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

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...

In this chapter we will prove Szegő’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 Szegő 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 iffis nonvanishing and analytic in 𝔻 and log(f(z)) is in some Hardy class$H^p (p \geqslant 1)$,then$f(0) = \exp (\smallint \log (f(e^{i\theta } ))\frac{{d\theta }}{{2\pi }})$.In later chapters, we have to use Blaschke products to accommodate poles and zeros that can occur.

Section 2.1 lays out the...

In this chapter, we focus on OPRL whose essential support is [−2, 2]. See the end of Section 3.1 for a summary of the chapter.

In this chapter, we turn to analogs of Szegő’s theorem for OPRL that are close to a free case. The main theorem is Theorem 3.1.1.

(Killip-Simon Theorem).$Let\{ a_n ,b_n \} _{n = 1}^\infty $be the Jacobi parameters of a Jacobi matrix, J. Then

\[\sum\limits_{n = 1}^\infty {{{({a_n} - 1)}^2} + b_n^2} < \infty \] (3.1.1)

if and only if

(a)

\[{\sigma _{{\text{ess}}}}(J) = {\sigma _{{\text{ess}}}}({J_0})\quad (Blumenthal - Weyl)\] (3.1.2)

(b)The eigenvalues$E_n \notin \sigma _{{\text{ess}}} (J_0 )$obey

\[\sum\limits_{n = 1}^\infty {{\text{dist}}({E_n},{\sigma _{{\text{ess}}}}({J_0})} {)^{3/2}} < \infty \quad (Lieb - Thirring)\] (3.1.3)

(c)The function f of (1.4.3) obeys

\[{\smallint _{\sigma ({J_0})}}{\text{dist}}{(x,\mathbb{R}\backslash \sigma ({J_0}))^{1/2}}\log f(x)dx > - \infty \quad (Quasi - Szeg\ddot o)\] (3.1.4)

Our proof will rely on a sum rule, namely, ifF,g, andQare given...

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.

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$J_0 $and$J_1 $.There is another place in...

Thus far we have been looking at perturbations of OPUC and OPRL with constant Jacobi parameters; specifically, we looked at perturbations of

\[a_n^{(0)} = a\quad b_n^{(0)} = b\] (5.1.1)

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

\[a_{n + p}^{(0)} = a_n^{(0)}\quad b_{n + p}^{(0)} = b_n^{(0)}\] (5.1.2)

for allnand some fixedp. (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${\text{drop}}^{(0)} $henceforth in this chapter since we are restricting...

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.

The structure that the spectra of periodic Jacobi matrices induce on Jacobi parameters is striking.$[(0,\infty ) \times \mathbb{R}]^p $,consisting of points$(a_n ,b_n )_{n = 1}^p $,is decomposed into its isospectral tori, generically of dimensionp– 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...

This chapter is an interlude. In our proof of Szegő asymptotics in Section 9.13, we will need the Denisov–Rakhmanov–Remling theorem, which we will prove below in Section 7.6. This leads us naturally to the notion of right limits. IfJis a (one-sided) Jacobi matrix with parameters$\{ a_n ,b_n \} _{n = 1}^\infty $and$J_r $a two-sided matrix with parameters$\{ a_n^{(r)} ,b_n^{(r)} \} _{n = - \infty }^\infty $,we say$J_r $is aright limit ofJif and only if for some subsequence$m_j \to \infty $,for all fixed n ∈ ℤ, as j → ∞,

\[{a_{n + {m_j}}} \to a_n^{(r)}\quad {b_{n + {m_j}}} \to b_n^{(r)}\] (7.1.1)

By compactness of product spaces, ifJis bounded, there exist right limits. If we...

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.

We have looked at four results on perturbations of the Jacobi matrix,$J_0 $,with$a_n \equiv 1,b_n \equiv 0$:

Weyl-type results that$a_n \to 1,b_n \to 0 \Rightarrow \sigma _{{\text{ess}}} (J) = [ - 2,2]$

Denisov–Rakhmanov-type results that$\sigma _{{\text{ess}}} (J) = \Sigma _{{\text{ac}}} (J) = [ - 2,2]$implies$a_n \to 1,b_n \to 0$

Szegő-Shohat-Nevai-type results relating a Szegő condition plus$\frac{1}{2}$eigenvalue bounds to boundedness of$\sum\nolimits_{n = 1}^N {\log (a_n )} $

Killip–Simon-type results relating a pseudo-Szegő condition plus$\frac{3}{2}$eigenvalue bounds to$\ell ^2 $conditions of the form

\[\sum_{n} (a_{n} -1 )^{2} + b_n^2 < \infty \caption{(8.1.1)}\]

In this chapter, we want to focus on perturbation results of...

In this chapter, we consider a general finite gap set, 𝔢, of the form

\[\mathfrak{e} = \bigcup\limits_{j=1}^{\ell + 1}[\alpha_{j}, \beta_{j}] \quad \alpha_{1} < \beta_{1} < \alpha_{2} < \cdots < \beta_{\ell +1} (9.1.1)\]

and prove a Szegő–Shohat–Nevai theorem and Szegő asymptotics for suitable measures, μ, with$\sigma _{{\text{ess}}} (\mu ) = \mathfrak{e}$.The key is to find an analog of the map$z \to z + z^{ - 1} $of 𝔻 to ℂ ⋃ {∞} \ [-2,2], which was central to Chapter 3. Thus, we seek an analytic map

\[{\mathbf{x}}:\mathbb{D} \to \mathbb{C} \cup \{ \infty \} \backslash \mathfrak{e}\](9.1.2)

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,${\mathbf{x}}^{ - 1} (w)$,of a single point will be a countable...

In this final chapter, we discuss some work of Denisov [108, 110] 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 one-dimensional 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.

We begin by describing the rooted Bethe–Cayley tree of degree...