FINE STRUCTURE OF THE ZEROS OF

ORTHOGONAL POLYNOMIALS,

IV. A PRIORI BOUNDS AND CLOCK BEHAVIOR

YORAM LAST^{1,3} AND BARRY SIMON^{2,3}

Dedicated to Percy Deift on the occasion of his sixtieth birthday

Abstract. We prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions (Jacobi parameters approach the free ones and are of bounded variation). We prove that for ergodic discrete Schrödinger operators, Poisson behavior implies positive Lyapunov exponent. Both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs.

1. Introduction

Our primary goal in this paper concerns the fine structure of the zeros of orthogonal polynomials on the real line (OPRL), although we will say something about zeros of paraorthogonal polynomials on the unit circle (POPUC) (see Section 10). Specifically, dμ will be a measure of compact support which is nontrivial (i.e., not supported on a finite set), usually a probability measure. P_{n}(x) or P_{n}(x; dμ) will be the monic orthogonal polynomials and p_{n}(x) = P_{n}/∥P_{n}∥ the orthonormal polynomials. The Jacobi parameters, {a_{n}, b_{n}}_{n=1}^{∞}, are defined by

xP_{n}(x) = P_{n+1}(x) + b_{n+1}P_{n}(x) + a_{n}^{2}P_{n−1}(x)  (1.1)

n = 0, 1, 2, …, where P_{−1}(x) ≡ 0. It follows that (when μ(ℝ) = 1)

∥P_{n}∥ = a_{1} … a_{n}  (1.2)

so

xp_{n}(x) = a_{n+1}p_{n+1}(x) + b_{n+1}p_{n}(x) + a_{n}p_{n−1}(x)  (1.3)

Date: May 16, 2006. 1 Institute of Mathematics, The Hebrew University, 91904 Jerusalem, Israel. E-

mail: ylast@math.huji.ac.il. Supported in part by The Israel Science Foundation (Grant No. 188/02).

2 Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125. E-mail: bsimon@caltech.edu. Supported in part by NSF grant DMS-0140592.

3 Research supported in part by Grant No. 2002068 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.

and x has a matrix representation in the orthonormal basis {p_{n}}_{n=0}^{∞},

J = (\begin{matrix}b_{1} & a_{1} & 0 & ⋯ \\ a_{1} & b_{2} & a_{2} & ⋯ \\ ⋮ & ⋮ & ... & ... \end{matrix})  (1.4)

called the Jacobi matrix. The finite Jacobi matrix, J_{n;F}, is the n × n submatrix of J in the upper left corner. It is easy to see that (see (b) of the Appendix)

P_{n}(x) = det(x1 − J_{n;F})  (1.5)

Let {x_{j}^{(n)}}_{j=1}^{n} be the zeros of P_{n}(x), so (1.5) says that the x_{j}^{(n)} are eigenvalues of J_{n;F}. Let dν_{n} be the pure point probability measure that gives weight 1/n to each x_{j}^{(n)}. We say the density of states exists if dν_{n} has a weak limit dν_{∞}. By (1.5), one sees that

∫ x^{k} dν_{n}(x) = \frac{1}{n} Tr(J_{n;F}^{k})  (1.6)

which is often useful.

The existence of the limit for a large class of regular measures on [−2, 2] goes back to Erdös-Turán [9]. Nevai [34] realized all that was used is

lim_{n→∞} |b_{n}| + |a_{n} − 1| = 0  (1.7)

(following the convention in the Schrödinger operator community, we use a_{n} ≡ 1 as a “free” case, while the OP community uses a_{n} ≡ \frac{1}{2}). Indeed,

Theorem 1.1 (known). If dμ is a measure where (1.7) holds, the density of states exists and is given by

dν = π^{−1}(\sqrt{4 − x^{2}} )^{−1}χ_{[−2,2]} dx  (1.8)

The modern proof is not hard. For a_{n} ≡ 1, b_{n} ≡ 0, the OPRL are explicitly given by

P_{n}(2 cos θ) = \frac{sin((n + 1)θ)}{sin θ}

from which one computes dν exactly for this case. If J_{0,n;F} is the corresponding cutoff J_{0}, then (1.7) implies

\frac{1}{n} Tr(J_{n;F}^{k} − J_{0,n;F}^{k}) → 0  (1.9)

for each k = 0, 1, 2, … which, by (1.6), implies ∫ x^{k} dν_{n} has the same limits as for the free case.

Another case where it is known that dν exists is when a_{n}, b_{n} are samples of an ergodic family, that is, a_{n}^{(ω)} = f(T^{n}ω), b_{n}^{(ω)} = g(T^{n}ω) with T: Ω → Ω an ergodic transformation on (Ω, dρ), a probability measure space. In that case, it is known (going back to the physics literature and proven rigorously by Pastur [38], Avron-Simon [1], and Kirsch-Martinelli [28]):

Theorem 1.2 (known). For ergodic Jacobi matrices, dν_{n,ω} has a limit dν for a.e. ω and dν is a.e. ω-independent.

Again, the proof uses (1.6) plus in this case that, by ergodicity, \frac{1}{n}Tr(J_{n;F}^{k}) has a limit a.e. by the Birkhoff ergodic theorem.

The most important examples of the ergodic case are periodic, almost periodic, and random.

One easily combines the two ideas to see that dν exists (and does not depend on δa_{n}, δb_{n}) if

a_{n} = a_{n}^{(ω)} + δa_{n} b_{n} = b_{n}^{(ω)} + δb_{n}

with a_{n}^{(ω)}, b_{n}^{(ω)} ergodic and |δa_{n}| + |δb_{n}| → 0.

These results describe the bulk features of the zeros. Here we are interested in the fine structure, on the level of individual eigenvalues; specifically, the focus in [41, 42, 43] and a main focus in this paper is what we call clock behavior, that the spacing locally is equal spacing. The term clock comes from the case of orthogonal polynomials on the unit circle (OPUC) where dν is typically Lebesgue measure on a circle and the equal space means the zeros look like the numbers on a clock.

In order for the density of zeros to be dν, the equal spacing must be 1/(dν/dE). The symmetric derivative dν/dE exists for a.e. E and, of course, (dν/dE) dE is the a.c. part of dν. It is known (see, e.g., Avron-Simon [1]) that dν has no pure points and, in many cases, it is known that dν/dE is a continuous function, or even C^{∞}. To be formal, we define first

Definition. Let E_{0} ∈ supp(dν). We let z_{n}^{(j)}(E_{0}) be the zeros nearest E_{0} so that

z_{n}^{(−2)}(E_{0}) < z_{n}^{(−1)}(E_{0}) ≤ E_{0} < z_{n}^{(1)}(E_{0}) < z_{n}^{(2)}(E_{0}) < ⋯

if such zeros exist. If E_{0} ∈ [supp(dν)]^{int}, then z_{n}^{j}(E_{0}) exists for j fixed and n large.

Definition. Let E_{0} ∈ supp(dν). We say there is weak clock behavior at E_{0} if dν/dE_{0} exists and

lim_{n→∞} n[z_{n}^{(1)}(E_{0}) − z_{n}^{(−1)}(E_{0})] \frac{dν}{dE} (E_{0}) = 1  (1.10)

We say there is strong clock behavior at E_{0} if dν/dE_{0} exists, (1.10) holds, and for j = 1, ±2, ±3, … fixed,

lim_{n→∞} n[z_{n}^{(j)}(E_{0}) − z_{n}^{(j+1)}(E_{0})] \frac{dν}{dE} (E_{0}) = 1  (1.11)

Definition. We say there is uniform clock behavior on [α, β] if dν/dE is continuous and nonvanishing on [α, β] and

lim_{n}[sup\{|n[E−E′]−(\frac{dν}{dE} )^{−1}| | E, E′ are successive zeros of p_{n} in [α, β]\}] = 0

(1.12)

It is obvious that uniform clock behavior implies strong clock behavior at each interior point.

In the earlier papers in this series that discussed clock behavior for OPRL [41, 43], there was a technical issue that severely limited the results in general situations. In all cases, a Jost function-type analysis was used to show that a suitably rescaled p_{n} converged, that is,

c_{n}p_{n}(E_{0} + n(x − E_{0}) \frac{dν}{dE} (E_{0}) + ξ_{n}) → f_{∞}(x)  (1.13)

where |ξ_{n}| ≤ c/n and f_{∞} has zeros at 0, ±1, ±2, . . .. This would naively seem to show that p_{n} has clock-spaced zeros and, indeed, it does imply at least one zero near E_{0} + ξ_{n} + \frac{j}{n}(\frac{dν}{dE_{0}} )^{−1} consistent with clock spacing.

The snag involves uniqueness, for the function

sin(πx) − \frac{1}{n} sin(n^{2}πx)  (1.14)

has a limit like f_{∞} but has more and more zeros near x = 0. That is, one needs to prove uniqueness of the zeros near E_{0} + ξ_{n} + \frac{j}{n}(\frac{dν}{dE_{0}} )^{−1}.

In previous papers in this series, two methods were used to solve this uniqueness problem. One relied on some version of the argument principle, essentially Rouché’s theorem. This requires analyticity which, typically, severely restricts what recursion coefficients are allowed. In the case of OPUC where one needs to control zeros in the complex plane, some kind of analyticity argument seems to be necessary. The second method relies on the fact that if (1.13) also holds for derivatives and f′_{∞}(j) ≠ 0, then there is a unique zero. This argument also requires extra restrictions on the recursion coefficients, albeit not as severe as the analyticity requirement. For example, in [41], one only needed ∑_{n=1}^{∞}|a_{n} − 1| + |b_{n}| < ∞ to get (1.13) for E ∈ [−2 + ε, 2 − ε]. But to control derivatives, we needed ∑_{n=1}^{∞} n(|a_{n} −1|+|b_{n}|) < ∞. In fact, the general argument failed to capture Jacobi polynomials (whose clock estimates were earlier obtained by Vértesi and Szabados [49, 52, 53, 54]) where separate arguments are needed. In [43], it was decided not to deal with asymptotically periodic OPRL since the derivative arguments looked to be tedious.

Note. The papers [41, 42, 44] discussed asymptotics of OPUC when the Verblunsky coefficients decay exponentially. We have discovered a paper of Pan [37] whose results overlap those in [41, 42, 44].

The key realization of this paper is that there is a more efficient way to eliminate pathologies like those in (1.14). Namely, we will seek a priori lower bounds on eigenvalue spacings. If we find any O(1/n) lower bound, that implies the rescaled p_{n} of (1.13) has at most one zero near any zero of f_{∞}. Such lower bounds are not new in suitable situations. Erdös-Turán [9] already have such bounds if the measure is purely absolutely continuous in an interval with a.c. weights bounded away from zero and infinity. These ideas were developed by Nevai [34] and Golinskii [14]. Under suitable hypotheses on the transfer matrix, lower bounds are known in the Schrödinger operator community; see, for example, Jitomirskaya et al. [21]. (Note: They use “equal spacing” for O(1/n) lower bounds and do not mean clock behavior by this term.)

While we could have used these existing bounds in the proofs of Theorems 1.3 and 1.4 below, we have found a new approach which allows us to also prove Theorem 1.5 below, and this approach is discussed in Section 2. With these lower bound ideas, we can prove:

Theorem 1.3 (≡ Theorem 3.2). Suppose that

∑_{n=1}^{∞}(|a_{n} − 1| + |b_{n}|) < ∞  (1.15)

Then there is uniform clock behavior on each interval [−2 + ε, 2 − ε] for any ε > 0.

Theorem 1.4 (≡ Theorem 3.4). Suppose that

lim_{n→∞} a_{n} = 1  lim_{n→∞} b_{n} = 0  (1.16)
∑_{n=1}^{∞}(|a_{n+1} − a_{n}| + |b_{n+1} − b_{n}|) < ∞  (1.17)

Then there is uniform clock behavior on each interval [−2 + ε, 2 − ε] for any ε > 0.

Remarks. 1. Theorem 1.4 implies Theorem 1.3, but we state them as separate theorems since the proof of the first is easier.

2. We will also prove results of this genre for perturbations of periodic recursion coefficients and for where n + 1 in (1.17) is replaced by n + p for some p.

3. We also obtain results (see Theorem 3.8) near ±2 if

∑_{n=1}^{∞} n(|a_{n} − 1| + |b_{n}|) < ∞  (1.18)

Using our strong lower bound, we will also prove the following:

Theorem 1.5. Let a_{n}^{(ω)}, b_{n}^{(ω)} be ergodic Jacobi parameters. Let E_{0} be such that

(i) The Lyapunov exponent γ(E_{0}) = 0. (ii) The symmetric derivative of ν exists at E_{0} and is finite and

nonzero. Then there exists C > 0 so that with probability 1,

lim inf_{n→∞} n[z_{n}^{(1)}(E_{0}) − z_{n}^{(−1)}(E_{0})] > C  (1.19)

This result is especially interesting because it implies that the zeros cannot have Poisson behavior. It implies that Poisson behavior and ergodicity require γ(E_{0}) > 0. We will say more about these issues in Sections 4 and 11.

While our initial interest in a priori lower bounds came from clock theorems and we could have finished the paper with Section 4 if our sole purpose was to prove Theorems 1.3–1.5, it seemed natural to also consider upper bounds. Moreover, in looking over the upper/lower bound results in the OP literature, we realized one could get more from these methods, so we discuss that also.

Broadly speaking, we have two sets of results and methods. The methods rely on either transfer matrices with hypotheses on recursion coefficients or on OP methods with hypotheses on the measure. We believe that the results are of interest to both the Schrödinger operator and OP communities. Because the OP methods are unfamiliar to many Schrödinger operator theorists and are easy to prove (albeit very powerful), we have included an appendix with some major OP methods.

The detailed plan of the paper is as follows: In Section 2, we prove our a priori lower bounds involving the transfer matrix (or, more precisely, the growth of subordinate and nonsubordinate solutions). In Section 3, we prove a variety of clock theorems, including Theorems 1.3 and 1.4. In Section 4, using ideas of Deift-Simon [8], we prove Theorem 1.5. In Section 5, we obtain upper bounds on eigenvalue spacing using the transfer matrix. Section 6 discusses using suitable Prüfer angles to control spacing of zeros.

In Section 7, we begin our discussion of OP methods with a technical result on L^{p} bounds on w^{−1} and bounds of the Christoffel function. These bounds, which we will need for examples later are local versions of some bounds of Geronimus [11] with a rather different method of proof. In Section 8, we discuss upper bounds on eigenvalue spacing using OP methods and, in particular, find a remarkable lower bound on the density of states that is a kind of microlocal version of some bounds of Deift-Simon [8]. In Section 9, we discuss lower bounds on eigenvalue spacing. The methods in Sections 8 and 9 are borrowed from Erdös-Turán [9, 51], Nevai [34], and Golinskii [14], but we show how to localize them and squeeze out stronger results. In Section 10, we briefly discuss the analogs of our results for zeros of POPUC, and in Section 11 discuss a number of examples, counterexamples, conjectures, and questions.

It is a pleasure to thank L. Golinskii, S. Jitomirskaya, R. Killip, P. Nevai, and M. Stoiciu for useful discussions. This research was begun during B. S.’s stay as a Lady Davis Visiting Professor at The Hebrew University of Jerusalem. He would like to thank H. Farkas for the hospitality of the Einstein Institute of Mathematics at The Hebrew University, and E. B. Davies and A. N. Pressley for the hospitality of King’s College, London where this was work was continued. Y. L. would like to thank G. A. Lorden and T. A. Tombrello for the hospitality of Caltech, where this work was completed.

Percy Deift has long been a player in spectral theory and more recently, a champion for orthogonal polynomials. In particular, this paper exploits the work of Deift-Simon [8]. It is a pleasure to dedicate this paper to Percy.

2. Variation of Parameters and Lower Bounds via Transfer Matrices

Our goal here is to use variation of parameters to study eigenvalue spacing. Variation of parameters has an ancient history going back to Lagrange [15] and it was extensively used to study variation of solution with change in potential, for example, to study asymptotics in tunnelling problems [16]. The usefulness of the method as a tool in spectral theory goes back at least to the work of Gilbert-Pearson [13] (also see [12, 23]) with significant later contributions by Jitomirskaya-Last [18, 19, 20] and Killip-Kiselev-Last [24]. It is essentially their equation we will use although, interestingly enough, the earlier applications are to E, E′ with E ∈ ℝ and E′ = E +iε, while our application is to E, E′ both in ℝ.

Given E ∈ ℂ, we consider solutions of

a_{n}u_{n+1} + (b_{n} − E)u_{n} + a_{n−1}u_{n−1} = 0  (2.1)

for n = 1, 2, . . .. Here {a_{n}, b_{n}}_{n=1}^{∞} are the Jacobi parameters of the measure we are considering and

a_{0} ≡ 1  (2.2)

For θ ∈ [0, π), we denote by u_{n}(E, θ) the solution of (2.1) with

u_{0}(E, θ) = sin(θ) u_{1}(E, θ) = cos(θ)  (2.3)

In particular,

u_{n}(E, θ = 0) = p_{n−1}(E) n = 1, 2, …  (2.4)

The transfer matrix is defined by

T (n, E) = (\begin{matrix}u_{n+1}(E, θ = 0) & u_{n+1}(E, θ = π/2) \\ u_{n}(E, θ = 0) & u_{n}(E, θ = π/2) \end{matrix})  (2.5)

so for any solution of (2.1),

\binom{u_{n+1}}{u_{n}} = T (n, E)\binom{u_{1}}{u_{0}}  (2.6)

Let K(n, m; E) be the kernel

K(n, m; E) = u_{n}(E, 0)u_{m}(E, π/2) − u_{n}(E, π/2)u_{m}(E, 0)  (2.7)

Define the operator A_{L}(E) on ℂ^{L} = {v_{·} = {v_{k}}_{k=1}^{L} | v_{k} ∈ ℂ} by

(A_{L}(E)v)_{n} = ∑_{m=1}^{n} K(n, m; E)v_{m}  (2.8)

(note that K(n, n) = 0, so the sum also goes to n − 1). The following summarizes results from [18, 19, 20, 24]:

Theorem 2.1. Let w_{n}(E), w_{n}(E′) solve (2.1) for E, E′, and suppose

w_{0}(E) = w_{0}(E′)  w_{1}(E) = w_{1}(E′)  (2.9)

Then

w_{n}(E′) = w_{n}(E) + (E′ − E) ∑_{m=1}^{n} K(n, m; E)w_{m}(E′)  (2.10)

that is,

w_{·}(E′) = w_{·}(E) + (E′ − E)(A_{L}(E)w(E′))_{·}  (2.11)

Moreover, with ∥·∥_{HS} the Hilbert-Schmidt norm on ℂ^{L} and ∥·∥_{L} defined by

∥v_{·}∥_{L} = ( ∑_{n=1}^{L} |v_{j}|^{2})^{1/2}  (2.12)

and ⟨ · , · ⟩_{L} the associated inner product, we have that

∥A_{L}(E)∥_{HS}^{2} = ∥u(E, 0)∥_{L}^{2}∥u(E, π/2/)∥_{L}^{2} − |⟨u(E, 0), u(E, π/2)⟩_{L}|^{2} (2.13)

= max_{θ} ∥u(E, θ)∥_{L}^{2} min_{θ} ∥u(E, θ)∥_{L}^{2}  (2.14)

In particular,

∥A_{L}(E)∥ ≤ ∑_{j=0}^{L−1} ∥T (j, E)∥^{2}  (2.15)

Remarks. 1. That (2.10)/(2.11) hold is either a direct calculation verifying the formula or a calculation obtained by expanding \binom{w_{n+1}(E′)}{w_{n}(E′)} in terms of \binom{u_{n+1}(E,θ)}{u_{n}(E,θ)} for θ = 0, π/2. (2.13) is a direct calculation from (2.7) and (2.14) is a clever observation [24].

2. Clearly, ∥T (n, E)\binom{1}{0}∥^{2} = |u_{n+1}(E)|^{2} + |u_{n}(E)|^{2} ≥ |u_{n}(E)|^{2}, so ∥u(E, 0)∥_{L}^{2} ≤ RHS of (2.15). Similarly for ∥u(E, π/2)∥, so ∥u(E, 0)∥_{L}∥u(E, π/2)∥ ≤ RHS of (2.15), so (2.13) implies (2.15).

3. ∥T (n)∥ measures the growth of the fastest growing solution, so the RHS of (2.15) in fact measures (max_{θ} ∥u(E, θ)∥_{L}^{2}) and thus, by (2.14), one could place min_{θ} ∥u(E, θ)∥/ max_{θ} ∥u(E, θ)∥ in front of the RHS of (2.15).

Here is the key lower bound:

Theorem 2.2. Let E′, E′′ be two distinct zeros of p_{L}(x). Then

|E′ − E_{0}| + |E′′ − E_{0}| ≥ ∥A_{L}(E_{0})∥^{−1}  (2.16)

In particular,

|z_{L}^{(+1)}(E_{0}) − z_{L}^{(−1)}(E_{0})| ≥ ∥A_{L}(E_{0})∥^{−1}  (2.17)

Proof. By (2.11), for E_{j} = E′, E′′,

p_{·−1}(E_{j}) = p_{·−1}(E_{0}) + (E_{j} − E_{0})A_{L}(E_{0})p_{·−1}(E_{j})  (2.18)

To say p_{L}(E′) = p_{L}(E′′) = 0 says E, E′′ are eigenvalues of J_{L;F} and p_{·−1}(·) are the eigenvectors. So, by orthogonality of eigenvectors,

⟨p_{·−1}(E′), p_{·−1}(E′′)⟩_{L} = 0  (2.19)

By interchanging E′ and E′′, if necessary, suppose

∥p_{·−1}(E′)∥_{L} ≥ ∥p_{·−1}(E′′)∥_{L}  (2.20)

Take (2.18) for E′ and E′′ and take the inner product with p_{·−1}(E′) and subtract to get

∥p_{·−1}(E′)∥^{2} ≤ |E′ − E_{0}| |⟨p_{·−1}(E′), A_{L}(E_{0})p_{·−1}(E′)⟩|
+ |E′′ − E_{0}| |⟨p_{·−1}(E′), A_{L}(E_{0})p_{·−1}(E′′)⟩|
≤ (|E′ − E_{0}| + |E′′ − E_{0}|)∥A_{L}(E_{0})∥ ∥p_{·−1}(E′)∥^{2}

on account of (2.20). (2.16) is immediate.

(2.13) follows from (2.16) and z^{(−1)} ≤ E_{0} ≤ z^{(+1)}. □

While our main applications are to clock theorems and Poisson statistics, there is a universal tunnelling bound.

Theorem 2.3. Let J_{L;F} be a finite Jacobi matrix with α_{−} = inf a_{n}, α_{+} = max a_{n}, β = max b_{n} − min b_{n}. Let

γ = α_{−}^{−1}|(β + 2α_{+})^{2} + α_{+}^{2} + 1|^{1/2}  (2.21)

Then any pair of eigenvalues, E, E′, of J_{L;F} obeys

|E − E′| ≥ \frac{γ^{2} − 1}{γ^{2L} − 1}  (2.22)

Remark. This bound is exponential, ∼ γ^{−2L}, for L large.

Proof. Adding a constant to b_{n} does not change eigenvalue differences, so we can suppose that

max b_{n} = − min b_{n} = \frac{β}{2}  (2.23)

Then any E_{0} in the convex hull of spec(J_{L;F}) obeys |E_{0}| ≤ \frac{β}{2} + 2α_{+}, so |E_{0}−b_{n}| ≤ β +2α_{+}. Thus, γ is an upper bound on the Hilbert-Schmidt norm of

\frac{1}{a_{n+1}} (\begin{matrix}E − b_{n+1} & −a_{n} \\ 1 & 0 \end{matrix})

and so on the norm.

Pick E between E′ and E′′. It follows that

∥T (j, E)∥ ≤ γ^{j}

so (2.21) follows from (2.17) and (2.15). □

One can also use our proof to see that one cannot have too many zeros near E_{0}.

Theorem 2.4. Define δ_{L,n}(E_{0}) to be the smallest number so that

#{z_{j}^{(n)} | |z_{j}^{(n)} − E_{0}| < δ} ≤ n − 1  (2.24)

Then, for n ≥ 2,

δ_{L,n}(E_{0}) ≥ \frac{1}{2} ∥A_{L}(E_{0})∥_{HS}^{−1} \sqrt{n}  (2.25)

Remarks. 1. If one has strong clock behavior, δ_{L,n} ∼ cn/L for n fixed and L large, so \sqrt{n} is worse than one expects in nice cases.

2. Our proof shows that (2.17) can be “improved,” if ∥ · ∥_{HS} is used, to

|z_{L}^{(+1)}(E_{0}) − z_{L}^{(−1)}(E_{0})| ≥ \frac{1}{\sqrt{2}} ∥A_{L}(E_{0})∥_{HS}^{−1}  (2.26)

Proof. There are at least n zeros, z_{1}, …, z_{n}, in {z | |z−E_{0}| ≤ δ_{L,n}}. Order them so that if φ_{j,m} = p_{m}(z_{j}), then ∥φ_{j}∥_{L} ≥ ∥φ_{2}∥_{L} ≥ ⋯ ≥ ∥φ_{n}∥_{L}. Let φ̃_{j} = φ_{j}/∥φ_{j}∥_{L}. Then the argument in the proof of Theorem 2.2 says that, for j ≤ k,

1 ≤ [δ_{L,n}(|A_{jj}| + |A_{jk}|)]^{2}
≤ 2δ_{L,n}^{2}(|A_{jj}|^{2} + |A_{jk}|^{2})  (2.27)

where

A_{jk} = ⟨φ_{j}, A_{L}(E_{0})φ_{k}⟩  (2.28)

and (2.27) comes from (a + b)^{2} ≤ 2(a^{2} + b^{2}). Summing over all pairs and noting that |A_{11}|^{2} occurs n − 1 times, we find that

\frac{n(n − 1)}{2} ≥ 2δ_{L,n}^{2}(n − 1)∥A∥_{HS}^{2}

which is (2.25). □

3. Clock Theorems for Bounded Variation Perturbations of Free and Periodic OPRL

The basic result from which we will derive all our clock theorems presupposes the existence of a complex solution to the difference equation (2.1) for which we have precise information on the phase. The model is the Jost solution which is asymptotic to e^{iθ(E)n}, where E = 2 cos(θ), 0 ≤ θ ≤ π.

Theorem 3.1. Let {a_{n}, b_{n}}_{n=1}^{∞} be a set of Jacobi parameters and Δ a closed interval in ℝ. Suppose there exists a solution u_{n}(E) of (2.1) for E ∈ Δ which obeys (i)

u_{0}(E) > 0  Im u_{1}(E) > 0  (3.1)

(ii)

u_{n}(E) = |u_{n}(E)| exp(i[nα(E) + β_{n}(E)])  (3.2)

where α is C^{1} with

−\frac{dα}{dE} > 0  all E ∈ Δ  (3.3)

each β_{n} continuous on Δ, and

lim_{n→∞} [ sup_{\begin{matrix}E′,E∈Δ \\ |E−E′|≤1/n \end{matrix}}|β_{n}(E′) − β_{n}(E)|] = 0  (3.4)

(iii) The transfer matrix T_{n}(E) of (2.5) obeys

τ ≡ sup_{n,E∈Δ} ∥T_{n}(E)∥ < ∞  (3.5)

Then the density of states exists on Δ,

dν(E) = −\frac{1}{π} \frac{dα}{dE} dE  (3.6)

and there is clock behavior uniformly in Δ.

Remarks. 1. α is, of course, a rotation number and (3.6) an expression of the connection between the density of states and rotation numbers; see Johnson-Moser [22].

2. (3.4) implies β_{n} is irrelevant for eigenvalue spacing comparable to 1/n. To control possible spacings with ΔE small compared to 1/n, one needs some Lipschitz control of β_{n}, that is,

sup_{E≠E′} \frac{|β_{n}(E′) − β_{n}(E)|}{n|E − E′|} → 0

which is where differentiability of β_{n} and so moment conditions on {a_{n}, b_{n}} came into [41]. We avoid this by using (3.5) to get a priori bounds.

3. (3.4) implies the same if 1/n is replaced by A/n for any fixed A. Define ζ_{n}(A) by

ζ_{n}(A) = sup_{\begin{matrix}E,E′∈Δ \\ |E−E′|≤A/n \end{matrix}}|β_{n}(E′) − β_{n}(E)|  (3.7)

4. (3.4) is implied by an equicontinuity assumption, for example, uniform convergence of β_{n} to a continuous limit.

Proof. By (3.1), u_{n} and ū_{n} are independent solutions of (2.1) and so cannot vanish at any points. Moreover,

p_{n−1}(E) = A(E)u_{n}(E) + \overline{A(E)} \overline{u_{n}(E)}  (3.8)

where

A(E) = \frac{W (p_{·−1}, ū)}{W (u, ū)}  (3.9)

Here, W is the Wronskian. Given sequences, f_{n}, g_{n},

W_{n}(f, g) = a_{n}(f_{n+1}g_{n} − f_{n}g_{n+1})  (3.10)

is constant (call it W (f, g)) if f, g both solve (2.1). Since p_{−1} = 0, p_{0} = 1, a_{0} = 1, we have W (p_{·−1}, ū) = u_{0}(E), and clearly, W (u, ū) = 2i Im(u_{1}ū_{0}) = 2iu_{0} Im u_{1}, so

A(E) = −\frac{i}{2} \frac{1}{Im u_{1}}  (3.11)

is pure imaginary. Thus, (3.8) says p_{n−1}(E) vanishes if and only if u_{n} is real, that is, by (3.2),

p_{n−1}(E) = 0 ⇔ nα(E) + β_{n}(E) = kπ (k ∈ ℤ)  (3.12)

Let

η = min_{E∈Δ} [−\frac{dα}{dE} ]  (3.13)

Pick N_{0} so that n > N_{0} implies

\frac{η}{τ^{2}} ≥ 4ζ_{n}(\frac{1}{τ^{2}} )  (3.14)

This can be done since ζ_{n}(A) → 0 as n → ∞ by hypothesis. Since ζ_{n}(A) is increasing in A and ζ_{n}(A + B) ≤ ζ_{n}(A) + ζ_{n}(B), we have ζ_{n}(xβ) ≤ ζ_{n}([x]β) + ζ_{n}(xβ − [x]β) ≤ ([x] + 1)ζ_{n}(β) ≤ 2xζ_{n}(β), if x ≥ 1, (3.14) implies

q ≥ \frac{1}{τ^{2}} ⇒ ηq ≥ 2ζ_{n}(q)  (3.15)

This in turn implies

E − E′ ≥ \frac{1}{nτ^{2}} ⇒ [n[α(E′) − α(E)] + [β_{n}(E′) − β_{n}(E)]] ≥ \frac{n}{2} η|E′ − E|

(3.16)

By (2.17) and (2.15), any two successive zeros obey

|E′ − E| ≥ \frac{1}{nτ^{2}}  (3.17)

Thus, (3.17) implies that for n > N_{0}, any two solutions of (3.12) have distinct values of k. We also see from (3.16) and continuity that if E is a solution of (3.12), there is another solution in (E − \frac{2π}{nη}, E) and it has the next larger value of k (i.e., k + 1).

Subtracting (3.12) for two successive values of (3.12) and using ζ_{n}(2π/η) → 0, we see that

sup_{E′<E successive eigenvalues in Δ} |n(α(E′) − α(E)) − π| → 0  (3.18)

Given the uniformity of convergence of the difference quotient to the derivative, (3.18) implies that

sup_{E′<E successive eigenvalues in Δ} |n(E′ − E) \frac{dα}{dE} − π| → 0  (3.19)

This implies the density of states exists and is given by (3.6) and that one has uniform clock behavior. □

Theorem 3.2 (≡ Theorem 1.3). Let {a_{n}, b_{n}}_{n=0}^{∞} be a set of Jacobi parameters obeying

∑_{n=1}^{∞} |b_{n}| + |a_{n} − 1| < ∞  (3.20)

so ess supp(dμ) = [−2, 2]. For any ε > 0, we have uniform clock behavior on [−2 + ε, 2 − ε].

Remarks. 1. This includes Jacobi polynomials (rescaled to [−2, 2]) for which

|b_{n}| + |a_{n} − 1| = O(n^{−2})

2. Of course, the density of states is the free one.

Proof. It is well-known (see, e.g., [26, 6]) that when (3.20) holds, there exists, for all x ∈ (−2, 2), a solution ũ_{n}(x) so that if

z + z^{−1} = x  (3.21)

(i.e., z = e^{iθ} with x = 2 cos θ and θ ∈ (0, π)), then

z^{−n}ũ_{n}(x) → 1  (3.22)

uniformly on compact subsets of (−2, 2). Moreover, ũ_{n}(x) is continuous on (−2, 2) for each fixed n. By evaluating the Wronskian near n = ∞, we see

W (ũ_{n}, ¯ ũ_{n}) = z − z^{−1}  (3.23)

Thus, if

u_{n}(E) = \frac{ũ_{n}(E)}{ũ_{0}(E)}  (3.24)

then (3.1) and (3.2) hold. If

u_{0}(E) = |u_{0}(E)|e^{iβ_{∞}(E)}  (3.25)

then we have that

β_{n}(E) → β_{∞}(E)  (3.26)

uniformly on [−2 + ε, 2 − ε]. By continuity, (3.4) holds.

α is given by

α(E) = arccos(\frac{E}{2} )  (3.27)

so

−\frac{1}{π} \frac{dα}{dE} = \frac{1}{π} \frac{1}{\sqrt{4 − E^{2}}}  (3.28)

and (3.3) holds.

Finally, standard variation of parameters about z^{n}, z^{−n} shows that (3.5) holds for each Δ = [2 + ε, 2 − ε].

Thus, Theorem 3.1 applies, and we have clock behavior. □

In the above, we used the fact that β_{n} → β_{∞} uniformly to obtain (3.4). In the bounded variation case, we will instead use:

Lemma 3.3. If β_{n} = β_{n}^{(1)} + β_{n}^{(2)} where β_{n}^{(1)} is C^{1} and

\frac{1}{n} sup_{E∈Δ} |\frac{∂β_{n}^{(1)}}{∂E} | → 0  (3.29)

and β_{n}^{(2)} → β_{∞}^{(2)} uniformly, then (3.4) holds.

Proof. Immediate since E, E′ ∈ Δ implies

|β_{n}(E) − β_{n}(E′)| ≤ |E − E′| sup_{E∈Δ} |\frac{∂β_{n}^{(1)}}{∂x} | + |β_{n}^{(2)}(E) − β_{n}^{(2)}(E′)| □

Theorem 3.4 (≡ Theorem 1.4). Let {a_{n}, b_{n}}_{n=0}^{∞} be a set of Jacobi parameters with

a_{n} → 1 b_{n} → 0 (3.30)

and

∑_{n=1}^{∞}(|a_{n+1} − a_{n}| + |b_{n+1} − b_{n}|) < ∞  (3.31)

then for any ε > 0, we have uniform clock behavior in [−2 + ε, 2 + ε].

Remark. Again, the density of states is the free one by (3.30).

In order to prove this theorem, we need the following result:

Theorem 3.5. Let B_{n}(θ) depend continuously on θ ∈ I, a compact subinterval of (0, π), and suppose

sup_{θ∈I} ∑_{n=1}^{∞} ∥B_{n+1}(θ) − B_{n}(θ)∥ < ∞  (3.32)

and

B_{n}(θ) → B_{∞}(θ)  (3.33)

uniformly where B_{∞}(θ) has eigenvalue e^{±iθ}. Explicitly for V_{∞}(θ) continuous and invertible:

B_{∞}(θ) = V_{∞}(θ) (\begin{matrix}e^{iθ} & 0 \\ 0 & e^{−iθ} \end{matrix}) V_{∞}(θ)^{−1}  (3.34)

Suppose each B_{n}(θ) has eigenvalues e^{±iφ_{n}(θ)} with φ_{n}(θ) ∈ (0, π). Let

T̃_{n}(θ) = B_{n}(θ) … B_{1}(θ)  (3.35)

Then (i)

sup_{θ∈I,n} ∥T̃_{n}(θ)∥ < ∞  (3.36)

(ii) There exists S_{∞}(θ) depending continuously on θ so that

∥T̃_{n}(θ) − V_{∞}(θ)D_{n}(θ)S_{∞}(θ)∥ → 0  (3.37)

where

D_{n}(θ) = (\begin{matrix}e^{iη_{n}(θ)} & 0 \\ 0 & e^{−iη_{n}(θ)} \end{matrix})  (3.38)

with

η_{n}(θ) = ∑_{η=1}^{n} φ_{n}(θ)  (3.39)

Remark. See the notes to Section 2.1 of [40] for a history of results on bounded variation.

Proof. This is a strong version of Kooman’s theorem [29]. In Section 12.1 of [40], (3.36) is proven, and in the notes to that section, it is noted that (3.37) holds. In those notes, there is no V_{∞} because the analog of D_{n} is not diagonal but can be diagonalized in a basis where B_{∞} is diagonal. □

Proof of Theorem 3.4. Define

B_{n}(θ) = (\begin{matrix}(a_{n}a_{n+1})^{−1/2}(2 cos(θ) − b_{n+1}) & −(a_{n}/a_{n+1})^{1/2} \\ (a_{n}/a_{n+1})^{−1/2} & 0 \end{matrix}) (3.40)

so the transfer matrix at E = 2 cos θ is

T_{n}(θ) = (\frac{a_{n}}{a_{n+1}} )^{1/2}B_{n}(θ)(\frac{a_{n−1}}{a_{n}} )^{1/2} … B_{1}(θ)
= a_{n+1}^{−1/2}T̃_{n}(θ)  (3.41)

Since det(B_{n}) = 1, B_{n}(θ) has eigenvalues e^{±iφ_{n}}, φ_{n} ∈ (0, π), if and only if

(a_{n}a_{n+1})^{−1/2}(2 cos θ − b_{n+1}) = 2 cos(φ_{n}) ∈ (−2, 2)  (3.42)

and this holds uniformly for θ ∈ (δ, π − ε) and n > N_{0} for some fixed N_{0}. Thus, for n ≥ N_{0}, we define

η_{n}(θ) = ∑_{j=N_{0}}^{n} φ_{j}(θ)  (3.43)

and define

D̃_{n}(θ) = (\begin{matrix}e^{iη_{n}(θ)} & 0 \\ 0 & e^{−iη_{n}(θ)} \end{matrix})  (3.44)

and

S̃_{∞}(θ) = S_{∞}(θ; B_{n} … B_{N_{0}})B_{N_{0}−1} … B_{1}  (3.45)

where S_{∞}(θ; B_{n} … B_{N_{0}}) is the S_{∞} for the sequence B_{N_{0}}, B_{N_{0}+1}, . . .. Thus, (3.37) and (3.41) show that

∥T_{n}(θ) − a_{n+1}^{−1/2}V_{∞}(θ)D̃_{n}(θ)S̃_{∞}(θ)∥ → 0  (3.46)

We now proceed to construct a solution u obeying the hypothesis of Theorem 3.1. Pick a two-vector x(θ) by x(θ) = e^{iφ_{0}(θ)}S̃_{∞}(θ)^{−1}\binom{1}{0} where φ_{0} is chosen below and u_{n} by

T_{n}(θ)x(θ) = \binom{u_{n+1}(θ)}{u_{n}(θ)}  (3.47)

then (3.46) says that

u_{n}(θ) = |u_{n}(θ)| exp(i[η_{n}(θ) + β_{n}^{(2)}(θ)])  (3.48)

where

β_{n}^{(2)}(θ) → β_{∞}^{(2)}(θ)  (3.49)

uniformly.

Here β_{∞}^{(2)}(θ) is φ_{0}(θ) plus the phase of the 21 element of V_{∞}(θ), and so i(η_{n}(θ) + β_{∞}^{(2)}(θ)) is the phase of the lower component of a_{n+1}^{−1/2}V_{∞}(θ)D̃_{n}(θ)S̃_{∞}(θ)x(θ). Since u_{n} is not real, u_{0}(θ) ≠ 0, and so φ_{0}(θ) can be chosen so that u_{0}(θ) > 0.

Since φ_{j}(θ) → θ, η_{n+1} − η_{n} → θ, and thus the imaginary part of the Wronskian of u and ū is positive, so u obeys (3.1).

By (3.42),

\frac{∂φ_{n}}{∂E} = \frac{sin(θ)}{sin(φ_{n})} (a_{n}a_{n+1})^{−1/2} \frac{∂θ}{∂E}  (3.50)

so

\frac{∂η_{n}}{∂E} = \frac{∂θ}{∂E} (n − N_{0}) + ∑_{j=N_{0}}^{n}(\frac{sin(θ)}{sin(φ_{n})} (a_{n}a_{n+1})^{−1/2} − 1)  (3.51)

Thus, if

β_{n}^{(1)} = η_{n} − n \frac{∂θ}{∂E}  (3.52)

then

\frac{1}{n} \frac{∂β^{(1)}}{∂E} = (−\frac{N_{0}}{n} + \frac{1}{n} ∑_{j=N_{0}}^{n}(\frac{sin(θ)}{sin(φ_{n})} (a_{n}a_{n+1})^{−1/2} − 1))\frac{∂θ}{∂E} (3.53)

converges uniformly to zero, since

\frac{sin(θ)}{sin(φ_{n})} → 1 (a_{n}a_{n+1})^{−1/2} → 1  (3.54)

uniformly in θ.

Lemma 3.3 applies, so condition (ii) of Theorem 3.1 holds with α(E) = θ(E) = arccos(\frac{E}{2} ). We thus have clock behavior with density of states the free one, that is, given by (3.28). □

Now consider the periodic case, that is,

a_{n+p} = a_{n} b_{n+p} = b_{p}  (3.55)

The spectrum now has bands (see, e.g., [30]). For any E in the interior of the bands, there is a Floquet solution with u_{n+p} = e^{iγ(E)}u_{n} with γ(E) ∈ (0, π) and \frac{∂γ}{∂E} < 0. Thus

u_{Lp+r} = |u_{r}|e^{i(Lγ(E)+β_{r}^{(∞)})}  (3.56)

where β_{r}^{(∞)} is the phase of u_{r}. Theorem 3.1 applies with α = γ(E)/p and

β_{Lp+r}(E) = β_{r}^{(∞)}(E) − \frac{r}{p} γ(E)  (3.57)

There are only r such functions so (3.4) holds, and we recover the zero spacing part of Theorem 2.6 of [43].

If a_{n}^{(0)}, b_{n}^{(0)} are periodic and a_{n} = a_{n}^{(0)} + δa_{n}, b_{n} = b_{n}^{(0)} + δb_{n} and

∑_{n=1}^{∞}(|δb_{n}| + |δa_{n}|) < ∞  (3.58)

then one can construct Jost solutions on the interiors of the bands. All that changes is that (3.57) is replaced by

lim_{L→∞} β_{Lp+r}(E) = β_{r}^{(∞)}(E) − \frac{r}{p} γ(E)  (3.59)

so Theorem 3.1 still applies. Similarly applying the ideas in the proof of Theorem 3.4, we obtain a bounded variation result. Since it includes the (3.58) result, we summarize in a single theorem:

Theorem 3.6. Let a_{n}^{(0)}, b_{n}^{(0)} obey

a_{n+p}^{(0)} = a_{n}^{(0)}  b_{n+p}^{(0)} = b_{n}^{(0)}  (3.60)

for some p. Let a_{n}, b_{n} obey

lim_{n→∞} |a_{n} − a_{n}^{(0)}| + |b_{n} − b_{n}^{(0)}| = 0  (3.61)
∑_{n=1}^{∞} (|a_{n+p} − a_{n}| + |b_{n+p} − b_{n}|) < ∞  (3.62)

Then, for any closed interval which is a subset of the interior of the bands (see Remark), we have uniform clock behavior with density of states −\frac{1}{p} \frac{dγ}{dE}.

Remark. There are p closed bands, B_{1}, …, B_{p}, generically disjoint but perhaps touching in a single point (closed gap). By the interior of the bands, we mean ∪_{j=1}^{p}B_{j}^{int} which will be smaller than (∪_{j=1}^{p}B_{j})^{int} if some gap is closed, that is, we must remove all of the gaps, including those that degenerate to single points.

p need not be the minimal period, so we have that

Corollary 3.7. Suppose

a_{n} → 1 b_{n} → 0

and, for some p, (3.62) holds. Then, in any closed interval in {E = 2 cos(θ) | pθ ≠ 0, \frac{π}{p}, …, \frac{(p−1)π}{p}\}, we have uniform clock behavior.

As a final topic, we want to discuss zeros very near E = 2 when

∑_{n=1}^{∞} n(|a_{n} − 1| + |b_{n}|) < ∞  (3.63)

It should be possible to extend this argument to get uniform clock behavior in [−2, 2], with a suitable modification to take into account the behavior exactly at ±2. When (3.63) holds, the Jost function, u, can de defined on [−2, 2]; see, for example, the appendix to [6]. If u(2) = 0, we say there is a resonance at 2, and if u(2) ≠ 0, we say that 2 is nonresonant.

Theorem 3.8. Let {a_{n}, b_{n}}_{n=1}^{∞} be a set of Jacobi parameters obeying (3.63). Define 0 ≤ θ_{1}^{(n)} < θ_{2}^{(n)} < ⋯ so E_{j}^{(n)} = 2 cos(θ_{j}^{(n)}) are the zeros of p_{n}(x) nearest to x = 2 and below. Then (a) If 2 is a resonance, then

nθ_{j}^{(n)} → (j − \frac{1}{2})π  (3.64)

(b) If 2 is nonresonant, then

nθ_{j}^{(n)} → jπ  (3.65)

Remarks. 1. The two simplest examples are the nonresonant a_{n} ≡ 1, b_{n} ≡ 0 where

p_{n}(2 cos θ) = c_{n} \frac{sin((n + 1)θ)}{sin θ}  (3.66)

and the resonant a_{n} ≡ 1 (n ≥ 2), a_{1} = \sqrt{2}, and b_{n} ≡ 0, where

p_{n}(2 cos θ) = d_{n} cos(nθ)  (3.67)

2. Notice that (for simplicity, consider the nonresonant case)

E_{j}^{(n)} − E_{j+1}^{(n)} ∼ \frac{(2j + 1)π^{2}}{n^{2}}

the eigenvalue space is O(n^{−2}) and not equal in E but has clock spacing in θ.

Remark. The key fact that at a zero energy resonance, the scattering phase is π/2 (mod π) and otherwise it is 0 (mod π) is well-known in the continuum case, for which there is extensive physics literature; see, for example, Newton [36].

Proof. By the theorems found in the appendix to [6] (which codifies well-known results), when (3.63) holds, one has ([6, eqn. (A.27)])

|p_{n}(e^{iθ})| ≤ C(n + 1)  (3.68)

and the existence of a solution u_{n}(e^{iθ}) with

e^{−inθ}u_{n}(e^{iθ}) → 1  (3.69)

uniformly on ∂𝔻. u_{0} is called the Jost function and

W (p_{·−1}, u) = u_{0}  (3.70)

We want to use (2.14) where there is a collision of notation, so we let v(θ, φ) be the solution at E = 2 cos θ and boundary condition φ. Then (2.14) becomes

∥A_{L}(2 cos θ)∥ ≤ max_{φ} ∥v(θ, φ)∥_{L} min_{φ} ∥v(θ, φ)∥_{L}  (3.71)

If u_{0}(θ = 0) ≠ 0, we get one solution for θ small, v(θ, φ_{0}(θ)) which is uniformly bounded in θ and n, and another solution v(θ, φ_{1} = 0) (= p_{·−1}) bounded by Cn. It follows that

min_{φ} ∥v(θ, φ)∥_{L} ≤ CL^{1/2}  max_{φ} ∥v(θ, φ)∥ ≤ CL^{3/2}  (3.72)

If u_{0}(θ = 0) = 0, we start at n = 1 since u_{1}(θ = 0) ≠ 0 and construct the bounded and linearly growing solution that way (in essence, the two solutions in this case are p_{·−1} and q, where q is the second kind polynomial), so (3.72) still holds.

We conclude, using Theorem 2.2, that when (3.63) holds, then

sup_{E,E′ successive zeros of p_{n} E,E′∈[2−ε,2]}|E − E′| ≥ \frac{c}{n^{2}}  (3.73)

Define φ(θ) by

u_{0}(e^{iθ}) = |u_{0}(e^{iθ})|e^{iφ(θ)}  (3.74)

φ can be defined by θ ≠ 0 since u_{0}(θ) is then nonzero. We can pick θ continuous on (0, ε). We claim

φ(0) ≡ lim_{θ↓0} φ(θ) = \begin{cases}0 & mod π if 2 is nonresonant \\ \frac{π}{2} & mod π if 2 is a resonance \end{cases}  (3.75)

Postponing the proof of this for now, let us complete the proof of the theorem.

By (3.70) and W (u, ū) = z − z^{−1} (since u ∼ z^{n}), we see that

p_{n}(2 cos θ) = \frac{\overline{u_{0}(e^{iθ})} u_{n}(e^{iθ}) − u_{0}(e^{iθ}) \overline{u_{n}(e^{iθ})}}{2i sin(θ)}  (3.76)
= \frac{|u_{0}(e^{iθ})| |u_{n}(e^{iθ})|}{2 sin θ} sin(nθ + β_{n}(θ))  (3.77)

where

β_{n}(θ) → −φ(θ)  (3.78)

as n goes to infinity uniformly in a neighborhood of θ = 0. Thus, zeros of p_{n}(2 cos θ) are given as solutions of

nθ + β_{n}(θ) = jπ  (3.79)

In the resonant case, since

sup_{|θ|≤ \frac{c}{n}} |β_{n}(θ) − \frac{π}{2} |_{mod π} → 0  (3.80)

there is at least one solution asymptotic with

nθ_{j}^{(n)} → (j − \frac{1}{2})π  (3.81)

If there were multiple solutions for some j, we would have two zeros with 0 < θ′ < θ,

n(θ − θ′) → 0 θ < \frac{π}{n}

for n large (π can be any number strictly larger than π/2),

|2 cos θ′ − 2 cos θ| ≤ |θ′ − θ| sin(θ)

= o(\frac{1}{n})O(\frac{1}{n})

violating (3.73). Thus, there are unique solutions and (3.64) holds. In the nonresonant case, (3.79) holds, but instead

sup_{|θ|≤ \frac{c}{n}} |β_{n}(θ)|_{mod π} → 0  (3.82)

which proves existence of solutions with

nθ_{j}^{(n)} → jπ  (3.83)

for j = 1, 2, . . .. We must prove uniqueness for j ≥ 1 and nonexistence for j = 0.

The uniqueness argument for j ≥ 1 is the same as in the resonant case. To show no solution with j = 0, we suppose that J has m eigenvalues above E = 2 (by Bargmann’s bound [17], the number is finite). Let J(λ) be the Jacobi matrix with

a_{n}(λ) = a_{n}
b_{n}(λ) = \begin{cases}b_{n} + λ & n ≤ m + 1 \\ b_{n} & n > m + 1 \end{cases}

It is easy to see that as λ → ∞, J(λ) has at least m + 1 eigenvalues. So pick λ_{0} = inf{λ | J(λ) has m + 1 eigenvalues in (2, ∞)}. Then λ_{0} > 0 and J(λ_{0}) has a resonance at 2. By the analysis of the resonant case, p_{n}^{(λ_{0})}(x) has m zeros in (2, ∞) and its (m + 1)st zero asymptotic to 2 − (\frac{1}{2} \frac{π}{n})^{2}, which means p_{n}(x), whose zeros are less than those of p_{n}^{(λ_{0})}, cannot have a zero asymptotic to θ^{(n)} → 0.

That proves the result subject to (3.75). In the nonresonant case, u_{0}(e^{iθ}) is continuous and nonvanishing at θ = 0, and u_{0}(1) is real, φ(0) ≡ 0 mod π, so continuity proves the top half of (3.75).

In the resonant case, we note that a Wronskian calculation (see [6, eqn. (A.49)]) shows that

Im(u_{1}(e^{iθ})u_{0}(e^{iθ})) = sin θ  (3.84)

Since u_{0}(1) = 0, u_{1}(1) ≠ 0, and u_{1}(1) is real, so

lim_{θ↓0} Im(\frac{u_{0}(e^{iθ})}{sin θ}) ≠ 0  (3.85)

On the other hand, u_{0}(e^{−iθ}) = \overline{u_{0}(e^{iθ})}, so φ(θ) + φ(−θ) ≡ 0 mod π, which means that any limit point of φ(θ) is a multiple of π/2. This is only consistent with (3.85) if the limit is congruent to π/2 mod π. □

4. Lower Bounds in the Ergodic Case

Our main goal in this section is to prove the following:

Theorem 4.1 (≡ Theorem 1.5). Let a_{n}^{(ω)}, b_{n}^{(ω)} be ergodic Jacobi parameters. Let E_{0} be such that

(i) The Lyapunov exponent γ(E_{0}) = 0. (ii) The symmetric derivative of ν exists at E_{0} and is finite and

nonzero. Then there exists C > 0 so that with probability 1,

lim inf_{n→∞} n[z_{n}^{(1)}(E_{0}) − z_{n}^{(−1)}(E_{0})] > C  (4.1)

This is particularly interesting because of the connection to Poisson behavior, where:

Definition. We say a probabilistic family of Jacobi matrices has Poisson behavior at E_{0} if and only if for some λ (normally λ = density of zeros) we have that for any α_{1} < β_{1} ≤ α_{2} < β_{2} ≤ ⋯ < β_{ℓ} and any k_{1}, k_{2}, …, k_{ℓ} ∈ {0, 1, 2, … },

\begin{aligned}Prob(#\{z_{n}^{(j)}(E_{0}) & ∈ [E_{0} + \frac{α_{m}}{n} , E_{0} + \frac{β_{m}}{n} ]\} = k_{m} for m = 1, …, ℓ) \\ & → ∏_{m=1}^{ℓ} \frac{[λ(β_{m} − α_{m})]^{k_{m}}}{k_{m}!} e^{−λ(β_{m}−α_{m})} \end{aligned} ; (4.2)

Remark. Poisson behavior was proven in certain random Schrödinger operators by Molchanov [33] and for random Jacobi matrices by Minami [32]. See Stoiciu [47, 48] and Davies-Simon [7] for related work on OPUC.

Corollary 4.2. Let a_{n}^{(ω)}, b_{n}^{(ω)} be ergodic Jacobi parameters and E_{0} ∈ ℝ so that the symmetric derivative of ν exists at E_{0} and is finite. Suppose there is Poisson behavior at E_{0}. Then γ(E_{0}) > 0.

Remarks. 1. Basically, (4.1) is a rigid level repulsion inconsistent with Poisson behavior.

2. Ergodicity is critical here. Killip-Stoiciu [27] have examples which are not ergodic for which there is Poisson behavior with γ = 0.

Proof of Corollary 4.2. Suppose first γ(E_{0}) = 0 so Theorem 4.1 applies. For each n, let f_{n}(ω) be the characteristic function of {ω | |z_{n}^{(1)}(E_{0}) − z_{n}^{(−1)}(E_{0})| ≤ \frac{1}{2}Cn^{−1}\}. By (4.1), f_{n}(ω) → 0 for a.e. ω, so

∫ f_{n}(ω) dω → 0  (4.3)

as n → ∞.

Clearly, if there is one z^{(j)} in [E_{0} − \frac{1}{4}Cn^{−1}, E_{0}] and one in [E_{0}, E_{0} + \frac{1}{4}Cn^{−1}], then f_{n}(ω) = 1. Thus, by the assumption of Poisson behavior,

lim ∫ f_{n}(ω) dω ≥ (\frac{λC}{4} e^{−λC/4})^{2}

which contradicts (4.3). □

Our proof of Theorem 4.1 will use the complex solutions constructed by Deift-Simon [8] and the estimate Theorem 2.2. It is thus important to be able to estimate A in terms of any pair of solutions with Wronskian 1.

Lemma 4.3. Let u^{(1)}, u^{(2)} be any pair of solutions of (2.1) with Wronskian 1. Then the kernel K of (2.7) has the form

K(n, m, E) = u_{n}^{(1)}u_{m}^{(2)} − u_{n}^{(2)}u_{m}^{(1)}  (4.4)

In particular,

∥A_{L}(E)∥_{HS}^{2} ≤ ∥u^{(1)}∥_{L}^{2}∥u^{(2)}∥_{L}^{2}  (4.5)

Proof. Noting that u_{n}^{(j)}u_{m}^{(j)} − u_{m}^{(j)}u_{n}^{(j)} = 0, we see that K is invariant under linear changes of the u’s of determinant 1. This proves (4.4). (4.5) follows as in Theorem 2.1. □

We need the following result of Deift-Simon [8]:

Theorem 4.4. Let a_{n}^{(ω)}, b_{n}^{(ω)} be ergodic Jacobi parameters and let (i)– (ii) of Theorem 4.1 hold for E_{0}. Then for a.e. ω, there exists a complexvalued solution u( · , ω) of (2.1) so that

(i) The Wronskian of u and ū is −2i. (ii) We have that

lim sup_{n→∞} |n|^{−1} ∑_{j=0}^{n−1} |u(j, w)|^{2} ≤ 2π \frac{dν}{dE}  (4.6)

Remarks. 1. In fact, |u(j, w)| = |u(0, T^{j}w)| and 𝔼(|u(0, w)|^{2}) < ∞, so by the Birkhoff ergodic theorem, the lim sup on the left of (4.6) can be replaced by a limit which is a.e. constant with a constant bounded by the right side of (4.6)

2. [8] states results for a.e. E with γ(E) = 0, but the proof shows that what is needed is (i)–(ii).

Proof of Theorem 4.1. By (4.5), taking u^{(1)} = u and u^{(2)} = (2i)^{−1}ū, we see

lim sup n^{−1}∥A_{n}(E)∥ ≤ π \frac{dν}{dE}  (4.7)

by (4.6). Thus, by Theorem 2.2,

lim inf_{n→∞} n[z_{n}^{(1)}(E_{0}) − z_{n}^{(−1)}(E_{0})] ≥ \frac{1}{π} [\frac{dν}{dE} ]^{−1}

which is (4.1). □

5. Upper Bounds via Transfer Matrices

Our goal in this section is to prove:

Theorem 5.1. Let {a_{n}, b_{n}}_{n=1}^{∞} be a set of Jacobi parameters. For any bounded interval I ⊂ ℝ, we have

sup_{E,E′ successive zeros of p_{n} in I}|E − E′| ≤ \frac{[8e sup_{E∈I} ∥T_{n}(E)∥](∏_{j=1}^{n} a_{j})^{1/n}}{n}  (5.1)

Corollary 5.2. Let {a_{n}, b_{n}}_{n=1}^{∞} be a set of Jacobi parameters and I = [α, β] a closed interval. Let

A = sup_{n} a_{n} < ∞  T = sup_{n; E∈I} ∥T_{n}(E)∥ < ∞  (5.2)

Let c = 8eAT. Then for any δ > 0, there is N_{0} so that if n > N_{0} and E ∈ [α + δ, β − δ] is a zero of p_{n}, then there are at least two additional zeros in [E − \frac{c}{n}, E + \frac{c}{n}], one above E and one below.

Proof of Corollary 5.2. It is known that any point in spec(J) is a limit point of zeros, so for large enough N_{0}, there are zeros in [α, α + δ) and (β − δ, β]. Thus, (5.1) implies the result. □

Example 5.3. Let a_{n} ≡ 1, b_{n} ≡ 0 so

p_{n}(2 cos θ) = \frac{sin((n + 1)θ)}{sin θ}  (5.3)

and for n odd, p_{n}(0) = 0. The next nearest zero is at θ = \frac{π}{2} − \frac{π}{n+1}, so at E ∼ \frac{2π}{n+1} ( \frac{d}{dθ}(2 cos θ)|_{θ= \frac{π}{2}} = −2). In this case, p_{n} at E = 0 (θ = \frac{π}{2}) is (1, 0, −1, 0, 1, 0, …) for n = 0, 1, 2, … and q_{n} = (0, −1, 0, 1, …), so T_{n odd} = ( \begin{matrix}0 & +1 \\ ∓1 & 0 \end{matrix} ) so ∥T_{n}(0)∥ = 1 and ∥T_{n}(E)∥ ∼ 1 for E near 0. Thus, the correct answer for the spacing is 2π ∼ 6.3 and our upper bound is 8e ∼ 21.7, a factor of about 3.5 too large. □

To get Theorem 5.1, we will use

Theorem 5.4. Let Q be a polynomial with all its zeros real. Let Q(E_{0}) = 0, Q′(E_{1}) = 0, with E_{0} < E_{1} and Q nonvanishing on (E_{0}, E_{1}). Then

|E_{1} − E_{0}| ≤ e \frac{|Q(E_{1})|}{|Q′(E_{0})|}  (5.4)

Proof. Since \frac{d^{2}}{dx^{2}} log(x − x_{0}) = −\frac{1}{|x−x_{0}|^{2}} < 0, we see that g(E) = log(|Q(E)|/|E − E_{0}|) is concave on [E_{0}, E_{1}]. Note that g(E_{0}) = log|Q′(E_{0})| and that the tangent to g at E_{1} is

log(\frac{|Q(E_{1})|}{|E_{1} − E_{0}|}) − \frac{1}{|E_{1} − E_{0}|} (E − E_{1})  (5.5)

Thus,

|Q′(E_{0})| ≤ (\frac{|Q(E_{1})|}{|E_{1} − E_{0}|})e

which is (5.4). □

The Q we will take to get (5.1) is not P, but the discriminant

Δ_{n}(E) = Tr(T_{n}(E))  (5.6)

associated to the periodic set of Jacobi parameters

a_{mn+q}^{(n)} = a_{q}  b_{mn+q}^{(n)} = b_{q}  q = 1, …, n; m ≥ 0  (5.7)

We have:

Lemma 5.5. The zeros of p_{n−1} and Δ_{n} interlace. Thus, if E_{1} < E_{2} < E_{3} are three successive zeros of Δ_{n}, then p_{n−1} has two zeros, E and E′, with

|E − E′| < |E_{3} − E_{1}|  (5.8)

Proof. We need the analysis of Δ_{n} as a periodic discriminant [30]. Δ_{n} has n bands given by α_{1} < β_{1} ≤ α_{2} < β_{2} ≤ ⋯ ≤ α_{n} < β_{n} and bands [α_{j}, β_{j}]. Δ^{−1}(−2, 2) = ∪_{j=1}^{n}(α_{j}, β_{j}) and, in particular, the zeros of Δ lie one per band. p_{n−1} has one zero in each gap [β_{1}, α_{2}], [β_{2}, α_{3}], …, [β_{n−1}, α_{n}]. That gives us the interlacing. (5.8) is an immediate consequence of this interlacing. □

To get a bound on Δ′_{n} at its zeros, we need a bound on the rotation number for ergodic Schrödinger operators found by Deift-Simon [8]. This rotation number is

α(E) = π(1 − ν(−∞, E))  (5.9)

where ν is the density of states. Thus, α runs from π to 0 as E runs from min spec(J) to max spec(J). cos(α) runs from −1 to 1.

Proposition 5.6. In the periodic case, on spec(J),

\frac{d cos(α(E))}{dE} ≥ \frac{1}{2}( ∏_{j=1}^{n} a_{j})^{−1/n}  (5.10)

Remarks. 1. (1.2) of [8] is an integrated form of (5.10). We can take derivatives since spec(J) is a union of intervals.

2. Deift-Simon assume a_{j} ≡ 1. By using the modification of the Thouless formula for general a_{j}, it is easy to see their proof yields (5.10).

3. In the free case (a_{n} constant, b_{n} = 0), one has equality in (5.10).

Proof of Theorem 5.1. Δ_{n}(E) is related to α(E) by

Δ_{n}(E) = 2 cos(nα(E))  (5.11)

Thus, at zeros of Δ_{n} where cos(nα(E)) = 0, we have |sin(nα(E))| = 1. So at such points,

|\frac{dΔ_{n}}{dE} | = 2n \frac{dα}{dE} ≥ 2n sin(α(E)) \frac{dα}{dE}

= 2n \frac{d cos(α(E))}{dE}

≥ n( ∏_{j=1}^{n} a_{j})^{−1/n}

by (5.10).

Since Δ_{n} = Tr(T_{n}), |Δ_{n}| ≤ 2∥T_{n}∥, so (5.4) becomes

|E_{1} − E_{0}| ≤ 2e \frac{∥T_{n}∥(∏_{j=1}^{n} a_{j})^{1/n}}{n}  (5.12)

Between the zeros of Δ are two (E_{0}, E_{1})-type intervals and so, between the first and third of three zeros are four such intervals. (5.8) and (5.12) imply (5.1). □

6. Prüfer Angles and Bounds on Zero Spacing

There are various possible Prüfer angles. We will exploit one that is ideal for studying the energy dependence of zeros of p_{n}.

Proposition 6.1. Fix Jacobi parameters {a_{n}, b_{n}}_{n=1}^{∞}. For each n = 1, 2, …, there is a unique continuous function θ_{n}(E) determined by

tan(θ_{n}(E)) = \frac{p_{n}(E)}{p_{n−1}(E)}  (6.1)
lim_{E→−∞} θ_{n}(E) = −\frac{π}{2}  (6.2)

Moreover,

\frac{dθ_{n}}{dE} = \frac{∑_{j=0}^{n−1} p_{j}(E)^{2}}{a_{n}(p_{n−1}(E)^{2} + p_{n}(E)^{2})} > 0  (6.3)

Proof. p_{n}(E)/p_{n−1}(E) → −∞ as E → −∞ and the ratio is continuous on ℝ ∪ {∞}, so existence and uniqueness are immediate, as is (6.2) since tan(−π/2) = −∞. Note next that since \frac{d}{dy} arctan(y) = \frac{1}{1+y^{2}} , we have

\frac{dθ_{n}}{dE} = \frac{p_{n−1}p′_{n} − p_{n}p′_{n−1}}{p_{n−1}^{2}(1 + p_{n}^{2}/p_{n−1}^{2})}  (6.4)

so that (6.3) follows from the CD formula (A.3). □

Remark. (6.3) is closely related to well-known formulae relating the derivative of projective angles in 𝕊𝕌(1, 1) or 𝕊𝕃(2, ℝ); see Theorem 10.4.11 of [40]. These formulae have been used by Furstenberg, Carmona, and others; see, for example, Carmona [5] or Simon [45].

The transfer matrix, T_{n}(E), is a 2 × 2 matrix of determinant 1/a_{n+1}, so

∥T_{n}(E)^{−1}∥ = a_{n+1}∥T_{n}(E)∥  (6.5)

Thus, since (p_{j+1}, p_{j})^{t} = T_{j}(E)(1, 0)^{t}, we have that

a_{j+1}^{−2}∥T_{j}(E)∥^{−2} ≤ p_{j+1}(E)^{2} + p_{j}(E)^{2} ≤ ∥T_{j}(E)∥^{2}  (6.6)

Since

\frac{1}{2} ∑_{j=0}^{n−2} (p_{j}^{2} + p_{j+1}^{2}) ≤ ∑_{j=0}^{n−1} p_{j}^{2} ≤ ∑_{j=0}^{n−2} (p_{j}^{2} + p_{j+1}^{2})  (6.7)

(6.3) immediately implies

Theorem 6.2. Let

t_{n}(E) = sup_{0≤j≤n−1} (1 + a_{j+1}^{2})∥T_{j}(E)∥^{2}  (6.8)

Then

\frac{n}{2t_{n}(E)t_{n−1}(E)} ≤ a_{n} \frac{dθ_{n}}{dE} ≤ nt_{n}(E)t_{n−1}(E)  (6.9)

This, in turn, implies 1/n upper and lower bounds on zero spacings sufficient for what we needed in Section 3:

Theorem 6.3. If Δ is an interval in ℝ on which τ_{Δ} ≡ sup_{E∈Δ,n} (1 + a_{n+1}^{2})∥T_{n}(E)∥^{2} < ∞, then

inf_{E,E′∈Δ E,E′ successive zeros of p_{n}(E)}|E − E′| ≥ \frac{a_{n}π}{τ_{Δ}^{2}n}  (6.10)

and

sup_{E,E′∈Δ E,E′ successive zeros of p_{n}(E)}|E − E′| ≤ \frac{2a_{n}πτ_{Δ}^{2}}{n}  (6.11)

Moreover, if Δ = [α, β], p_{n} has zeros in [β−\frac{2a_{n}πτ_{Δ}^{2}}{n} , β] and [α, α+\frac{2a_{n}πτ_{Δ}^{2}}{n}] once |β − α| ≥ \frac{2a_{n}πτ_{Δ}^{2}}{n}.

Proof. Since θ_{n} is monotone in E and p_{n}(E) = 0 if and only if θ_{n}(E) = ℓπ for some ℓ ∈ ℤ, we have at successive zeros, E < E′, that

π = θ_{n}(E′) − θ_{n}(E) = ∫_{E}^{E′} \frac{dθ_{n}}{dE} dE  (6.12)

(6.10)/(6.11) are then immediate from (6.9). The final assertion comes from the fact that θ(E_{1}) − θ(E_{0}) ≥ π implies that tan(θ(E)) has a zero in [E_{0}, E_{1}]. □

7. Relations of the Weight to the Christoffel Function

The previous sections were dominated by the transfer matrix. In this section, we shift to the weight where the CD kernel (see (A.1)) will play a major role. This section is a technical interlude: a detailed result that will be useful in the analysis of examples in later sections. Our main result in this section is

Theorem 7.1. Suppose that

dμ = w(x) dx + dμ_{s}  (7.1)

with dμ_{s} singular, and that for some x_{0}, a > 0, and some r > 0,

∫_{x_{0}−a}^{x_{0}+a} w(x)^{−r} dx < ∞  (7.2)

Then

K_{n}(x_{0}, x_{0}) ≤ C_{r}n^{1+r^{−1}}  (7.3)

where C_{r} only depends on r, a and the integral in (7.2).

This result generalizes one of Geronimus (see [11, Remark 3.3 and Table II]) in two ways. His estimate is on |φ_{n}|^{2} not K_{n} and, more importantly, his estimates require global estimates on w in the context of OPUC rather than just our local estimate. One reason we can go beyond Geronimus is that he uses the Szegő function and we just use the Christoffel variational principle. Another reason is that we have a powerful result of Nevai [34]:

Proposition 7.2 ([34]). For any p in (0, ∞), there is a constant D_{p} so

∫_{−1}^{1}|π_{n}(x)|^{p} dx ≥ D_{p}n^{−1}|π_{n}(0)|^{p}  (7.4)

for any polynomial π_{n} of degree n.

Proof. Since this is a special case of Nevai’s result which depends on several arguments, for the reader’s convenience, we extract exactly what is needed for (7.4).

Let

dμ_{0}(x) = \frac{1}{π} (1 − x^{2})^{−1/2} dx  (7.5)

on [−1, 1], so if x = cos(θ), then

dμ_{0} = \frac{dθ}{π}

on [0, π] which implies, as is well known, that the OPs for (7.5) are given by

p_{n}(cos(θ)) = \begin{cases}1 & n = 0 \\ \sqrt{2} cos(nθ) & n ≥ 1 \end{cases}  (7.6)

the Chebyshev polynomial (of the first kind).

It follows that for x = cos(θ) ∈ [−1, 1],

K_{n}(x, x; dμ_{0}) = 1 + ∑_{j=1}^{n} 2 cos^{2}(jn) ≤ 2n + 1  (7.7)

Thus, by (A.7),

sup_{|x|≤1} |π_{n}(x)|^{2} ≤ (2n + 1) ∫_{−1}^{1}|π_{n}(x)|^{2}(1 − x^{2})^{−1/2} \frac{dx}{π}  (7.8)

If m is an integer, (π_{n})^{m} is a polynomial of degree at most mn, so (7.8) implies

sup_{|x|≤1} |π_{n}(x)|^{2m} ≤ (2mn + 1) ∫_{−1}^{1}|π_{n}(x)|^{2m}(1 − x^{2})^{−1/2} \frac{dx}{π}  (7.9)

If 2m − 2 < p ≤ 2m, we write

|π_{n}|^{2m} ≤ |π_{n}|^{p}( sup_{|x|≤1} |π_{n}(x)|)^{2m−p}

to deduce

sup_{|x|≤1} |π_{n}(x)|^{p} ≤ \{2n([\frac{p}{2}] + 1)\} ∫_{−1}^{1}|π_{n}(x)|^{p}(1 − x^{2})^{−1/2} \frac{dx}{π} (7.10)

Given p, pick ℓ from 1, 2, … so ℓp ≥ \frac{1}{2} and apply (7.10) to the polynomial π_{n}(x)(1 − x^{2})^{ℓ} which has degree n + 2ℓ and get

|π_{n}(0)|^{p} ≤ sup_{|x|≤1} |(1−x^{2})^{ℓ}π_{n}(x)|^{p} ≤ \{2(n+2ℓ)([\frac{p}{2}]+1)\} ∫_{−1}^{1}|π_{n}(x)|^{p} \frac{dx}{π}

(7.11)

since (1 − x^{2})^{ℓp− \frac{1}{2}} ≤ 1.

Find D_{p} so for n ≥ 1,

2(n + 2ℓ)([\frac{p}{2}] + 1) ≤ D_{p}^{−1}n

and (7.11) implies (7.4). □

Proof of Theorem 7.1. By (A.16), we can suppose dμ_{s} = 0. By scaling and translation, we can suppose x_{0} = 0, a = 1. By Theorem A.2, we need to get lower bounds on ∫_{−1}^{1}|π_{n}(x)|^{2}w(x) dx. By Hölder’s inequality, for any α, β, p ∈ (1, ∞), and q dual to p,

∫_{−1}^{1}|π_{n}(x)|^{α} dx = ∫_{−1}^{1}|π_{n}(x)|^{α}w(x)^{β}w(x)^{−β} dx

≤ (∫_{−1}^{1}|π_{n}(x)|^{αp}w(x)^{βp})^{1/p}(∫_{−1}^{1} w(x)^{−βq} dx)^{1/q} (7.12)

We want to pick β, q, α so βq = r, αp = 2, βp = 1, that is,

q = 1 + r  p = \frac{1 + r}{r}  α = \frac{2r}{1 + r}  β = \frac{r}{1 + r}  (7.13)

The result is that

∫_{−1}^{1}|π_{n}(x)|^{2}w(x) dx ≥ (∫_{−1}^{1} w(x)^{−r} dx)^{−1/r}(∫_{−1}^{1}|π_{n}(x)|^{2r/(1+r)} dx)^{(1+r)/r}
≥ C|π_{n}(0)|^{2}n^{−1−r^{−1}}  (7.14)

Taking the inf over all π_{n}’s with π_{n}(0) = 1 and using (A.5), we get

K_{n}(0, 0)^{−1} ≥ Cn^{−1−r^{−1}}

which is (7.3). □

Example 7.3. Let dμ be the measure on [−1, 1] given by

dμ(x) = C_{a,b}|x|^{a}(1 − |x|^{2})^{b} dx  (7.15)

where a ≥ 0, b ≥ −1, and C is a normalization constant.

This is an even measure so p_{2n−1}(0) = 0. Moreover,

p_{2n}(x) = q_{n}(x^{2})  (7.16)

where q_{n} are the OPs for the measure obtained from an x → y = x^{2} change of variables. Since dx = (dy)/y^{1/2}, we see q_{n} are the orthogonal polynomials for the measure

C̃_{a,b}|y|^{a/2−1/2}(1 − y)^{b} dy  (7.17)

Thus, up to a constant,

K_{n}(0, 0; dμ) = K̃_{n,α.β}(1, 1)

where K̃_{n;α,β} is the CD kernel for the Jacobi polynomial associated to (1 − x)^{α}(1 + x)^{β} dx with α = \frac{1}{2}a − \frac{1}{2}, β = b. If we call these orthogonal polynomials j_{α,β}, and J_{α,β} the conventional normalization, then [51, 31]

∥J_{α,β}∥_{2} ∼ C_{α,β}^{(1)}n^{−1}
J_{α,β}(1) ∼ C_{α,β}^{(2)}n^{α}

so

j_{α,β}(1) ∼ (C_{α,β}^{(1)})^{−1/2}(n^{1/2})J_{α,β}(1) = C_{α,β}^{(3)}n^{α+1/2}

and

K̃_{n;α,β}(1, 1) ∼ n^{2α+2}

Taking α = \frac{1}{2}(a − 1), we get

K_{n}(0, 0; dμ) ∼ n^{1+a}  (7.18)

We can take r in Theorem 7.1 arbitrary with ra < 1, so (7.3) cannot be improved. □

The following shows that in some cases the power of n in Theorem 7.1 is optimal:

Theorem 7.4. Let dμ(x) = w(x) dx where supp(dμ) ⊂ [−1, 1] and

|w(x)| ≤ C|x|^{α}  (7.19)

for some α < 1. Then

|K_{n}(0, 0)| ≥ C_{1}n^{1+α}  (7.20)

Remark. For w(x) = C_{α}|x|^{α} on [−1, 1], (7.2) holds for any r < 1/α, so (7.20) says (7.3) cannot hold for any smaller power of n in case r > 1.

Proof. Let π_{n} be the polynomial of Theorem A.7 where x_{0} = 0, a = 1. On |x| ∈ [\frac{j}{n}, \frac{j+1}{n}], j = 0, 1, 2, …, n − 1,

|π_{n}(x)| ≤ \begin{cases}1 & j = 0 \\ \frac{1}{2n} + \frac{1}{2j} & j = 1, 2, … \end{cases}

so, by (7.19),

∫ |π_{n}(x)|^{2} dμ ≤ 2c \{\frac{1}{n^{1+α}} + ∑_{j=1}^{n−1}(\frac{j + 1}{n})^{α}[\frac{1}{2n^{2}} + \frac{1}{2j^{2}} ]\frac{1}{n}\}

≤ 2c \{\frac{1}{2n^{2}} + \frac{1}{n^{1+α}} (1 + ∑_{j=1}^{∞} \frac{(j + 1)^{α}}{j^{2}} )\}

≤ C_{1}^{−1}n^{−(1+α)}

since n^{2} ≥ n^{1+α} and ∑_{j=1}^{∞} \frac{(j+1)^{α}}{j^{2}} < ∞ since α < 1.

By deg(π_{n}) = 2n − 2, and Theorem A.2,

K_{2n−2}(0, 0)^{−1} ≤ C_{1}^{−1}n^{−(1+α)}

which is (7.20). □

8. Upper Bounds via OP Methods

Our main purpose in this section is to note that the upper bounds produced by the method of Erdös-Turan [9] provide a universal bound. So long as dμ_{s} = 0 near x_{0} ∈ supp(dμ) and w(x) is continuous and nonvanishing at x_{0}, the bound is independent of the value of w at x_{0}! Upper bounds on spacing imply lower bounds on the density of zeros. Deift-Simon [8] obtained universal lower bounds on the density of zeros, so the bounds we find are a kind of microscopic analog of theirs. One key to the Erdös-Turan method is

Lemma 8.1. Let x_{1} < x_{2} < ⋯ < x_{n} in ℝ and let 1 ≤ j ≤ n − 1. Then there exists a polynomial π of degree at most n − 1 so

π(x_{ℓ}) = 0  1 ≤ ℓ ≤ n; ℓ ≠ j, j + 1  (8.1)
π(x_{j}) = π(x_{j+1}) = 1  (8.2)
π(y) ≥ 1 in [x_{j}, x_{j+1}]  (8.3)

Proof. Let

π_{0}(x) = ∏_{ℓ≠j,j+1} (x − x_{j})  (8.4)

If

π_{0}(x_{j}) = π_{0}(x_{j+1})  (8.5)

take

π(x) = \frac{π_{0}(x)}{π_{0}(x_{j})}  (8.6)

so (8.1)/(8.2) hold (we will look at (8.3) shortly). If (8.5) fails, for y ∈ ℝ\[x_{j}, x_{j+1}], let

π_{y}(x) = (x − y)π_{0}(x)  (8.7)

As y runs through (−∞, x_{j}], \frac{y−x_{j}}{y−x_{j+1}} runs from 1 down to 0, and as y runs from ∞ to x_{j+1}, the ratio runs from 1 to ∞. Since (8.5) fails and π_{0}(x_{j}) and π_{0}(x_{j+1}) have the same sign, there is a unique y with

π_{y}(x_{j}) = π_{y}(x_{j+1})  (8.8)

so take

π(x) = \frac{π_{y}(x)}{π_{y}(x_{j})}  (8.9)

In any event, π obeys (8.1) and (8.2). By Snell’s theorem, π′ has a zero between any two zeros of π, and so by counting degrees, exactly 1. It follows that π′ has a local maximum in [x_{j}, x_{j+1}] and no local minimum, so (8.3) holds. □

Let {x_{j}}_{j=1}^{n} be the zeros of the OP, P_{n}, associated to a measure dμ. Recall (see Theorem A.4) that there are positive weights {λ_{j}}_{j=1}^{n} so

∫ π̃(x) dμ(x) = ∑_{ℓ=1}^{n} λ_{ℓ}π̃(x_{ℓ})  (8.10)

for any polynomial π̃ with deg π̃ ≤ 2n − 1.

Theorem 8.2. For any j = 1, 2, …, n − 1,

μ([x_{j}, x_{j+1}]) ≤ λ_{j} + λ_{j+1}  (8.11)

Proof. Let π be the polynomial of degree n − 1 or less given by Lemma 8.1. Let π̃ = π^{2}, so deg π̃ ≤ 2n − 2 ≤ 2n − 1. Since π̃(x_{ℓ}) = 0, ℓ ≠ j, j + 1, and π̃(x_{j}) = π̃(x_{j+1}) = 1,

RHS of (8.10) = λ_{j} + λ_{j+1}  (8.12)

Since π̃ ≥ 0 and π̃ ≥ 1 on [x_{j}, x_{j+1}],

LHS of (8.10) ≥ μ([x_{j}, x_{j+1}])  (8.13)

so (8.10) implies (8.11). □

To exploit (8.11), we need upper bounds on λ_{j}.

Suppose

E_{±} = _{inf}^{sup} supp(dμ)  (8.14)

and for E ∈ [E_{−}, E_{+}],

d(E) = max(E_{+} − E, E − E_{−}) ≤ E_{+} − E_{−}  (8.15)

Theorem 8.3. Suppose I is a closed interval on which dμ is purely a.c. and max_{x∈I} w(x) = w_{+} < ∞. Then for each δ > 0 and all weights λ_{j} associated with x_{j} ∈ I and dist(x_{j}, ℝ\I) ≥ δ, we have with m = [\frac{n}{2}] and n the number of zeros,

λ_{j} ≤ \frac{w_{+}π max_{I} d(E)}{m} + O(\frac{1}{n^{2}} )  (8.16)

where the O(\frac{1}{n^{2}} ) is uniform in all λ’s with the given δ (and depends on max_{I} d(E)).

Proof. Let a = max_{I} d(E) and x_{0} be the x_{j} for λ_{j}, and let π̃(x) = π_{m}(x; x_{0}, a) given by Theorem A.7. Since π̃(x) = 1 and deg(π̃^{2}) ≤ 2n − 1, we have

λ_{j} ≤ ∑ λ_{ℓ}π̃(x_{ℓ})^{2}
= ∫ dμ(x)π̃(x)^{2} = K_{1} + K_{2}

where K_{1} is the integral over (x_{0} − δ, x_{0} + δ), and K_{2} the integral over all other x. By (A.34),

K_{2} = O(\frac{1}{n^{2}} )

with estimates only dependent on a and δ. For π̃(x)^{2} = O(\frac{1}{n^{2}} ) on the region of integration and μ(ℝ) = 1.

If (x_{0} −δ, x_{0} +δ) ⊂ I, dμ ≤ w_{+} dx, so (8.16) follows from (A.35). □

Theorem 8.4. Suppose I is a closed interval on which dμ is purely a.c. and

0 < w_{−} ≡ min_{x∈I} w(x) ≤ max_{x∈I} w(x) ≡ w_{+} < ∞  (8.17)

Then for any E ∈ I^{int},

lim sup_{n→∞} n[z_{n}^{(1)}(E) − z_{n}^{(−1)}(E)] ≤ 4π d(E) lim_{δ↓0} \frac{max\{w(x) | |x − E| < δ\}}{min\{w(x) | |x − E| < δ\}}

(8.18)

In particular, if E is a point of continuity of w,

LHS of (8.18) ≤ 4π d(E)

independently of the value of w(E).

Proof. Clearly,

\begin{aligned}|z_{n}^{(1)}(E) & − z_{n}^{(−1)}(E)| \\ & ≤ [min\{w(x) | z_{n}^{(−1)}(E) ≤ x ≤ z_{n}^{(1)}(E)\}]^{−1}μ(z_{n}^{(−1)}(E), z_{n}^{(1)}(E)) \end{aligned} ; (8.19)

From this, (8.11), (8.16), and lim n/m = 2, we get (8.18) by using the fact that since E is a limit point of an infinity of zeros, we have lim_{n→∞}|z_{n}^{±1}(E) − E| = 0. □

This is the promised universal lower bound on the density of zeros. The method is flexible enough to say something if w(x) has a zero of a fixed order.

Theorem 8.5. Suppose dμ is purely absolutely continuous in a neighborhood of E_{0}, and for some q > 0,

0 < γ_{−} = lim inf_{x→E_{0}} \frac{w(x)}{|x − E_{0}|^{q}} ≤ lim sup_{x→E_{0}} \frac{w(x)}{|x − E_{0}|^{q}} = γ_{+} < ∞ (8.20)

Then

lim sup n|z_{n}^{(1)}(E_{0}) − z_{n}^{(−1)}(E_{0})| < ∞  (8.21)

Proof. By (8.20) for any δ, there is N so for n ≥ N,

μ([z_{n}^{(−1)}(E_{0}), z_{n}^{(1)}(E_{0})]) ≥ (γ_{−} − δ)2^{−q}(q + 1)^{−1}(z_{n}^{(1)}(E_{0}) − z_{n}^{(−1)}(E_{0}))^{q+1}

(8.22)

By the proof of (8.16), the λ’s associated to z_{n}^{±}(E) obey

λ_{j} ≤ \frac{C_{1}(γ_{+} + δ)[max(C_{2}n^{−1}, |z_{n}^{±}(E_{0}) − E_{0}|)]^{q}}{n}  (8.23)

for constants C_{1}, C_{2}. (8.11),(8.22), and (8.23) imply (8.21). □

Given our bounds in Section 7, we can also say something when the singularity of the weight is not as regular as some power. The key is an abstraction of an argument of Nevai [34] (see also Golinskii [14]).

Theorem 8.6. Let a = max(supp(dμ)) − min(supp(dμ)). Fix integers p, q so that

(2p − 2)^{2q} ≤ 2n − 1  (8.24)

Then for any successive zeros E, E′ of p_{n}, we have

|E − E′| ≤ \frac{a}{p} [K_{p}(\frac{1}{2} (E + E′), \frac{1}{2} (E + E′))]^{1/2q}  (8.25)

Proof. Let π̃ be defined in terms of the π of Theorem A.7 by Theorem A.4,

π̃(x) = [π_{p}(x; \frac{1}{2} (E + E′), a)]^{q}  (8.26)

By (8.24), deg[π̃]^{2} ≤ 2n − 1 so, by (8.10),

∫ |π̃(x)|^{2} dμ(x) = ∑_{j=1}^{n} λ_{j}|π̃(E_{j})|^{2}  (8.27)

≤ (\frac{1}{2p} + \frac{a}{2p|E − E′|})^{2q}  (8.28)

since ∑ λ_{j} = 1, λ_{j} ≥ 0, and min|E_{j} − \frac{1}{2}(E + E′)| = \frac{1}{2}|(E − E′)|. Since π̃(\frac{1}{2}(E + E′)) = 1, by Theorem A.2,

K_{n}(\frac{1}{2} (E + E′), \frac{1}{2} (E + E′))^{−1} ≤ ∫ |π̃(x)|^{2} dμ(x)  (8.29)

Since |E − E′| ≤ a,

\frac{1}{2p} + \frac{a}{2p|E − E′|} ≤ \frac{a}{p|E − E′|}  (8.30)

(8.25) is immediate from (8.28), (8.29), and (8.30). □

The following abstracts an argument of Golinskii, who needed to make global hypotheses since he relied on estimates of Geronimus:

Corollary 8.7. Suppose that for some interval I, A > 0 and C, we have

sup_{E∈I} |K_{n}(E, E)| ≤ C(n + 1)^{A}

Then for any δ > 0,

lim_{n→∞} sup_{E,E′ successive zeros dist(E,ℝ\I)>δ}|E − E′|[\frac{n}{log n}] < ∞  (8.31)

Proof. Pick q = [log n] and p as large as possible so that (8.24) holds. Since

(n + 1)^{A} = exp(A log(n + 1))

[K_{p}]^{1/2q} is bounded and (8.25) implies (8.31). □

Combining this corollary and Theorem 7.1, we obtain a local version of Golinskii’s [14] result:

Corollary 8.8. If (7.1) and (7.2) hold, then we have (8.31) for I = (x_{0} − a, x_{0} + a).

We also have the following (a local version of results of Nevai [34] and Golinskii [14]):

Theorem 8.9. Suppose for some interval I we have that

dμ = w dx + dμ_{s}  (8.32)

where

∫_{I} log w dx > −∞  (8.33)

Then for any δ > 0,

lim_{n→∞} sup_{E,E′ successive zeros dist(E,ℝ\I)>δ}|E − E′|n^{1/2} < ∞  (8.34)

We need the following lemma:

Lemma 8.10. If J has Jacobi parameters obeying

∑_{n=1}^{∞} |a_{n} − 1|^{2} + |b_{n}|^{2} < ∞  (8.35)

then for any δ > 0, there is C_{δ} so

sup_{E∈[−2+δ,2−δ]} ∥T_{n}(E)∥ ≤ exp(C_{δ}\sqrt{n + 1} )  (8.36)

Proof. Define u_{n}^{±}(E) = e^{inθ} where 2 cos θ = E and 0 < θ < π. By standard variation of parameters about u_{n}^{±}, one proves

LHS of (8.36) ≤ ∏_{j=1}^{n}{1 + C(|b_{j}| + |a_{j} − 1|)}

Since 1 + x ≤ exp(x) and

∑_{j=1}^{n}(|b_{j}| + |a_{j} − 1|) ≤ [ ∑_{j=1}^{n} (|b_{j}|^{2} + |a_{j} − 1|^{2})]^{1/2}[2n]^{1/2}

(8.36) is immediate. □

Proof of Theorem 8.9. By scaling, we suppose I = [−2, 2]. By Corollary A.3,

K_{n}(x, x; dμ) ≤ K_{n}(x, x; dν)  (8.37)

where

dν = χ_{[−2,2]}w dx  (8.38)

Let dν̃ be the normalized dν. By the theorem of Killip-Simon [25], the Jacobi parameters obey (8.35), so by (8.37),

sup_{x∈[−2+δ,2−δ]} K_{n}(x, x; dμ) ≤ exp(C_{δ}\sqrt{n + 1} )

In Theorem 8.6, take q = [\sqrt{n}] and p as large as can be so (8.24) holds. Then p ∼ c\sqrt{n} and (8.25) implies (8.34). □

9. Lower Bounds via OP Methods

In this section, we will get lower bounds in terms of the CD kernel alone. The basic method is due to Golinskii [14], but when he applied the method, he made global assumptions on the measure, and we want to note that local assumptions suffice. Other OP lower bound methods are due to Erdös-Turan [9] and Nevai [34].

Theorem 9.1. If E, E′ are distinct zeros of P_{n}(x), if Ē = \frac{1}{2}(E + E′) and δ > \frac{1}{2}|E − E′|, then

|E − E′| ≥ \frac{[δ^{2} − (\frac{1}{2}|E − E′|)^{2}]}{3n} [\frac{K_{n}(E, E)}{sup_{|y−Ē|≤δ} K_{n}(y, y)}]^{1/2}  (9.1)

Remarks. 1. In most applications, δ is fixed and |E − E′| → 0, so δ^{2} −(\frac{1}{2}|E −E′|)^{2} ∼ δ^{2} > 0. In typical cases, the inf and sup of K_{n}(y, y) for |y − Ē| < δ are comparable and (9.1) gives an 1/n lower bound.

2. This theorem also yields a result with the same asymptotics as (2.22) for K_{n}(E, E) ≥ 1, while the sup is bounded exponentially in n.

3. Interestingly enough, the proof here depends on (2.19) written as (9.2).

4. It is interesting to compare Theorems 9.1 and 2.2. (2.17) only depends on information at E_{0} while (9.1) has a sup K_{n}(y, y) over a neighborhood, but (2.17) requires information on both solutions of (2.1) while (9.1) only on p_{n}.

Proof. Since p_{n}(E) = p_{n}(E′) = 0 and E ≠ E′, we have

K_{n}(E, E′) = 0  (9.2)

by (A.3) (the Christoffel-Darboux formula). Thus, (supposing E < E′ for notational convenience),

K_{n}(E, E) = K_{n}(E, E) − K_{n}(E, E′)  (9.3)
≤ |E − E′| sup_{|y−Ē|≤ \frac{1}{2} |E−E′|}|\frac{∂}{∂y} K_{n}(E, y)|  (9.4)
≤ |E − E′|\{δ^{2} − [\frac{1}{2} (E − E′)]^{2}\}^{−1}
sup_{|y−Ē|≤δ} [δ^{2} − (y − Ē)^{2}]^{1/2}|\frac{∂}{∂y} K_{n}(E, y)| (9.5)

K_{n}(E, y) is a polynomial in y of degree n, so by (A.29) and (9.5),

K_{n}(E, E) ≤ |E −E′|\{δ^{2} −[\frac{1}{2} (E −E′)]^{2}\}^{−1}(3n) sup_{|y−Ē|<δ} |K_{n}(E, y)| (9.6)

By the Schwartz inequality,

sup_{|y−Ē|<δ} |K_{n}(E, y)| ≤ K_{n}(E, E)^{1/2} sup_{|y−Ē|<δ} |K_{n}(y, y)|^{1/2}  (9.7)

(9.6) plus (9.7) imply (9.1). □

Corollary 9.2. Let I be an interval on which

t = sup_{n,y∈I} ∥T_{n}(y)∥ < ∞  (9.8)

Then, for any E ∈ I^{int},

lim inf_{n→∞} n|z_{n}^{(+1)}(E) − z_{n}^{(−1)}(E)| ≥ \frac{t^{−2}}{3} dist(E, ℝ\I)^{2}  (9.9)

Remark. This should be compared with what follows from (2.15) and (2.17) which implies

LHS of (9.9) ≥ t^{−2}

Proof. (9.9) follows from (9.1) if one notes that for y ∈ I,

(n + 1)t^{−2} ≤ K_{n}(y, y) ≤ (n + 1)t^{2}  □

We can also use Theorem 9.1 to get a lower bound in terms of local bounds on the weights.

Theorem 9.3. Suppose dμ = w dx + dμ_{s} where dμ_{s}(x_{0} − δ, x_{0} + δ) = 0 and

0 < inf_{|y−x_{0}|≤δ} w(x) ≤ sup_{|y−x_{0}|≤δ} w(x) < ∞  (9.10)

Then for any ε < δ,

inf_{|y−x_{0}|<ε} lim inf_{n→∞} n|z_{n}^{(+1)}(y) − z_{n}^{(−1)}(y)| > 0  (9.11)

Proof. By (9.1), it suffices to prove

sup_{|y−x_{0}|<ε} [n^{−1}K_{n}(y, y)] < ∞  (9.12)

and

inf_{|y−x_{0}|<ε} [n^{−1}K_{n}(y, y)] > 0  (9.13)

By (9.10), for ε fixed, uniformly in y with |y − x_{0}| < ε, we can find a fixed scaling and some translate of c(4 − x^{2})^{1/2}χ_{[−2,2]} dx lying below dμ(x − y), so using (A.14) and the explicit K_{n} for Chebyshev polynomials of the second kind (i.e., the free K_{n}), we get (9.12).

On the other hand, by (A.4)/(A.5),

K_{n}(x_{0}, x_{0}) ≥ (∫ |π_{n}(x)|^{2} dμ(x))^{−1}  (9.14)

for any π_{n} of degree n with π_{n}(x_{0}) = 1. Using a suitable π_{n/2} of the form given by Theorem A.7 and estimates we used earlier in this paper, we get an O(n) lower bound on K_{n}, that is, (9.13) holds. □

10. Zeros of POPUC

While we have discussed OPRL up to now, virtually all the ideas extend to POPUC. POPUC are defined by taking the first n − 1 recursion parameters (Verblunsky coefficients), α_{0}, …, α_{n−2} in 𝔻, and picking β ∈ ∂𝔻 and letting

φ̃_{n}(z) = zφ_{n−1}(z) − β̄φ_{n−1}^{∗}(z)  (10.1)

Since φ_{n−1}^{∗} is nonvanishing on 𝔻 and |φ_{n−1}| = |φ_{n−1}^{∗}| on ∂𝔻, we have |φ_{n−1}/φ_{n−1}^{∗}| < 1 on 𝔻 by the maximum principle. Thus, φ̃_{n}(z) is nonvanishing on 𝔻 and, by symmetry (\overline{φ̃_{n}(1/z̄)} = z^{−n}[φ_{n−1}^{∗} −βzφ_{n−1}]), nonvanishing on ℂ\\overline{𝔻}. Thus, the zeros of φ̃_{n} lie on ∂𝔻; indeed, they are the zeros of a finite unitary matrix (see Theorem 8.2.7 of [39]). Zeros of POPUC are discussed extensively in Golinskii [14], Cantero-Moral-Velázquez [3, 4], Simon [46], and Wong [55].

As explained in Section 10.8 of [40], there is an OPUC analog of (2.7)–(2.8) (namely, (10.8.3)–(10.8.5) of [40]) which immediately leads to an analog of Theorem 2.1. While (10.8.3)–(10.8.5) are stated for the solutions ψ_{·} + F φ_{·}, they also hold for ψ_{·} and φ_{·}. Key to this analog is the orthogonality of φ_{·}(z_{1}) and φ_{·}(z_{2}) for two zeros of φ̃_{n}(z). This follows from the CD formula for OPUC (see Theorem 2.2.7 of [39]) for φ̃_{n}(z_{1}) = φ̃_{n}(z_{2}) = 0 implies z_{j}φ_{n}(z_{j}) = β̄φ_{n}^{∗}(z_{j}), so

\overline{φ_{n}^{∗}(z_{1})} φ_{n}^{∗}(z_{2}) − z̄_{1}z_{2} \overline{φ_{n}(z_{1})} φ_{n}(z_{2}) = 0

and thus, by (2.2.42) of [39],

∑_{j=0}^{n} \overline{φ_{j}(z_{1})} φ_{j}(z_{2}) = 0  (10.2)

Combined with the techniques of Section 12.1 of [40] and our proof of Theorem 3.4, we get

Theorem 10.1. Let {α_{n}}_{n=0}^{∞} be a set of Verblunsky coefficients that obeys

∑_{n=0}^{∞} |α_{n+1} − α_{n}| < ∞  (10.3)

and

|α_{n}| → 0  (10.4)

Then the zeros of the POPUC, φ̃_{n}(z), for any choice of β have uniform clock behavior on any compact subset of ∂𝔻\{1}.

Remarks. 1. An interesting example is α_{n} = (n + 2)^{−β} for any β > 0. This is related to a conjecture of [41], albeit the conjecture there is for OPUC, not POPUC.

2. The density of zeros in this case is dθ/2π on ∂𝔻; see Theorem 8.2.7 and Example 8.2.8 of [39].

3. If (10.3)/(10.4) are replaced by

∑_{n=0}^{∞} |α_{n}| < ∞  (10.5)

then ∂𝔻\{1} can be replaced by ∂𝔻. This is a result of [41]. Because we have global control in this case, one does not need a priori 1/n bounds on zero spacing.

There are also analogs of the bounds of Sections 5, 6, 8, and 9:

1. One has that

\frac{d^{2}}{dθ^{2}} log|e^{iφ} − e^{iθ}| = −\frac{1}{|e^{iφ} − e^{iθ}|^{2}}  (10.6)

so there is a bound like (5.4) for POPUC (all of whose zeros lie on ∂𝔻), and thus, there is an analog of Theorem 5.1.

2. If one defines η_{n}(z) by

e^{iη_{n}(θ)} = \frac{e^{iθ}φ_{n−1}(e^{iθ})}{φ_{n−1}^{∗}(e^{iθ})}  (10.7)

then

\frac{dη_{n}}{dθ} = \frac{[∑_{j=0}^{n−1}|φ_{j}(e^{iθ})|^{2}]}{|φ_{n−1}(e^{iθ})|^{2}}  (10.8)

This follows from (2.2.71) of [39] which implies

\frac{∂}{∂r} log|φ_{n+1}(re^{iθ})|^{2}|_{r=1} = (n + 1) + |φ_{n+1}(e^{iθ})|^{−2} ∑_{j=0}^{n} |φ_{j}(e^{iθ})|^{2} (10.9)

By the Cauchy-Riemann equations

LHS of (10.9) = 2 \frac{∂}{∂θ} arg[φ_{n+1}(e^{iθ})]  (10.10)

Since (10.7) implies

η_{n} = θ − (n − 1)θ + 2 arg[φ_{n−1}(e^{iθ})]  (10.11)

we obtain (10.8).

(10.8) implies dη_{n}/dθ > 0 and, given that zeros of φ̃_{n} occur when η_{n} = arg β̄ (mod 2π), bounds like those of Theorem 6.3 on zero spacing for POPUCs.

3. Since all the techniques of the Appendix extend to OPUC, the estimates of Sections 8 and 9 extend to POPUC; indeed, somewhat weaker variants occur already in Golinskii [14].

11. Examples, Counterexamples, Conjectures, and Questions

(a) Clock behavior based only on local behavior. Suppose (7.1) holds on for some (c, d) ⊂ ℝ, we have dμ_{s}([c, d]) = 0, w > 0 on (c, d) and w is C^{∞} there. We have proven O(1/n) upper and lower bounds in this case. This leads to the natural question:

Open Question 11.1. Under the above hypothesis, does one have clock behavior on (c, d)?

This is a very subtle question because clock behavior involves the density of states, and it is not even clear that exists on (c, d) only under the above hypothesis. What is clear is that if the density of states exists, it is a global quantity and not just dependent on w on (c, d). We want to demonstrate this by example. We will need the following:

Proposition 11.2. Let dμ_{0} be given by

dμ_{0} = (2π)^{−1}\sqrt{4 − x^{2}} χ_{[−2,2]} dx  (11.1)

Suppose f is a C^{2} function on [−2, 2] with f′(2) = f′(−2) = 0, f′′(−2) = f′′(2), and

f ≥ α > 0  ∫ f dμ_{0} = 1  (11.2)

where α > 0 is a positive real. Let

dμ(x) = f(x) dμ_{0}(x)  (11.3)

Then the density of states exists for dμ and is given by (1.8) and there is clock behavior uniformly on each interval [−2 + ε, 2 − ε].

Proof. By Theorem 13.2.1 of [40], there is a map Sz_{2} of real measures, dρ, on ∂𝔻 (i.e., those measures with real Verblunsky coefficients) to those measures dμ on [−2, 2] which are of the form f dμ_{0} where

∫_{−2}^{2} f(x)(4 − x^{2})^{−1/2} dx < ∞  (11.4)

and under this map, dρ(θ) = w(θ) dθ where

w(θ) = c f(2 cos θ)  (11.5)

Thus, w is C^{2} (including at θ = 0, π) and so ∑_{n=−∞}^{∞} n^{2}|ŵ_{n}|^{2} < ∞, which implies ∑_{n=−∞}^{∞}|ŵ_{n}| < ∞. Thus, by Baxter’s theorem (Theorem 5.2.1 of [39]), the Verblunsky coefficients are in ℓ^{1}. By (13.2.20)/(13.2.21) of [40], (1.15) holds, which implies the claimed result. □

Example 11.3. Let dμ̃_{0} be dμ_{0} scaled to [−1, 1]. We pick f_{1} obeying the hypothesis of Proposition 11.2 and f_{2} scaled to [−1, 1], and so that dμ_{1} = f_{1} dμ_{0} and dμ_{2} = f_{2} dμ̃_{0} obey dμ_{1} ≡ dμ_{2} on [−\frac{1}{2}, \frac{1}{2}]. Both have clock behavior on [−\frac{1}{2}, \frac{1}{2}] but with different density of states, namely (1.8) and (1.8) scaled. □

(b) Pointwise upper bounds. We obtained lower bounds on z_{n}^{(1)}(E_{0}) − z_{n}^{(−1)}(E_{0}) if T_{n}(E_{0}) is bounded, but our upper bounds required control of T_{n}(E) for E in a neighborhood of E_{0}.

Open Question 11.4. Are there upper bounds on spacing if we only know that T_{n}(E_{0}) is bounded?

(c) Improved spacing estimates.

Open Question 11.5. Can \sqrt{n} in (2.25) be improved?

(d) More on spacing and γ(E). We saw that γ(E_{0}) = 0 plus some regularity of ν at E_{0} implies an O(1/n) lower bound. Does it imply clock spacing? In particular,

Open Question 11.6. Is there local clock behavior for a.e. E_{0} with γ(E_{0}) = 0 in the case of almost periodic Jacobi parameters?

Example 11.7. [27] has proven, for α < \frac{1}{2}, the OPUC analog of Poisson behavior for a_{n} ≡ 1, b_{n} independent random variable of the form b_{n} = Cn^{−α}w_{n} where w_{n} is uniformly distributed in [−1, 1]. We assume their result is true in the Jacobi case. Of course, γ(E) = 0 in this case. We do not have a contradiction with Corollary 4.2 since this model is not ergodic. The example does show though that ergodicity is a necessary hypothesis. □

Example 11.8. Corollary 4.2 shows that ergodicity along with Poisson behavior imply positive Lyapunov exponent. This raises the natural question: Does ergodicity along with a positive Lyapunov exponent imply Poisson behavior? The answer is negative, as can be shown by the following example: Consider the Jacobi matrix with a_{n} = 1, b_{n} = λ cos(2παn+θ), where |λ| > 2 and α is a (Liouville) irrational for which there is a sequence of rationals {p_{n}/q_{n}}_{n=1}^{∞} such that |α−p_{n}/q_{n}| < n^{−q_{n}}. This is an ergodic Jacobi matrix and it is well known (see, e.g., [1]) that its Lyapunov exponent γ(E) is positive for any E. By using the results of Avron-van Mouche-Simon [2] and considering scales of the form mq_{n}, where m > 2 is an integer, one can show that, for each θ, there would be clusters of m − 1 zeros each of which is contained in an interval whose length is of order (2/|λ|)^{q_{n}/2}. As θ is varied, these clusters will move over regions whose size is roughly of order 1/q_{n}. This behavior contradicts Poisson behavior. More precisely, it is possible to show that Poisson behavior does not occur for Lebesgue a.e. E in the spectrum.

(e) Zero spacing and the Szegő condition.

Open Question 11.9. Does one have O(n^{−1}) bounds (upper and lower) when a Szegő or quasi-Szegő condition holds?

(f) Spacing at zeros of w(x).

Open Question 11.10. What can one say at zero spacing at points x_{0} where w(x) has a “regular” zero, that is, w(x) ∼ |x − x_{0}|^{α} for some α > 0?

(g) Edge zeros when a_{n} = 1 − n^{−γ}. The following illuminates Theorem 3.8.

Example 11.11. Let b_{n} ≡ 0, a_{n} = 1 − n^{−γ} for γ > 0. Then Theorem 3.4 applies and there is clock behavior away from −2 and 2. If γ > 2, Theorem 3.8 applies and the largest E_{j} has E_{j} = 2 − Cn^{−2} + o(n^{−2}). We claim for γ in general

2 − C_{2}n^{−γ} + O(n^{−1}) ≤ E_{j}^{max} ≤ 2 − C_{1}n^{−γ}  (11.6)

capturing the leading behavior for γ ∈ (0, 1). The upper bound in (11.6) comes from monotonicity of E_{j}^{max} in the a’s and the fact that for J_{n;F} max_{1≤j≤n}|a_{j}| ∼ 1 − n^{−γ}. The lower bound comes with a trial vector that lives in [n/2, 2] with maximum at 3n/4 and constant slope in between. □

(h) OPUC. This paper has a fairly complete analysis of OPRL and POPUC. Many questions remain for general OPUC.

Appendix A. Tools of the OP Trade

As explained in the introduction, this paper is intended for two audiences, so we include here a summary of tools well known to the OP community but not so well to the Schrödinger operator community. Because the tools, while powerful, are simple, we can even give complete proofs. We will discuss the Christoffel variational principle, Gauss-Jacobi quadrature, Bernstein’s inequality, and Dirichlet-Fejér trial polynomials.

(a) Christoffel variational principle. We will define OPRL for arbitrary positive measures (with finite moments) even if μ(ℝ) ≠ 1. The monic polynomials, P_{n}, are independent of normalization, but the orthonormal polynomials, p_{n}, are not. For example, p_{0}(x, dμ) = dμ(ℝ)^{−1/2}. The Christoffel-Darboux (a.k.a. CD) kernel or reproducing kernel is defined by

K_{n}(x, y) = ∑_{j=0}^{n} \overline{p_{j}(y)} p_{j}(x)  (A.1)

We will use K_{n}(x, y; dμ) if the measure needs to be more explicit. The name reproducing kernel comes from

(P_{n}f)(x) = ∫ K_{n}(x, y)f(y) dμ(y)  (A.2)

where P_{n} is the projection in L^{2}(ℝ, dμ) onto the space of polynomials of degree n.

We need the following in Section 6:

Theorem A.1 (CD formula). We have

K_{n}(x, y) = a_{n+1}[\frac{\overline{p_{n+1}(y)} p_{n}(x) − \overline{p_{n}(y)} p_{n+1}(x)}{ȳ − x}]  (A.3)

Proof. This is a discrete version of integrating a Wronskian. Take the equation (1.3) at ȳ, multiply by p_{n}(x) and subtract (1.3) at x, multiplied by \overline{p_{n}(y)}, and obtain

Q_{n+1}(x, y) = \overline{p_{n}(y)} p_{n}(x) + Q_{n}(x, y)

where Q_{n+1} is the right side of (A.3). Since p_{−1}(x) = 0, Q_{0} ≡ 0, so (A.3) follows by iteration. □

Remark. For x, y real, p_{n}(x), p_{n}(y), are real so the bars are not needed. Indeed, one can drop all the bars for complex x, y, but given the OPUC analogs, it is natural to use the bars.

Theorem A.2 (Christoffel variational principle). Let

λ_{n}(x_{0}; dμ) = inf(∫ |π_{n}(x)|^{2} dμ(x) | deg π_{n} ≤ n; π_{n}(x_{0}) = 1) (A.4)

Then

λ_{n}(x_{0}; dμ) = K_{n}(x_{0}, x_{0}; dμ)^{−1}  (A.5)

Remarks. 1. λ_{n} are called Christoffel numbers. More generally, we have p-Christoffel numbers defined, for 0 < p < ∞, by

λ_{n}(x_{0}, p; dμ) = inf(∫ |π_{n}(x)|^{p} dμ(x) | deg π_{n} ≤ n; π_{n}(x_{0}) = 1) (A.6)

2. Another way of writing (A.4) is that λ_{n} is the optimal constant in

|π_{n}(x_{0})|^{2} ≤ λ_{n}(x_{0}; dμ)^{−1} ∫ |π_{n}(x)|^{2} dμ(x)  (A.7)

or

|π_{n}(x_{0})|^{p} ≤ λ_{n}(x_{0}, p; dμ)^{−1} ∫ |π_{n}(x)|^{p} dμ(x)  (A.8)

3. Our proof shows the inf in (A.4) is a min and the minimizing π is given by

π(x) = K(x, x_{0})  (A.9)

Proof. Expand π_{n} in terms of the orthonormal basis {p_{j}}_{j=0}^{n}:

π_{n}(x) = ∑_{j=0}^{n} a_{j}p_{j}(x)  (A.10)

π_{n}(x_{0}) = 1 is equivalent to

∑_{j=0}^{n} a_{j}p_{j}(x_{0}) = 1  (A.11)

By the Schwartz inequality

1 ≤ K(x_{0}, x_{0}) ∑_{j=0}^{n} a_{j}^{2}  (A.12)

= K(x_{0}, x_{0}) ∫ π_{n}(x)^{2} dμ(x)  (A.13)

where equality occurs in (A.12) if a_{j} = \overline{p_{j}(x_{0})}/K(x_{0}, x_{0}), that is, if π_{n} is given by (A.10). (A.5) is immediate from this case of equality and (A.13). □

Christoffel numbers have been a critical tool in OP theory for over a century, with important uses by Erdös-Turán [9] and turned to high art by Freud and Nevai; see Nevai [35]. They can be used for lower bounds on K, that is, upper bounds for λ_{n}, by using any convenient trial polynomial for π_{n} (see (d) below). One gets upper bounds on K, that is, lower bounds for λ_{n}, by the immediate

Corollary A.3. If dμ ≥ dν, then

K_{n}(x_{0}, x_{0}; dμ) ≤ K_{n}(x_{0}, x_{0}; dν)  (A.14)

Remarks. 1. This shows the true power of Theorem A.2 and the need to allow μ(ℝ) ≠ 1.

2. In particular, if

dμ = f(x) dx + dμ_{s}  (A.15)

then

K_{n}(x_{0}, x_{0}; dμ) ≤ K_{n}(x_{0}, x_{0}; f dx)  (A.16)

(b) Gauss-Jacobi quadrature. The main result here is

Theorem A.4 (Gauss-Jacobi quadrature). Let μ be an arbitrary positive nontrivial measure on ℝ with finite moments. Fix n and define dμ_{n} to be the point measure with weights only at the zeros {x_{j}^{(n)}}_{j=1}^{n} of p_{n}(x) and weights

dμ_{n}({x_{j}^{(n)}}) = λ_{n}(x_{j}^{(n)}; dμ)  (A.17)

the Christoffel numbers of dμ. Then, if π is a polynomial of degree 2n − 1 or less, we have

∫ π(x) dμ(x) = ∫ π(x) dμ_{n}(x)  (A.18)

Remark. In our applications, we will care much more that the masses are at the zeros than the variational formulae for the weights.

Sketch of Proof. Here is a proof intended for Schrödinger operator experts. (For the more usual OP proof, see Freud’s book [10].) Let J_{n;F} be an n × n matrix in the upper corner of the Jacobi matrix, (1.4), associated to dμ. Then the recursion (1.3) implies that if

u_{j}(z) = p_{j−1}(z) j = 1, 2, …, n  (A.19)

then

[(J_{n;F} − z)u]_{j} = −a_{n+1}δ_{jn}p_{n}(z)  (A.20)

Thus the eigenvalues of J_{n;F} are the zeros of p_{n} and the normalized eigenvectors are p_{j−1}(z)/K(z, z)^{1/2}. It follows that dμ_{n} is the spectral measure of J_{n;F} with eigenvector (1 0 … 0)^{t}.

For any measure dη and associated Jacobi matrix, J, since dη is the spectral measure for δ ≡ (1 0 …)^{t}, we have

∫ x^{ℓ} dη = ⟨δ, J^{ℓ}δ⟩  (A.21)

so ∫ x^{ℓ+k} dη = ⟨J^{k}δ, J^{ℓ}δ⟩ for k, ℓ = 0, …, n − 1 depends only on {J^{k}δ}_{k=0}^{n−1} and so on {a_{k}, b_{k}}_{k=0}^{n−1}. Moreover,

∫ x^{2n−1} dη = ⟨J^{n−1}δ, JJ^{n−1}δ⟩  (A.22)

only depends on {a_{k}, b_{k}}_{k=0}^{n−1} ∪ {b_{n}}. Thus, J and J_{n;F}, which have the same set of these parameters, have the same moments of order up to 2n − 1, that is,

∫ x^{k} dμ = ∫ x^{k} dμ_{n;F}  0 ≤ k ≤ 2n − 1  (A.23)

which is (A.18). □

(c) Bernstein’s inequality. These inequalities control π′_{n} in terms of n and π_{n} for polynomials π_{n} of degree at most n.

Theorem A.5. Let ∂𝔻 be the unit circle in ℂ, ∂𝔻 = {z | |z| = 1}. Let π_{n} be a polynomial of degree n. Then

sup_{z∈∂𝔻}|π′_{n}(z)| ≤ n sup_{z∈∂𝔻}|π_{n}(z)|  (A.24)

Remark. If π_{n}(z) = z^{n}, one has equality in (A.24).

Proof. (Szegő [50]) Since π_{n}(e^{iθ}) = ∑_{j=0}^{n} a_{j}e^{ijθ}, we have

π_{n}(e^{iθ}) = ∫_{0}^{2π} ∑_{j=0}^{n} e^{ij(θ−φ)}π_{n}(e^{iφ}) \frac{dφ}{2π}  (A.25)

so

−iπ′_{n}(e^{iθ}) = ∫_{0}^{2π} ∑_{j=1}^{n} je^{ij(θ−φ)}π_{n}(e^{iθ}) \frac{dρ}{2π}

= ∫ F_{n}(θ − φ)e^{in(θ−φ)}π_{n}(e^{iφ}) \frac{dρ}{2π}  (A.26)

where

F_{n}(θ) = ∑_{j=−n+1}^{n−1}(n − |j|)e^{ijθ}  (A.27)

(for the j > 0 terms in (A.26), integrate to zero).

By cancellation,

(1 − cos θ)F_{n}(θ) = 1 − cos(nθ)  (A.28)

so F_{n}(θ) ≥ 0 and, by (A.27), ∫ F_{n}(θ)\frac{dθ}{2π} = n. Thus,

|π′_{n}(e^{iθ})| ≤ ∥π_{n}∥_{∞} ∫ |F_{n}(θ − φ)| \frac{dφ}{2π} = n∥π_{n}∥_{∞}  □

Theorem A.6. Let π_{n} be an arbitrary polynomial of degree n. Then

sup_{x∈[−a,a]} [|π′_{n}(x)|(a^{2} − x^{2})^{1/2}] ≤ 3n sup_{x∈[−a,a]} |π_{n}(x)|  (A.29)

Proof. By scaling, we need only check the case a = 2. Define

π̃_{n}(z) = z^{n}π_{n}(z + \frac{1}{z})  (A.30)

π̃_{n} is a polynomial of degree 2n so, by (A.24),

sup_{θ} |π̃′_{n}(e^{iθ})| ≤ 2n sup_{x∈[−2,2]} |π_{n}(x)|  (A.31)

since e^{iθ} → e^{iθ} + e^{−iθ} = 2 cos θ maps ∂𝔻 to [−2, 2].

By (A.30),

π′_{n}(z + \frac{1}{z})(1 − z^{−2}) = \frac{d}{dz} z^{−n}π̃_{n}(z)
= −nz^{−n−1}π̃_{n}(z) + z^{−n}π̃′_{n}(z)

so, by (A.31),

sup_{e^{iθ}∈∂𝔻}|π′_{n}(2 cos θ)2 sin θ| ≤ 3n∥π̃_{n}∥_{∞}

which is (A.29) for a = 2. □

(d) Dirichlet trial polynomials. For use in both (A.4) and (A.18), we want a rich set of trial polynomials, π_{n}(x). In particular, we want π_{n}’s concentrated near x = x_{0} and otherwise small in some interval [x_{0} − a, x_{0} + a]. By scaling, we may as well consider x_{0} = 0, a = 1. An analyst might try (1 − x^{2})^{n}, but that has width n^{−1/2} — and we will see that one can do better. We will get width n^{−1}. One can’t do better than this, by Bernstein’s inequality, if π_{n}(θ) = 1 and ∥π_{n}∥_{∞} = 1, then π_{n}(x) ≥ \frac{1}{2} for |x| ≤ \frac{1}{2n} − O(\frac{1}{n^{2}} ).

Our choice is related to Dirichlet and Fejér kernels and is, in fact, essentially the minimizer for the Christoffel problem with x_{0} = 0 and dμ = χ_{[−1,1]}(1 − x^{2})^{−1/2} dx.

Theorem A.7. For any x_{0} ∈ ℝ and a > 0, there exist, for each n, polynomials π_{n}(x; x_{0}, a) so that (i)

deg π_{n} = 2n − 2  (A.32)

(ii)

π_{n}(x_{0}) = 1  (A.33)

(iii)

|π_{n}(x)| ≤ min(1, \frac{1}{2n} + \frac{a}{2n|x − x_{0}|})  if |x − x_{0}| ≤ a (A.34)

(iv) For any δ ≤ a,

∫_{x_{0}−δ}^{x_{0}+δ}|π_{n}(x)|^{2} dx = \frac{πa}{n} + O(\frac{1}{n^{2}} )  (A.35)

Remark. (A.34) implies |π_{n}(x)| ≤ C_{δ}/n if |x − x_{0}| > δ, and for any ε, |π_{n}(x)| < ε if |x − x_{0}| ≥ C_{ε}/n.

Proof. By scaling, we can suppose that x_{0} = 0, a = 1, in which case we will call the polynomials D_{n}, that is,

π_{n}(x; x_{0}, a) = D_{n}(\frac{x − x_{0}}{a} )  (A.36)

Recall there are polynomials T_{n}(x) (Chebyshev of the first kind) with

deg T_{n} = n  (A.37)

so that

T_{n}(cos θ) = cos(nθ)  (A.38)

Define D_{n} by

D_{n}(x) = \frac{1}{n} ∑_{j=0}^{n−1}(−1)^{j}T_{2j}(x)  (A.39)

By (A.37), (A.32) holds for D_{n}. By (A.38) (and cos θ = 0 ⇔ θ = \frac{π}{2} mod π), T_{2j}(0) = (−1)^{j}, so D_{n} obeys (A.33) for x_{0} = 0.

By (A.38), |T_{n}(x)| ≤ 1 on [−1, 1], so

|D_{n}(x)| ≤ 1 on [−1, 1]

which is half of (A.34). For the other half, sum the geometric series to see that

D_{n}(cos θ) = \frac{1}{2n} + \frac{(−1)^{n−1}}{2n} \frac{cos((2n − 1)θ)}{cos θ}  (A.40)

which implies the other half of (A.34).

Since

∫_{−π}^{π} cos(2kθ) cos(2jθ) \frac{dθ}{2π} = \begin{cases}1 & if k = j = 0 \\ \frac{1}{2} & if k = j ≠ 0 \\ 0 & if k ≠ j \end{cases}

we have

∫_{−π}^{π} D_{n}^{2}(cos θ) \frac{dθ}{2π} = \frac{1}{n^{2}} [1 + \frac{1}{2} (n − 1)]

= \frac{1}{2n} + O(\frac{1}{n^{2}} )

Since dθ = (1 + O(x^{2})) dx near x = 0 and D_{n}^{2} = O(\frac{1}{n^{2}} ) away from x = 0, we obtain (A.35) when a = 1. □

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