets of C \ P. It follows that the product converges uniformly on compacts of the open set C \ P, which includes ∂D \ Λ(Γ). Since |bz0 (eiθ )| = 1 and |bγ(z0 ) (eiθ )| = 1, the uniform convergence implies (9.7.6). By Hurwitz’s theorem, the only zeros in C \ P are at {γ(z0 )}γ∈Γ . From (9.7.6) and the fact that ∂D \ Λ(Γ) is open in ∂D and nonempty, we get, by the reflection principle, −1

z , z0 ) B(z, z0 ) = B(1/¯

(9.7.10)

initially for z ∈ (C \ P) ∪ C \ D. This then implies the claim about poles. Since the set of zeros of B( · , z0) is invariant under all γ ∈ Γ, one might guess that this is true of B itself. We will see this is true for |B( · , z0)| but not for the phase. Definition. A character of a Fuchsian group, Γ, is group homomorphism of Γ to ∂D viewed as a multiplicative group. Γ∗ is the group of all characters of Γ under pointwise multiplication. Given ω ∈ Γ∗ , a function f on D is called character automorphic with character ω if f (γ(z)) = ω(γ)f (z)

(9.7.11)

for all γ ∈ Γ, z ∈ D. f is called character automorphic if and only if it is character automorphic for some ω ∈ Γ∗ . For a finite gap set, Γ is generated by {γj }ℓj=1. So, since ∂D is abelian, {ω(γj )}ℓj=1 determine ω. Since Γ is free, any values in ∂D are allowed, that is, if (α1 , . . . , αℓ ) ∈ ∂Dℓ , then there is a unique character with ωα (γj ) = αj (9.7.12) and this describes all characters. Thus, Γ∗ ∼ = (∂D)ℓ , a torus of the same dimension as the the isospectral torus. We will eventually see that this is no coincidence! Theorem 9.7.2. For any z0 ∈ D, there is a character ωz0 ∈ Γ∗ so B(γ(z), z0 ) = ωz0 (γ)B(z, z0 )

(9.7.13)


595

z0 → ωz0 is continuous in z0 and obeys ωγ(z0 ) = ωz0

(9.7.14)

Proof. We claim first that for any z1 ∈ D and γ, there is αγ,z1 ∈ ∂D with b(γ(z), z1 ) = αγ,z1 b(z, γ −1 (z1 )) (9.7.15) For g(z) =

b(γ(z), z1 ) b(z, γ −1 (z1 ))

(9.7.16)

is a ratio of functions analytic in a neighborhood of D, each with a single simple zero at γ(z) = z1 , that is, z = γ −1 (z1 ). Thus, g is analytic and nonvanishing on D. Since |g(z)| = 1 on ∂D, g has a meromorphic continuation to C ∪ {∞} given by g(z) = (g(1/¯ z))−1

(9.7.17)

outside D. But g is nonvanishing on D, so g is entire and bounded, hence a constant αγ,z1 . But |g(z)| = 1 on ∂D, so αγ,z1 ∈ ∂D. Now fix γ0 ∈ Γ. Then, by (9.7.15), b(γ0 (z), γ(z0 )) = αγ0 ,γ(z0 ) b(z, γ0−1 γ(z0 ))

(9.7.18)

As γ runs through all of Γ, γ0−1 γ does also. So, by uniform convergence of the product, B(γ0 (z), z0 ) = ωz0 (γ0 )B(z, z0 ) (9.7.19) where for now ωz0 (γ0 ) is just some number in ∂D. But B(γ0 γ1 (z), z0 ) = ωz0 (γ0 )B(γ1 (z), z0 ) = ωz0 (γ0 )ωz0 (γ1 )B(z, z0 )

(9.7.20)

proving that ω ∈ Γ∗ . Since z0 → B(z, z0 ) is continuous for any z ∈ D and B(z, γ(z0 )) = B(z, z0 ), we see that z0 → ωz0 is continuous and that (9.7.14) holds. We want to note a corollary of (9.7.15): Proposition 9.7.3. For any type 2 Fuchsian group, one has Y |B(z)| = |γ(z)| (9.7.21) γ∈Γ

for all z ∈ (C ∪ {∞}) \ Λ(Γ).

596


Proof. By convergence of the product defining B and analyticity, it suffices to prove this for z ∈ D. By (9.7.15), |b(z, γ(0))| = |b(γ −1 (z), 0)| = |γ −1 (z)|

(9.7.22)

Since γ −1 runs through Γ as γ does, (9.7.21) follows (using 1 − |w| ≤ |1 − w| for |w| < 1). One might worry that B is really fully automorphic and it is just our proof that is lacking. After some notation, we will show that is an unfounded worry. Henceforth, we suppose Γ is the Fuchsian group of a finite gap covering map. Define Q1 , . . . , Qℓ+1 arcs on ∂D ∩ C+ as follows: Q1 runs from 1 to the right endpoint of Cℓ+ , Q2 from the left + endpoint of Cℓ+ to the right of Cℓ−1 , . . . Qℓ from C2+ to C1+ and Qℓ+1 + from the left endpoint of C1 to −1. Proposition 9.7.4. Fix z0 ∈ (−1, 1). Let ∆1 , ∆2 , . . . , ∆ℓ+1 be the change of arg B(eiθ , z0 ) as eiθ runs counterclockwise along Q1 , Q2 , . . . , Qℓ+1 . Then (i) (ii)

0 < ∆j < π ℓ+1 X

(9.7.23)

∆j = π

(9.7.24)

j=1

(iii)

ℓ+1−j X ∆k ωz0 (γj ) = exp 2i

(9.7.25)

k=1

Remark. In particular, by (9.7.23)/(9.7.24), ωz0 (γj ) 6= 1, so B(z, z0 ) is not automorphic. Proof. We first claim that for any z1 ∈ D, b(z, z¯1 ) = b(¯ z , z1 )

(9.7.26)

as follows from the definition or by noting, as in the proof of (9.7.15), that the two are equal up to phase but both are positive at z = 0. Secondly, since cγj c = γj−1 , we see {cγc}γ∈Γ runs through Γ as γ runs through Γ. Thus, if z0 ∈ (−1, 1), {γ(z0 )}γ∈Γ and {γ(¯ z0 )}γ∈Γ is the same. In particular, for such z0 , B(z, z0 ) = B(¯ z , z0 ) Thus, for z ∈ (−1, 1), B(z, z0 ) is real. By (9.7.13), B(γj (z), z0 ) = ωz0 (γj )B(z, z0 )

(9.7.27)

(9.7.28)


597

By (9.6.7), this implies that if x is real, then B(rj+ (x), z0 ) = ωz0 (γj ) B(x, z0 )

(9.7.29)

(since B(x, z0 ) = B(x, z0 )). This in turn implies that for all z ∈ D, B(rj+ (z), z0 ) = ωz0 (γj ) B(z, z0 )

for both sides are anti-analytic in z and agree if z ∈ (−1, 1). Suppose for z ∈ / {γ(z0 )}γ∈Γ we write B(z, z0 ) = |B(z, z0 )|A(z, z0 )

(9.7.30)

(9.7.31)

Then, by (9.7.30), if rj (z) = z, that is, z ∈ Cj+ , A(z, z0 ) is constant, and for such z, A(z, z0 )2 = ωz0 (γj ) (9.7.32) Consider tracking arg B(z, z0 ), as z follows a path from 1 to −1, + going successively through Q1 , Cℓ+ , Q2 , Cℓ−1 , . . . , C1+ , Qℓ+1 on a curve we call η. On each Qj , arg B is increasing, for |B(eiθ , z0 )| = 1 and ∂ |B(reiθ , z0 )| < 1 for r < 1 implies ∂r |B(reiθ , z0 )| < 0 which, by the Cauchy Riemann equations, imply ∂ arg B(eiθ , z0 ) > 0 (9.7.33) ∂θ Thus, ∆j > 0, and since arg B is constant on each Cj+ , the change of arg B along the curve η is ∆1 + · · · + ∆ℓ+1 . If we follow η by η¯ run backwards, the change is the same by (9.7.27), so the closed curve running from 1 to 1 along ∂F is 2(∆1 + · · · + ∆ℓ+1 ). By the argument principle, the change is also 2π× number of zeros in F int which is 2π since the only zero is at z0 . This proves (9.7.24), which in turn implies ∆j < π since ∆j > 0. P By construction, the constant argument on Cj+ is ℓ+1−j k=1 ∆k , so by (9.7.32), we obtain (9.7.25).

Our next topic concerns the connection of B(z) to the potential theorist’s Green’s function, Ge(z), discussed in Section 5.5 (see (5.5.111))—recall for e, a finite gap set, it is the unique positive harmonic function of C \ e so that limz→e Ge(z) = 0 and Ge(z) = log(|z|) + O(1) as |z| → ∞; indeed (see (5.5.111)), with C(e) the capacity of e, 1 Ge(z) = log|z| − log(C(e)) + O (9.7.34) z We will also need a symbol for limz→0, z6=0 zx(z), so we define x∞ by requiring x∞ x(z) = + O(1) (9.7.35) z


598

near z = 0. Theorem 9.7.5. Let e be a finite gap set and B(z) the associated Blaschke product for z0 = 0. Then |B(z)| = e−Ge (x(z))

(9.7.36)

In particular, (i) C(e) z + O(z 2 ) (9.7.37) x∞ (ii) For z0 = 0, the numbers ∆j of Proposition 9.7.4 are given by B(z) =

∆j = πρe(eℓ+1−j )

(9.7.38)

where ρe is the equilibrium measure, and ej = [αj , βj ] is the jth interval in e. Proof. By (9.7.19), |B(z)| is automorphic, so there exists a realvalued function β on (C ∪ {∞}) \ e with values in [0, 1) β(x(z)) = |B(z)| For z1 6= 0, b(0, z1 ) = |z1 | and b(z, 0) = z, so Y B(z) = |γ(0)| z + O(z 2 )

(9.7.39)

(9.7.40)

γ6=1

which implies that near x = ∞ in C, Y 1 − log(β(x)) = log|x| − log x∞ |γ(0)| + o x γ6=1

(9.7.41)

Away from z ∈ {γ(0)}γ∈Γ , |B(z)| is nonvanishing, so − log(β(x)) is a positive harmonic function on C \ e. Since |B(reiθ )| → 1 as r ↑ 1 with eiθ ∈ ∂F ∩ ∂D, as x → e, − log(β(x)) → 0

(9.7.42)

Thus, by the unique specification of Ge, we have − log(β(x)) = Ge(x)

(9.7.43)

which is (9.7.36). (9.7.34), (9.7.40), and (9.7.41) then imply (9.7.37) as well as Y C(e) |γ(0)| = (9.7.44) x∞ γ6=1


599

Finally, by looking at the curve in Figure 9.6.2 and (9.7.25), we see that ℓ+1−j X 2 ∆k (9.7.45) k=1

is the change of the argument of the multivalued analytic function whose magnitude is e−Ge (x) under the curve in the lower half of Figure 9.6.2. This implies, using a Cauchy Riemann equation, that Z βℓ+1−j ∂Ge ∆j = (x) dx (9.7.46) ∂n αℓ+1−j (the 2 in (9.7.45) and the two sides of the contour cancel to give a single integral over the top of the cut). By (5.6.7) for x ∈ eint , ∂Ge (x) = πρe(x) (9.7.47) ∂n with ρe the density of dρe and thus, (9.7.46) is (9.7.38). This will let us compute integrals of automorphic functions over ∂D! Theorem 9.7.6. Let e be a finite gap set and dρe its equilibrium measure. Then Z Z dθ iθ f (x(e )) = f (x) dρe(x) (9.7.48) 2π ∂D e where this holds for any continuous function, f , on e and also for any positive measurable function (with integrals allowed to be infinite). This dθ implies f (x(eiθ )) ∈ Lp (∂D, 2π ) if and only if f (x) ∈ Lp (e, dρe). Remark. The explicit formula (5.4.96) for dρe/dx (which only deR e (x) pends on the fact that dρx−z has pure imaginary boundary values on e and so works in all finite gap situations) and (9.7.48) implies Z Z dθ iθ |f (x(e ))| < ∞ ⇔ |f (x)|dist(x, R \ e)−1/2 dx < ∞ (9.7.49) 2π ∂D e Proof. If we prove it for continuous f ’s, we get it for characteristic functions of open sets by taking decreasing monotone limits, and then for general positive functions by taking increasing monotone limits. Let A = ∂F ∩ ∂D and Aγ = γ[A] so ∂D \ Λ(F ) is the disjoint union of Aγ over γ ∈ Γ, that is, Z Z dθ X dθ iθ f (x(e )) = f (x(eiθ )) (9.7.50) 2π 2π ∂D Aγ γ


600

Since γ is a smooth function from A to Aγ and f (x(γ(eiθ ))) = f (x(eiθ )), we see Z Z dθ dθ iθ f (x(e )) = f (x(eiθ ))|γ ′ (eiθ )| (9.7.51) 2π 2π Aγ A Since |γ(eiθ )| = 1, we see

∂ arg γ(eiθ ) |γ (e )| = ∂θ ′

iθ

(9.7.52)

where we use

∂ arg γ(eiθ ) ≥0 ∂θ because ∂ iθ |γ(re )| ≥0 ∂r r=1 (since |γ(reiθ )| < 1 = |γ(eiθ )| if r < 1). By (9.7.21), X ∂ ∂ log|B(reiθ )| = log|γ(reiθ )| ∂r ∂r γ∈Γ which leads, via a Cauchy Riemann equation, to X ∂ arg B(eiθ ) = |γ ′ (eiθ )| ∂θ γ∈Γ

From (9.7.50), (9.7.51), and (9.7.56), we deduce Z Z d arg B(eiθ ) dθ dθ iθ f (x(e )) = f (x(eiθ )) 2π dθ 2π ∂D ZA −1 d arg B(x (u)) du = f (x) du π e

(9.7.53) (9.7.54)

(9.7.55)

(9.7.56)

(9.7.57)

(2π)−1 becomes (π)−1 because x−1 maps the u+i0 to A∩C+ and u−i0 to A ∩ C− , so the single integral over e gets counted twice when we integrate over A. By a Cauchy Riemann equation, d arg B(x−1 (u)) ∂ =− log|B(x−1 (u))| du ∂n ∂ = Ge(u) ∂n by (9.7.36). By (9.7.47), ∂ du Ge(u) = ρe(u) du ∂n π

(9.7.58)

(9.7.59)


so RHS of (9.7.57) = proving (9.7.48).

Z

f (u) dρe(u)

601

(9.7.60)

There is a version of (9.7.48) that holds for noninvariant functions. Namely, given any function g ∈ L1 (∂D, dθ/2π), we define P iθ ′ iθ γ∈Γ g(γ(e ))|γ (e )| iθ P g˜(e ) = (9.7.61) ′ iθ γ∈Γ |γ (e )| which is invariant under γ, so there is h on e with

h(x(eiθ )) = 12 [˜ g (eiθ ) + g˜(e−iθ )] and then

(9.7.62)

Z dθ = h(x) dρe(x) (9.7.63) g(e ) 2π e Note that if g ∈ C(∂D), then h ∈ C(e). As a final topic, we want to consider when infinite products and alternating products of B(z, zk ) converge. Since B(z, P P γ(0)) = B(z, 0) and γ∈Γ (1 − |γ(0)|) < ∞, we cannot hope that (1 − |zk |) < ∞ is enough with no restrictions on zk . But if we restrict to zk ∈ F , it is sufficient. Here is a pair of relevant theorems: Z

iθ

Theorem 9.7.7. Let {zk }∞ k=1 all lie in F . If, X (1 − |zk |) < ∞

(9.7.64)

k

QK

then k=1 B(z, zk ) is absolutely convergent as K → ∞ for all z ∈ D, that is, X (1 − |B(z, zk )|) < ∞ (9.7.65) k

uniformly on compact subsets of D. If (9.7.64) fails, then uniformly on Q compact subsets of D, K k=1 B(z, zk ) → 0. Proof. Since |b(z, γ(zk ))| ≤ 1 for z ∈ D, we have |B(z, zk )| ≤ |b(z, zk )|

Q Thus, by Proposition 2.3.16(i), if (9.7.64) fails, k |B(z, zk )| → 0. Conversely, by Proposition 2.3.16(iv), we need only prove that X (1 − |γ(zk )|) < ∞ (9.7.66) zk γ∈Γ

P to imply the absolutely convergence of the product. Since γ∈Γ (1 − |γ(0)|) < ∞, we can drop any zk = 0 terms and so suppose the sum


602

is over those zk with zk = 6 0. Then inf k,γ |γ(zk )| > 0, so (9.7.66) is equivalent to Y |γ(zk )| > 0 (9.7.67) zk 6=0 γ∈Γ

or equivalently, by (9.7.21), to Y

|B(zk )| > 0

(9.7.68)

(1 − |B(zk )|) < ∞

(9.7.69)

zk 6=0

or equivalently to

X k

B is analytic in a closed neighborhood, N, of F¯ (closure in D). We can suppose this neighborhood has the property that for some ε > 0, ω ∈ D ∩ N and |ω| > 1 − ε implies ω/|ω| ∈ N. Since B is analytic on N, supω∈N |B ′ (ω)| < ∞, so for some C and all ω with |ω| > 1 − ε, 1 − |B(ω)| ≤ |B(ω/|ω|) − B(ω)| ≤ C(1 − |ω|)

(9.7.70) (9.7.71)

In proving (9.7.70), we used |B(ω/|ω|)| = 1. Since only finitely many zk have 1 − |zk | > ε, we have, by the hypothesis zk ∈ F , X X (1 − |B(zk )|) ≤ const + (1 − |B(zk )|) zk

|zk |>1−ε

≤ const + C proving (9.7.69).

ζ1 > ρ1 > ζ2 > · · · > βj

(9.7.80)

αj+1 > ρ1 > ζ2 > ρ2 > · · · > βj {pj }∞ j=1 be the unique points in F with

(9.7.81)

or Let n,

{zj }∞ j=1 ,

x(n) = η

Then, as N → ∞,

x(zj ) = ζj

x(pj ) = ρj

N Y B(z, zj ) → B∞ (z) B(z, p ) j j=1

uniformly on compact subsets of [ ∞ −1 ∞ C ∪ {∞} L∪ ({γ(pj )}j=1 ∪ {γ(zj )}j=1 ∪ γ(n))

(9.7.82)

(9.7.83)

(9.7.84)

γ∈Γ

to an analytic function with only simple poles at [ −1 ∞ {γ(pj )}∞ ∪ {γ(z )} j=1 j=1 j

(9.7.85)

γ∈Γ

B∞ is nonvanishing in (9.7.84) except at [ −1 ∞ {γ(zj )}∞ j=1 ∪ {γ(pj )}j=1

(9.7.86)

γ∈Γ

Moreover, on ∂D \ [L ∪ ∪γ∈Γ {γ(n)}],

|z| = 1 ⇒ |B∞ (z)| = 1

(9.7.87)

604


Finally, we have that if arg B∞ is defined on F by requiring arg B∞ (0) = 0, then there is a Γ-dependent constant, CΓ , so that z ∈ F ⇒ |arg B∞ (z)| ≤ CΓ

(9.7.88)

If we place a cut along the orthocircle which contains the {zj }∞ j=1 from z1 (in case (9.7.78) or (9.7.80)) or p1 (in case (9.7.79) or (9.7.81)) to n and all its images under γ ∈ Γ to get a region B which is simply connected and on which B∞ is analytic and nonvanishing, then z ∈ B \ Rn+1 ⇒ |arg B∞ (z)| ≤ (2n + 1)CΓ

(9.7.89)

Remarks. 1. In the case of Theorem 3.3.2, we have zj real, so we could replace z¯j−1 by zj . Here zj may lie on some Ck± so not be real. However, by (9.6.9), (γk± )−1 (zj ) = z¯j (9.7.90) −1 −1 so {γ(zj )}γ∈Γ = {γ(¯ zj )}γ∈Γ . 2. For simplicity of notation, we henceforth restrict to the case (9.7.78) or (9.7.80). x-ref?

To begin the proof, we need an analog of the functions ˜b(z, x) that will be useful also in TK. ∞ Proposition 9.7.10. (i) Let {aj }∞ j=1 , {bj }j=1 be sets in C with no aj equal to a bk and ∞ X |aj − bj | < ∞ (9.7.91) j=1

Then uniformly on compact subsets of C ∪ {∞} \ {bj }∞ j=1 , we have that N Y z − aj (9.7.92) z − bj j=1

converges uniformly and absolutely. The only zeros are at {aj }∞ j=1 . (ii) For ζ, ω ∈ C \ L distinct so ∞ ∈ / {γ(ζ)}γ∈Γ ∪ {γ(ω)}γ∈Γ and all z ∈ C ∪ {∞} \ [L ∪ {γ(ω)}γ∈Γ], Y z − γ(ζ) (9.7.93) z − γ(ω) γ∈Γ ω(γ)≤n

converges uniformly and absolutely as n → ∞. We write Ξ(z; ζ, ω) for the limit Y z − γ(ζ) Ξ(z; ζ, ω) = (9.7.94) z − γ(ω) γ∈Γ


(iii) For any z0 ∈ D, z0 6= 0, Y −1 |γ(z0 )| Ξ(z; z0 , z¯0−1 ) B(z, z0 ) =

605

(9.7.95)

γ∈Γ

(iv) For ζ, ω ∈ C ∪ [L ∪ {γ(0}γ∈Γ ∪ {γ(∞)}γ∈Γ ], Ξ(z; ζ, ω) is jointly meromorphic in z, ζ, ω. Remark. By (9.7.21), the product in (9.7.95) is |B(z0 )|. Proof. (i) Since |aj − bj | (z − a ) |aj − bj | j 1 − = ≤ (z − bj ) |z − bj | mink |z − bk |

(9.7.96)

|c(t) − γ −1 (∞)| = Q > 0

(9.7.97)

we get the absolute convergence by (9.7.91). (ii) Since ζ, ω ∈ / {γ −1 (∞)}γ∈Γ , we can find a smooth curve c(t) with c(0) = ζ, c(1) = ω so inf

γ6=1; γ∈Γ t∈[0,1]

By (9.4.26), for γ 6= 1,

|γ ′ (c(t))| |γ −1 (∞)|2 = |γ ′ (0)| |c(t) − γ −1 (∞)|2 supγ6=1 |γ −1 (∞)|2 ≡ Q1 ≤ Q2

Thus,

so with Q2 = Q1 Since

P

γ∈Γ |γ

R1 0

′

d γ(c(t)) ≤ |c′ (t)|Q1 |γ ′ (0)| dt

|c′ (t)| dt,

|γ(ζ) − γ(ω)| ≤ Q2 |γ ′ (0)|

(0)| < ∞, (9.7.101) implies X |γ(ζ) − γ(ω)| < ∞

(9.7.98) (9.7.99)

(9.7.100)

(9.7.101)

(9.7.102)

γ∈Γ

that is, (9.7.91), so (i) ⇒ (ii). (iii) We have z0 − (1 − z¯0 z) = |z0 |(z − z¯0−1 ) |z0 | so 1 z − z0 bz0 (z) = |z0 | z − z¯0−1

(9.7.103) (9.7.104)


606

which leads to (9.7.95). (iv) This is clearly true for finite products, and so for the limit. Lemma 9.7.11. Fix Q a compact subset of a single Cℓ± (closure in ∂D) or (0, 1] or [−1, 0) and K a compact subset of C with [ K∩ γ[Q] ∪ γ[Q−1 ] ∪ L = ∅ (9.7.105) γ∈Γ

Then there is a C so that for all ζ, ω ∈ Q ∩ D and z ∈ K, 1 − B(z, ζ) ≤ C|ζ − ω| B(z, ω)

(9.7.106)

Proof. By (9.7.105),

inf |B(z, ω)| > 0

z∈K ω∈Q

so it suffices to prove |B(z, ζ) − B(z, ω)| ≤ C1 |ζ − ω|

(9.7.107)

inf |B(ω)| > 0

(9.7.108)

||B(ζ)| − |B(ω)|| ≤ C2 |ζ − ω|

(9.7.109)

which, by (9.7.104) and

ω∈Q

is implied by ¯−1

|Ξ(z, ζ, ζ

−1

) − Ξ(z, ω, ω ¯ )| ≤ C3 |ζ − ω|

(9.7.110)

To prove (9.7.109), we use the fact that ||B(ζ)| − |B(ω)|| ≤ |B(ζ) − B(ω)|

(9.7.111)

and that B is analytic in a neighborhood of Cℓ± . For (9.7.110), we use the fact that when (9.7.90) holds, Ξ(z, ζ, η) is jointly analytic in all variables in a neighborhood of z ∈ K, ζ ∈ Q, ¯ −1 , so (9.7.110) holds. η∈Q

Lemma 9.7.12. Let C be a circle, {z = z0 + reiθ }, in C and f a smooth function on C. Define Z 2π d f (z) dθ VarC (f ) = (9.7.112) dθ 0

be the total variation of f over C. If w ∈ / closed disk surrounded by C and r is the radius of C and fw (z) = arg(w − z)

(9.7.113)


607

then (i)

VarC (fw ) ≤ 2π

(9.7.114)

(ii)

VarC (fw ) ≤

(9.7.115)

4r dist(w, C)

Proof. Let z0 , z1 be the two points on C where the lines from w through zj are tangent to C. Order them so the clockwise arc from z0 to z1 goes through the point, z2 , on C closest to w (see Figure 9.7.1 TK). Let θ0 be the angle between the lines from w to the center of C x-ref? and the line from w to z1 . Let θ1 be arg(w − z2 ). Then arg(w−z) goes from θ1 −θ0 to θ1 +θ0 , monotonically increasing as z runs from z0 to z1 and monotonically decreasing from θ1 + θ0 to θ1 − θ0 as z completes the circuit, that is, VarC (fw ) = 4θ0

(9.7.116)

Since θ0 ≤ π/2, (9.7.114) is immediate. Let z˜ = 12 (z0 + z1 ). Then |z1 − z˜| |w − z˜| r ≤ dist(w, C)

tan(θ0 ) =

(9.7.117) (9.7.118)

(9.7.115) follows from this, (9.7.116), and (for y > 0) Z y dx −1 tan (y) = ≤y 2 0 1+x

(9.7.119)

It will be useful to discuss total variations over arcs of C also. Recall in Theorem 9.6.13, we used rγ for the radius of the orthocircle Cγ . Lemma 9.7.13. For any z ∈ F and ζ in some Cj± , let fz (ζ) = arg(B(z, ζ)) Then VarC ± (fz ) ≤ 4ℓπ + j

where

X

γ : w(γ)≥2

(9.7.120) 4rγ d

[ d = min |z − w| z ∈ F , w ∈ Cγ w(γ)=2

(9.7.121)

(9.7.122)


608

If A± = ±(0, 1), Im z > 0, z ∈ F , and fz (ζ) is given by (9.7.120) for ζ ∈ A± , then VarA± (fz ) ≤ π + RHS of (9.7.121)

(9.7.123)

Proof. As ζ runs through the part of some orthocircle, C, inside D, ζ¯−1 runs through the part of the same orthocircle outside D. Thus, for z fixed in D outside C, if z−ζ gz (ζ) = arg (9.7.124) z − ζ¯−1 and hz (ζ) = arg(z − ζ), then Since

Q

γ∈Γ |γ(z0 )|

VarC∩D (gz ) ≤ VarC (hz )

(9.7.125)

−1

is positive, by (9.7.95) and (9.7.125), X VarCj± (B(z, · )) ≤ Var ] (9.7.126) ± (arg(z − · )) γ

(Cj )

± ± ^ where γ(C Since j ) is the complete orthocircle containing γ(Cj ). ± γ(Cj ) is inside Cγ , its radius and its distance ffrom z ∈ F is bounded by the same for Cγ . Thus, 4ℓπ bounds the 2ℓ terms in (9.7.126) with w(γ) = 1 and, by (9.7.115), the sum over w(γ) ≥ 2 is bounded by the sum in (9.7.121). For A± , we have γ(A± ) is inside Cγ for γ 6= 1, so the sum over γ’s with γ 6= 1 is bounded by the right side of (9.7.121). The A± term is bounded by π as in Theorem 3.3.6.

Proof of Theorem 9.7.9. By Lemma 9.7.11, for any compact K in the set (9.7.84), X X 1 − B(z, zj ) ≤ C(K)|zj − pj | (9.7.127) B(z, p ) j j j

Since |zj − pj | ≤ arclength on that Ck± which contains all the zj and pj . But, by the interlacing property, these arcs are disjoint, so their sum is bounded by the total arclength of Ck± . Thus, the sum converges uniformly on K and so, all the analyticity properties and also (9.7.87) hold. Thus we need only prove the statements about arg B∞ . (9.7.88) follows from Lemma 9.7.13 since the arg of a finite product is bounded by a sum of args of single ratios—which is precisely what a bounded variation condition bounds. Thus, (9.7.88) holds with CΓ given by the RHS of (9.7.123).

9.8. CONTINUITY OF THE COVERING MAP

609

Because each B( · , zj ) is character automorphic, so is B∞ (z) as a uniform limit. Thus, max |arg B∞ (z) − arg B∞ (w)|

z,w∈γ[F ]

is γ-independent, and so bounded by 2CΓ . If z ∈ B ∩ (Rn \ Rn+1 ), there is a path from 0 to z that goes through part of F , γ (1) (F ), γ (2) (F ), . . . , γ (n) (F ) where w(γ (j) ) = j successively. The change of arg B∞ is at most CΓ in F and 2CΓ in γ (j) (F ), so at most (2n + 1)CΓ. Remarks and Historical Notes. The connection between s = 1 Poincaré convergence and convergence of Blaschke products is classical. Indeed, Poincaré used his series to construct automorphic functions. The connection of B(z) to the potential theorist’s Green’s function is also part of standard lore; see, for example, Tsuji [435]. Theorem 9.7.7 is from the work of Sodin–Yuditskii [404] and Peherstorfer–Yuditskii [336, 337] who also have calculations similar to (9.7.48). The present proof we give of Theorem 9.7.6 and Theorem 9.7.9 are from the work of Christiansen–Simon–Zinchenko [82, 83, 84, 85]. One can use (9.7.48) to define a natural map that is an analog of the Szeg˝o mapping of Section 1.9. The idea is that, under this map, Sz : dρe goes to dθ/2π and g(x) dρe(x) goes to g(x(eiθ ))dθ/2π. This plus continuity determines this mapping. Put differently, there is a map, x∗♯ : M+,1 (∂D) → M+,1(e) by ρ = x∗♯ (µ) given by Z Z h(x) dρ = h(x(eiθ )) dµ(θ (9.7.128) This map is many-to-one. But it is one-one if we restrict to quasiinvariant measures, that is, measures with and for all γ ∈ Γ,

µ(−θ) = µ(θ)

(9.7.129)

dµ(arg(γ(eiθ ))) = |γ ′ (eiθ )| dµ(θ)

(9.7.130)

9.8. Continuity of the Covering Map Fix ℓ and let Qℓ ⊂ R2ℓ+2 be all (2ℓ+2)-tuples (α1 , . . . , βℓ+1) obeying (5.12.2). In this section, we want to consider the dependence of the basic objects of this chapter, the covering map, x, the Fuchsian group generators, {γj }ℓj=1 , and the Blaschke factors, B(z, w), on q ∈ Qℓ . So (q) we will often write xq (z), γj (z), Bq (z), Bq (z, w). Our main goal in this section is to prove that

610


Theorem 9.8.1. (i)

q 7→ xq ( · )

(9.8.1)

(ii)

q 7→ γj ( · )

(q)

(9.8.2)

q 7→ Bq ( · , w)

(9.8.3)

(iii)

are continuous as maps in q ∈ Qℓ to analytic functions in the topology of uniform convergence on compact subsets of D. Remark. γj , B( · , w) have values in D but xq has values in C∪{∞}, so we mean uniform in the proper local coordinates on C ∪ {∞} (to handle poles). This is the kind of result that one is tempted to prove via Goldberger’s method (see the Notes): “The argument is via the method of reductio ad absurdum—suppose the result is false. Why, that’s absurd!” The proof, while not difficult, is not so short. Two keys will be that if fn is a sequence of analytic functions from D → D, then there is a subsequence, n(j), so fn(j) converges uniformly on compact subsets of D either to another analytic function of D to D or to a constant function with value in ∂D (Montel’s theorem). The second is that if fn → f uniformly on compact subsets of a region Ω and if zn → z in Ω, then fn (zn ) → f (z) (because Cauchy estimates imply equicontinuity). (n) We will let qn → q in Qℓ and use xn for xqn , x∞ for xq∞ , γj for (q ) γj n , en , e∞ for the associated subsets of R, etc. The idea of the proof ˜ ∞ of the xn is x∞ . To do will involve showing that any limit point x this, we will need a way of identifying covering maps. Here is the result we will use: Proposition 9.8.2. Let x : D → C ∪ {∞} have the following properties: (a) x′ (z) 6= 0 for all z with x(z) 6= ∞, and at any point with x(z0 ) = ∞, the pole is simple. (b) x(0) = ∞; the residue at 0 is in (0, ∞). (c) There is a Fuchsian group, Γ, with x(z) = x(w) ⇔ ∃γ ∈ Γ so that w = γ(z)

(9.8.4)

Then x is the covering map of D onto Ran(x) and Γ the associated Fuchsian group. Proof. By (a) and (b), x is locally one-one and x has the normalization we have demanded for covering maps, so we need only confirm for any z0 ∈ D, x(z0 ) has an open neighborhood, N, so x−1 [N] is a disjoint union of open sets on which x is one-one.


611

Since x is locally one-one, for no γ ∈ Γ, γ 6= 1, can we have γ(z0 ) = z0 . Thus, r = min ρ(γ(z0 ), z0 ) > 0 (9.8.5) γ∈Γ

where ρ is the hyperbolic metric. For γ ∈ Γ, let r Mγ = w ρ(w, γ(z0 )) < 2 Since

(9.8.6)

ρ(γ −1 (w), z0 ) = ρ(w, γ(z0 ))

(9.8.7)

γ −1 [Mγ ] = M1

(9.8.8)

x[Mγ ] = x[M1 ] = N

(9.8.9)

x−1 [N] = ∪γ Mγ

(9.8.10)

M1 ∩ Mγ = ∅

(9.8.11)

so, by (9.8.4), Also, by (9.8.4) Next, if γ 6= 1, since w ∈ M1 ∩ Mγ implies

ρ(z0 , γ(z0 )) ≤ ρ(z0 , w) + ρ(w, γ(z0 )) < r

violating (9.8.5). Since

Mγ ∩ Mγ ′ = γ[M1 ∩ Mγ −1 γ ′ ] we see the Mγ are disjoint. Thus, N is the required neighborhood. Finally, (9.8.8) and (9.8.11) imply that if w ∈ M1 and γ 6= id, then γ(w) ∈ / M1 , so w, w1 ∈ M1 , w 6= w1 implies γ(w) 6= w1 , and thus, by (9.8.4), x(w) 6= x(w ′ ), that is x is one-one on M1 , and so on each Mγ . Next, we want to construct limits of xn . Fix an interval [c, d] ∈ eint ∞. For n large, [c, d] ⊂ en also. Let G : D → C∪{∞}\[c, d] be the standard conformal bijection with G(0) = ∞ and the residue at ∞ positive (i.e., G(z) = C(z + z −1 ) + D for suitable C, D) and G−1 is its inverse. Let gn (z) = G−1 (xn (z))

(9.8.12)

which maps D to D and 0 to 0. By compactness, {gn } have a limit point, g∞ , in the topology of uniform convergence on compact subsets of D, and since gn (0) = 0, we have that g∞ (0) = 0. Thus, g∞ maps to D. We therefore define ˜ ∞ (z) = G(g∞ (z)) x

(9.8.13)

612


˜ ∞ = x∞ , then by compactness again, we have convergence If we prove x of the original sequence. We will abuse notation by still using xn for the subsequence picked to converge. ˜ ∞ (0) = ∞. Proposition 9.8.3. (i) x ˜ ∞ (z) ≡ ∞ or else x ˜ ∞ is locally one-one (in the sense that (ii) Either x ˜ ′∞ 6= 0 at nonpoles and all poles are simple. x ˜ ∞ 6≡ ∞, then the residue at z = 0 is strictly positive. (iii) If x (iv) Ran(˜ x∞ ) ⊂ C ∪ {∞} \ e∞ (9.8.14) Proof. (i)–(iii) It is immediate from g∞ (0) = 0, that either g∞ is locally one-one or identically 0 by applying Hurwitz’s theorem to gn′ → ′ ′ ˜ ∞ has a positive residue at 0 if and only if g∞ g∞ and that x (0) > 0. ˜ be (iv) Since Ran(g∞ ) ⊂ D, Ran(˜ x∞ ) ⊂ C ∪ {∞} \ [c, d]. Let [˜ c, d] int ˜ the associated conformal map from D to another interval in e∞ and G −1 ˜ ˜ ˜ −1 (˜ C ∪ {∞} \ [˜ c, d]. Then G (xn ) → G x∞ ) near z = 0, and so on all −1 ˜ ˜ of D. Thus, Ran(G (˜ x∞ )) ⊂ D, so [˜ c, d] ∩ Ran(˜ xn ) = ∅. It follows that Ran(˜ x∞ ) ⊂ C ∪ {∞} \ eint ∞

(9.8.15)

˜ ∞ 6≡ ∞, its range is open, so either way, (9.8.14) holds. But if x

Proposition 9.8.4. We have (∞)

(∞)

Ran(˜ x∞ ) ⊃ C ∪ {∞} \ [α1 , βℓ+1 ]

(9.8.16)

˜ ∞ 6≡ ∞, so x ˜ ∞ is locally one-one with positive residue In particular, x at ∞. Proof. As constructed in Section 9.6, xn has a unique inverse, yn , (n) (n) from Xn ≡ C ∪ {∞} \ [α1 , βℓ+1 ] into D with yn (∞) = 0

(9.8.17)

xn (yn (z)) = z

(9.8.18)

It is inverse in the sense that for all z ∈ Xn . Of course, Ran(yn ) = Fnint . Since {yn } are uniformly bounded, and for any z ∈ X∞ ≡ C ∪ (∞) (∞) {∞} \ [α1 , βℓ+1 ], eventually z ∈ Xn , by passing to a subsequence, we can suppose yn has a limit, y∞ , uniformly on compact subsets of X∞ . By (9.8.17), y∞ maps to D and, by the uniform convergence within D, ˜ ∞ (y∞ (z)) = z x which proves (9.8.16).

(9.8.19)


613

˜ ∞ 6≡ ∞ and so, by Proposition 9.8.3, completes the proof Clearly, x of the last statement. Remark. While it is not essential (since passing to subsequences ˜ ∞ is finitely many times is harmless), we note that once we see that x locally one-one near ∞, we see all solutions of (9.8.19) with y∞ (∞) = 0 are equal near ∞, and so equal. Thus, yn → y∞ without the need to (∞) pass to a subsequence. The same is true of the γj discussed below. Proposition 9.8.5. (i) (ii) We have that

Ran(˜ x∞ ) = C ∪ {∞} \ e

(9.8.20)

(n)

sup |γj (0)| < 1

(9.8.21)

j,n

(∞)

Proof. If [c, d] ⊂ Gj , for some j, for n large, [c, d] ∩ e(n) = ∅. ˜ n be Xn with [α(n) , β (n) ] replaced by ([α(n) , β (n) ] \ [c, d]) ∪ {w | Let X 1 1 1 1 |w − 12 [c + d]| = 12 |d − c|; Im w ≥ 0}, that is, the interval pushed into ˜ n is simply connected, a semicircle in the upper half-plane. Because X ˜ ˜ n : Xn → D, so y ˜ n obeys (9.8.17) and (9.8.18). there is a unique map, y ˜ ∞ , which ˜ n converges to y ˜ ∞ so y ˜ n converges to y ˜ ∞ on X Near infinity, y 1 1 ˜ agrees with y∞ in X∞ \ {w | |w − 2 (c + d)| ≤ 2 |d − c|; Im w ≥ 0}. Since ˜∞ ◦ y ˜ ∞ (z) = z, we see [c, d] ⊂ Ran(x∞ ). Since [c, d] is an arbitrary x interval in any gap and we have (9.8.14) and (9.8.16), we conclude (9.8.20). (n)+ (n) Since Cj is the hyperbolic perpendicular bisector of 0, γj (0), we have (n)+ w ∈ Cj ⇒ ρ(0, γ(0)) ≤ 2ρ(0, w) (9.8.22) (∞)

By construction, if [c, d] ⊂ Gj (n)

(n)+

, yñ ( 12 (c + d)) ∈ Cj

, so

lim ρ(0, γj (0)) ≤ 2ρ(0, y˜∞( 12 (c + d)))

n→∞

This holds for each j and proves (9.8.20).

(9.8.23)

Let Γ(n) be the Fuchsian group associated to C ∪ {∞} \ e(n) . We will need to look at limits of Γ(n) as n → ∞. For this, the following will be useful: ˜∞ (z) Proposition 9.8.6. (i) As n → ∞, Bn (z) has a limit B (uniformly on compact subsets of D), which is not identically 0. (ii) X (1 − |γ(0)|) < ∞ (9.8.24) sup n

γ∈Γ(n)

614


Proof. (i) We can find ε > 0, so for large n, Ran(yn ) ⊃ {z | |z| < 2ε}. Since xn is one-one on Ran(yn ), we see Bn is nonvanishing on {z | 0 ≤ |z| ≤ ε}. Since Bn′ (0) > 0, this implies for |z| < 1 that Z iθ e +z dθ iθ Bn (zε) = z exp log|Bn (εe )| (9.8.25) eiθ − z 2π By (9.7.36) and Proposition 5.6.2, |Bn (εeiθ )| = exp(−Gen (xn (εeiθ ))) → exp(−Ge(˜ x∞ (εeiθ ))

(9.8.26)

so, by (9.8.25), Bn (z) converges for |z| < ε. By boundedness of Bn (z) uniformly in n and |z| < 1, we get convergence on all D. Since (9.8.26) ˜∞ is nonvanishing on {z | 0 < |z| < ε}, we see B ˜∞ implies the limit B is not identically zero. (ii) By Hurwitz’s theorem, B∞ (z)/z has a nonzero value at z = 0, so by (9.7.40), Y inf |γ(0)| > 0 (9.8.27) n

γ∈Γ(n) γ6=1

For all real y, ey ≥ 1 + y (by convexity), so e(w−1) ≥ w. So for 0 < w < 1, e(1−w) ≤ w −1 and X Y exp (1 − |γ(0)|) ≤ |γ(0)|−1 (9.8.28) γ∈Γ γ6=1

γ∈Γ γ6=1

and (9.8.27) implies (9.8.24).

By Corollary 9.4.2 and (9.8.20), by passing by a subsequence, we (∞) can suppose for each j = 1, . . . , ℓ that there is γ˜j ∈ M so (n)

γj

(∞)

→ γj

(9.8.29)

˜ (∞) be the free group generated by {γ (∞) }ℓj=1. By (9.8.24), Let Γ j X (1 − |γ(0)|) < ∞ (9.8.30) ˜ (∞) γ∈Γ

˜ (∞) is Fuchsian. so Γ ˜ (∞) , there exist γn ∈ Γ(n) so γn → γ. Proposition 9.8.7. If γ ∈ Γ (∞) Conversely, if γn is a sequence in Γ(n) and γn has a limit in D, then ˜ (∞) . γn(j) → γ for some γ ∈ Γ


615

(∞)±1

˜ (∞) is a finite word in {γ Proof. If γ ∈ Γ }ℓj=1 , it is a limit of j (n)±1 ℓ the same word in {γj }j=1. For the converse, we note that, by Corollary 9.3.14, if w(˜ γ ) is the word length of γ˜ ∈ Γ(n) , we can find 1, γ(1) , . . . , γ(w−1) ∈ Γ(n) with |γ(j)(0)| ≤ |˜ γ (0)| and w(γ(j) ) = j for we can write γ˜ (0) = r1 . . . rw(γ) (0) where rk is a reflection in a Cj+ and the rk+1 . . . rw(γ) (0) is outside the circle in which rk is a reflection. Thus, X |1 − γ(0)| ≥ w(˜ γ )(1 − |˜ γ (0)|) (9.8.31) γ∈Γ(n)

Now suppose γn → γ so γn (0) → z∞ ∈ D. By (9.8.24) and (9.8.31), supn w(γn ) ≡ W < ∞. There are only finitely many word patterns of length W or less, so one must get repeated infinitely often, and that provides the subsequence. ˜ ∞ has the property that Proposition 9.8.8. x ˜ (∞) ˜ ∞ (z) = x ˜ ∞ (w) ⇔ ∃ γ ∈ Γ x

so that w = γ(z)

(9.8.32)

˜ (∞) , there exist γn ∈ Γ(n) so γn → γ. Thus, Proof. If γ ∈ Γ ˜ ∞ (γ(z)) and xn (γn (z)) = xn (z) γn (z) → γ(z) ∈ D so xn (γn (z)) → x implies ˜ ∞ (γ(z)) = x ˜ ∞ (z) x (9.8.33) Conversely, let z, w be such that LHS of (9.8.32) holds. Since xn (w) − xn (z) → 0 and xn (w), xn (z) have locally one-one limits, so xn is uniformly locally invertible, there exists wn → w so xn (wn ) = xn (z). Thus, there is γn ∈ Γ(n) with wn = γ(z) → w. By Proposition 9.8.7, ˜ (n) with γ(z) = w. there is γ ∈ Γ ˜ ∞ be a limit point of the xn ’s. Proof of Theorem 9.8.1. Let x ˜ ∞ obeys all the hypotheses of Proposition 9.8.2 As discussed above, x ˜ ∞ = x∞ and with Ran(˜ x∞ ) = C ∪ {∞} \ e∞ and Γ = Γ(∞) . Thus, x (n) (∞) (∞) (∞) ˜ Γ = Γ . By compactness, we conclude xn → x∞ and γj → γj uniformly on compacts. This implies convergence of finite Blaschke products associated to a set of words in Fn . By (9.8.24), these finite Blaschke products converge to B(z, w) uniformly in n. Thus, Bn (z, w) → B∞ (z, w). Remarks and Historical Notes. Theorem 9.8.1 is a special case of a result of Hejhal [189] who noted that one could also base a proof on ideas of Ahlfors–Bers [8]. Hejhal’s method is different from the one in this section which describes joint work with Jacob Christiansen and


616

Maxim Zinchenko. M. Goldberger is a distinguished theoretical physicist with a running gag about “Goldberger’s method” as we have quoted it—it expressed the notion of many theoretical physicists that mathematical statements that are “obviously” true do not require proof! 9.9. Step-by-Step Sum Rules for Finite Gap Jacobi Matrices With the covering map in hand, we can follow the by now standard path to get nonlocal step-by-step sum rules and from that the step-bystep sum rule that will yield a Szeg˝o–Shohat–Nevai-type theorem in the next section. The disappointment is that we do not know how to get a Killip–Simon-type sum rule. Theorem 9.9.1 (Nonlocal finite gap step-by-step sum rule). Let e be a finite gap set and J a Jacobi matrix with σess (J) = e. Let x be the N2 1 covering map for C ∪ {∞} \ e and let {pj }N j=1 , {zj }j=1 be a counting of the points in F which go, under x, into the eigenvalues of J (for pj ) and J1 (for zj ). Let M(z) be given by (9.7.1) and let B∞ be the alternating Blaschke product for the z’s and p’s given by Theorem 9.7.9. Then up dθ to sets of 2π measure zero, and

{θ | Im M(eiθ ) 6= 0} = {θ | Im M1 (eiθ ) 6= 0}

Im M(eiθ ) log Im M1 (eiθ )

Moreover, a1 M(z) = B(z)B∞ (z) exp

Z

dθ ∈ L ∂D, 2π p −∞

(9.10.27)

(ii)

1 2

1 2

Z(J | J1 ) = S(ρe | µ1 ) − S(ρe | µ)

(9.10.28)

where µ1 is the spectral measure for J1 , the once stripped Jacobi matrix.


622

Remarks. 1. (9.10.28) is only true if S(ρe | µ) > −∞. More properly, it should say S(ρe | µ) > −∞ ⇔ S(ρe | µ1 ) > −∞, and then (9.10.28) holds. 2. In (3.6.8), we had an extra − 12 log 2 because we really had two reference measures in mind, dρe and the free Jacobi measure, dµ0 . Proof. (i) By Theorem 5.5.22 and (5.5.139), we have R

C dist(x, R \ e)−1/2 ≤ ρe(x) ≤ D dist(x, R \ e)−1/2

(9.10.29)

so |log(ρe(x))| dρe(x) < ∞ and (9.10.26) implies (9.10.27). (ii) By (2.3.56), Z 1 Im m1 (x + i0) Z(J | J1 ) = log dρe(x) (9.10.30) 2 e Im m(x + i0) which, by

Im mµ (x + i0) = πwµ (x) and (9.10.26), implies (9.10.28).

(9.10.31)

We can now rewrite (9.9.17). Proposition 9.10.5. If S(ρe | µ) > −∞, then a1 . . . an = K exp( 12 S(ρe | µ) − 12 S(ρe | µn )) n C(e)

(9.10.32)

where µn is the spectral measure of Jn , the n times stripped J, and X K = exp [Ge(Ej (J)) − Ge(Ej (Jn ))] (9.10.33) j

Remarks. 1. Included is S(ρe | µ) > −∞ ⇔ S(ρe | µn ) > −∞

(9.10.34)

2. Ej (J) are the eigenvalues of J outside e. The sum in (9.10.33) may only be conditionally convergent. Proof. For n = 1, immediate from (9.9.17) and (9.10.28). For general n, we iterate and take products. (9.10.34) follows from |Z(J | J1 )| < ∞ always. Proposition 9.10.6. If (9.10.5) holds, there is a constant, C1 , depending only on e and the sum in (9.10.5) so that a1 . . . an lim sup ≤ C1 exp( 12 S(ρe | µ)) (9.10.35) C(e)n In particular, ⇒ holds in (9.10.6).

9.10. THE SSN THEOREM FOR FINITE GAP JACOBI MATRICES

623

Proof. Let Je be the Jacobi matrix whose spectral measure is the (e) (e) equilibrium measure dρe and let {an , bn }∞ n=1 be its Jacobi parameters. Define J (n) to be the Jacobi matrix with parameters ( aj j = 1, . . . , n (n) (9.10.36) aj = (e) aj−n j = n + 1, . . . ( bj j = 1, . . . , n (n) bj = (9.10.37) (e) bj−n j = n + 1, . . . We claim that C1 = sup exp n

X

n

Ge(Ej (J ))

j

−∞. Since S(ρe | µn ) ≤ 0, exp(− 12 S(ρe | µn )) ≥ 1 so, by (9.10.32)/ (9.10.35), a1 . . . an ≥ C2 exp( 12 S(ρe | µ)) n C(e) where (since Ge ≥ 0)

(9.10.48)

so (9.10.47) is equivalent to X sup Ge(Ej (Jn )) < ∞

(9.10.49)

C2 = exp − sup

n

n

X

Ge(Ej (Jn )

j

j

By (9.10.42), this follows if we show that for each of ℓ + 2 intervals (ℓ gaps plus intervals adjacent to α1 and βℓ+1 ) that sup Σ(a,b) (Jn ) < ∞ n

But if Pn is the projection onto {δj }∞ j=n+1 , then Jn = Pn JPn

(9.10.50)

9.11. THETA FUNCTIONS AND ABEL’S THEOREM

Since Pn J(1 − Pn ) is rank 1, (9.10.12) implies 1/2 |b − a| Σ(a,b) (Jn ) ≤ Σ(a,b) (J) + 2 proving (9.10.50).

625

(9.10.51)

Proof of Theorem 9.10.1. We have proven (9.10.6), which is equivalent to (9.10.4). As noted in (9.10.27), the left side of (9.10.6) is equivalent to S(ρe | µ) > −∞. By (9.10.35)/(9.10.47), this implies (9.10.5). Remarks and Historical Notes. The approach in this section is from Christiansen–Simon–Zinchenko [84], but large parts predate their work. With no bound states that LHS of (9.10.4) implies (9.10.5) is due to Widom [447]. Widom considered general sets of finitely many arcs and Szeg˝o proved asymptotics of polynomials. Aptekarev [21] specified the impact on Jacobi parameters in the OPRL case. Peherstorfer– Yuditskii [336], using the framework of Sodin–Yuditskii [404], recovered Widom’s results and extended them to certain infinite gaps sets. In [337], they considered the general condition (9.10.5) on bound states. 1 ...an While [84] were the first to state S(ρe | µ) = −∞ ⇒ aC(e) → n 0, Peherstorfer noted that one can obtain it also from the results of [336, 337] (see [84] for details). 1 ...an [84] also prove that S(ρe | µ) > −∞ and lim supn→∞ aC(e) n < −∞ implies (9.10.5). The proof uses the same sum rule ideas we discuss in this section. 9.11. Theta Functions and Abel’s Theorem Blaschke products allow us to specify arbitrary points z0 ∈ F and find f analytic in D with zeros only at {γ(z0 )}γ∈Γ . The resulting functions can be meromorphically continued to C ∪ {∞} \ Λ(Γ) and they still have zeros only at {γ(z0 )}γ∈Γ . But the poles lie at {γ(¯ z0 )−1 }γ∈Γ . In this section, one of our main goals will be to break this rigid connection between zeros and poles and allow poles instead at {γ(z1 )}γ∈Γ where z1 may not be z¯0−1 . We will only accomplish this when both z0 e ± . This will suffice for and z1 lie in the same complete orthocircles C j our applications. The corresponding “theta functions” will be character automorphic. If a product of these theta functions has trivial character, that is, is character automorphic, then it defines a memomorphic function on S and this will give us a handle on the existence part of Abel’s theorem.


626

We will only get Abel’s theorem when the zeros and poles lie in the Gj , but that suffices for the application in Theorem 5.12.10. e± for How might one get rid of the poles at {γ(¯ z0 )−1 }? If z0 ∈ C j

some j, then x(z0 ) = x(z0−1 ), so B(z, z0 )(x(z) − x(z0 )) has no pole at z¯0−1 (it has a pole at z = 0 and its images—we will worry about that soon), but it has a double zero at each {γ(z0 )}γ∈Γ . Thus, we need to be able to take square roots of functions with only double zeros and poles. Lemma 9.11.1. Let f be a character automorphic meromorphic function on C ∪ {∞} \ Λ(Γ) that obeys e± and its images under Γ. (i) The only zeros and poles lie on ∪ℓj=1 C j (ii) Every zero and pole of f has even order. e± (e.g., a circle (iii) If Dj± is a counterclockwise contour just outside C j with the same center and a slightly longer radius), then Z f ′ (z) dz = 0 (9.11.1) Dj± f (z) (iv) f (0) > 0 (9.11.2) Then there is a unique character automorphic function g (denoted √ by f) so g(z)2 = f (z) g(0) > 0 (9.11.3) Proof. By (9.11.1),

Z z ′ f (w) h(z) = log(f (0)) + dw (9.11.4) 0 f (w) defines a single-valued function on Feint where any contour staying in Feint can be used in (9.11.4). On Feint , define g(z) = exp( 21 h(z)) (9.11.5) e± are of even order, which obeys (9.11.3). Since all zeros and poles on C j

e g can be meromorphically continued to a neighborhood, N, of F. For each j, Sj ≡ {z ∈ Feint | γj (z) ∈ N} (9.11.6) is nonempty and open, and by decreasing N, we can suppose it is connected. By hypothesis, g(γj (z))2 = cf (γj )g(z)2 for all z ∈ Sj . By continuity and compactness of Sj , we can find a square root, cg (γj ), of cf (γj ) so that for z ∈ Sj , g(γj (z)) = cg (γj )g(z). This allows a unique character automorphic extension of g to C ∪ {∞} \ Λ(Γ).


627

e + where we will place the pole. We need a base point on each C j Once we have a function with an arbitrary zero and such a pole, we can take a ratio of two such to move the pole. For each j = 1, . . . , ℓ, ζj e+ with will be the unique point in C j x(ζj ) = βj

(9.11.7)

ζj lies in ∂D. Theorem 9.11.2. Let y ∈ Gj , some gap in S. Let ζ be the unique point in Cj+ with x(ζ) = y (9.11.8) Then there exists a unique function Θ0 ( · ; y) meromorphic on C∪{∞}\ Λ(Γ) so that (i) Θ0 has simple zeros at {γ(ζ)}γ∈Γ = {z ∈ C | x(z) = y} and simple poles at {γ(ζj )}γ∈Γ = {z ∈ C | x(z) = βj } and no other zeros and poles. (ii) Θ0 is character automorphic. (iii) Θ0 (0; y) = 1 (9.11.9) Moreover, Θ0 is continuous in y as a function from C ∪ {∞} \ Λ(Γ) to C ∪ {∞} in the topology of uniform convergence on compacts. Remarks. 1. Of course, if y = βj , the conditions on zeros and poles conflict. We set Θ0 (z; βj ) ≡ 1. 2. We use Θ0 since we will define a slightly different Θ below (see (9.12.24)). Proof. We will prove existence and continuity now and defer the proof of uniqueness. Define η(z) by   if ζ ∈ D B(z, ζ) η(z) = 1 (9.11.10) if ζ ∈ ∂D  B(z, ζ¯−1 )−1 if ζ ∈ C \ D and f by

f (z) =

x(z) − x(ζ) η(z)η(0)−1 x(z) − βj

(9.11.11)

with f ≡ 1 if ζ = ζj . It is easy to see that f is continuous in y. Moreover, we claim that f obeys all the hypotheses of Lemma 9.11.1. Indeed, it has double poles at {γ(ζj )}γ∈Γ and nowhere else since the pole of η at ζ¯−1 is cancelled


628

by the zeros of x(·) − x(ζ) there and it has double zeros at {γ(ζ)}γ∈Γ. Thus, conditions (i) and (ii) hold. (9.11.2) holds since f (0) = 1

(9.11.12)

on account of x(0) = ∞. To check (9.11.1), let f˜(z) = (x(z) − x(ζ))/(x(z) − x(ζj )). Then f′ f˜′ η˜′ + = f η f˜

(9.11.13)

So we need only prove (9.11.2) for f replaced by f˜ and by η. f˜ is real on ∂D and Dj± are conjugate symmetric, so the f˜ integral is Q zero. For η, it suffices to prove it for B replaced by a finite product γ∈G b( · , γ(ζ)), and then (9.11.1) follows by noting the number of zeros and poles inside Dj+ cancel. Finally, f is clearly character automorphic since B is character automorphic and x is automorphic. Thus, we can apply Lemma 9.11.1 and define p (9.11.14) Θ0 (z; y) = f (z)

It has the required properties and is continuous in y since f is.

Let Aj (yj ) ∈ Γ∗ be the character of Θ0 ( · ; y), that is, Aj (yj )(γ) =

Θ0 (γ(0); y) = Θ0 (γ(0); y) Θ0 (0; y)

(9.11.15)

Recall that in Section 5.12, we defined Te = G1 × · · · × Gℓ . Define by and

e : Te → Γ∗ A

(9.11.16)

e y) = A1 (y1 ) . . . Aℓ (yℓ ) A(~

(9.11.17)

Θ0 (z; yj )

(9.11.18)

e 0 (z;~y) = Θ

ℓ Y j=1

Note that in Section 5.12, we used A and a ˜ for different maps with the same significance. e is a real analytic homeomorphism of the ℓTheorem 9.11.3. A dimensional tori Te and Γ∗ . Remark. By real analytic, we mean given locally by convergent Taylor series in real coordinates describing the tori.


629

Proof. Θ0 (z; y) is real analytic in y, so by (9.11.15), Aj (·)(γk ) is e By degree theory as explained in real analytic, and so therefore is A. e is one-one, then it is onto, and the Section 5.12, if we prove that A theorem is proven. Suppose ~y and ~w are in Te so e y) = A(~ e w) A(~ (9.11.19) and that k = #{j | yj 6= wj }. Consider g(z) =

e 0 (z;~y) Θ e 0 (z; ~w ) Θ

(9.11.20)

By (9.11.19), g is automorphic, so there is a meromorphic function G on S, so g(z) = G(x♯ (z)) (9.11.21) The poles at ζj cancel in g, so g has exactly k zeros and k poles on ∪ℓj=1 Cj+ and thus, G has exactly k ≤ ℓ zeros and poles. By the theory in Section 5.12, G is root free and so it must have an even number of zeros and poles on each Gj . Since g has exactly zero or one zero or pole on each Cj+ , we see that G has no zeros and poles, that is, k = 0, so ~y = ~w. As an immediate consequence, we get that Theorem 9.11.4. Let f be analytic and nonvanishing on C∪{∞}\ Λ(Γ) and suppose that f is character automorphic. Then f is constant. e is onto, we can find ~y ∈ Te so that A(~y) is the Proof. Since A character of f . Then g = f Θ−1 y) is automorphic and the function 0 ( · ,~ G of (9.11.21) has at most ℓ zeros and poles, so it is square root free— and that is impossible by the same argument as above, unless ~y = (β1 , . . . , βℓ ) and g = f is constant. Corollary 9.11.5 (Uniqueness part of Theorem 9.11.2). The function Θ0 of Theorem 9.11.2 is the unique function obeying (i)–(iii) of that theorem. Moreover, Θ0 is real, that is, Θ0 (z; yj ) = Θ0 (¯ z ; yj )

(9.11.22)

Proof. Let h obey (i), (ii), (iii) and let f = h/Θ0 . Then f has no zeros and poles, is character automorphic, and thus constant by the above theorem. Since f (0) = 1, we see h = Θ0 . ¯ and {¯ If ζ ∈ Cj+ , γj− (ζ) = ζ, γ (ζ)}γ∈Γ = {γ(ζ)}γ∈Γ. It follows that Θ0 (¯ z ; yj ) has the same zeros and poles as Θ0 (z; yj ). So, by the first part of the corollary, it must equal Θ0 (z; y).

630


We are now ready to prove the special case of Abel’s theorem as used in Section 5.12. Definition. By a divisor, we mean a finite subset ∆ ⊂ ∪ℓj=1 Gj and an assignment of a nonzero, nx , to each x ∈ ∆ plus an assignment of integers, n∞± to ∞± . We require and, for j = 1, . . . , ℓ,

n∞− = −n∞+ X

(9.11.23)

nx = 0

(9.11.24)

x∈Gj

We write

n∞+ δ∞+ + n∞− δ∞− + as the formal divisor.

X

nx δx

(9.11.25)

x∈∆

Definition. By a special meromorphic function, we mean a meromorphic function, f , on S, all of whose zeros and poles lie on ∪ℓj=1 Gj ∪ {∞+ } ∪ {∞− }, and if nx is the order of the zero at x (nx < 0 means a pole), then nx obeys (9.11.23) and (9.11.24). We define A(∞± ) by letting ω0 ∈ Γ∗ be the character of B(·) and ω ∈ Γ∗ a solution of ω 2 = ω0 (there are 2ℓ such solutions) and setting A(∞± ) = ω ±1

Theorem 9.11.6 (Abel’s Theorem for Spectral Meromorphic Functions). If f is a special meromorphic function and nx is the order of its poles and zeros, then Y (9.11.26) A(x)nx = 1 x∈∆∪{∞± }

Conversely, if nx , x ∈ ∆ ∪ {∞± }, where ∆ ⊂ ∪ℓj=1 Gj is finite, obeys (9.11.26), then there is a unique (up to a multiplicative constant) special meromorphic function, f , whose divisor is (9.11.25).

Proof. Given nx obeying (9.11.23)/(9.11.24), let g be the meromorphic function on C ∪ {∞} \ Λ(Γ) Y g(z) = B(z)n∞+ Θ0 (z; ζ(x))nx (9.11.27) x∈∆

where ζ(x) is the unique ζ ∈ ∪ℓj=1 Cj+ with x(ζ) = x. Then g is character isomorphic with character Y Ag ≡ A(x)nx (9.11.28) x∈∆∪{∞± }


631

(since n∞+ = −n∞− and ω 2 = ω0 , we get the character of B to the n∞+ power). If (9.11.26) holds, then g is automorphic and there is a special meromorphic function, f , with g(z) = f (x♯ (z))

(9.11.29)

proving existence. Uniqueness is obvious, since the ratio of two functions with the same nx is an analytic function on S with no zeros and poles, hence constant. If f is a special meromorphic function with divisor (9.11.25) and g is given by (9.11.27), then g(z) ≡ h(z) f (x♯ (z)) is character automorphic with no zeros and poles, hence constant by Theorem 9.11.4. Thus, g is automorphic, that is, (9.11.26) holds. Remark. The existence proof is constructive, that is, we have shown f (x♯ (z)) = LHS of (9.11.27) (9.11.30) We can thus write down the m-function for elements of the isospectral torus in terms of theta functions. Theorem 9.11.7. Let (p1 , . . . , pℓ ) ∈ Te. Let (z1 , . . . , zℓ ) ∈ Te be determined by e z) = A(~ e p)A(∞− )A(∞+ )−1 A(~ (9.11.31) Then, mp (z), the minimal Herglotz function with poles at (p1 , . . . , pℓ ), is given by e 0 (ζ;~z) B(ζ)Θ mp (x(ζ)) = −C(e)−1 (9.11.32) e 0 (ζ;~p) Θ

Proof. The two sides of (9.11.32) have the same zeros and poles, so by Theorem 9.11.6, they agree up to a constant. Near ζ = 0, mp (x♯ (ζ)) = −

ζ + O(ζ 2) x∞

(9.11.33)

while, by (9.7.37) and (9.11.9),

so (9.11.32) holds.

e 0 (ζ;~z) B(ζ)Θ C(e) = ζ + O(ζ 2) e x ∞ Θ0 (ζ;~p)

632


In the next section, we will use U(~p) = ~z for the map~p → ~z given by e U is just multiplication by the character (9.11.31). Note that under A, of B. We also want to note the following: Theorem 9.11.8. Parametrize yj ∈ Gj by a point in ∂D by lete+ farthest from 0, cj the center of C e+ , and ting pj be the point in C j j ζj = cj + eiθj (pj − cj ). We can use these as uniform parameters as we vary α1 , β1 , . . . , αℓ+1 , βℓ+1 , the edges of the bands. Then as functions of q = (α1 , . . . , βℓ+1 ) in the set Qℓ of Section 9.8 and θ1 , . . . , θℓ ∈ Dℓ , Θ0 (z;~y; q) is continuous in θ and q. Proof. Direct from the construction and the continuity results of Section 9.8. Remarks and Historical Notes. The history of elliptic functions involved the study of doubly periodic entire functions on C, that is, functions on the torus which is our S with ℓ = 1. Jacobi constructed elliptic functions as products and quotients of building blocks we now call Jacobi theta functions. These were only doubly periodic up to a phase (not actually a character)—the condition on the zeros and poles we call Abel’s theorem came from a requirement that these phases cancel. Thus, building blocks like those in this section have come to be called theta functions. Poincaré also had products related to our Blaschke products that he called theta functions. So the name is quite natural. For their Riemann surface, which could be infinite genus, Sodin– Yuditskii [404] defined theta functions for zeros in the gap. Our construction here of Θ0 , following Christiansen–Simon–Zinchenko [83], is motivated by, but distinct from, the Sodin–Yuditskii construction. The A(∞) of [83] is our A(∞+ )A(∞− )−1 . Widom [448] and Aptekarev [21] use Riemann theta functions which are distinct from what we use here and which do not lead to character automorphic functions when lifted. They allow poles and zeros off the real axis on S but are not as natural from the covering space point of view we use. See the Notes to Section 5.12 for a discussion of references for Abel’s theorem. 9.12. Jost Functions and the Jost Isomorphism In this section, we will associate to each measure, µ ∈ Sz(e), which obeys the Szeg˝o condition, a character automorphic function, u(z; Jµ ),

9.12. JOST FUNCTIONS AND THE JOST ISOMORPHISM

633

on D which we will call the Jost function. In the Notes, we discuss the connection to the Jost function of Section 3.7. This function will allow us to define a modified Weyl solution called the Jost solution whose asymptotics, as in Section 3.7, will be important in establishing Szeg˝o asymptotics. In the case of µ in the isospectral torus, we will see that the Jost function can be expressed in terms of theta functions and that it continues to a meromorphic function on C ∪ {∞} \ Λ(Γ). Moreover, we will prove the important fact that the map from the isospectral torus to Γ∗ obtained by mapping ~y ∈ Te into the character of its Jost function is an isomorphism which we will call the Jost isomorphism. As a bonus, we will prove that, for {an , bn }∞ n=1 in the isospectral torus, n n → a1 . . . an /C(e) is almost periodic. Jost functions require the choice of a base point whose eigenvalues determine poles of the Jost function. It will be natural to make these poles as far from z = 0 as possible and to pick the base point in the e+ be the isospectral torus. Thus, for each j = 1, . . . , ℓ, we let ρj ∈ C j point farthest from 0, that is, |ρj | = supζ∈Ce+ |ζ|. We let rj ∈ Gj be j defined by rj = x♯ (ρj ) (9.12.1) rj lies in S− . Let ~ρ = (ρ1 , . . . , ρℓ ) and ~r = (r1 , . . . , rℓ ) ∈ Te. For ~y ∈ Te, we let m~y be the m-function for the associated Jacobi matrix, J~y . Definition. Let µ ∈ Sz(e). For each eigenvalue (= point mass), pj ∈ R \ e, let πj ∈ (−1, 1) ∪ ∪ℓj=1 Cj+ be the point with x(πj ) = pj

(9.12.2)

The Jost function for µ is defined, for z ∈ D, by Z iθ Y 1 e +z w~r (x(eiθ )) B(z, πj ) exp u(z; µ) = log dθ 4π eiθ − z w(x(eiθ )) j

(9.12.3)

where w is the weight for µ and w~r for m~r . Notice that since µ ∈ Sz(e), the πj obey a Blaschke condition and the product in (9.12.3) converges by Theorem 9.7.7. Since µ ∈ Sz(e) and the measures in the isospectral torus obey a Szeg˝o condition (by Theorem 5.13.6), iθ log w~r (x(e )) ∈ L1 dθ (9.12.4) w(x(eiθ )) 2π by Theorem 9.7.6.


634

Theorem 9.12.1. For any µ ∈ Sz(e), the Jost function, u(z; µ), is analytic in D, character automorphic, and its zeros lie exactly at {γ(πj )}j;γ∈Γ. Moreover, u is real on (−1, 1) so u(¯ z ; µ) = u(z; µ)

(9.12.5)

Proof. Analyticity and the zero position is immediate from the L1 condition (9.11.4) and the Blaschke condition. Reality on (−1, 1) is immediate from symmetry of the set of zeros and w·(x(eiθ )) = w·(x(eiθ )) for · = ~r or nothing. The Blaschke factors are character automorphic and the product is convergent, so that u is character automorphic follows from the lemma below. Lemma 9.12.2. Let f be a real-valued L1 function on ∂D so that f (γ(eiθ )) = f (eiθ ) for all γ ∈ Γ. Define for z ∈ D, Z iθ e +z iθ dθ Sf (z) = exp f (e ) eiθ − z 2π Then Sf is character automorphic. Proof. Suppose first f ∈ L∞ , so eiθ +z

exp(−kf k∞ ) ≤ |Sf (z)| ≤ exp(kf k∞ )

(9.12.6)

(9.12.7)

(9.12.8)

since Re( eiθ −z ) is a Poisson kernel which is positive and whose integral is 1. Fix γ ∈ Γ and let Sf (γ(z)) h(z) = (9.12.9) Sf (z) so that, by (9.12.8), exp(−2kf k∞ ) ≤ |h(z)| ≤ exp(2kf k∞). By (9.12.7) and Proposition 2.3.11 for a.e. θ, lim |Sf (reiθ )| = exp(f (eiθ )) r↑1

(9.12.10)

so, by (9.12.6), for a.e. θ, lim |h(reiθ )| = 1 r↑1

(9.12.11)

Thus, log|h(z)| is a bounded harmonic function whose boundary values vanish on ∂D, so |h(z)| = 1, that is, h(z) = eiψγ for some ψγ , and thus, Sz (γ(z)) = eiψγ Sf (z)

(9.12.12)

which shows Sf is character isomorphic. For general f ∈ L1 , we can find fn → f in L1 with fn ∈ L∞ . Then, Sfn → Sf uniformly on compact subsets of D. By the compactness of Γ∗ , this implies Sf is also character automorphic.


635

The following is a rewriting of Theorem 9.9.1: Theorem 9.12.3. Let µ ∈ Sz(e) and let M(z; µ) be its M-function, {aj , bj }∞ j=1 its Jacobi parameters, and u(z; µ) its Jost function. Let µ1 be the first stripped measure, that is, the measure with Jacobi parameters {aj+1, bj+1 }∞ j=1 . Then a1 M(z; µ) =

B(z)u(z; µ1 ) u(z; µ)

In particular, if J (µ) is the character of u( · ; µ), then J (µ1 ) = J (µ)A(∞)−2

(9.12.13)

(9.12.14)

Note that the reference measure drops out of the ratio (9.12.13). Recall (see Proposition 3.2.5) that the Weyl solution is given by (n = 0, 1, . . . ) gn−1 (x; µ) = hδn , (Jµ − x)−1 δ1 i (9.12.15) We define the Jost solution by lifting to D and multiplying by u(z; µ) (to cancel the poles of g by zeros of u!). Definition. Let µ ∈ Sz(e). The Jost solution, un (z; µ), for n = 0, . . . is defined for z ∈ D by where

un (z; µ) = −u(z; µ)gn−1 (x(z); µ)

(9.12.16)

gn=−1(z) = −a−1 0

(9.12.17)

Remarks. 1. gn has poles at eigenvalues of Jµ and u has zeros; in (9.12.16) we cancel the poles by the zeros. 2. Of course, a0 is not defined by µ and is only relevant to the extent we want (9.12.19) below to hold at n = 1. One often takes a0 ≡ 1, but for elements of the isospectral torus, there is another natural definition. Theorem 9.12.4. Let µ ∈ Sz(e). Let µn be the n-stripped measures (i.e., with Jacobi parameters {aj+n , bj+n }∞ j=1 ). Then n un (z; µ) = a−1 n B(z) u(z; µn )

For all z ∈ D,

{un (z; µ)}∞ n=1

which is ℓ2 at infinity.

(9.12.18)

is a nonzero solution of

(J − x(z))u = 0

(9.12.19)

Proof. By Theorem 3.7.6 (or (5.4.57)), m(x; Jµn ) = −

gn (z) an gn−1(x)

(9.12.20)

636


which, taking into account the minus sign in M(z) = −m(x(z)) leads to (for n ≥ 1) gn (x(z)) = an M(z; µn )gn−1(x(z))

= −an . . . a1 M(z; µn ) . . . M(z; µ1 )M(z; µ)

(9.12.21)

where the minus sign comes from M(z; µ) = −m(x(z)) = −g0 (x(z))). Thus, by the definition (9.12.16), an un (z; µ) = an an−1 . . . a1 M(z; µn−1 ) . . . M(z; µ)u(z; µ) u(z; µn ) u(z; µn−1) n = B(z) · · · u(z; µ) u(z; µn−1 ) u(z; µn−2) = B(z)n u(z; µn ) (9.12.22) proving (9.12.18). Since un is a multiple of gn−1 , it solves the difference equation. If u0 (z) = 0, then M(z) has a pole, so u1(z) 6= 0, showing that we have a nonzero solution of the difference equation. Remark. (9.12.21) is essentially an extension of Corollary 3.7.7. Having presented the Jost function and solution for general µ ∈ Sz(e) (something we will return to in the next section), we turn to µ associated to an element of the isospectral torus. We use u(z;~y) for u(z; µ~y ) when ~y ∈ Te. Initially, via (9.12.16), un (z; µ) is only defined for n ≥ 0. But by (9.12.18) which says, for ~y ∈ Te, that n n un (z;~y) = a−1 y)) n B(z) u(z; U (~

(9.12.23)

allows one to define un for all n ∈ Z and it obeys (J(~y) − x(z))u = 0 as a difference equation on all of Z. Here the theta functions enter. Having moved the base points for u, we want to move them for Θ, so we define for yj ∈ Gj , Θ(z; yj ) =

Θ0 (z; yj ) Θ0 (z; pj )

(9.12.24)

which has its zeros still at ζj (with x(ζj ) = yj ) but its pole now at πj . Of course, we also define for ~y ∈ Te, e y) = Θ(z;~

ℓ Y

Θ(z; yj )

(9.12.25)

j=1

Theorem 9.12.5. For ~y ∈ Te, define U(~y) as the point of Te corresponding to the once-stripped Jacobi matrix of J~y (i.e., if {aj (~y), bj (~y)}∞ y)) = aj+1(~y), j=1 are the Jacobi parameters of J~y , aj (U(~


637

bj (U(~y)) = bj+1 (~y)). Then there is a continuous strictly positive function, ϕ, on Te so that Moreover, ϕ obeys

e y) u(z;~y) = ϕ(~y)Θ(z;~

(9.12.26)

ϕ(~p) = 1

(9.12.27)

ϕ(U(~y)) a1 (~y) = ϕ(~y) C(e)

(9.12.28)

Before turning to the proof of this theorem, we note: Corollary 9.12.6. For any fixed ~y ∈ Te, n→

a1 (~y) . . . an (~y) C(e)n

(9.12.29)

is almost periodic. Proof. By (9.12.28), RHS of (9.12.29) =

ϕ(U n (~y)) ϕ(~y)

(9.12.30)

is almost periodic since, under the Abel map, U is translation on the torus by a fixed amount. We also note that (9.12.26) says that for points in the isospectral torus, u(z) has an analytic continuation from D to a neighborhood of D \ Λ(Γ); indeed, a meromorphic continuation to all of C ∪ {∞} \ Λ(Γ) with poles only at images of {πj }ℓj=1 under Γ. Proof of Theorem 9.12.5. We will first note that (9.12.26) holds at ~y = ~p, then on the orbit {U n (~p)}n∈Z , and then, everywhere, by a double density theorem. Since J~y=~p has no eigenvalues in gaps and ~p is the base point, e p) = 1, so (9.12.26) with u(z;~y = ~p) = 1. By construction, Θ(z;~ (9.12.27) holds. Next, we claim that if (9.12.26) holds at some point ~y, it holds at U(~y) with ϕ(U(~y)) given by (9.12.28). For, by Theorem 9.11.7, e U(~y)) Θ(z; = M~y (z)B(z)−1 C(e) (9.12.31) e y) Θ(z;~ On the other hand, by Theorem 9.12.3,

u(z; U~y ) = a1 M~y (z)B(z)−1 u(z;~y)

(9.12.32)

638


Thus, a1 (~y) u(z;~y) u(z; U(~y)) = (9.12.33) e U(~y)) e y) C(e) Θ(z;~ Θ(z; This proves our claim that if (9.12.26) holds at ~y and ϕ(U(~y)) is defined by (9.12.28), then (9.12.26) holds at U(~y). It also proves that if (9.12.26) holds at U(~y), then it holds at ~y with ϕ(~y) defined by (9.12.28). Now suppose q ∈ Qℓ so that all the harmonic measures of {[αj , βj ]}ℓj=1 are rationally independent. Then {U n (~p)}n∈Z are distinct and dense in Te. By the above, it follows that (9.12.26) holds at all these points. Define ϕ on Te by ϕ(~y) ≡ u(z = 0;~y)

(9.12.34)

Since u is continuous in ~y, ϕ is continuous on Te and, by (9.12.28), it agrees with the previous definition on {U n (~p)}n∈Z . By continuity of ϕ, u, and Θ in ~y, and the density of {U n (~p)}n∈Z , we get (9.12.26) on all of Te. (0) Next, note that, by our proof of Theorem 5.6.1, Qℓ ≡ {q ∈ Qℓ | {[αj , βj ]}ℓj=1 are rationally independent} is dense in Qℓ . Since ϕ is e are all continuous in q (if we parametrize given by (9.12.34), u and Θ yj by θj as in Theorem 9.11.8), we get (9.12.26) for all of Qℓ from the (0) formula for Qℓ . Theorem 9.12.7. The map J from Te to Γ∗ , given by taking ~y ∈ Te to the character of u( · ;~y), is an isomorphism called the Jost isomorphism. e multiplication Proof. By (9.12.26) and (9.12.24), J differs from A by a fixed group element, that is, e y)A(~ e p)−1 J (~y) = A(~ (9.12.35)

Thus, this theorem is a restatement of Theorem 9.11.3.

We can use this to specify the asymptotics of pn (z;~y), the OPs for µ, when ~y ∈ Te and z ∈ / σ(J~y ).

Theorem 9.12.8. Let ~y ∈ Te. Then for any z with x(z) ∈ / σ(J~y ), n pn (x(z);~y)B(z) is asymptotically almost periodic. In addition, for any compact K ⊂ C \ [α1 , βℓ+1 ], there is a constant C > 0 so that for any n and all z ∈ D with x(z) ∈ K, C|B(z)|n ≤ |pn (x(z);~y)| ≤ C −1 |B(z)|n

(9.12.36)

Remark. This replaces Szeg˝o asymptotics, which says in the case e = [−2, 2] that pn (x(z))/B(z)n has a nonzero limit as n → ∞.


639

Proof. Let y~˜ be the element in Te (see the remark after the theorem) with bn (~y˜) = b−n (~y) an (~y˜) = a1−n (~y) (9.12.37) and let uñ (z;~y) = u−n (z; ~y˜) (9.12.38) Then uñ is also a solution of (J~y − x(z))u = 0—it is also the Weyl solution at −∞. Since pn−1 solves the half-line case and un , uñ are independent (since one is ℓ2 at +∞ and one at −∞ and x(z) ∈ / σ(J~y )), we see pn−1 (x(z)) = α(z)un (z) + β(z)˜ un (z) (9.12.39) where α, β are given by Wronskians and are analytic in z ∈ D. Since only un is ℓ2 at +∞, the zeros of β are exactly z’s with x(z) ∈ σ(J~y ). As n → ∞, un B n → 0, while uñ B −n has an almost periodic limit. So if β(z) 6= 0 asymptotically, −n ~ pn−1 (x(z))B(z)n−1 ∼ B(z)−1 β(z)a−1 (y˜)) n u(z; U

(9.12.40)

is asymptotically almost periodic and (9.12.36) holds. Since u(z;~y) has zeros only in x−1 (∪Gj ), we get (9.12.36). Remarks. 1. Given that u(z;~y) can be written in terms of theta functions, (9.12.40) gives “explicit” asymptotics for pn (x) to the extent that theta functions are explicit. 2. If pn (x(z)) is replaced by |pn (x(z))|2 + |pn−1 (x(z))|2 , we can replace K ⊂ C \ [α1 , βℓ+1 ] by K ⊂ C \ σ(J~y ). The fact that u can be analytically continued from D to a neighborhood of D \ Λ(Γ) allows one to control the limits of the Jost solutions as one approaches the spectrum, and thereby the spectral theorists’ Green’s function. The following are proven in [83]: Theorem 9.12.9 ([83]). For x ∈ e and ~y ∈ Te, define u+ y) = un (z(x);~y) n (x;~

(9.12.41)

where z(x) ∈ F¯ is determined by x(z(x)) = x. Let u− y) = u+ y) n (x;~ n (x;~

Then there are constant C1 and C2 so that (i) Uniformly in ~y ∈ Te, x ∈ eint , and n,

|u± y)| ≤ C1 dist(x; R \ e)−1/2 n (x;~

(9.12.42)

|u± y)| ≤ C2 (|n + 1|) n (x;~

(9.12.43)

(ii) Uniformly in ~y ∈ Te, x ∈ e, and n,

640


Remark. Similar bounds hold for pn (x). For ~y ∈ Te, we use J~y for the half-line Jacobi matrix and J˜~y for the whole-line Jacobi matrix (using the almost periodic construction of the Jacobi parameters). The spectral theorists’ Green’s functions are given by Gnm (z) = hδn (J~y − z)−1 δm i enm (z) = hδn , (J˜~y − z)−1 δm i G

n, m = 1, 2, . . . n, m ∈ Z

Theorem 9.12.10 ([83]). There is a constant C3 so that uniformly in ~y ∈ Te and x ∈ R \ e, enm (x)| ≤ C3 exp(−Ge(x)|n − m|)dist(x, e)−1/2 |G (9.12.44) where Ge is the potential theorists’ Green’s function.

We say a point, x0 ∈ {αj , βj }ℓ+1 j=1 , is a resonance for J~y if m~y (z) has a pole there in the sense of the Riemann surface (i.e., |m(z)| ∼ C|x0 − z|−1/2 for z near x0 ). If x0 is not a resonance, we say it is nonresonant. If x0 is nonresonant for x0 at an edge of a gap, we let I be the open interval that has one end at x0 and the other at half the distance to any pole of my (z) in the gap, to the other end of the gap if there is no pole, and one unit from x0 if x0 is α1 or βℓ+1 .

Theorem 9.12.11 ([83]). Let ~y ∈ Te be fixed and x0 ∈ {αj , βj }ℓ+1 j=1 nonresonant. There are constants C4 , C5 so that for x ∈ I, the interval above, and for all n, m ∈ {1, 2, . . . }, |Gnm (x)| ≤ C4 min(n, m)

|Gnm (x)| ≤ C5 |x − x0 |−1/2

(9.12.45)

(9.12.46)

Remarks and Historical Notes. This section describes material from [83, 84], mainly [83]. If e = [−2, 2], the isospectral torus has a single point with Jacobi parameters given by an ≡ 1, bn ≡ 0. (3.7.42) shows that our Jost function specialized to this case agrees with the Jost function of Section 3.7. 9.13. Szeg˝ o Asymptotics One of our main goals in this section is to prove the following: Theorem 9.13.1 ([84]). Let µ be a measure in Sz(e) for a finite gap set, e, with Jacobi parameters {an , bn }∞ an , ˜bn }∞ n=1 . Let {˜ n=1 be the Jacobi parameters of the unique element of Te whose Jost function has the same character as the Jost function of µ. Then, as n → ∞, |an − a ñ | + |bn − ˜bn | → 0 (9.13.1)

˝ ASYMPTOTICS 9.13. SZEGO

641

Remark. In this form, the theorem is due to Christiansen–Simon– Zinchenko [84]; but, as we will explain in the Notes, the essence is an earlier result of Widom [448] and Peherstorfer–Yuditskii [336, 337]. It is the proof here that is different from the earlier work. The key to the proof of this is Theorem 9.13.2. Let J be a Jacobi matrix whose spectral measure, µ, is in Sz(e) and let u(z; µ) be its Jost function and µn the measure of the n-times stripped Jacobi matrix. Suppose for some {a♯n , b♯n }∞ n=1 in the isospectral torus, u♯ (z) is its Jost function, and for some nk → ∞, we have that an+nk → a♯n bn+nk → b♯n (9.13.2) as k → ∞. Then, uniformly on compact subsets of D, u(z; µnk ) → u♯ (z)

(9.13.3)

We will prove this later in the section—we want to show it implies Theorem 9.13.1 and explore the consequences of these two theorems for further asymptotic results, including Szeg˝o asymptotics for the OPs. Proof of Theorem 9.13.1 given Theorem 9.13.2. Suppose (9.13.1) fails. By compactness of bounded sequences in the product topology, there is a sequence {a♯n , b♯n }∞ n=1 and some nk so that (9.13.2) holds, and for some n0 , |a♯n0 − a ñ0 +nk | − |b♯n0 − ˜bn0 +nk | → d 6= 0

(9.13.4)

By the Denisov–Rakhmanov–Remling theorem, {a♯n , b♯n }∞ n=1 lies in the isospectral torus. Since Γ∗ is compact, without loss, we can pass to a further subsequence and suppose A(∞)−2nk has a limit c ∈ Γ∗ . By the Abel theorem analysis of the isospectral torus, that means {˜ an+nk , ˜bn+nk }∞ n=1 also has ♮ ♮ ∞ a limit, call it {an , bn }n=1 . By Theorems 9.13.2 and 9.12.3, if J ({an , bn }∞ n=1 ) is the character of the associated Jost function, then ♯ ♯ ∞ J ({an , bn }∞ n=1 )c = J ({an , bn }n=1 ) J ({˜ an , ˜bn }∞ )c = J ({a♮ , b♮ }∞ )

(9.13.5)

J ({a♯n , b♯n }) = J ({a♮n , b♮n }∞ n=1 )

(9.13.7)

n=1

n

n n=1

(9.13.6)

Thus, ♮ ♮ ∞ Thus, by Theorem 9.12.7, {a♯n , b♯n }∞ n=1 = {an , bn }n=1 , violating (9.13.4). This contradiction shows that (9.13.1) holds.

642


Remark. We can summarize the proof by saying the Denisov– Rakhmanov–Remling theorem implies the Jacobi parameters must approach the isospectral torus and the character of the Jost function specifies which orbit. We can use these theorems to deduce Szeg˝o asymptotics for any pn (x; µ) with µ ∈ Sz(e): Theorem 9.13.3 (Szeg˝o Asymptotics). Let µ ∈ Sz(e). Let ~y ∈ Te be the point in the isospectral torus with lim |an (µ) − an (~y)| + |bn (µ) − bn (~y)| = 0

(9.13.8)

n→∞

Then for all x ∈ C \ [α1 , βℓ+1 ], we have pn (x; µ) u(z(x); µ) → pn (x;~y) u(z(x);~y)

(9.13.9)

as n → ∞, where z(x) is the unique point in F int with x(z(x)) = x. In particular, pn (x)B(z(x)) is asymptotically almost periodic as n → ∞. Remark. Given (9.12.40), (9.13.9) gives explicit asymptotics for pn (x; µ). Proof [84]. Since x ∈ / [α1 , βℓ+1 ], the Green’s functions hδj , (J − x)−1 δj i and hδj , (J˜y − x)−1 δj i are nonvanishing (if Im x 6= 0, Im(J − x)−1 = Im x[(J − x¯)−1 (J − x)−1 ] and k(J − x)ϕk ≥ Im xkϕk, so |Imhδj , (J − x)−1 δj i| ≥ |Im x|−1 , and if x ≤ α1 or x ≥ βℓ+1 , k(J − x)−1 ϕk ≥ dist(x, [αj , βj ])−1 ). It follows that lim

n→∞

hδn , (J − x)−1 δn i =1 hδn , (J~y − x)−1 δn i

(9.13.10)

But, since an (pn un − pn−1 un+1 )|n=0 = a0 u(z(x); µ), we see hδn , (J − x)−1 δn i =

pn−1 (x; µ)un (z(x); µ) a0 u(z(x); µ)

so (9.13.10) and (9.13.3) imply (9.13.9),

(9.13.11)

Corollary 9.13.4. Let µ ∈ Sz(e) and ~y ∈ Te with (9.13.8). Then, as n → ∞, a1 (~y) . . . an (~y) u(0; µ) → (9.13.12) a1 (µ) . . . an (µ) u(0;~y) In particular (by Corollary 9.12.6), a1 (µ) . . . an (µ)/C(e)n is asymptotically almost periodic.


Proof. For z ∈ D, let fn (z) =

pn (x(z); µ) pn (x(z);~y)

643

(9.13.13)

The last theorem implies that for some ε > 0 and 0 < |z| < ε, we have fn (z) →

u(z; µ) u(z;~y)

(9.13.14)

and the proof shows the convergence is uniform on {z | 2ε < |z| < ε}. By the maximum principle, (9.13.14) holds for z = 0. But fn (0) = LHS of (9.13.12) so (9.13.14) for z = 0 is (9.13.12).

We now turn to the proof of Theorem 9.13.2. We write the Jost function (9.12.3) as the product β(z, µ) of the Blaschke factors and ε(z, µ), the exponential, which we call the Blaschke part and the entropy part of the Jost function, respectively. We will prove (9.13.2) by proving convergence of the two parts separately. By Corollary 9.10.3, one has uniform control over the tail of the contributions to the Blaschke part, so it suffices to prove convergence of individual eigenvalues, that is, the first part of the following implies (9.13.15). Proposition 9.13.5. Let (9.13.2) hold. Let (βj , αj+1) be a gap in e and let ε > 0 be such that neither βj + ε nor αj+1 − ε is an eigenvalue of J ♯ . Then, as k → ∞, the number of eigenvalues of Jnk in Iε ≡ (βj + ε, αj+1 − ε) is the number of eigenvalues of J ♯ in Iε and the eigenvalues converge. In particular, uniformly for compact subsets of D, β(z; µnk ) → β ♯ (z) (9.13.15) Remark. Of course, since J ♯ is in the isospectral torus, it has either 0 or 1 eigenvalue in Iε . Proof. Let λ♯ be an eigenvalue of J ♯ in Iε and u♯ a unit vector so that J ♯ u ♯ = λ♯ u ♯ (9.13.16) Then lim k(Jnk − λ♯ )u♯ k = 0

k→∞

(9.13.17)

so if εk is the norm on the left, Jnk has spectrum in (λ♯ − εk , λ♯ + εk ). Since Jnk only has eigenvalues in Iε , for k large, Jnk has an eigenvalue near λ♯ .


644

Let λnk be eigenvalues of Jnk in Iε so that λnk → λ♯ . We will prove that λ♯ is an eigenvalue of J ♯ and that the corresponding onedimensional spectral projections converge. It is easy to see that this and the initial part of this proof shows, for large k, that Jnk has exactly one eigenvalue in Iε for each eigenvalue of J ♯ , and we have the claimed convergence. Pick the unique ℓ2 element, u(k) , with Jnk u(k) = λnk u(k)

(9.13.18)

and (k)

uj=1 > 0

ku(k) k = 1

(9.13.19)

Let u(♯) be a weak limit point of u(k) ; we will continue to use u(k) for the subsequence that converges. Then J ♯ u ♯ = λ♯ u ♯

(9.13.20)

ku♯ k = 1

(9.13.21)

Suppose we prove ♯

♯

♯

Then u 6= 0, so λ is an eigenvalue of J , and since (9.13.21) holds, ku(k) − u♯ k2 = 2 − 2 limhu(k) , u♯ i = 0

(9.13.22)

so we have the claimed convergence of spectral projections. Thus, we need only verify that (9.13.21) is true. If it is false, then for ε > 0, ku♯ k = 1 − ε

(9.13.23)

and for each m, with χm the characteristic function of {1, . . . , m}, lim kχm u(k) k ≤ 1 − ε

k→∞

(9.13.24)

so we can find mk → ∞ and δ > 0 so that for all k, k(1 − χmk )u(k) k ≥ δ

(9.13.25) (k)

(k)

Picking ℓk with m2k ≤ ℓk ≤ mk with minimum value of |uℓ | + |uℓ+1| in the range, we see that (k)

(k)

|uℓk | + |uℓk +1 | → 0 Let (k) vj

( (k) uj = 0

j ≥ ℓk + 1 j < ℓk

(9.13.26)

(9.13.27)

By (9.13.26), (k)

k(Jnk − λnk )vj k → 0

(9.13.28)


and if (k)

wj

( (k) vj−nk /kv (k)k j ≥ nk = 0 j ≤ nk

645

(9.13.29)

then, by (9.13.25) and (9.13.28), k(J − λnk )w (k)k → 0

(9.13.30)

Since w (k) → 0 weakly and kw (k)k = 1, we conclude that λ♯ ∈ σess (J), contrary to σess (J) ⊂ e. This contradiction proves (9.13.21) and completes the proof of the theorem. To control the entropy part, we need the following: Proposition 9.13.6 (Simon–Zlatoˇs [401]). Let X be a compact Hausdorff space and S given by (2.2.1). Let µ be a fixed probability measure, νn → ν∞ weakly for probability measures, and let dνn = fn dµ + dνn;s for νn;s singular with respect to dµ. Suppose S(µ | νn ) → S(µ | ν)

(9.13.31)

w

(9.13.32)

Then log(fn ) dµ −→ log(f ) dµ

Proof. Suppose first that w is continuous and strictly positive on X. Then by upper semicontinuity (Theorem 2.2.3), lim sup S(wµ | νn ) ≤ S(wµ | ν) or lim sup

Z

log(fn )w dµ ≤

Z

log(f )w dµ

(9.13.33) (9.13.34)

For arbitrary real-valued continuous g, let w = 2kgk∞ ± g in (9.13.34) to conclude the claimed weak convergence. Proposition 9.13.7. To prove convergence of the entropy part, ε(z; µnk ), to ε♯ (z) uniformly on compact subsets of D, it suffices to prove that Z Z dθ dθ iθ lim log(|Im Mµk (e )|) = log(|Im Mµ (eiθ )|) (9.13.35) k→∞ 2π 2π Proof. By (9.10.28), (9.13.35) is equivalent to lim S(ρe | µk ) = S(ρe | µ♯ )

k→∞

(9.13.36)

646


By Proposition 9.13.6, this implies for any continuous function, h, on e that with wk and w ♯ the weights of µk and µ♯ , Z Z w~r (x) w~r (x) h(x) log dρe → h(x) log dρe(x) (9.13.37) wk (x) w ♯ (x) By (9.7.63), this implies that, uniformly for compact subsets of z ∈ D, ε(z; µnk ) converges to ε♯ (z).

As noted, (9.13.35) is equivalent to (9.13.36). By upper semicontinuity of the entropy, we have lim sup S(ρe | µk ) ≤ S(ρe | µ♯ )

(9.13.38)

k→∞

so it suffices to prove that lim inf S(ρe | µk ) ≥ S(ρe | µ♯ ) k→∞

(9.13.39)

By passing to a subsequence where S(ρe | µk ) converges to the lim inf, we can also suppose a1 . . . ank (9.13.40) τk ≡ C(E)nk has a limit τ∞ (since {τk }∞ k=1 are bounded by Theorem 9.10.1). Define, for k < ℓ, a Jacobi matrix J (k,ℓ) by ( ank +m 1 ≤ m ≤ nℓ − nk a(k,ℓ) = (9.13.41) m a♯m m > nℓ − nk ( bn +m 1 ≤ m ≤ nℓ − nk b(k,ℓ) = ♯k (9.13.42) m bm m > nℓ − nk that is, first replace (a, b) by (a♯ , b♯ ) for indices nℓ + 1 or more, and then strip off nk rows and columns. By iterating the step-by-step sum rule to go from J (nj ,nk ) to J ♯ (by stripping off nℓ − nk rows and columns), one gets, via (9.9.17) and (9.10.28), τℓ β(0; µ♯) = exp[ 12 S(ρe | µk,ℓ) − 12 S(ρe | µ♯ )] τk β(0; µk,ℓ)

(9.13.43)

Lemma 9.13.8. lim β(0; µk,ℓ) = β(0; µk )

ℓ→∞

(9.13.44)

Proof. By Corollary 9.10.3, we need only prove convergence of eigenvalues. The proof follows that of Proposition 9.13.5. The elimination of (9.13.24) is even easier than in that Proposition. By (9.13.2), any eigenvector that lives far from m = 0 gives essential spectral values for Jnk .


647

The following completes the proof of Theorem 9.13.2: Proposition 9.13.9. (9.13.39) holds. Proof. By upper semicontinuity of the entropy, lim sup S(ρe | µk,ℓ ) ≤ S(ρe | µk )

(9.13.45)

ℓ→∞

Thus, taking ℓ → ∞ in (9.13.43) and using (9.13.44) gives τ∞ β(0; µk ) (9.13.46) exp[ 12 S(ρe | µk ) − 12 S(ρe | µ♯ )] ≥ τk β(0; µ♯) Picking a subsequence so S(ρe | µk ) goes to the lim inf and using τk → τ∞ and (9.13.14) at z = 0, we get exp[ 12 lim inf S(ρe | µk ) − 12 S(ρe | µ♯ )] ≥ 1

which is (9.13.39).

Remarks and Historical Notes. This section follows Christiansen– Simon–Zinchenko [84]. In 1967, Widom [448] proved Szeg˝o asymptotics for orthogonal polynomials for measures in the complex plane supported on a finite union of analytic curves obeying a Szeg˝o condition. This was made precise for finite gap sets on D by Aptekarev [21], who noted the consequences for the Jacobi parameters. Peherstorfer– Yuditskii [336, 337] extended this to certain infinite gap sets and to allow eigenvalues outside e obeying a Blaschke condition. Widom, Aptekarev, and Peherstorfer–Yuditskii all use a variational approach with a character-dependent variational principle. For problems with varying but smooth weights, Riemann–Hilbert problem asymptotics have been used by Deift et al. [99] and Aptekarev– Lysov [22] to get Szeg˝o asymptotics. The variable weights are technically harder to control and Riemann–Hilbert methods can also be used for fixed analytic weights, as is made explicit in [22].

CHAPTER 10

A.C. Spectrum for Bethe–Cayley Trees In this final chapter, we discuss some work of Denisov [104, 105] about the use of sum rules in the study of perturbed Laplacians on what physicists call Bethe lattices and mathematicians call Cayley trees—so we will call them Bethe–Cayley trees. These have a kind of coefficient stripping, so a one-dimensional aspect, but they are not fully onedimensional in part because a single spectral measure does not describe them. For this reason, the results will be one-sided: from coefficient information to spectral data and not vice-versa. 10.1. Overview We begin by describing the rooted Bethe–Cayley tree of degree d = 1, 2, . . . , which is a graph that is homogeneous and then cut in half. We have a root, which we will denote by φ. It has d neighbors, which we denote (0), . . . , (d − 1). Each of these d neighbors, (j), has d + 1 neighbors, φ, and d neighbors (j1), . . . , (jd − 1). At level k, there are dk elements (we have described levels k = 0, 1, 2 above) described by (j1 . . . jk ) with jℓ ∈ {0, . . . , d − 1}. Two elements are neighbors if one is obtained from the other by deleting the last j or by adding a j at the end. The root has d neighbors and all other vertices have d + 1 neighbors. For a vertex α, we will use #(α) for the level of α, that is, its distance from φ. If β is obtained from α by adding extra j’s at the end, we write α β or β α (with the convention that α α). If neither holds, we write α ≁ β. We use |α − β| for the length of the shortest path from α to β. Thus, #(α) = |α − φ|. d = 1 is, of course, the half-line. Figure 10.1 shows the case d = 2 displaying levels 0–3. Given such a tree, B, we let ℓ2 (B) be the sequences labelled by vertices, α, in B. The free Hamiltonian is defined by X (H0 u)(α) = u(β) (10.1.1) |β−α|=1

649

650

10. A.C. SPECTRUM FOR BETHE–CAYLEY TREES 000

001

010

00

011

100

01

101

110

10

0

111

11

1

Φ

Figure 10.1.1. A Bethe–Cayley tree of degree two Given a function V on B, we let (Hu)(α) = (H0 u)(α) + V (α)u(α)

(10.1.2)

Thus, we have an analog of a Jacobi matrix where an ≡ 1, and we use V (α) instead of bn . We will only consider the case where V is a bounded function so H is a bounded operator on ℓ2 (B). By the spectral measure, dµ, we mean the one given by Z dµ(x) −1 hδφ , (H − z) δφ i = (10.1.3) x−z We will see (in Section 10.2) that for H0 , √ √ σ(H0 ) = −2 d, 2 d

(10.1.4)

with spectrum that is purely absolutely continuous of infinite multiplicity. Moreover, 1 dµ0(x) = (4d − x2 )1/2 dx (10.1.5) 2dπ which is just a scaling of the free Jacobi matrix. The main theorem we will focus on in this chapter is: Theorem 10.1.1 (Denisov [104]). Let V be bounded and obey ∞ X 1 dn n=0

X

α|#(α)=n

|V (α)|2 < ∞

(10.1.6)

√ √ Then H has a.c. spectrum of infinite multiplicity on [−2 d, 2 d]. Remarks. 1. We are not claiming that H only has spectrum in this interval. 2. We will see in Section 10.2 that this is optimal in the sense that if |V (α)|2 is replaced by |V (α)|p with any p > 2, then it can happen that H has no a.c. spectrum.

10.1. OVERVIEW

651

We have simplified the presentation by taking the analog of the case an ≡ 1. We will also simplify things by henceforth taking d = 2 in the text, leaving general d to the Notes. We begin the sketch of the strategy of the proof of Theorem 10.1.1 by noting the following: Suppose we prove that H has a.c. spectrum of multiplicity at least k under (10.1.6). In H, drop the links of φ to (0) and (1). This is a rank four perturbation, so by Corollary 7.3.4, the a.c. spectrum and its multiplicity is unchanged. But if B0 (resp. B1 ) is the subset of B with #(α) ≥ 1 and α1 = 0 (resp. α1 = 1), e is a direct ℓ2 (B) = C ⊕ ℓ2 (B0 ) ⊕ ℓ2 (B1 ) and the modified H, call it H (0) (1) (j) sum of V (φ) ⊕ H ⊕ H where H is an operator like H but on ℓ2 (Bj ). However, the potentials for H (j) also satisfy (10.1.6). Thus, e has a.c. multiplicity at least 2k. Therefore, it suffices to show the H a.c. multiplicity is at least 1, for then, iterating this argument, we get multiplicities at least 2, 4, 8, . . . , and so infinite. In terms of the relative entropy, (2.2.1), we will prove that ∞ 1 X 1 S(µ0 | µ) ≥ − 4 n=0 dn

X

α|#(α)=n

|V (α)|2

(10.1.7)

so (10.1.6) implies this entropy is finite, so if dµ = W dµ0 + dµs √ √ then W (x) > 0 for a.e. x ∈ [−2 2, 2 2], which implies a.c. spectrum of multiplicity at least 1. In fact, we will eventually prove a much stronger result than Theorem 10.1.1. Consider all paths out to infinity, that is, sequences ω ≡ (ω1 , ω2 , . . . ) ∈ {0, 1}N ≡ Ω. Associate such an α with the walk through nearest neighbors ( φ j=0 αω (j) = (10.1.8) (ω1 . . . ωj ) j ≥ 1 Put the infinite product measure, ν, on Ω with ν(ωj = 0) = ν(ωj = 1) = 12 . Moreover, define 2

kVω k =

∞ X j=0

|V (αω (j))|2

Then it is not hard to see that Z ∞ X X 1 2 |V (n)| = kVω k2 dν(ω) n d n=0 α|#(α)=n

(10.1.9)

(10.1.10)

652

10. A.C. SPECTRUM FOR BETHE–CAYLEY TREES

The following improves Theorem 10.1.1: Theorem 10.1.2 (Denisov–Kiselev [105]). Let V be bounded and suppose ν({ω | kVω k√< ∞}) √ > 0. Then H has a.c. spectrum of infinite multiplicity on [−2 2, 2 2]. Again, the key will be an improved sum rule. Instead of (10.1.7), we will prove Z (10.1.11) exp(S(µ0 | µ)) ≥ exp(− 14 kVω k2 ) dν(ω)

R R By Jensen’s inequality (log ef dη ≥ f dη for real-valued f and dη a probability measure), (10.1.11) implies (10.1.7). In Section 10.2, we will discuss the free Laplacian, while in Section 10.3, we will find the coefficient stripping formula which will lead to a step-by-step sum rule in Section 10.4. We will use the step-by-step sum rule to prove Theorem 10.1.1 in Section 10.5 and Theorem 10.1.2 in Section 10.6.

Remarks and Historical Notes. As noted, Theorem 10.1.1 is from Denisov [104] and Theorem 10.1.2 from Denisov–Kiselev [105]. Higher-order sum rules for trees have been found by Kupin [255]. Bethe lattices were introduced by Bethe [44]. The name Cayley tree comes from the fact that the Cayley graph of a free nonabelian group on p-generators is the unrooted Cayley tree with degree 2p. Other than [104, 105], most of the spectral theory literature on Bethe–Cayley trees concerns Anderson localization; see [9, 10, 17, 138, 147, 223, 224]. For other papers on spectral theory on trees, see [57, 58, 405]. 10.2. The Free Hamiltonian and Radially Symmetric Potentials The lattice B has a huge symmetry group. The map Tφ interchanges the two branches coming out of φ, that is, Tφ (φ) = φ, and for any α 6= φ, Tφ (α) = (t(α1 ), α0 , . . . , α#(α) ) where t(0) = 1, t(1) = 0. Similarly, one can define Tα for any α ∈ B which leaves invariant α and any point not comparable or smaller than α and interchanges the two branches coming out of α. Let Tα generate a map Uα on ℓ2 (B) (by (Uα ϕ)(β) = ϕ(Tα (β))). All the Uα ’s commute with H0 , the free Hamiltonian. In this section, we will use this symmetry to do a complete spectral analysis of H0 and, as a bonus, reduce any H0 + V , where V (α) = v(#(α)) is radially symmetric, to a direct sum of Jacobi matrices.

(10.2.1)

10.2. THE FREE HAMILTONIAN

653

Let Hs be the totally symmetric functions in ℓ2 (B), that is, there is f : {0, 1, . . . } → C so ϕ(α) = f (#(α)) (10.2.2) Clearly, for ϕ ∈ Hs , ∞ X kϕk2 = 2n |f (n)|2 (10.2.3) n=0

Fix α0 ∈ B. Let Hα0 be the odd there is f : {0, 1, . . . } → C so   0 ϕ(β) = f (#(β) − #(α0 ) − 1)  −f (#(β) − #(α ) − 1) 0

functions based on α0 , that is, if β ≁ α0 or β α0 if β (α0 , 0) if β (α0 , 1)

(10.2.4)

that is, f lives on the two trees coming out of the node α0 and is antisymmetric under the switching of these two trees. Of course, if ϕ ∈ Hα , then ∞ X kϕk2 = 2 2n |f (n)|2 (10.2.5) n=0

Define

Us : Hs → ℓ2 (N) (10.2.6) (where N = {0, 1, 2, . . . }) by the inverse of the map g 7→ ϕ with ϕ(α) = 2−n/2 g(#(α))

(10.2.7)

and similarly, Uα : Hα → ℓ2 (N) as the inverse map to ϕ given by (10.2.4) with g(n) = 2(n+1)/2 f (n) Theorem 10.2.1. We have ℓ2 (B) = Hs ⊕

M α∈B

Hα

(10.2.8) (10.2.9) (10.2.10)

that is, Hs ⊥ Hα ⊥ Hβ for all α and all β 6= α, and these spaces span ℓ2 (B). Proof. The orthogonality to Hs is a simple calculation (given the sign flip in (10.2.4)). When Hα and Hβ have disjoint subspaces if α ∼ β and when α β, the same sign flip yields orthogonality. To get the spanning fact, note first that we can write M ℓ2 (B) = ℓ2 (B(n) ) (10.2.11) n

654


where so

B(n) = {α | #(α) = n}

(10.2.12)

dim(ℓ2 (B(n) )) = 2n

(10.2.13)

On the other hand,

( 0 if #(α) ≥ n dim(Hα ∩ ℓ2 (B(n) )) = 1 if #(α) < n

(10.2.14)

and that (the 1 comes from dim(Hs ∩ ℓ2 (B(n) )) = 1) 1 + #(α | #(α) < n) = 1 + 1 + · · · + 2n−1 = 2n , accounting for (10.2.13). Theorem 10.2.2. Let V obey (10.2.1). Then H0 and V leave Hs and each Hα invariant. Moreover, on ℓ2 (N), √ Uα H0 Uα−1 = 2 J0 (10.2.15) where J0 is the matrix



0 1 J0 =  0 .. .

and Uα V Uα−1

1 0 1 .. .

0 1 0 .. .

 ... . . .  . . . .. .

 v(#(α) + 1) v(#(α) + 2)  = √

(10.2.16)

..

.

 

(10.2.17)

Moreover, Us H0 Us−1 = 2J0 and Us V Us has the form (10.2.17) where #(α) is replaced by −1. Proof. Because V acts pointwise and, by (10.2.1), preserves the symmetries, the results for V are immediate. As for H0 , using the formula for ϕ in terms of f ((10.2.2) or (10.2.4)), we see H0 ϕ has the same form for some f˜ (by the minus symmetry, the bottom point α has (H0 ϕ)(α) = 0 for ϕ ∈ Hα ) and f˜(n) = 2f (n + 1) + f (n − 1) (10.2.18)

(with f (−1) ≡ 0). Since g and f are related by (10.2.9) (or for H2 , g(n) = 2n/2 f (n)), we see √ (10.2.19) (Uα H0 Uα−1 g)(n) = 2 (g(n + 1) + g(n − 1)) which is (10.2.15).

10.2. THE FREE HAMILTONIAN

655

Corollary 10.2.3. √ √ σ(H0 ) = − 2 2, 2 2

(10.2.20)

and is purely a.c. spectrum of infinite multiplicity. Proof. As we have seen in (1.10.4), σ(J0 ) = [−2, 2] and is purely a.c. Thus, since each H0 ↾ Hα and H0 ↾ Hs is unitarily equivalent to √ 2J0 , we get the result. Corollary 10.2.4. There exist V (α)’s obeying |V (α)| ≤ C(1 + #(α))−1/2

(10.2.21)

and, in particular, for all p > 2, ∞ X n=0

2−n

X

α|#(α)=n

|V (α)|p < ∞

(10.2.22)

√ √ so that H0 + V has only pure point spectrum in [−2 2, 2 2]. Proof. Immediate from the last theorem and Theorem 3.5.6. Finally, we can compute dµ0 and its Stieltjes transform directly: Theorem 10.2.5. The spectral measure, dµ0 , for H0 is given by (10.1.5) (with d = 2). In particular, √ −z + z 2 − 8 −1 hδφ , (H0 − z) δφ i = (10.2.23) 4 Remark. We will give another proof of this in Corollary 10.3.3. Proof. Since δφ ∈ Hs and H0 leaves Hs invariant, we have √ −1 Us H0 Us = 2J0 and Us δφ = δ1 , this follows by scaling (1.10.3) and the fact that the m-function for J0 , by (3.2.28), solves m(z) =

1 −z − m(z)

with m(z) = −z −1 + O(z −2 ) at infinity.

Remarks and Historical Notes. The form of σ(H0 ) and m (Theorem 10.2.5) goes back at least to Acosta–Klein [5]. Corollary 10.2.4 is noted by Denisov [104]. For an interesting application of the reduction to one-dimensional problems, see [58].

656


10.3. Coefficient Stripping for Trees As we have seen in Chapters 2 and 3, a key first step is coefficient stripping—and we consider that in this section. For Jacobi matrices, removing the root and the link to it yields another Jacobi matrix; but for our Bethe–Cayley tree with d = 2, removing φ and links to it yields two operators, H (0) on ℓ2 (B0 ) and H (1) on ℓ2 (B1 ). Let m(z) be the function in (10.1.3) and m(0) (z), m(1) (z) the analogs for H (0) and H (1) . Then we will prove Theorem 10.3.1. We have for any V that 1 m(z) = −z + V (φ) − m(0) (z) − m(1) (z)

(10.3.1)

e 0 = φ ∪ B0 , To prove this, it is useful to consider extended lattices B e 1 = φ ∪ B1 , and B e = ψ ∪ B, which is B with a node, ψ, added B e (H e (0) , H e (1) ) like H but with a that connects only to φ. Define H e coupling to the extra node added (i.e., (Hg)(φ) = g(ψ) + (H g)(φ) ˙ and e (Hg)(ψ) = g(φ)). We have that e by Lemma 10.3.2. Define u on B

u(α) = hδα , (H − 1)−1 δφ i

(10.3.2)

e Hu(α) = zu(α)

(10.3.4)

u(ψ) = −1

(10.3.3)

Then u obeys for α ∈ B and is the unique such function obeying (10.3.3) and u ∈ e ℓ2 (B). Proof. uˇ ≡ u ↾ B obeys

H uˇ(α) = z uˇ(α) + δαφ

(10.3.5)

which is (10.3.4), given (10.3.3). Moreover, given (10.3.3), (10.3.4) is equivalent to (10.3.5), so the uniqueness follows from the invertibility of (H − z), which implies there is a unique ℓ2 solution of (H − z)ˇ u = δφ

(10.3.6)

First Proof of Theorem 10.3.1. Let u be given by (10.3.2). e 0 solves (for α ∈ B0 ) Then u(0) ≡ u ↾ B e (0) u(0) (α) = zu(0) (α) H

(10.3.7)

10.3. COEFFICIENT STRIPPING FOR TREES

657

e 0 ), but it does not obey the analog of (10.3.3). However, and is in ℓ2 (B −1 (0) e 0 ). It follows −u(φ) u does and still solves (10.3.7) and is in ℓ2 (B that u((0)) m(0) (z) = − (10.3.8) u(φ) and similarly, m(1) (z) = −

u((1)) u(φ)

(10.3.9)

Moreover, u(φ) = m(z)

(10.3.10)

Thus, (V (φ) − z)u(φ) + u((0)) + u((1)) − 1 = 0

(10.3.11)

V (φ) − z − m(0) (z) − m(1) (z) = m(z)−1

(10.3.12)

becomes (dividing by u(φ)) which is (10.3.1).

Second Proof of Theorem 10.3.1. Here is a more operator theoretic argument. Let Γ be the part of H0 that links φ to (0) and (1), that is, Γδφ = δ(0) + δ(1)

Γδ(0) = δφ

Γδ(1) = δφ

(10.3.13)

and otherwise Γ = 0. Thus, e =H−Γ H

= V (0) ⊕ H

(10.3.14) (0)

acting on C + ℓ2 (B0 ) ⊕ ℓ2 (B1 ). By the resolvent identity,

⊕H

(1)

e − z)−1 − (H − z)−1 Γ(H e − z)−1 (H − z)−1 = (H

(10.3.15)

(10.3.16)

and, of course, by (10.3.15),

e − z)−1 = (V (0) − z)−1 ⊕ (H (0) − z)−1 ⊕ (H (1) − z)−1 (H

(10.3.17)

Apply (10.3.16) to δφ and to δ(0) to get

(H − z)−1 δφ = (V (0) − z)−1 [δφ − (H − z)−1 [δ(0) + δ(1) ]]

(H − z)−1 δ(0) = −m(0) (z)(H − z)−1 δφ

and similarly for δ(1) .

(10.3.18) (10.3.19)

658


Taking inner products with δφ , we get from the second equation for j = 0, 1, hδφ , (H − z)−1 δ(j) i = −m(j) (z)m(z)

(10.3.20)

and then from the first equation that m(z) = (V (0) − z)−1 [1 + m(z)(m(0) (z) + m(1) (z))]

(10.3.21)

Multiplying by m(z)−1 (V (0) − z), we get V (0) − z = m(z)−1 + m(0) (z) + m(1) (z) which is equivalent to (10.3.1).

(10.3.22)

Remark. Both these methods can be used to prove Theorem 3.2.4. As an application of Theorem 10.3.1, we get a second proof of (10.2.21): Corollary 10.3.3. The free m-function is given by √ −z + z 2 − 8 m0 (z) = 4

(10.3.23)

Proof. In this case, V (φ) = 0 and m(z) = m(1) (z) = m(2) (z), so solves 1 m(z) = (10.3.24) −z − 2m(z) which gives a quadratic equation solved by (10.3.23).

From (10.3.1), using m(j) (z) = −z −1 + O(z −2 ), m(z) = (−z)−1 (1 − V (φ)z −1 + 2z −2 + O(z −3 ))

= −z −1 − V (φ)z −2 − (2 + V (φ)2 )z −3 + O(z −4 )

(10.3.25)

Remarks and Historical Notes. Theorem 10.3.1 goes back at least to Klein [224]; see also [138]. Klein gave what we call the second proof. For general degrees, there are d subtrees with m-function {m(j) (z)}d−1 j=0 and the coefficient stripping formula is m(z)

−1

= −z + V (φ) −

d−1 X j=0

m(j) (z)

(10.3.26)

10.4. A STEP-BY-STEP SUM RULE FOR TREES

659

10.4. A Step-by-Step Sum Rule for Trees Our goal in this section is to prove a step-by-step inequality: Theorem 10.4.1 (Denisov [104]). Let V be a bounded function on B with |V (α)| → 0 as #(α) → ∞. Let µ be the spectral measure for H0 + V and µ(j) , j = 1, 2, for H (j) and ℓ2 (Bj ). Then S(µ0 | µ) ≥ S(µ0 | 12 (µ(0) + µ(1) )) − 14 V (φ)2

(10.4.1)

This will come from a sum rule:

Theorem 10.4.2. Under the hypotheses of Theorem 10.4.1, X X ek ) + 2 Y (E Y (Ek ) S(µ0 | µ) = S(µ0 | 21 µ(0) + 12 µ(1) ) − 14 V (φ)2 −2 k

k

(10.4.2) √ √ ek } 2, 2 2) and {E where {Ek } are the eigenvalues of H outside √ (−2 √ are the eigenvalues of H (0) ⊕ H (1) outside (−2 2, 2 2) and where √ Y (E) = F E/ 2 (10.4.3) and F is given by (1.10.9).

Remarks. 1. Since |V (α)| 0 as #(α) → ∞, σess (H) = σess (H0 ), √ →√ so the spectrum outside [−2 2, 2 2] is discrete. √ e ± with E + > 2 2 and E − < 2. As usual, we should label Ej± , E j √ e+, E − < E e− , and so Y (E e ± ) ≤ Y (E ± ). −2 2. As we will see, Ej+ > E j j j j j P P ek ), we mean By k Y (Ek ) − j Y (E X e± )) (Y (Ej± ) − Y (E j j,±

which is a sum of pointwise terms which could be ∞. Actually, we only apply the step-by-step sum rule directly √ to√cases where V has finite support and the spectrum outside [−2 2, 2 2] is finite.

Theorem 10.4.2 ⇒ Theorem 10.4.1. H (0) ⊕ H (1) = P HP where P is the projection onto ℓ2 (B(0) ) ⊕ ℓ2 (B(1) ). So by the min-max principle, e± ≤ ±E ± ±E (10.4.4) j j F , and so Y, is monotone (see (3.5.10)), so e ± ) ≤ Y (E ± ) Y (E (10.4.5) j

and so

X k

Y (Ek ) −

so (10.4.2) implies (10.4.1).

j

X k

ek ) ≥ 0 Y (E

(10.4.6)

660

x-ref?


Remark. It may appear that we have thrown away something useful but, as we will explain in the Notes to Section 10.5, we have not. Proof of Theorem 10.4.2. This closely follows the proof of Corollary 3.4.7. Define √ M(z) = −m( 2 (z + z −1 )) (10.4.7)

In comparison with (3.4.26), we have, by (10.3.25), M(z) 1 V (φ) V (φ)2 2 log = − log(2) + √ z + z + O(z 3 ) z 2 4 2

(10.4.8)

On the other hand, by (10.3.1),

Im(M(eiθ )) = Im(M (0) (eiθ ) + M (1) (eiθ )) (10.4.9) iθ |M(e )| By directly following the proof of the P2 sum rule, that is, combining the zeroth- and second-order Poisson–Jensen formulae, one gets Z 2π 1 Im(M (0) ) + Im(M (1) ) (eiθ )(1 − cos 2θ) dθ − log 4π 0 Im(M) (10.4.10) X V (φ)2 X 1 e = − 2 log(2) − + Y (Ek ) − Y (Ek ) 8 k k Since

1 − 4π

Z

0

2π

log( 21 )(1 − 2 cos θ) dθ =

1 2

log(2)

(10.4.11)

we can drop the − 12 log(2) from the right side of (10.4.10) if we replace Im(M (0) + M (1) ) by Im((M (0) + M (1) )/2). Following the identification of energies in Section 3.4, we get (10.4.2). Remarks and Historical Notes. This section follows Denisov [104]. 10.5. The Global ℓ2 Theorem In this section, we will prove Theorem 10.1.1 by proving (10.1.7). Recall (Theorem 2.2.3) that S(µ | ν) is jointly concave, so S(µ0 |

1 2

(µ(0) + µ(1) )) ≥ 21 S(µ0 | µ(0) ) + 12 S(µ0 | µ(1) )

(10.5.1)

Proposition 10.5.1. Let V have compact support. Then ∞ 1 X 1 X S(µ0 | µ) ≥ − V (α)2 (10.5.2) n 4 n=0 2 #(α)=n

10.5. THE GLOBAL ℓ2 THEOREM

661

Remark. Since V has compact support, the sum in (10.5.2) is finite. Proof. By (10.4.1) and (10.5.1), S(µ0 | µ) ≥ − 14 V (φ)2 − 12 S(µ0 | µ(0) ) − 12 S(µ0 | µ(1) )

For α ∈ B, let

(10.5.3)

Bα = {β | α β}

(10.5.4)

H (α) = Pα HPα ↾ ℓ2 (Bα )

(10.5.5)

and H (α) , the operator on ℓ2 (Bα ) obtained by letting Pα be the projection of ℓ2 (Bα ) to ℓ2 (B) and setting that is, H with all links from Bα to R \ Bα dropped. Let µ(α) be the spectral measure for H (α) and δ[α] . By induction, from (10.5.3) we get for any k, k−1 1 X 1 X 1 (α) S(µ0 | µ) ≥ − k S(µ0 | µ ) − 2 4 n=0 2n #(α)=k

X

V (α)2 (10.5.6)

#(α)=n

Since V has compact support, we can find k so V (β) = 0 if #(β) ≥ k. For such k, µ(α) = µ0 if #(α) = k, so (10.5.6) is (10.5.2). Theorem 10.5.2 (implies Theorem 10.1.1). (10.5.2) holds for any bounded V. In particular, if ∞ X 1 X V (α)2 < ∞ n 2 n=0

(10.5.7)

#(α)=n

√ √ then H has a.c. spectrum of infinite multiplicity on [−2 2, 2 2]. Proof. For any k, define V [k] by ( V (α) if #(α) ≤ k V [k] (α) = 0 if #(α) > k

(10.5.8)

and let µ[k] be the measure for H [k] = H0 + V [k]. For any η ∈ ℓ2 (B), we have k(V − V [k] )ηk → 0, so H [k] → H

strongly

(10.5.9)

Since the H’s are uniformly bounded, (H [k] −z)−1 → (H −z)−1 strongly, so in the weak (vague) topology on measures, w

µ[k] −→ µ

(10.5.10)

662


By the weak upper semicontinuity of S (see Theorem 2.2.3), we have S(µ0 | µ) ≥ lim sup S(µ0 | µ[k] ) (10.5.11) k→∞

so (10.5.2) for V [k] implies it for V. Given this, if (10.5.7) holds, then S(µ0 | µ) > −∞, so Σac (µ) > √ √ [−2, 2, 2 2]. By the argument in Section 10.1, we get infinite multiplicity. Remarks and Historical Notes. The results of this section are from Denisov [104]. One can see why this method does not yield much information about eigenvalues. For after one step, one finds X 2 Y (Ej± ) − S(µ0 | µ) ≤ 14 V (φ)2 j,±

X X ± (1) (2) e ) − S(µ0 | µ ) − S(µ0 | µ ) + e±) + 2 Y (E Y (E j j 1 2

j,±

j,±

(10.5.12)

e ± ), 1 of the Y ’s at the next level, One can iterate this but 12 of the Y (E j 4 P etc. will be left over. So one does not get a bound on j,± Y (Ej± ) in P P −n 2 terms of ∞ n=0 2 α|#(α)=n |V (α)| . However, since S ≤ 0, we have S(µ0 | µ) ≥ S(µ0 | µ(0) ) + S(µ0 | µ(1) ) and this can be iterated to get X X Y (Ej± ) ≤ 14 |V (α)|2 j,±

(10.5.13)

(10.5.14)

α

It remains to be seen if this can be improved. 10.6. The Local ℓ2 Theorem In this section, we will prove Theorem 10.1.2 by proving (10.1.11). The key is the following, whose proof we defer: Theorem 10.6.1. Consider S(µ | ν) where µ and ν are probability measures. For µ fixed, the map η → eS(µ|η) is concave, that is, for η0 , η1 probability measures and 0 < θ < 1, we have eS(µ|θη1 +(1−θ)η0 ) ≥ θ eS(µ|η1 ) + (1 − θ)eS(µ|η0 )

(10.6.1)

10.6. THE LOCAL ℓ2 THEOREM

663

Theorem 10.6.2 (implies Theorem 10.1.2). (10.1.11) holds for any bounded V. In particular, if ν({ω | kVω k < ∞}) > 0

(10.6.2) √ then H has a.c. spectrum of infinite multiplicity on [−2 2, 2 2]. √

Proof. By (10.6.1) and (10.4.1), we have 1

2

(0) )

eS(µ0 |µ) ≥ 12 e− 4 V (φ) [eS(µ0 |µ

(1) )

+ eS(µ0 |µ

]

(10.6.3)

So, by iterating, k 1 X 1X (α) 2 exp((S(µ0 | µ)) = k exp S(µ0 | µ ) − V ((α1 , . . . , αj )) 2 4 j=0 #(α)=k

(10.6.4) where the j = 0 term in the sum is V (φ)2 . If V has compact support, by taking large k, we get (10.1.11) and then we get (10.1.11) in general by taking limits using upper semicontinuity of S. If (10.6.2) holds, then the right side of √ (10.1.11) √ is strictly positive, so S(µ0 | µ) > 0, which implies that [−2 2, 2 2] ⊂ Σac (µ), showing that the a.c. spectrum of H contains that interval with multiplicity 1. The proof of infinite multiplicity has to be modified slightly compared to the argument in Section 10.1. Removing the connections to φ breaks the lattice in two. However, (10.6.2) does not necessarily imply that for both j = 0 and 1 that 2ν({ω | ω (j) and kVω k < ∞}) > 0

(10.6.5)

so one cannot conclude each of H (0) and H (1) has a.c. spectrum. But if ν({ω | kVω k < ∞}) > 2−k (10.6.6)

removing connections to nodes, α, with #(α) ≥ k − 1 breaks B into 2k lattices, and at least two of them must have paths with kVω k < ∞ and positive ν measure (since all the paths in a single sublattice only have measure 2−k ). Thus, we see that if (10.6.2) implies a.c. spectrum of multiplicity at least ℓ, it implies multiplicity at least 2ℓ. So, as in Section 10.1, we get infinite multiplicity. We turn to the proof of Theorem 10.6.1: First Proof of Theorem 10.6.1. If (ε)

νj = (1 + ε)−1 (νj + εµ)

(10.6.7)

664


then, by using the monotone convergence theorem, (ε)

(ε)

S(µ | θν1 + (1 − θ)ν0 ) = lim S(µ | θν1 + (1 − θ)ν0 ) ε↓0

(10.6.8)

so, without loss, we can suppose dνj = wj dµ + dµ(j) s

(10.6.9)

with inf wj (x) > 0 x

In that case, S(θ) ≡ S(µ | θν1 + (1 − θ)ν2 ) =

Z

log(θw1 + (1 − θ)w2 ) dµ (10.6.10)

is a C ∞ function of θ and Z (w1 − w2 ) ′ S (θ) = dµ θw1 + (1 − θ)w2 Z (w1 − w2 )2 ′′ S (θ) = − dµ (θw1 + (1 − θ)(w2 ))2 By the Schwarz inequality,

(10.6.11) (10.6.12)

−S ′′ (θ) − S ′ (θ)2 ≥ 0

(10.6.13)

g(θ) = eS(θ)

(10.6.14)

g ′′(θ) = [S ′′ (θ) + S ′ (θ)2 ]eS(θ) ≤ 0

(10.6.15)

Let Then, by (10.6.13),

so g is concave in θ, as claimed.

Our second proof depends on a variational principle for S that complements (2.2.4)/(2.2.5): Proposition 10.6.3. Let µ, ν be probability measures. Then the relative entropy obeys (see (2.2.20)) Z g S(µ | ν) = inf log e dν − g dµ (10.6.16) g∈C(X)

Proof. Let S(f ; µ, ν) be given by (2.2.5) and let Z Z g G(g) = log e dν − g dµ

(10.6.17)

We claim that

S(eg ; µ, ν) ≥ G(g) ≥ S(µ | ν) from which (10.6.16) is immediate from (2.2.4).

(10.6.18)

10.6. THE LOCAL ℓ2 THEOREM

665

By concavity of log(y), we have that so

log(y) ≤ y − 1 Z

g

e dν − 1 ≥ log

Z

(10.6.19) g

e dν

(10.6.20)

which implies the first inequality in (10.6.18). For the second inequality, note it is trivial if S(µ | ν) = −∞. If S(µ | ν) > −∞, then µ is ν a.c. So write −1 dµ dν = dµ + dµs (10.6.21) dν R R Thus, by Jensen’s inequality (i.e., log eh dη ≥ h dη), we have that Z Z −1 Z g g dµ g log e dν = log e dµ + e dµs dν Z dµ dµ ≥ log exp g − log dν Z ≥ g dµ + S(µ | ν) (10.6.22) completing the proof of (10.6.18).

Second Proof of Theorem 10.6.1. By (10.6.16), Z Z S(µ|ν) g e = inf e dν exp g dµ g∈C(X)

is an inf of linear functionals of ν, and so concave.

(10.6.23)

Remarks and Historical Notes. The results of this section are from Denisov–Kiselev [105]. They have a different proof of Theorem 10.6.1 using Young’s inequality that for x, y > 0 and p−1 + q −1 = 1, p, q > 1, we have xp y q xy ≤ + p q They optimize over p.

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Szegos Theorem and its Descendants

Helly's theorem and its relatives

Szego's Theorem and Its Descendants: Spectral Theory for L2 Perturbations of Orthogonal Polynomials