EQUILIBRIUM MEASURES AND CAPACITIES IN SPECTRAL THEORY

BARRY SIMON^{∗}

Abstract. This is a comprehensive review of the uses of potential theory in studying the spectral theory of orthogonal polynomials. Much of the article focuses on the Stahl–Totik theory of regular measures, especially the case of OPRL and OPUC. Links are made to the study of ergodic Schrödinger operators where one of our new results implies that, in complete generality, the spectral measure is supported on a set of zero Hausdorff dimension (indeed, of capacity zero) in the region of strictly positive Lyapunov exponent. There are many examples and some new conjectures and indications of new research directions. Included are appendices on potential theory and on Fekete–Szegő theory.

1. Introduction

This paper deals with applications of potential theory to spectral and inverse spectral theory, mainly to orthogonal polynomials especially on the real line (OPRL) and unit circle (OPUC). This is an area that has traditionally impacted both the orthogonal polynomial community and the spectral theory community with insufficient interrelation. The OP approach emphasizes the procedure of going from measure to recursion parameters, that is, the inverse spectral problem, while spectral theorists tend to start with recursion parameters and so work with direct spectral theory.

Potential theory ideas in the orthogonal polynomial community go back at least to a deep 1919 paper of Faber [35] and a seminal 1924 paper of Szegő [107] with critical later developments of Kalmár [63] and

Date: August 23, 2007. 2000 Mathematics Subject Classification. Primary: 31A15, 05E35, 34L05. Sec-

ondary: 31A35, 33D45, 34P05.

Key words and phrases. Potential theory, spectral theory, regular orthogonal polynomials.

∗ Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125. E-mail: bsimon@caltech.edu. Supported in part by NSF grants DMS-0140592 and DMS-0652919 and U.S.–Israel Binational Science Foundation (BSF) Grant No. 2002068.

Erdös–Turán [34]. The modern theory was initiated by Ullman [114] (see also [115, 116, 117, 118, 112, 119, 123] and earlier related work of Korovkin [68] and Widom [122]), followed by an often overlooked paper of Van Assche [120], and culminating in the comprehensive and deep monograph of Stahl–Totik [105]. (We are ignoring the important developments connected to variable weights and external potentials, which are marginal to the themes we study; see [91] and references therein.)

On the spectral theory community side, theoretical physicists rediscovered Szegő’s potential theory connection of the growth of polynomials and the density of zeros—this is called the Thouless formula after [110], although discovered slightly earlier by Herbert and Jones [51]. The new elements involve ergodic classes of potentials, especially Kotani theory (see [69, 96, 70, 71, 30, 27]).

One purpose of this paper is to make propaganda on both sides: to explain some of the main aspects of the Stahl–Totik results to spectral theorists and the relevant parts of Kotani theory to the OP community. But this article looks forward even more than it looks back. In thinking through the issues, I realized there were many interesting questions to examine. Motivated in part by the remark that one can often learn from wrong conjectures [56], I make several conjectures which, depending on your point of view, can be regarded as either bold or foolhardy (I especially have Conjectures 8.7 and 8.11 in mind).

The potential of a measure μ on ℂ is defined by

Φ_{μ}(x) = ∫ log|x − y|^{−1} dμ(y)  (1.1)

which, for each x, is well defined (although perhaps ∞) if μ has compact support. The relevance of this to polynomials comes from noting that if P_{n} is a monic polynomial,

P_{n}(z) = ∏_{j=1}^{n}(z − z_{j})  (1.2)

and dν_{n} its zero counting measure, that is,

ν_{n} = \frac{1}{n} ∑_{j=1}^{n} δ_{z_{j}}  (1.3)

the point measure with nν_{n}({w}) = multiplicity of w as a root of P_{n}, then

|P_{n}(z)|^{1/n} = exp(−Φ_{ν_{n}}(z))  (1.4)

If now dμ is a measure of compact support on ℂ, let X_{n}(z) and x_{n}(z) be the monic orthogonal and orthonormal polynomials for dμ, that is,

X_{n}(z) = z^{n} + lower order  (1.5)

with

∫ \overline{X_{n}(z)} X_{m}(z) dμ(z) = ∥X_{n}∥_{L^{2}}^{2}δ_{nm}  (1.6)

and

x_{n}(z) = \frac{X_{n}(z)}{∥X_{n}∥_{L^{2}}}  (1.7)

Here and elsewhere ∥·∥ without a subscript means the L^{2} norm for the measure currently under consideration.

When supp(dμ) ⊂ ℝ, we use P_{n}, p_{n} and note (see [108, 39]) there are Jacobi parameters {a_{n}, b_{n}}_{n=1}^{∞} ∈ [(0, ∞) × ℝ]^{∞}, so

xp_{n}(x) = a_{n+1}p_{n+1}(x) + b_{n+1}p_{n}(x) + a_{n}p_{n−1}(x)  (1.8)
∥P_{n}∥ = a_{1} … a_{n}μ(ℝ)^{1/2}

and if supp(dμ) ⊂ ∂𝔻, the unit circle, we use Φ_{n}, φ_{n} and note (see [108, 44, 98, 99]) there are Verblunsky coefficients {α_{n}}_{n=0}^{∞} ∈ 𝔻^{∞}, so

Φ_{n+1}(z) = zΦ_{n} − ᾱ_{n}Φ_{n}^{∗}(z)  (1.9)
∥Φ_{n}(z)∥ = ρ_{0} … ρ_{n−1}μ(∂𝔻)^{1/2}  (1.10)

where

Φ_{n}^{∗}(z) = z^{n} \overline{Φ_{n}(1/z̄)} ρ_{j} = (1 − |α_{j}|^{2})^{1/2}  (1.11)

As usual, we will use J for the Jacobi matrix formed from the parameters in the OPRL case, that is, J is tridiagonal with b_{j} on diagonal and a_{j} off-diagonal.

The X_{n} minimize L^{2} norms, that is,

∥X_{n}∥_{L^{2}(dμ)} = min{∥Q_{n}∥_{L^{2}} | Q_{n}(z) = z^{n} + lower order}  (1.12)

Given a compact E ⊂ ℂ, the Chebyshev polynomials are defined by (L^{∞} is the sup norm over E)

∥T_{n}∥_{L^{∞}(E)} = min{∥Q_{n}∥_{L^{∞}} | Q_{n}(z) = z^{n} + lower order}  (1.13)

These minimum conditions suggest that extreme objects in potential theory, namely, the capacity, C(E), and equilibrium measure, dρ_{E}, discussed in [50, 73, 81, 88, 91] and Appendix A will play a role (terminology from Appendix A is used extensively below). In fact, going back to Szegő [107] (we sketch a proof in Appendix B), one knows Theorem 1.1 (Szegő [107]). For any compact E ⊂ ℂ with Chebyshev polynomials, T_{n}, one has

lim_{n→∞} ∥T_{n}∥_{L^{∞}(E)}^{1/n} = C(E)  (1.14)

This has an immediate corollary (it appears with this argument in Widom [122] and may well be earlier):

Corollary 1.2. Let μ be a measure of compact support, E, in ℂ. Let X_{n}(z; dμ) be its monic OPs. Then

lim sup_{n→∞} ∥X_{n}∥_{L^{2}(ℂ,dμ)}^{1/n} ≤ C(E)  (1.15)

Proof. By (1.12),

∥X_{n}∥_{L^{2}(ℂ,dμ)}^{1/n} ≤ ∥T_{n}∥_{L^{2}(ℂ,dμ)}^{1/n}  (1.16)

where T_{n} are the Chebyshev polynomials for E. On E, |T_{n}(z)| ≤ ∥T_{n}∥_{L^{∞}(E)} so, since supp(dμ) = E,

∥X_{n}∥_{L^{2}(ℂ,dμ)}^{1/n} ≤ ∥T_{n}∥_{L^{∞}(E)}^{1/n}μ(E)^{1/2n}  (1.17)

and (1.15) follows from (1.14). □

For OPRL and OPUC, the relation (1.15) says

lim sup(a_{1} … a_{n})^{1/n} ≤ C(E) (OPRL)  (1.18)
lim sup(ρ_{1} … ρ_{n})^{1/n} ≤ C(E) (OPUC)  (1.19)

(1.18) is a kind of thickness indication of the spectrum of discrete Schrödinger operators (with a_{j} ≡ 1) where it is not widely appreciated that C(E) ≥ 1.

In many cases that occur in spectral theory, one considers discrete and essential spectrum. In this context, σ_{ess}(dμ) is the nonisolated points of supp(dμ). σ_{d}(dμ) = supp(dμ)\σ_{ess}(dμ) is a countable discrete set. If dν is any measure with finite Coulomb energy ν(σ_{d}(dμ)) = 0, thus C(supp(dμ)) = C(σ_{ess}(dμ)); so we will often consider E = σ_{ess}(dμ) in (1.15). In fact, as discussed in Appendix A after Theorem A.13, we should take E = σ_{cap}(dμ).

The inequality (1.16) suggests singling out a special case. A measure dμ of compact support, E, on ℂ is called regular if and only if

lim_{n→∞} ∥X_{n}∥_{L^{2}(dμ)}^{1/n} = C(E)  (1.20)

For E = [−1, 1], this class was singled out by Ullman [114]; the general case is due to Stahl–Totik [105].

Example 1.3. The Nevai class, N(a, b), with a > 0, b ∈ ℝ, is the set of probability measures on ℝ whose Jacobi parameters obey

a_{n} → a b_{n} → b  (1.21)

The Jacobi matrix with a_{n} ≡ a, b_{n} ≡ b is easily seen to have spectrum

E(a, b) = [b − 2a, b + 2a]  (1.22)

so, by (A.8),

C(E(a, b)) = a  (1.23)

By Weyl’s theorem, if μ ∈ N(a, b), σ_{ess}(μ) = E(a, b), so μ is regular. □

Example 1.4. Here is an example of a regular measure on ℝ not in a Nevai class. Let dμ be the measure with Jacobi parameters b_{n} ≡ 0 and

a_{n} = \begin{cases}1 & n ≠ k^{2} for all k \\ \frac{1}{2} & n = k^{2} for some k \end{cases}  (1.24)

Clearly, lim(a_{1} … a_{n})^{1/n} = 1, so if supp(dμ) = [−2, 2] (which has capacity 1 by (1.23)), we will have a regular measure not in a Nevai class. Since ( \begin{matrix}0 & c \\ c & 0 \end{matrix} ) ≥ ( \begin{matrix}−c & 0 \\ 0 & −c \end{matrix} ) for any c > 0, the Jacobi matrix J associated to dμ is bounded below by a diagonal matrix with elements either −\frac{1}{2}, −1, −\frac{3}{2}, or −2 (for n = 1, 2, k^{2} or k^{2} + 1 and otherwise), so J ≥ −2. Similarly, J ≤ 2. Thus, σ(J) = supp(dμ) ⊂ [−2, 2]. On the other hand, since lim(a_{1} … a_{n})^{1/n} = 1, (1.18) implies C(E) ≥ 1 where E = supp(dμ). If E ⫋ [−2, 2], it is missing an open subset and so |E| < 4 and C(E) < 1 (by (A.57)). Thus C(E) ≥ 1 implies E = [−2, 2]. Alternatively, using plane wave trial functions cut off to live in [k^{2} + 2, (k + 1)^{2} − 1], we easily see directly that [−2, 2] ⊂ σ(J).

This example has no a.c. spectrum by results of Remling [90]. In Section 8 (see Example 8.12), we have models which are regular, not in Nevai class with nonempty a.c. spectrum. □

Example 1.5. The CN (for Cesàro–Nevai) class was introduced by Golinskii-Khrushchev [46] for OPUC and it has an OPRL analog. For OPRL, it says

\frac{1}{n} [ ∑_{j=1}^{n} |a_{j} − 1| + |b_{j}|] → 0  (1.25)

Example 1.4 is in this class and is regular, but it is not true that every element of CN for OPRL is regular; for example, if a_{n} ≡ 1 and each b_{j} is 0 or 1 so (1.25) holds but with arbitrarily long strings of only 0’s and also of only 1’s, e.g., b_{j} = 1 if n^{2} ≤ j ≤ n^{2} + n and b_{j} = 0 otherwise. Then σ(J) = [−2, 3] with C(σ(J)) = \frac{5}{4}, but (a_{1} … a_{n})^{1/n} → 1. (To see that σ(J) = [−2, 3], let J_{+}, J_{−} be the matrices with a_{n} ≡ 1 and b_{n} ≡ 1 (for J_{+}) and b_{n} ≡ 0 (for J_{−}). Then −2 ≤ J_{−} ≤ J ≤ J_{+} ≤ 3, so σ(J) ⊂ [−2, 3]. On the other hand, because of the long strings, a variational argument shows σ(J_{+}) ∪ σ(J_{−}) ⊂ σ(J).) However, for OPUC, the subset of CN class with sup_{n}|α_{n}| < 1 consists of only regular measures. For by Theorem 4.3.17 of [98], σ(dμ) = ∂𝔻 with capacity 1. On the other hand, if A = sup_{n}|α_{n}| and

L(A) = −\frac{log(1 − A)}{2A}

then (since log|1 − x| is convex)

− log ρ_{j} ≤ L(A)|α_{j}|^{2}

so

\frac{1}{n} ∑_{j=0}^{n−1} log ρ_{j} ≤ L(A) \frac{1}{n} ∑_{j=0}^{n−1} |α_{j}|^{2} ≤ L(A) \frac{1}{n} ∑_{j=0}^{n−1} |α_{j}|

goes to zero so (ρ_{0} … ρ_{n−1})^{1/n} → 1. It is easy to see that if sup_{n}|α_{n}| < 1 is dropped, regularity can be lost. □

Example 1.6. Let a_{n} ≡ \frac{1}{2} and let b_{n} = ±1 chosen as identically distributed random variables. As above, all these random J’s have −2 ≤ J ≤ 2, so supp(dμ) ⊂ [−2, 2]. Since there will be, with probability 1, long stretches of b_{n} ≡ 1 or b_{n} ≡ −1, it is easy to see supp(dμ) ⊃ ([−1, 1] + 1) ∪ ([−1, 1] − 1) = [−2, 2]. Thus, a typical random dμ has support [−2, 2] with capacity 1, but obviously lim(a_{1} … a_{n})^{1/n} = \frac{1}{2}. This shows there are measures which are not regular. By Example 1.3, random slowly decaying Jacobi matrices are regular, so neither randomness nor pure point measures necessarily destroy regularity. We return to pure point measures in Theorem 5.5 and Corollary 5.6. □

Section 8 has many more examples of regular measures. Regularity is important because of its connections to zero distributions and to root asymptotics. Let dν_{n} denote the zero distribution for X_{n}(z) defined by (1.3). Then

Theorem 1.7. Let dμ be a measure on ℝ with σ_{ess}(dμ) = E compact and C(E) > 0. If μ is regular, then dν_{n} converges weakly to dρ_{E}, the equilibrium measure for E.

Remarks. 1. For E = [0, 1], ideas close to this occur in Erdös–Turán [34]. The full result is in Stahl–Totik [105] who prove a stronger result. Rather than E ⊂ ℝ, they need that the unbounded component, Ω, of ℂ \ E is dense in ℂ. We will prove Theorem 1.7 in Section 2.

2. The result is false for measures on ∂𝔻. Indeed, it fails for dμ = \frac{dθ}{2π}, that is, α_{n} ≡ 0 and Φ_{n}(z) = z^{n}. However, there is a result for paraorthogonal polynomials and for the balayage of dν_{n}. Theorem 1.7 is true if supp(dμ) ⊂ ∂𝔻 but is not all of ∂𝔻. We will discuss this further in Section 3 where we also prove a version of Theorem 1.7 for OPUC.

For later purposes, we note

Proposition 1.8. Let dμ be a measure on ℝ with σ_{ess}(dμ) = E compact. Then any limit of dν_{n} is supported on E.

Proof. It is known (see [98, Sect. 1.2]) that if (a, b) ∩ supp(dμ) = ∅, then P_{n}(x) has at most one zero in (a, b). It follows that if e is an isolated point of supp(dμ), then (e − δ, e + δ) has at most three zeros for δ small (with more argument, one can get two). Thus, points not in E have neighborhoods, N, with ν_{n}(N) ≤ \frac{3}{n}. □

Stahl–Totik [105] also have the following almost converse (their Sect. 2.2)—for E ⊂ ℝ, we prove a slightly stronger result; see Theorem 2.5.

Theorem 1.9. Let dμ be a measure on ℝ with σ_{ess}(dμ) = E compact and C(E) > 0. Suppose that dν_{n} → dρ_{E}, the equilibrium measure. Then either dμ is regular or there exists a Borel set, X, with dμ(ℝ \ X) = 0 and C(X) = 0.

Remarks. 1. As an example where such an X exists even though C(supp(μ)) > 0, consider a μ which is dense pure point on [−2, 2].

2. In Section 8, we will see explicit examples where dν_{n} → dρ_{E} but dμ is not regular.

The other connection is to root asymptotics of the OPs. Recall the Green’s function, G_{E}(z), is defined by (A.40); it vanishes q.e. (quasieverywhere, defined in Appendix A) on E, is harmonic on Ω, and asymptotic to log|z| − log C(E) + o(1) as |z| → ∞. The main theorem on root asymptotics is:

Theorem 1.10. Let E ⊂ ℂ be compact and let μ be a measure of compact support with σ_{ess}(μ) = E. Then the following are equivalent:

(i) μ is regular, that is, lim_{n→∞} ∥X_{n}∥_{L^{2}(dμ)}^{1/n} = C(E). (ii) For all z in ℂ, uniformly on compacts,

lim sup|x_{n}(z)|^{1/n} ≤ e^{G_{E}(z)}  (1.26)

(iii) For q.e. z in ∂Ω (with Ω the unbounded connected component of ℂ \ E), we have

lim sup|x_{n}(z)|^{1/n} ≤ 1  (1.27)

Moreover, if (i)–(iii) hold, then (here cvh = closed convex hull) (iv) For every z ∈ ℂ \ cvh(supp(dμ)), we have

lim_{n→∞} |x_{n}(z)|^{1/n} = e^{G_{E}(z)}  (1.28)

(v) For q.e. z ∈ ∂Ω,

lim sup_{n→∞} |x_{n}(z)|^{1/n} = 1  (1.29)

(vi) For any sequence Q_{n}(z) of polynomials of degree n, for all z ∈ ℂ,

lim sup|\frac{Q_{n}(z)}{∥Q_{n}∥_{L^{2}(dμ)}} |^{1/n} ≤ e^{G_{E}(z)}  (1.30)

Remark. It is easy to see that (iv), (v) or (vi) are equivalent to (i)–(iii).

For E ⊂ ℝ, we will prove this in Section 2. For E ⊂ ∂𝔻, we prove it in Section 3.

The original result asserting cases where regularity holds was proven in 1940!

Theorem 1.11 (Erdös–Turán [34]). Let dμ be supported on [−2, 2] and suppose

dμ(x) = w(x) dx + dμ_{s}  (1.31)

with dμ_{s} singular. Suppose w(x) > 0 for a.e. x in [−2, 2]. Then μ is regular.

Remarks. 1. Erdös–Turán [34] worked on [−1, 1] and had dμ_{s} = 0.

2. We now have a stronger result than this—namely, Rakhmanov’s theorem (see [99, Ch. 9]). If w(x) > 0 for a.e. x, one knows a_{n} → 1 (and b_{n} → 0) much more than (a_{1} … a_{n})^{1/n} → 1 (equivalently, we have ratio asymptotics on the p’s and not just root asymptotics). Regularity is a “poor man’s” Rakhmanov’s theorem. But unlike Rakhmanov’s theorem which is only known for a few other E’s (see [29, 90] and the discussion in Section 8), this weaker version holds very generally.

3. In this case, dρ_{E} is equivalent to dx, so (1.31) and W (x) > 0 for a.e. x is equivalent to saying that ρ_{E} is μ-a.c.

In Section 4, we will prove the following vast generalization of the Erdös–Turán result:

Theorem 1.12 (Widom [122]). Let μ be a measure on ℝ with compact support and E = σ_{ess}(dμ) and C(E) > 0. Suppose dρ_{E} is the equilibrium measure for E and

dμ = w(x) dρ_{E}(x) + dμ_{s}  (1.32)

where dμ_{s} is dρ_{E}-singular. Suppose w(x) > 0 for dρ_{E}-a.e. x. Then μ is regular.

Remarks. 1. As above, (1.32) + w(x) > 0 is equivalent to saying that dρ_{E} is absolutely continuous with respect to dμ.

2. Widom’s result is much more general than what we have in this theorem. His E is a general compact set in ℂ. His polynomials are defined by general families of minimum conditions, for example, L^{p} minimizers. Most importantly, he has a general family of support conditions that, as he notes in a one-sentence remark, include the case where dρ_{E} is a.c. with respect to dμ. Because of its spectral theory connection, we have focused on the L^{2} minimizers, although it is not hard to accommodate more general ones. We focus on the w > 0 case because if one goes beyond that, it is better to look at conditions that depend on weights and not just supports of the measure as Widom (and Ullman [114]) do (see Theorem 1.13 below).

3. In Section 4, we will give a proof of this theorem due to Van Assche [120] who mentions Widom’s paper but says it is not clear his hypotheses apply despite an explicit (albeit terse) aside in Widom’s paper. Stahl–Totik [105] state Theorem 1.12 explicitly. They seem to be unaware of Van Assche’s paper or Widom’s aside.

It was Geronimus [43] who seems to have first noted that there are non-a.c. measures which are regular (and later Widom [122] and Ullman [114]). Of course, with the discovery of Nevai class measures which are not a.c. [95, 32, 111, 78, 79, 80, 97, 67], there are many examples, but given a measure, one would like to know effective criteria. Stahl–Totik [105, Ch. 4] have many, of which we single out:

Theorem 1.13 (Stahl–Totik [105]). Let E be a finite union of disjoint closed intervals in ℝ. Suppose μ is a measure on ℝ with σ_{ess}(dμ) = E, and for any η > 0 (|·| is Lebesgue measure),

lim_{m→∞} |\{x | μ([x − \frac{1}{m} , x + \frac{1}{m}]) ≤ e^{−mη}\}| = 0  (1.33)

Then μ is regular.

Theorem 1.14 (Stahl–Totik [105]). Let E be a finite union of disjoint closed intervals in ℝ. Suppose μ is a measure on ℝ and that μ is regular. Then for any η > 0,

lim_{m→∞} C(\{x ∈ E | μ([x − \frac{1}{m} , x + \frac{1}{m}]) ≤ e^{−mη}\}) = 0  (1.34)

Remarks. 1. We will prove Theorem 1.13 in Section 5.

2. Stahl–Totik [105] state these results for E = [−1, 1], but it is easy to accommodate finite unions of disjoint closed intervals; see Corollary 6.6.

In Section 6, we turn to structural results (all due to Stahl–Totik [105]) connected to inheritance of regularity when measures have a relation, for example, when restrictions of regular measures are regular.

Section 7 discusses relations of potential theory and ergodic Jacobi matrices. This theory concerns OPRL (or OPUC) whose recursion coefficients are samples of an ergodic process—as examples, totally random or almost periodic cases. In that case, various ergodic theorems guarantee the existence of lim(a_{1} … a_{n})^{1/n}, of dν_{∞} ≡ lim dν_{n}, and of a natural Lyapunov exponent, γ(z), which off of supp(dμ) is lim|p_{n}(z; dμ)|^{1/n} and is subharmonic on ℂ. In that section, we will prove some of the few new results of this paper:

Theorem 1.15. Let dμ_{ω} be the measures associated to an ergodic family of OPRL, dν_{∞} and γ its density of states and Lyapunov exponent. Let E = supp(dν_{∞}). Then the following are equivalent:

(a)  γ(x) = 0 for dρ_{E}-a.e. x
(b)  lim_{n→∞} (a_{1}(ω) … a_{n}(ω))^{1/n} = C(E)  (1.35)

for a.e. ω. Moreover, if (a) and (b) hold, then

dν_{∞}(x) = dρ_{E}(x)  (1.36)

with dρ_{E} the equilibrium measure for E. Conversely, if (1.36) holds, either (a) and (b) hold, or else for a.e. ω, dμ_{ω} is supported on a set of capacity zero.

Remarks. 1. We will prove that for a.e. ω,

lim_{n→∞} (a_{1}(ω) … a_{n}(ω))^{1/n} = C(E) exp(− ∫ γ(x) dρ_{E}(x)) (1.37)

2. In Section 8, we will see examples where dν_{∞} = dρ_{E} but (1.35) fails. Of course, (a) also fails.

The following is an ultimate version of what is sometimes called the Pastur–Ishii theorem (see Section 7).

Theorem 1.16. Let dμ_{ω} be a family of measures associated to an ergodic family of OPRL and let γ be its Lyapunov exponent. Let S ⊂ ℝ be the Borel set of x ∈ ℝ with γ(x) > 0. Then for a.e. ω, there exists Q_{ω} of capacity zero so dμ_{ω}(S \ Q_{ω}) = 0. In particular, dμ_{ω} ↾ S is of local Hausdorff dimension zero.

We should explain what is really new in this theorem. It has been known since Pastur [86] and Ishii [53] that for ergodic Schrödinger operators, the spectral measures are supported on the eigenvalues union the bad set where Lyapunov behavior fails (this bad set actually occurs, e.g., [12, 60]). The classic result is that the bad set has Lebesgue measure zero. The new result here (elementary given a potential theoretic point of view!) is that the bad set has capacity zero.

Section 8 describes examples, open questions, and conjectures. Section 9 has some remarks on the possible extensions of these ideas to continuum Schrödinger operators. Appendix A is a primer of potential theory and Appendix B proves Theorem 1.1 on Chebyshev polynomials.

It is a pleasure to thank Jonathan Breuer, Jacob Christiansen, David Damanik, Svetlana Jitomirskaya, Yoram Last, Christian Remling, Vilmos Totik, and Maxim Zinchenko for useful discussions. I would also like to thank Ehud de Shalit and Yoram Last for the hospitality of the Einstein Institute of Mathematics of the Hebrew University where part of this paper was written.

2. Regular Measures for OPRL

In this section, our main goal is to prove Theorems 1.7 and 1.10 for OPRL. The key will be a series of arguments familiar to spectral theorists as the Thouless formula, albeit in a different (nonergodic) guise. The key will be an analog of positivity of the Lyapunov exponent off the spectrum.

Lemma 2.1. (a) Let J be a bounded Jacobi matrix and let H be the convex hull of the spectrum of J. For any φ ∈ L^{2}(ℝ, dμ),

|⟨φ, (J − z)φ⟩| ≥ dist(z, H)∥φ∥^{2}  (2.1)

(b) The Jacobi parameters a_{n} obey (recall a_{n} > 0)

a_{n} ≤ \frac{1}{2} diam(H)  (2.2)

Proof. In a spectral representation, J is multiplication by x. If d = dist(z, H), there is ω ∈ ∂𝔻 with Re[(x − z)ω] ≥ d for all x ∈ H. Thus Re(⟨φ, (J − z)φ⟩ω) ≥ d∥φ∥^{2}, which yields (2.1).

Let D = \frac{1}{2} diam(H) and c = center of H so H = [c−D, c+D]. Then

a_{n} = ∫ xp_{n}p_{n−1} dμ

= ∫ (x − c)p_{n}p_{n−1} dμ

≤ sup_{σ(J)} |x − c| = D

proving (2.2). □

The following is related to the proof of Theorem 4.3.15 in [98]:

Proposition 2.2. Let dμ be a measure on ℝ of compact support, p_{n}(x, dμ) the normalized OPRL, and H the convex hull of the support of dμ. For z ∉ H, let d(z) = dist(z, H). Let D = \frac{1}{2} diam(H). Then for such z,

|p_{n}(z, dμ)|^{2} ≥ (\frac{d}{D})^{2}(1 + (\frac{d}{D})^{2})^{n−1}  (2.3)

In particular, p_{n}(z) ≠ 0 for all n and

lim inf|p_{n}(z, dμ)|^{1/n} ≥ (1 + (\frac{d}{D})^{2})^{1/2} > 1  (2.4)

Remark. Of course, it is well known that p_{n} has all its zeros on H.

Proof. Let φ_{n}(x) be the function

φ_{n}(x) = ∑_{j=0}^{n} p_{n}(z)p_{n}(x)  (2.5)

which has components φ_{n} = ⟨p_{0}(z), …, p_{n}(z), 0, 0, … ⟩ in p_{n}(x) basis. Then, by the recursion relation,

[(J − z)φ_{n}]_{j} = \begin{cases}0 & j ≠ n, n + 1 \\ −a_{n+1}p_{n+1}(z) & j = n \\ a_{n+1}p_{n}(z) & j = n + 1 \end{cases}  (2.6)

(a version of the CD formula!). Thus,

⟨φ_{n}, (J − z)φ_{n}⟩ = −a_{n+1}p_{n+1}(z) \overline{p_{n}(z)}  (2.7)

and (2.1) becomes

a_{n+1}|p_{n+1}(z)p_{n}(z)| ≥ d ∑_{j=0}^{n} |p_{j}(z)|^{2}  (2.8)

By (2.2),

|p_{n+1}(z)p_{n}(z)| ≥ \frac{d}{D} ∑_{j=0}^{n} |p_{j}(z)|^{2}  (2.9)

Next use 2|xy| ≤ αx^{2} + α^{−1}y^{2} for any α to see

|p_{n+1}(z)p_{n}(z)| ≤ \frac{1}{2} \frac{d}{D} |p_{n}(z)|^{2} + \frac{1}{2} \frac{D}{d} |p_{n+1}(z)|^{2}  (2.10)

which, with (2.9), implies

|p_{n+1}(z)|^{2} ≥ (\frac{d}{D})^{2} ∑_{j=0}^{n} |p_{j}(z)|^{2}  (2.11)

This implies that

∑_{j=0}^{n+1} |p_{j}(z)|^{2} ≥ [1 + (\frac{d}{D})^{2}] ∑_{j=0}^{n} |p_{j}(z)|^{2}  (2.12)

so, since p_{0}(z) = 1, we obtain

∑_{j=0}^{n} |p_{j}(z)|^{2} ≥ [1 + (\frac{d}{D})^{2}]^{n}  (2.13)

(2.13) plus (2.11) imply (2.3), and that implies (2.4). □

Remark. (2.4) is also related to Schnol’s theorem (see [92, 93] and [98, Lemma 4.3.13]) and to Combes–Thomas estimates [23, 1].

This yields the key estimate, given the following equality:

Theorem 2.3. Let dμ be a measure of compact support on ℝ with H the convex hull of supp(dμ). Let n(j) be a subsequence (i.e., n(1) < n(2) < n(3) < … in {0, 1, 2, … }) so that the zero counting measures dν_{n(j)} have a weak limit dν_{∞} and so that (a_{1} … a_{n(j)})^{1/n(j)} has a nonzero limit A. Then, for any z ∉ H,

lim_{j→∞} |p_{n(j)}(z)|^{1/n(j)} = A^{−1} exp(−Φ_{ν_{∞}}(z))  (2.14)

where Φ_{ν} is the potential of ν. In particular,

exp(−Φ_{ν_{∞}}(z)) > A  (2.15)

Proof. (1.4) says that

|p_{n(j)}(z)|^{1/n(j)}(a_{1} … a_{n(j)})^{1/n(j)} = exp(−Φ_{ν_{n(j)}}(z))  (2.16)

For z ∉ H, log|z − y|^{−1} is continuous on H so since ν_{n} and so ν_{∞} are supported on H (indeed, ν_{∞} is supported on σ_{ess}(dμ)), Φ_{ν_{n(j)}}(z) → Φ_{ν_{∞}}(z) and (2.16) implies (2.14). By (2.4), LHS of (2.14) > 1, which implies (2.15). □

Note: (2.15) implies that Φ_{ν_{∞}} is bounded above which, by arguments of Craig–Simon [25], implies ν_{∞}((−∞, E]) is log-Hölder continuous. This is a new result, although the fact for ergodic Jacobi matrices is due to Craig–Simon [25].

This yields an independent proof of Corollary 1.2 for OPRL, and more:

Theorem 2.4. Under the hypotheses of Theorem 2.3, if E = σ_{ess}(dμ), then A ≤ C(E), and if A = C(E), then dν_{∞} = dρ_{E}, the equilibrium measure for E. In particular, if μ is regular (i.e., lim(a_{1} … a_{n})^{1/n} = C(E)), then dν_{n} → dρ_{E} and (1.28) holds for z ∉ H.

Remark. Thus, we have proven Corollary 1.2 again, Theorem 1.7, and one part of Theorem 1.10.

Proof. By (2.15) for z ∉ H,

Φ_{ν_{∞}}(z) ≤ log(A^{−1})  (2.17)

By lower semicontinuity, this also holds on H. Integrating dν_{∞} using (A.6), we obtain

E(ν_{∞}) ≤ log(A^{−1})  (2.18)

Since inf_{ν}(E(ν)) = log(C(E)^{−1}), we obtain log(C(E)^{−1}) ≤ log(A^{−1}), that is, A ≤ C(E). By uniqueness of minimizers, if A = C(E), dν_{∞} = dρ_{E} and regularity implies dρ_{E} is the only limit point. By compactness, dν_{n} → dρ_{E}, and then, by (2.14), we obtain (1.28) for z ∉ H. □

Completion of the Proof of Theorem 1.10 for OPRL. We proved above that (i) ⇒ (iv) and so, by the submean property of |f(z)|^{1/n} (alternatively, by the subharmonicity of log|f(z)|) for analytic functions, we get (ii) also on H and thus have (i) ⇒ (ii) in full. (ii) ⇒ (iii) is trivial. \underline{(iii) ⇒ (i).} Pick a subsequence n(j) so that (a_{1} … a_{n(j)})^{1/n(j)} → lim inf(a_{1} … a_{n(j)})^{1/n} = A, and so ν_{n(j)} → ν_{∞}. By (1.4) and Theorem A.7, (iii) implies for q.e. x in E we have that Φ_{ν_{∞}}(x) ≥ log(A^{−1}). Thus, since dρ_{E} gives zero weight to zero capacity sets (see Proposition A.6) and (A.2),

log(A^{−1}) ≤ ∫ log Φ_{ν_{∞}}(x) dρ_{E}(x)
= ∫ Φ_{ρ_{E}}(x) dν_{∞}(x)
≤ log(C(E)^{−1})  (2.19)

by (A.23). Thus A^{−1} ≤ C(E)^{−1}, so C(E) ≤ A. By Theorem 2.4 (or Corollary 1.2), we see A = C(E), that is, μ is regular.

The reader may be concerned about this argument if A = 0. But in that case, Theorem A.7 and (1.27) imply that q.e. on E, Φ_{ν}(x) = ∞ which is inconsistent with Theorem A.11 (or with Corollary A.5). Thus (1.27) implies A > 0. \underline{(i) ⇒ (v).} This is immediate from Theorem 2.4, the fact that equality holds q.e. in (A.15) and in (A.23). \underline{(ii) ⇒ (vi).} Without loss, we can redefine Q_{n} so ∥Q_{n}∥_{L^{2}(dμ)} = 1. Then

Q_{n}(z) = ∑_{j=0}^{n} c_{j,n}p_{j}(z, dμ)  (2.20)

where

∑_{j=0}^{n} |c_{j,n}|^{2} = 1 ⇒ |c_{j,n}| ≤ 1  (2.21)

Thus

|Q_{n}(z)| ≤ n sup_{0≤j≤n} |p_{j}(z, dμ)|  (2.22)

and (1.26) ⇒ (1.30). □

This completes the proof of Theorem 1.10 for OPRL and our presentation of the key properties of regular measures for OPRL. We turn to relations between the support of dμ and regularity of the density of zeros that will include Theorem 1.9.

Theorem 2.5. Let dμ be a measure of compact support, E, with Jacobi parameters, {a_{n}, b_{n}}_{n=1}^{∞}. Let n(j) be a subsequence so that dν_{n(j)} has a limit, dρ_{E}, the equilibrium measure for E. Then either (a)

lim_{j→∞} (a_{1} … a_{n(j)})^{1/n(j)} = C(E)  (2.23)

or

(b) μ is carried by a set of capacity zero, that is, there is X ⊂ E of capacity zero so μ(ℝ \ X) = 0.

Proof. Let A be a limit point of (a_{1} … a_{n(j)})^{1/n}. If A = 0, interpret A^{−1} as ∞. By (2.16) and the upper envelope theorem (Theorem A.7), we see for some subsubsequence ñ(j),

lim_{j→∞} |p_{ñ(j)}(x)|^{1/ñ(j)} = A^{−1} exp(−Φ_{ρ_{E}}(x))  (2.24)

for q.e. x. By Theorem A.10, Φ_{ρ_{E}}(x) = log(C(E)^{−1}) for q.e. x. So for q.e. x ∈ E,

lim_{j→∞} |p_{ñ(j)}(x)|^{1/ñ(j)} = A^{−1}C(E)  (2.25)

On the other hand (see (4.14) below), for μ-a.e. x, we have

|p_{n}(x)| ≤ C(x)(n + 1)  (2.26)

so for such x,

lim sup|p_{n}(x)|^{1/n} ≤ 1  (2.27)

If A < C(E), then \frac{C(E)}{A} > 1, so (2.27) can only hold on the set of capacity zero where (2.25) fails, that is, either A = C(E) (since it is always true that A ≤ C(E)) or μ is carried by a set of capacity zero. □

Before leaving the subject of OPRL, we want to say something about nonregular situations:

Theorem 2.6. Let μ be a fixed measure of compact support on ℝ.

(a) The set of limit points of (a_{1} … a_{n})^{1/n} is always a closed interval.

(b) The set of limits of zero counting measures dν_{n} is always a closed compact set.

Remarks. 1. As quoted in [119], where the first proof of (a) appeared,

(a) is a theorem of Freud and Ziegler.

2. Part (b) was conjectured in Ullman [117] and is proven in Stahl– Totik [105] (see Theorem 2.1.4 of [105]).

3. Stahl–Totik [105] also prove (their Theorem 2.2.1) that so long as no carrier of μ has capacity zero, the existence of a limit for dν_{n(j)} implies the existence of a limit for (a_{1} … a_{n(j)})^{1/n(j)}. However, as we will see (Example 2.7), the converse is false.

Proof. We sketch the proof of (a); the proof of (b) can be found in [105] and is similar in spirit. The set of limit points is a closed subset of [0, C(E)]. If it is not connected, we can find limit points A < B and c ∈ (A, B) which is not a limit point.

Thus, there are N and ε so for n > N,

Γ_{n} ≡ (a_{1} … a_{n})^{1/n} ∉ (c − ε, c + ε)  (2.28)

Suppose Γ_{n} < c − ε and let D = \frac{1}{2} diam(cvh(supp(μ))) ≥ a_{n} by (2.2). Then

Γ_{n+1} = a_{n+1}^{1/n+1}Γ_{n}^{n/n+1} ≤ D^{1/n+1}(c − ε)^{n/n+1}  (2.29)

Since RHS of (2.29) converges to c − ε, we can find N_{1} so

n ≥ N_{1} ⇒ RHS of (2.29) ≤ c  (2.30)

Thus n ≥ N, n ≥ N_{1}, and Γ_{n} ≤ c − ε implies Γ_{n+1} ≤ c − ε (by (2.28)). It follows that Γ_{n} cannot have both A and B as limit points.

This contradiction proves the set of limit points is an interval. □

Example 2.7. This example shows that (a_{1} … a_{n})^{1/n} may have a limit (necessarily strictly less than C(E)) but dν_{n} does not. A more complicated example appears as Example 2.2.7 in [105]. Let a_{n} ≡ 1 (so (a_{1} … a_{n})^{1/n} → 1) and

b_{n} = \begin{cases}1 & N_{2ℓ} ≤ n < N_{2ℓ+1} \\ −1 & N_{2ℓ+1} ≤ n < N_{2ℓ+2} \end{cases}

where N_{ℓ} = 2^{3^{ℓ}}. It is easy to see by looking at traces of powers of the cutoff Jacobi matrix that dν_{N_{2ℓ}^{2}} → dρ_{[−1,3]} and dν_{N_{2ℓ+1}^{2}} → dρ_{[−3,1]}. □

There is another result about the set of limit points that should be mentioned in connection with work of Ullman and collaborators. Define c_{μ} to be inf of the capacity of Borel sets, S, which are carriers of μ in the sense that μ(ℝ \ S) = 0. For example, if μ is a dense pure point measure with support E = [−2, 2], μ is supported on a countable set, so c_{μ} = 0 even though C(E) = 1. Then, in general, Ullman shows that any limit point of (a_{1} … a_{n})^{1/n} lies in [c_{μ}, C(supp(dμ))], and Wyneken [123] proved that given any μ and any [A, B] ⊂ [c_{μ}, C(supp(dμ))], there is η mutually equivalent to μ so the set of limit points of Γ_{n}(η) is [A, B] (see also Theorem 5.4 below).

In particular, these results show that if c_{μ} = C(supp(dμ)), then μ is regular—a theorem of Ullman [114], although Widom [122] essentially had the same theorem (this oversimplifies the relation between Widom [122] and Ullman [114]; see [105, Ch. 4]). We have not discussed this result in detail because the Stahl–Totik criterion of Theorem 1.13 essentially subsumes these earlier works (at least for E a finite union of closed intervals) and we will prove that in Section 5.

3. Regular Measures for OPUC

In this section, we will prove Theorem 1.10 for OPUC and an analog of Theorem 1.7. Here one issue will be that if E = ∂𝔻, the zero density may not converge to a measure on ∂𝔻. The key step concerns Proposition 2.2, which essentially depended on the CD formula which is only known for OPRL and OPUC, and where the OPUC version is not obviously relevant. Instead, we will see, using operator theoretic methods [101], that there is a kind of “half CD formula” that suffices. We begin with an analog of Lemma 2.1:

Lemma 3.1. (a) Let μ be a measure of compact support on ℂ and H the convex hull of the support of μ. Let M_{z} be multiplication by z on L^{2}(ℂ, dμ). Then for any z_{0} ∈ ℂ and φ ∈ L^{2}(ℂ, dμ), we have

|⟨φ, (M_{z} − z_{0})φ⟩| ≥ dist(z_{0}, H)∥φ∥^{2}  (3.1)

(b) Let D be defined by

D = min_{w} [ max_{z∈H} |z − w|]  (3.2)

(which lies between \frac{1}{2} diam(H) and diam(H)). Then

∥X_{n+1}∥_{L^{2}(dμ)} ≤ D∥X_{n}∥_{L^{2}(dμ)}  (3.3)

Proof. (a) Let ω ∈ ∂𝔻. Then

|⟨φ, (M_{z} − z_{0})φ⟩| ≥ ∫ Re((z − z_{0})ω̄)|φ(z)|^{2} dμ(z)

≥ min_{z∈H} Re((z − z_{0})ω̄)∥φ∥^{2}

Maximizing over ω yields (3.1).

(b) Since (z − w)X_{n} is a monic polynomial of degree n + 1,

∥X_{n+1}∥ ≤ ∥(z − w)X_{n}∥ ≤ max_{z∈H} |z − w|∥X_{n}∥

Minimizing over w yields (3.3). □

To get the analog of (2.7), we need

Proposition 3.2. Let dμ be a measure of compact support on ℂ and let M_{z} be multiplication by z on L^{2}(ℂ, dμ). Let K be the orthogonal projection in L^{2}(ℂ, dμ) onto the n+1-dimensional subspace polynomials of degree at most n. Then

[M_{z}, K]K = \frac{∥X_{n+1}∥}{∥X_{n}∥} [⟨x_{n}, · ⟩x_{n+1}]  (3.4)

Remark. This is essentially “half” the CD formula; operator theoretic approaches to the CD formula are discussed in [101].

Proof. For any φ,

[M_{z}, K]Kφ = (1 − K)z(Kφ)  (3.5)

This clearly vanishes if Kφ = 0 or if φ ∈ ranK_{n−1}. Thus, it is a rank one operator. Moreover, since (1 − K)zX_{n} = X_{n+1}, we see

[M_{z}, K]KX_{n} = X_{n+1}

Since X_{n+1} = ∥X_{n+1}∥x_{n+1} and X_{n} = ∥X_{n}∥x_{n}, we see that (3.4) holds. □ Proposition 3.3. Let dμ be a measure of compact support on ℂ, x_{n}(z; dμ) the normalized OPs, and H the convex hull of the support of dμ. For z_{0} ∉ H, let d(z_{0}) = dist(z_{0}, H) and let D be given by (3.2). Then for such z_{0},

|x_{n}(z_{0}; dμ)|^{2} ≥ (\frac{d}{D})^{2}(1 + (\frac{d}{D})^{2})^{n−1}  (3.6)

In particular, x_{n}(z_{0}) ≠ 0 for all n and

lim inf|x_{n}(z_{0}; dμ)|^{1/n} ≥ (1 + (\frac{d}{D})^{2})^{1/2} > 1  (3.7)

Remark. Again, it is well known (a theorem of Fejér) that zeros of x_{n} lie in H.

Proof. Define

φ_{n}(w) = ∑_{j=0}^{n} \overline{x_{j}(z_{0})} x_{j}(w)  (3.8)

We claim

⟨φ_{n}, (M_{z} − z_{0})φ_{n}⟩ = −\frac{∥X_{n+1}∥}{∥X_{n}∥} \overline{x_{n}(z_{0})} x_{n+1}(z_{0})  (3.9)

This is precisely an analog of (2.7). Given this and Lemma 3.1, the proof is identical to that of Proposition 2.2.

To prove (3.9), we note the integral kernel of K_{n} is

K_{n}(s, t) = ∑_{j=0}^{n} p_{j}(s) \overline{p_{j}(t)}  (3.10)

and that (3.4) says

∫ (s − w)K_{n}(s, w)K_{n}(w, t) dμ(w) = \frac{∥X_{n+1}∥}{∥X_{n}∥} x_{n+1}(s) \overline{x_{n}(z_{0})} (3.11)

(3.11) originally holds for a.e. s, t in supp(dμ), but since both sides are polynomials in s and t̄, for all s, t. Setting s = t = z_{0}, (3.11) is just (3.9). □

Now we want to specialize to OPUC. The zeros in that case lie in 𝔻. One defines the balayage of the zeros measure, dν_{n}, on ∂𝔻 by

P(dν) = F (θ) \frac{dθ}{2π}  (3.12)

where

F (θ) = ∫ \frac{1 − |z|^{2}}{|e^{iθ} − z|^{2}} dν_{n}(z)  (3.13)

It is the unique measure on ∂𝔻 with

∫ z^{k}P(dν_{n}) = ∫ z^{k} dν_{n}(z)  (3.14)

for k ≥ 0 (see [98, Prop. 8.2.2]).

Since |z| > 1 ≥ |w| implies

log|z − w|^{−1} = − log|z| + Re( ∑_{j=1}^{∞} \frac{1}{j}(\frac{w}{z})^{j})  (3.15)

by (3.14), we have

|z| > 1 ⇒ Φ_{ν_{n}}(z) = Φ_{P(dν_{n})}(z)  (3.16)

If dν_{n} → dν_{∞}, then P(dν_{n}) → P(dν_{∞}), and this equals dν_{∞} if dν_{∞} is a measure on ∂𝔻. If supp(dμ) ⫋ ∂𝔻, then it is known that the bulk of the zeros goes to ∂𝔻 (Widom’s zero theorem; see [98, Thm. 8.1.8]), so dν_{∞} is a measure on ∂𝔻. It is also known (see [98, Thm. 8.2.7]) that the zero counting measures for the paraorthogonal polynomials (POPUC) have the same weak limits as P(dν_{n}). The analogs of Theorems 2.3 and 2.4 are thus:

Theorem 3.4. Let dμ be a measure on ∂𝔻, the unit circle. Let n(j) be a subsequence with n(1) < n(2) < … so that (ρ_{1} … ρ_{n(j)})^{1/n(j)} has a nonzero limit A and so that there is a measure dν_{∞} on ∂𝔻 which is the weak limit of P(dν_{n(j)}) (equivalently, of dν_{n(j)} if supp(dμ) ≠ ∂𝔻; equivalently, of the zero counting measures of POPUC). Then for any |z| > 1 or z ∉ ∂𝔻 \ supp(dμ),

lim_{j→∞} |φ_{n(j)}(z)|^{1/n(j)} = A^{−1} exp(−Φ_{ν_{∞}}(z))  (3.17)

In particular,

exp(−Φ_{ν_{∞}}(z)) ≥ A  (3.18)

It follows that if E = σ_{ess}(dμ), then A ≤ C(E), and if μ is regular (i.e., (ρ_{1} … ρ_{n})^{1/n} → C(E)), then every limit point of P(dν_{n(j)}) is the equilibrium measure dν_{E}. So P(dν_{n}) → dν_{E} (and if E ≠ ∂𝔻, dν_{n} → dν_{E}).

Proof. Given the above discussion and results, this is identical to the proofs of Theorems 2.3 and 2.4. □

By mimicking the proof we give for Theorem 1.10 for OPRL, we obtain the same result for OPUC.

4. Van Assche’s Proof of Widom’s Theorem

In this section, we will prove Theorem 1.12 using part of Van Assche’s approach [120]. The basic idea is simple: By a combination of Chebyshev’s inequality and the Borel–Cantelli lemma, if ∥P_{n(j)}∥_{L^{2}(dμ)}^{1/n(j)} → A, then for dμ-a.e. x, we have lim sup_{j→∞}|P_{n(j)}(x)|^{1/n(j)} ≤ A. By using some potential theory, we will find that the density of zeros measure, dν, supported on E obeys for q.e. x, Φ_{ν}(x) ≥ log(A^{−1}) a.e. dμ. Since dρ_{E} is a.c. with respect to dμ, this will imply ∫ Φ_{ρ_{E}}(x) dν(x) ≥ log(A^{−1}). But by potential theory again, Φ_{ρ_{E}}(x) ≤ log(C(E)^{−1}), so we will have A^{−1} ≤ C(E)^{−1}, that is, C(E) ≤ A.

Lemma 4.1. Let dμ be a probability measure on a measure space X. Let f_{n(j)} be a sequence of functions indexed by integers 1 ≤ n(1) < n(2) < . . .. Suppose for some 1 ≤ p < ∞,

lim sup_{j→∞} ∥f_{n(j)}∥_{L^{p}}^{1/n(j)} = A  (4.1)

Then for dμ-a.e. x,

lim sup|f_{n(j)}(x)|^{1/n(j)} ≤ A  (4.2)

Proof. Fix B > A. Then

μ(S_{j}(B)) ≡ μ(\{x | |f_{n(j)}(x)| > B^{n(j)}\}) ≤ \frac{∥f_{n(j)}∥_{L^{p}}^{p}}{B^{n(j)p}}  (4.3)

By (4.1) and B > A, we see

∑_{j} μ(S_{j}(B)) < ∞

so for μ-a.e. x, there is J(x) with x ∉ S_{j} for all j > J(x). Thus, for μ-a.e. x,

lim sup|f_{n(j)}(x)|^{1/n(j)} ≤ B

Since B is arbitrary, we have (4.2). □

Proof of Theorem 1.12. Let A be a limit point of ∥P_{n(j)}∥_{L^{2}(dμ)}^{1/n(j)}. By passing to a subsequence, we can suppose the zero counting measure dν_{n(j)} has a limit dν_{∞} which, by Proposition 1.8, is supported on E.

By Lemma 4.1 for a.e. x(dμ),

lim sup|P_{n(j)}(x)|^{1/n(j)} ≤ A  (4.4)

By (1.4) for such x,

lim sup exp(−Φ_{ν_{n(j)}}(x)) ≤ A  (4.5)

By the upper envelope theorem (Theorem A.7) for q.e. x ∈ ℂ,

Φ_{ν_{∞}}(x) = lim inf Φ_{ν_{n(j)}}(x)  (4.6)

Thus, there exist sets S_{1} and S_{2} so that μ(S_{1}) = 0 and C(S_{2}) = 0, so that for x ∈ ℂ \ (S_{1} ∪ S_{2}),

Φ_{ν_{∞}}(x) ≥ log(A^{−1})  (4.7)

We can now repeat the argument that led to (2.19). By hypothesis, dρ_{E} is dμ-a.c. So ρ_{E}(S_{1}) = 0 and, of course, since E(ρ_{E}) < ∞, ρ_{E}(S_{2}) = 0. Thus, (4.7) holds a.e. dρ_{E}.

Therefore, by (A.2),

log(A^{−1}) ≤ ∫ Φ_{ν_{∞}}(x) dρ_{E}(x)
= ∫ Φ_{ρ_{E}}(x) dν_{∞}(x)
≤ log(C(E)^{−1})

by (A.23).

Thus, A^{−1} ≤ C(E)^{−1} or C(E) ≤ A. Thus, lim inf ∥P_{n}∥^{1/n} ≥ C(E). Since (see (1.15)), lim sup ∥P_{n}∥^{1/n} ≤ C(E), we have regularity. □

The above proof is basically a part of Van Assche’s argument [120] which can be simplified since he proves that dν_{∞} = dρ_{E} by a direct argument using similar ideas, and we can avoid that because of the general argument in Section 2.

This argument can also prove a related result—we will see examples of this phenomenon at the end of the next section.

Theorem 4.2. Suppose μ is a measure of compact support on ℝ so E ⊂ supp(dμ) for an essentially perfect compact set E with C(E) > 0. Suppose dρ_{E} is a.c. with respect to dμ, and for some n(1) < n(2) < …, we have

∥P_{n(j)}∥_{L^{2}(ℝ,dμ)}^{1/n(j)} → C(E)  (4.8)

for the monic P_{n}(x, dμ). Let dν_{n(j)} be the corresponding zero counting measure. Then dν_{n(j)} ⟶^{w} dρ_{E}.

Remarks. 1. We have in mind cases where E is a proper subset of supp(dμ). There will be many subsets with the same capacity, but there can only be one that has dρ_{E} a.e. with respect to dμ.

2. Since dμ ↾ E is regular (by Theorem 1.12) and ∥P_{n(j)}( · , μ)∥_{L^{2}(dμ)} ≥ ∥P_{n(j)}( · , μ ↾ E)∥_{L^{2}(dμ↾E)}, we see that

lim inf ∥P_{n}∥_{L^{2}(ℝ,dμ)}^{1/n} ≥ C(E)

so (4.8) is equivalent to a lim sup assumption.

Proof. Let dν_{∞} be a limit point of dν_{n(j)}. As in the proof of Theorem 1.12, there exist sets S_{1} with μ(S_{1}) = 0 and S_{2} with C(S_{2}) = 0, so for x ∈ ℂ \ (S_{1} ∪ S_{2}), we have

Φ_{ν_{∞}}(x) ≥ log(C(E)^{−1})  (4.9)

Since ρ_{E}(S_{1}) = 0 by the assumption and ρ_{E}(S_{2}) = 0 since C(E) > 0, (4.9) holds for ρ_{E}-a.e. x. Moreover, since

Φ_{ρ_{E}}(z) ≤ log(C(E)^{−1})  (4.10)

for all z,

log(C(E)^{−1}) ≤ ∫ Φ_{ν_{∞}}(x) dρ_{E}(x)
= ∫ Φ_{ρ_{E}}(x) dν_{∞}(x)
≤ log(C(E)^{−1})  (4.11)

Thus, using (4.10), we see

Φ_{ρ_{E}}(x) = log(C(E)^{−1})

for ν_{∞}-a.e. x. But Φ_{ρ_{E}}(z) < log(C(E)^{−1}) for all z ∉ E, so ν_{∞} is supported on E. By (4.11) and (4.9), Φ_{ν_{∞}}(x) = log(C(E)^{−1}) for dρ_{E}- a.e. x. By Theorem A.14, ν_{∞} = ρ_{E}. □

There is an alternate way to prove (4.4) without Lemma 4.1 that links it to ideas more familiar to spectral theorists. It is well known that for elliptic PDEs, there are polynomially bounded eigenfunctions for a.e. energy with respect to spectral measures. This is called the BGK expansion in [94] after Berezanskiĭ[14], Browder [20], Gårding [40], Gel’fand [41], and Kac [62]. The translation to OPRL is discussed in Last–Simon [75]. Since ∫ |p_{n}(x)|^{2} dμ = 1, we have

∑_{n=0}^{∞} (n + 1)^{−2} ∫ |p_{n}(x)|^{2} dμ < ∞  (4.12)

and thus, for dμ-a.e. x,

∑_{n=0}^{∞} (n + 1)^{−2}|p_{n}(x)|^{2} < ∞  (4.13)

so

|P_{n}(x)| ≤ C(x)(n + 1)∥P_{n}∥_{L^{2}}  (4.14)

which implies (4.4).

It is interesting to note that if E is such that it is regular and dρ_{E} is purely absolutely continuous on E = supp(dρ_{E}), one can use these ideas to provide an alternate proof (see Simon [100] for still another alternate proof in this case). For in that case, the measure associated to the second kind polynomials, q_{n}(x), also has a.c. weight w̃(x) > 0 for a.e. x in E, and thus

|q_{n}(x)| ≤ C(x)(n + 1)  (4.15)

which, by constancy of the Wronskian, implies

|p_{n}(x)|^{2} + |p_{n+1}(x)|^{2} ≥ C̃(x)(n + 1)^{−1}  (4.16)

If dν_{n(j)} → dν_{∞}, so does dν_{n(j)+1} (by interlacing of zeros), and thus, by (4.14) and (4.16), if lim(a_{1} … a_{n(j)})^{1/n(j)} → A, then

− log(A) + ∫ log(|x − y|^{−1}) dν_{∞}(y) = 0  (4.17)

for x ∈ E but with a set of Lebesgue measure zero and of capacity zero removed. By Theorem A.14, we conclude that A = C(E) and dν_{∞} = dρ_{E}.

Remark. We note that (4.17) holds a.e. on the a.c. spectrum and by the above arguments, a.e. on that spectrum, \frac{1}{n} log ∥T_{n}(x)∥ → 0, a deterministic analog of the Pastur–Ishii theorem.

5. The Stahl–Totik Criterion

In this section, we will present an exposition of Stahl–Totik’s proof [105] of their result, our Theorem 1.13. As a warmup, we prove

Theorem 5.1. Let dμ be a measure on ∂𝔻 obeying

inf_{θ_{0}} μ(\{e^{iθ} | |θ − θ_{0}| ≤ \frac{1}{m}\}) ≥ C_{δ}e^{−δm}  (5.1)

for all δ > 0. Then μ is regular.

Proof. We will use Bernstein’s inequality that for any polynomial, P_{n}, of degree n,

sup_{z∈𝔻}|P′_{n}(z)| ≤ n sup_{z∈∂𝔻}|P_{n}(z)|  (5.2)

Szegő’s simple half-page proof of this can be found, for example, in Theorem 2.2.5 of [98].

Applying this to the monic polynomials Φ_{n}(z; dμ), we see that if θ_{n} is chosen with |Φ_{n}(e^{iθ_{n}}; dμ)| = ∥Φ_{n}∥_{∂𝔻}, the sup norm, and |θ − θ_{n}| ≤ \frac{1}{2n}, then

|Φ_{n}(e^{iθ}; dμ)| ≥ \frac{1}{2} ∥Φ_{n}∥_{∂𝔻}  (5.3)

Thus, by (5.1) with m = 2n,

∥Φ_{n}( ·; dμ)∥_{L^{2}(dμ)}^{2} ≥ (\frac{1}{4} ∥Φ_{n}∥_{∂𝔻}^{2})C_{δ}e^{−2δn}  (5.4)

Since Φ_{n}( ·; dμ) is monic,

∫ Φ_{n}(e^{iθ}; dμ)e^{−inθ} \frac{dθ}{2π} = 1  (5.5)

so the sup norm obeys

∥Φ_{n}∥_{∂𝔻} ≥ 1  (5.6)

and so (5.4) implies

lim inf ∥Φ_{n}( ·; dμ)∥_{L^{2}}^{1/n} ≥ e^{−2δ}

Since δ is arbitrary, the lim inf is larger than or equal to 1. Since C(E) = 1, μ is regular. □

There are two issues with just using these ideas to prove Theorem 1.13. While (5.5) is special for ∂𝔻, its consequence, (5.6), is really only an expression of ∥T_{n}∥_{E} ≥ C(E)^{n} (see (B.8)), so it is not an issue.

However, (5.2) only holds because a circle has no ends. The analog for, say, [−1, 1] is Bernstein’s inequality

|p′(x)| ≤ \frac{n}{\sqrt{1 − x^{2}}} ∥p∥_{[−1,1]}  (5.7)

or (Markov’s inequality)

|p′(x)| ≤ n^{2}∥p∥_{[−1,1]}  (5.8)

Either one can be used to obtain a theorem like Theorem 5.1 on [−1, 1] but e^{−δm} needs to be replaced by e^{−δ\sqrt{m}}—interesting, but weaker than Theorem 1.13.

The other difficulty is that (5.1) is global, requiring a result uniform in θ_{0}, and (1.33) needs only a result for most θ_{0}. The problem with using bounds on derivatives is that they only get information on a single set of size O(\frac{1}{n}) at best. They get |p_{n}(x)| ≥ \frac{1}{2}∥p_{n}∥_{E} there, but that is overkill—we only need |p_{n}(x)| ≥ e^{−δ′n}∥p_{n}∥_{E}, and that actually holds on a set of size O(1)! The key will thus be a variant of the Remez inequality in the following form:

Proposition 5.2. Fix E a finite union of closed bounded intervals in ℝ. Then there is c(δ) > 0 with c(δ) → 0 as δ ↓ 0, so that for any F ⊂ E with |E \ F | < δ, we have

∥Q_{n}∥_{E} ≤ e^{c(δ)n}∥Q_{n}∥_{F}  (5.9)

for any polynomial, Q_{n}, of degree n.

Remarks. 1. This is a variant of an inequality of Remez [89]; see the proof for his precise result.

2. The relevance of Remez’s inequality to regularity appeared already in Erdös–Turán [34] and was the key to the proof in Freud [39] of the Erdös–Turán theorem, Theorem 1.11. Its use here is due to Stahl–Totik [105].

Proof. If E = I_{1} ∪ ⋯ ∪ I_{ℓ} disjoint intervals and |E \ F | ≤ δ, then |I_{j} \I_{j} ∩F | ≤ δ for all j, so it suffices to prove this result for each single interval and then, by scaling, for E = [−1, 1].

In that case, Remez’s inequality (due to Remez [89]; see Borwein– Erdélyi [15] for a proof and further discussion) says that if F ⊂ [−1, 1] and |[−1, 1] \ F | ≤ δ, then with T_{n} the classical first kind Chebyshev polynomials,

∥Q_{n}∥_{E} ≤ T_{n}(\frac{2 + δ}{2 − δ} )∥Q_{n}∥_{F}  (5.10)

(This can be proven by showing the worst case occurs when F = [−1, 1 − δ] and Q_{n}(x) = T_{n}(\frac{2x+δ}{2−δ}).)

Since

T_{n}(cosh(x)) = cosh(nx) ≤ e^{nx}  (5.11)

and cosh(ε) = 1 + \frac{ε^{2}}{2} + O(ε^{4}), we have

T_{n}(\frac{2 + δ}{2 − δ} ) ≤ exp(n[\sqrt{2δ} + O(δ^{3/2})])  (5.12)

so for E = [−1, 1], (5.9) holds with c(δ) = \sqrt{2δ} + O(δ^{3/2}). □

Lemma 5.3. If P_{n} is a real polynomial of degree n, and a > 0, S ≡ {λ ∈ ℝ | |P_{n}(x)| > a} is a union of most (n + 1) intervals.

Proof. ∂S is the finite set of points where P_{n}(x) = ±a. If all the zeros of P_{n} ± a are simple, these boundary points are distinct. Including ±∞ so each interval has two “endpoints,” these intervals have at most 2n+2 distinct endpoints (and exactly that number if all roots of P_{n} ±a are real). If some root of P_{n} ± a is double, two intervals can share an endpoint but that endpoint counts twice in the zeros. □

Proof of Theorem 1.13. By Proposition B.3, if P_{n}(x) = P_{n}(x; dμ), then

sup_{x∈E} |P_{n}(x)| ≥ c(E)^{n}  (5.13)

Fix δ_{1} and let

F = {x | |P_{n}(x)| ≤ c(E)^{n}e^{−2c(δ_{1})n}}  (5.14)

If |E \F | < δ_{1}, then (5.9) would imply ∥P_{n}∥_{E} ≤ c(E)^{n}e^{−c(δ_{1})n}, violating (5.13). So

|E \ F | ≥ δ_{1}  (5.15)

By Lemma 5.3, ℝ \ F is a union of at most n + 1 intervals, so if E is a union of ℓ intervals, E \ F consists of at most ℓ(n + 1) intervals (a very crude overestimate that suffices for us!).

Some of these intervals may have size less than \frac{δ_{1}}{4nℓ}, but the total size of those is at most \frac{δ_{1}}{2} , so we can find disjoint intervals I_{1}^{(n)}, …, I_{k(n)}^{(n)} in E \ F, so

|I_{j}^{(n)}| ≥ \frac{δ_{1}}{4nℓ}  |⋃_{j=1}^{k(n)} I_{j}^{(n)}| ≥ \frac{δ_{1}}{2}  (5.16)

Let Ĩ_{j}^{(n)} be the interval of size \frac{1}{2}|I_{j}^{(n)}| and the same center. Then with L^{(n)}(δ_{1}) = ∪_{j=1}^{k(n)}Ĩ_{j}^{(n)}, we have

|L^{(n)}(δ_{1})| ≥ \frac{δ_{1}}{4}  (5.17)
|P_{n}(y)| ≥ c(E)^{n}e^{−2c(δ_{1})n} if dist(y, L^{(n)}(δ_{1})) ≤ \frac{δ_{1}}{16nℓ}  (5.18)

Now define for any δ_{2} > 0 and m,

J(m, δ_{2}) = \{x | μ(x − \frac{1}{m}, x + \frac{1}{m}) ≥ e^{−δ_{2}m}\}

By hypothesis, for any fixed δ_{2},

lim_{m→∞} |E \ J(m, δ_{2})| = 0

and, in particular, for any fixed integer M, for all large n,

|E \ J(Mn, δ_{2})| < \frac{δ_{1}}{4}  (5.19)

so, in particular,

J(Mn, δ_{2}) ∩ L^{(n)}(δ_{1}) ≠ ∅  (5.20)

Given δ_{1}, pick M so large that

M^{−1} ≤ \frac{δ_{1}}{16ℓ}  (5.21)

If x lies in the set on the left side of (5.20), let I = \{y | |x − y| ≤ \frac{1}{Mn}\}. Then since \frac{1}{Mn} ≤ \frac{δ_{1}}{16nℓ}, for y ∈ I,

|P_{n}(y)| ≥ c(E)^{n}e^{−2c(δ_{1})n}  (5.22)

since x ∈ L^{(n)}(δ_{1}) and (5.18) holds. By x ∈ J(Mn, δ_{2}),

μ(I) ≥ e^{−Mδ_{2}n}  (5.23)

Thus,

∥P_{n}∥_{L^{2}} ≥ c(E)^{n}e^{−2c(δ_{1})n}e^{−MNδ_{2}/2}  (5.24)

so

lim inf ∥P_{n}∥_{L^{2}}^{1/n} ≥ c(E)e^{−2c(δ_{1})}e^{−Mδ_{2}/2}  (5.25)

First pick δ_{1}, then fix M by (5.21) (recall ℓ is fixed as the number of intervals in E) and let δ_{2} = \frac{δ_{1}}{M}. Then take δ_{1} ↓ 0 and get lim inf ∥P_{n}∥_{L^{2}}^{1/n} ≥ c(E), proving regularity. □

Here is a typical application of the Stahl–Totik criterion. It illustrates the limitations of regularity criteria like those of [114, 122] that only depend on what sets are carriers for μ. This result is a special case of a theorem of Wyneken [123].

Theorem 5.4. Let μ be a measure whose support is E, a finite union of closed intervals. Then there exists a measure η equivalent to μ which is regular.

Proof. For any n, define

μ_{n} = ∑_{\{j|μ(( \frac{j}{n} , \frac{j+1}{n}])>0\}} μ((\frac{j}{n} , \frac{j + 1}{n}])^{−1}μ ↾ (\frac{j}{n} , \frac{j + 1}{n}]  (5.26)

Then μ_{n} has total mass at most n(|E| + ℓ) where ℓ is the number of intervals. Let

η = ∑_{n=1}^{∞} n^{−3}μ_{n}  (5.27)

which is easily seen to be equivalent to μ.

Notice that if dist(x, ℝ \ E) > \frac{1}{n}, then [x − \frac{1}{n}, x + \frac{1}{n}] contains an interval of the form (\frac{j}{2n}, \frac{j+1}{2n}], so μ_{2n}([x − \frac{1}{n}, x + \frac{1}{n}]) ≥ 1. Thus

|\{x | η([x − \frac{1}{n} , x + \frac{1}{n}]) ≤ 8n^{−3}\}| ≤ \frac{2ℓ}{n}  (5.28)

and (1.33) holds. □

By using point measures, it is easy to construct nonregular measures, including ones that illustrate how close (1.34) is to being ideal. The key is

Theorem 5.5. Let {x_{j}}_{j=1}^{∞} be a bounded sequence in ℝ and {a_{j}}_{j=1}^{∞} an ℓ^{1} sequence of positive numbers. Let

μ = ∑_{j=1}^{∞} a_{j}δ_{x_{j}}  (5.29)

Let

d = max_{j,k} |x_{j} − x_{k}|  (5.30)

Then

∥P_{n}(x, dμ)∥_{L^{2}(ℝ,dμ)} ≤ d^{n}( ∑_{j=n+1}^{∞} a_{j})^{1/2}  (5.31)

Proof. Let Q_{n}(x) = ∏_{j=1}^{n}(x − x_{j}), which kills the contributions of the pure points at {x_{j}}_{j=1}^{n}, so

∥Q_{n}∥^{2} ≤ ∑_{j=n+1}^{∞} d^{2n}a_{j}  (5.32)

by (5.30). Since ∥P_{n}∥ ≤ ∥Q_{n}∥, (5.31) is immediate. □

Corollary 5.6. Let {x_{j}}_{j=1}^{∞} be an arbitrary bounded subset of ℝ. Then there exists a pure point measure dμ with precisely this set as its set of pure points, so that ∥P_{n}∥^{1/n} → 0. In particular, if E is any compact set with C(E) > 0, there is a measure μ with supp(μ) = E and μ not regular.

Proof. Pick a_{j} = e^{−j^{2}} so (∑_{j=n+1}^{∞} a_{j})^{1/2n} → 0. □

Example 5.7. The following illuminates (1.33). For 2^{n} ≤ k < 2^{n+1}, let x_{k} = \frac{k−2^{n}}{2^{n}} and let 0 < y < 1. Define

dμ = ∑_{k=1}^{∞} y^{k}δ_{x_{k}}  (5.33)

The x_{k} are not distinct, but that does not change the bound (5.31). Thus

lim sup ∥P_{n}∥^{1/n} ≤ y  (5.34)

Since C([0, 1]) = \frac{1}{4}, the measure is not regular if y < \frac{1}{4}. On the other hand, if 2^{n} ≤ m ≤ 2^{n+1} and x_{0} ∈ [0, 1], there is an x_{k} with |x_{k}−x_{0}| ≤ \frac{1}{m} and 2^{n} ≤ k < 2^{n+1}. Thus

μ([x_{0} − \frac{1}{m} , x_{0} + \frac{1}{m}]) ≥ y^{k} ≥ y^{2^{n+1}} ≥ y^{2m}

so (1.33) holds for η = − log y^{2}, that is, for some but not all η. This shows the exponential rate in Theorem 1.13 cannot be improved. □

Example 5.8. We will give an example of a measure dμ on [−2, 2] which is a.c. on [−2, 0] and so that among the limit points of the zero counting measures, dν_{n} are both dρ_{[−2,2]} and dρ_{[−2,0]}, the equilibrium measure for [−2, 2] and for [−2, 0]. This will answer a question asked me by Yoram Last, in reaction to Remling [90], whether a.c. spectra force the existence of a density of states and also show that bounds on limit points of dν_{n} of Totik–Ullman [112] and Simon [100] cannot be improved.

We define dμ by

dμ ↾ [−2, 0] = (−x(x + 2))^{−1/2} dx  (5.35)

picked so the OPRL for the restriction are multiples of the Chebyshev polynomials for [−2, 0]

dμ ↾ [0, 2] = ∑_{n=1}^{∞} a_{n} dη_{n}  (5.36)

where dη_{n} is concentrated uniformly at the dyadic rationals of the form k/2^{n} not previously “captured,” that is,

dη_{n} = ∑_{j=0}^{2^{n}−1} \frac{1}{2^{n}} δ_{(2j+1)/2^{n−1}}  (5.37)

The a_{n}’s are carefully picked as follows. Define N_{j} inductively by

N_{1} = 1 N_{j+1} = 2^{N_{j}^{3}}  (5.38)

and

a_{n} = \begin{cases}\frac{1}{n^{2}} & N_{2k−1} < N ≤ N_{2k} \\ 2^{−n^{4}} & N_{2k} < n ≤ N_{2k+1} \end{cases}

Our goal will be to prove that

dν_{2^{N{2 \atop 2k}}} → dρ_{[−2,0]} dν_{2^{N{2 \atop 2k+1}}} → dρ_{[−2,2]}  (5.39)

Intuitively, for m = 2^{N_{2k+1}^{2}}, the measures at level 1/m will be uniformly spaced out (on an exponential scale), so by the Stahl–Totik theorem, the zeros will want to look like the equilibrium measure for [−2, 2]. But for m = 2^{N_{2k}^{2}}, most intervals of size 1/m in [0, 2] will have tiny measure, so the zeros will want to almost all lie on [−2, 0], where the best strategy for these (to minimize ∫ P_{m}^{2} dμ) will be to approximate the equilibrium measure for [−2, 0].

As a preliminary, we will show

lim sup ∥P_{n}∥^{1/n} = 1  lim inf ∥P_{n}∥^{1/n} = \frac{1}{2}  (5.40)
lim_{k→∞} ∥P_{2^{N{2 \atop 2k}}} ∥^{1/2^{N{2 \atop 2k}}} = \frac{1}{2}  lim_{k→∞} ∥P_{2^{N{2 \atop 2k+1}}}∥_{1/2^{N{2 \atop 2k+1}}} = 1  (5.41)

We begin with

lim sup ∥P_{n}∥^{1/n} ≤ 1  lim inf ∥P_{n}∥^{1/n} ≥ \frac{1}{2}  (5.42)

The first is immediate from (1.15) and C([−2, 2]) = 1; the second from ∥P_{n}(x, dμ)∥ ≥ ∥P_{n}(x, dμ ↾ [−2, 0])∥ (by (1.12)), regularity of dμ ↾ [−2, 0], and C([−2, 0]) = \frac{1}{2}.

Next, we turn to

lim sup_{k→∞} ∥P_{2^{N{2 \atop 2k}}} ∥^{1/2^{N{2 \atop 2k}}} ≤ \frac{1}{2}  (5.43)

Let T_{n}(x; [−2, 0]) be the Chebyshev polynomials for [−2, 0] (which are just affinely related to the classic Chebyshev polynomials of the first kind) and let

Q_{2^{N{2 \atop 2k}}} (x) = T_{2^{N{2 \atop 2k}−N_{2k}}} (x; [−2, 0]) ∏_{ℓ=1}^{2^{N_{2k}}}(x − \frac{ℓ}{2^{N_{2k}−1}} )  (5.44)

so by (1.12),

∥P_{2^{N{2 \atop 2k}}} ∥ ≤ ∥Q_{2^{N{2 \atop 2k}}} ∥ ≤ 1 + 2  (5.45)

where 1 is the contribution of the integral from [−2, 0] and 2 from (0, 2).

Since cos ℓx = 2^{ℓ−1}(cos x)^{ℓ}+ lower order and the average of cos^{2} x is \frac{1}{2}, for any [a, b],

∥T_{m}(x; [a, b])∥ = \sqrt{2} C([a, b])^{m}  (5.46)

where the norm is over L^{2}(ℝ, dρ_{[a,b]}). Since the product in (5.44) is bounded by 4^{2^{N_{2k}}} on [−2, 2], we have

1 ≤ \sqrt{2} (\frac{1}{2})^{2^{N{2 \atop 2k}−N_{2k}}} 4^{2_{2k}^{N}}  (5.47)

On the other hand, there is a constant K so

∥T_{m}(x; [−2, 0])∥_{L^{∞}([−2,2])} ≤ K^{m}  (5.48)

and the product in (5.44) kills all the pure points up to level N_{2k}:

2 ≤ K^{2^{N_{2k}}} ∑_{n=N_{2k}}^{∞} a_{n}
≤ K^{2^{N_{2k}}} [N_{2k+1}2^{−N_{2k}^{4}} + (N_{2k+1})^{−1}]

is much smaller than the right side of (5.47) for k large. Thus, by (5.47),

lim sup ( 1 + 2 )^{1/2^{N{2 \atop 2k}}} ≤ \frac{1}{2}  (5.49)

proving (5.43).

In verifying (5.40), we finally prove that

lim inf_{k→∞} ∥P_{2^{N{2 \atop 2k+1}}}∥^{1/2^{N{2 \atop 2k+1}}} ≥ 1  (5.50)

By the fact that the Chebyshev polynomials for [−2, 2] obey

T_{n}(2 cos x; [−2, 2]) = 2 cos nx  (5.51)

has ∥T_{n}∥_{L^{∞}([−2,2])} = 2, we see

∥P_{m}∥_{L^{∞}([−2,2])} ≥ 2  (5.52)

By Markov’s inequality (5.8), we have

∥P′_{m}∥_{L^{∞}([−2,2])} ≤ \frac{m^{2}}{2} ∥P_{m}∥_{L^{∞}([−2,2])}  (5.53)

so there is an interval of size 4/m^{2} where P_{m}(x) ≥ 1, that is,

∥P_{m}(x)∥_{L^{2}(dμ)}^{2} ≥ inf_{y∈[−2,2]} μ([y − \frac{2}{m^{2}} , y + \frac{2}{m^{2}} ])  (5.54)

which implies that

∥P_{2^{N{2 \atop 2k+1}}}∥_{L^{2}(μ)} ≥ a_{2^{N{2 \atop 2k+1}}}2^{−N_{2k+1}^{2}}  (5.55)

so it is bounded from below by a power of 2^{−N_{2k+1}^{2}}. Since m^{−ℓ/m} → 1 for any fixed ℓ, we obtain (5.50).

Clearly, (5.42), (5.43), and (5.50) imply (5.40) and (5.41). We now only need to go from there to results on limits of dν_{n}. By Theorem 2.4, the second equality in (5.41) implies the second limit result in (5.39). By Theorem 4.2, the first equality in (5.41) implies the first limit result in (5.39). □

Example 5.9. Here is an example of a measure dμ on [0, 1] where the density of zeros has a limit singular relative to the equilibrium measure for [0, 1]. Such examples are discussed in [105] and go back to work of Ullman. Let Σ be the classical Cantor set and dρ_{Σ} its equilibrium measure. Let

dμ = dρ_{Σ} + ∑_{n=1}^{∞} 2^{−n^{4}}( ∑_{j=0}^{2^{n−1}} \frac{1}{2^{n}} δ_{(2j+1)/2^{n}})  (5.56)

As in the above construction, one shows ∥P_{n}∥^{1/n} → C(Σ) and then Theorem 4.2 implies that dν_{n} → dρ_{Σ} which is singular with respect to Lebesgue measure, and so relative to dρ_{[0,1]} ≡ dρ_{supp(dμ)}. □

6. Structural Results

In this section, we will focus on the mutual regularity of related measures. There are three main theorems, all from Stahl–Totik [105]:

Theorem 6.1. Let μ, η be two measures of compact support whose supports are equal up to sets of capacity zero. If μ ≥ η and η is regular, then so is μ.

Theorem 6.2. Let {E_{n}}_{n=1}^{∞} and E_{∞} be compact subsets of ℂ so that E_{∞} and ∪_{n=1}^{∞}E_{n} agree up to sets of capacity zero and C(E_{∞}) > 0. Let μ be a measure with supp(dμ) = E_{∞} so that each μ ↾ E_{j} which is nonzero is regular. Then μ is regular.

Remark. By μ ↾ K, we mean the measure

(μ ↾ K)(S) = μ(K ∩ S)  (6.1)

To understand why the next theorem is so restrictive compared to Theorem 6.2, consider

Example 6.3. Let E be the standard Cantor set in [0, 1]. Let η be a measure on E which is not regular (see Corollary 5.6) and let

dμ = dη + dx ↾ [0, 1]  (6.2)

By Theorem 6.1, dμ is regular. But dμ ↾ E = dη is not regular. □

Theorem 6.4. Let I = [a, b] be a closed interval with I ⊂ E ⊂ ℝ and E compact. Let μ be a regular measure with support in E so C(supp(μ ↾ I)) > 0. Then μ ↾ I is regular.

Remarks. 1. We do not require that supp(dμ) = E (nor that I ⊂ supp(dμ)) but only that supp(dμ) ⊂ E and that μ is regular in the sense that C(supp(dμ)) > 0 and ∥P_{n}( · , dμ)∥_{L^{2}(dμ)}^{1/n} → C(supp(dμ)).

2. The analog of the sets I in [105] must have nonempty twodimensional interior. Our I obviously has empty two-dimensional interior, but if I = [a, b] ⊂ E ⊂ ℝ and if D is the disk \{z | |z − \frac{1}{2}(a + b)| ≤ \frac{1}{2}|b − a|\}, then μ ↾ D = μ ↾ I.

The proofs of Theorems 6.1 and 6.2 will be easy, but Theorem 6.4 will be nontrivial. Here are some consequences of these results:

Corollary 6.5. Let μ, ν be two regular measures (with different supports allowed). Then their max, μ ∨ ν, and sum, μ + ν, are regular.

Remark. See Doob [33] for the definition of μ ∨ ν.

Proof. μ + ν and μ ∨ ν have the same support and μ + ν ≥ μ ∨ ν so, by Theorem 6.1, we only need the result for μ ∨ ν. Let E_{1} = supp(μ) and E_{2} = supp(ν), so supp(μ∨ν) = E_{1}∪E_{2}. By definition, (μ∨ν) ↾ E_{1} ≥ μ and they have the same supports. So, by Theorem 6.1, μ ∨ ν ↾ E_{1} is regular. Similarly, μ ∨ ν ↾ E_{2} is regular. By Theorem 6.2, μ ∨ ν is regular. □

Corollary 6.6. Let E = I_{1}∪⋯∪I_{ℓ} be a union of finitely many disjoint closed intervals. Let μ be a measure on E. Then μ is regular if and only if each μ ↾ I_{j} is regular.

Proof. Immediate from Theorems 6.2 and 6.4. □

Proof of Theorem 6.1. Since (1.12) holds,

∥X_{n}∥_{L^{2}(dη)} ≤ ∥X_{n}∥_{L^{2}(dμ)}  (6.3)

Given (1.15), we have (with E = supp(dμ))

lim ∥X_{n}∥_{L^{2}(dη)}^{1/n} = C(E) ⇒ lim ∥X_{n}∥_{L^{2}(dμ)}^{1/n} = C(E)  □

Proof of Theorem 6.2. Let μ_{j} = μ ↾ E_{j} and let x_{n}(z) be the x_{n}’s for dμ. Then

∥x_{n}∥_{L^{2}(dμ_{j})} ≤ ∥x_{n}∥_{L^{2}(dμ)} = 1  (6.4)

so

|x_{n}(z)| ≤ \frac{|x_{n}(z)|}{∥x_{n}∥_{L^{2}(dμ_{j})}}  (6.5)

By regularity and Theorem 1.10(vi), for q.e. z ∈ E_{j} (using G_{E_{j}}(z) = 0 for q.e. z ∈ E_{j} by Theorem A.10(b) and (A.40)), we see for q.e. x ∈ E_{j},

lim sup_{n→∞} |x_{n}(z)|^{1/n} ≤ 1  (6.6)

Since ∪_{j=1}^{∞}E_{j} is q.e. E, we have (6.6) q.e. on all of E. By (iii) ⇒ (i) in Theorem 1.10, μ is regular. □

To prove Theorem 6.4, we first make a reduction:

Proposition 6.7. Suppose there is I = [a, b] ⊂ E ⊂ ℝ (with a < b), μ regular, C(supp(μ ↾ I)) > 0 but

lim inf ∥P_{n}( · , μ ↾ I)∥_{L^{2}(μ↾I)}^{1/n} < C(supp(μ ↾ I))  (6.7)

Then there exists a μ, perhaps distinct but also regular and supported on E, so that (6.7) holds and

lim inf ∥P_{n}( · , μ ↾ I)∥_{L^{2}(μ↾I)}^{1/n} > 0  (6.8)

Proof. If (6.8) holds for the initial μ, we can stop. Otherwise, we will take

μ̃ = μ + ρ_{F}  (6.9)

where F = [x_{0} − δ, x_{0} + δ] ⊂ I with x_{0} = \frac{1}{2}(a + b) and δ sufficiently small chosen later.

By Corollary 6.5, μ̃ is regular and, by (6.3),

∥P_{n}( · , μ̃ ↾ I)∥_{L^{2}(μ̃↾I)}^{1/n} ≥ ∥P_{n}( · , ρ_{F})∥_{L^{2}(ρ_{F})}^{1/n}

so (6.8) holds since

lim_{n→∞} ∥P_{n}( · , ρ_{F})∥_{L^{2}(ρ_{F})}^{1/n} = C(F) > 0

Thus, we need only prove that (6.7) holds for suitable δ. Since we are supposing (6.8) fails for μ, pick n(j) → ∞ so

∥P_{n(j)}( · , μ ↾ I)∥_{L^{2}(μ↾I)}^{1/n(j)} → 0  (6.10)

Define

Q_{2n(j)}(x) = P_{n(j)}(x, μ ↾ I)(x − x_{0})^{n(j)}  (6.11)

Let d = diam(E) and note that on E, since x_{0} ∈ I ⊂ E,

|x − x_{0}|^{n(j)} ≤ d^{n(j)}  (6.12)

and since P_{n(j)} has all its zeros in cvh(E), for x ∈ E,

|P_{n(j)}(x)| ≤ d^{n(j)}  (6.13)

Thus,

∥Q_{2n(j)}∥_{L^{2}(μ̃↾I)}^{2} = ∥P_{n(j)}(· − x_{0})^{n(j)}∥_{L^{2}(μ↾I)}^{2} + ∥P_{n(j)}(· − x_{0})^{n(j)}∥_{L^{2}(dρ_{E})}^{2}
≤ d^{2n(j)}∥P_{n(j)}∥_{L^{2}(μ↾I)}^{2} + d^{2n(j)}δ^{2n(j)}  (6.14)

Using (6.10), we see

lim sup ∥Q_{2n(j)}∥_{L^{2}(μ̃↾I)}^{1/2n(j)} ≤ d^{1/2}δ^{1/2} < C(supp(μ ↾ I)) ≤ C(supp(μ̃ ↾ I))

if we take δ small. Since (1.12),

∥P_{2n(j)}( · , μ̃ ↾ I)∥_{L^{2}(μ̃↾I)} ≤ ∥Q_{2n(j)}∥_{L^{2}(μ̃↾I)}  (6.15)

we see μ̃ ↾ I is not regular. □

Proof of Theorem 6.4. By Proposition 6.7, we can find μ so μ is regular, μ ↾ I is not regular but for some a > 0 (with μ_{I} = μ ↾ I),

∫_{I} |P_{n}(x, μ_{I})|^{2} dμ ≥ a^{n}  (6.16)

Fix x_{0} ∈ I_{int}. Let d = diam(E) and for ℓ to be picked shortly, let

Q_{n(2ℓ+1)}(x) = P_{n}(x, μ_{I})(1 − \frac{(x − x_{0})^{2}}{d^{2}} )^{ℓn}  (6.17)

Since P obeys (6.13), if we define

η = max_{x∈E\I} (1 − \frac{(x − x_{0})^{2}}{d^{2}} ) < 1  (6.18)

we have for x ∉ I,

|Q_{n(2ℓ+2)}(x)| ≤ η^{ℓn}d^{n}  (6.19)

Choose ℓ so

(η^{ℓ}d)^{2} < a  (6.20)

Then, by (6.17), (6.19), and (6.20),

∫_{K}|Q_{n(2ℓ+1)}(x)|^{2} dμ ≤ 2 ∫ |P_{n}(x, μ_{I})|^{2} dμ  (6.21)

so

|p_{n}(x_{0}; dμ_{I})|^{1/n} ≤ \frac{2^{1/2n}|Q_{n(2ℓ+1)}(x_{0})|^{1/n}}{∥Q_{n(2ℓ+1)}∥_{L^{2}(K,dμ)}^{1/n}}  (6.22)

so, by (1.30) and regularity of μ, for μ-a.e. x_{0} in I^{int} ∩ supp(dμ),

lim sup|p_{n}(x_{0}; dμ_{I})|^{1/n} ≤ 1

But then Theorem 1.10 implies dμ_{I} is regular. This contradiction proves the theorem. □

7. Ergodic Jacobi Matrices and Potential Theory

In this section, we will explore regularity ideas for ergodic half- and whole-line Jacobi operators and see this is connected to Kotani theory (see [99, Sect. 10.11] and [27] as well as the original papers [69, 96, 42, 70]). A main goal is to prove Theorems 1.15 and 1.16.

Let (Ω, dσ) be a probability measure space. Let T: Ω → Ω be an invertible ergodic transformation. Let Ã, B̃ be measurable functions from Ω to ℝ with B̃ bounded, Ã positive, and both Ã and Ã^{−1} bounded. For ω ∈ Ω and n ∈ ℤ, define a_{n}(ω), b_{n}(ω) by

a_{n}(ω) = Ã(T^{n}ω) b_{n}(ω) = B̃(T^{n}ω)  (7.1)

By J(ω), we mean the Jacobi matrix with parameters {a_{n}(ω), b_{n}(ω)}_{n=1}^{∞}. By J̃(ω), we mean the two-sided Jacobi matrix with parameters {a_{n}(ω), b_{n}(ω)}_{n=−∞}^{∞}. Occasionally we will use J_{k}^{+}(ω) for the one-sided matrix with parameters {a_{k+n}(ω), b_{k+n}(ω)}_{n=1}^{∞} and J_{k}^{−}(ω) for the one-sided matrix with parameters {a_{k−n}(ω), b_{k+1−n}(ω)}_{n=1}^{∞}.

Spectral measures for one-sided matrices (and vector δ_{1}) are dμ_{ω}, dμ_{ω}^{±k} and for J̃(ω), we use dμ̃_{ω;k} for vector δ_{k}.

For spectral theory, the transfer matrix is basic. Define for n ∈ ℤ,

A_{n}(x, ω) = \frac{1}{a_{n+1}(ω)} (\begin{matrix}x − b_{n+1}(ω) & −1 \\ a_{n+1}(ω)^{2} & 0 \end{matrix})  (7.2)

then

a_{n+1}u_{n+1} + (b_{n+1} − x)u_{n} + a_{n}u_{n−1} = 0  (7.3)

is equivalent to

(\begin{matrix}u_{n+1} \\ a_{n+1}u_{n} \end{matrix}) = A_{n} (\begin{matrix}u_{n} \\ a_{n}u_{n−1} \end{matrix})  (7.4)

We define for n < m,

T (m, n; x, ω) = A_{m}(x, ω)A_{m−1}(x, ω) … A_{n+1}(x, ω)  (7.5)

and T (n, n; x, ω) = 1 and, for m < n, T (m, n; x, ω) = T (n, m; x, ω)^{−1}. Thus, solutions of (7.3) obey

(\begin{matrix}u_{m+1} \\ a_{m+1}u_{m} \end{matrix}) = T (m, n; x, ω) (\begin{matrix}u_{n+1} \\ a_{n+1}u_{n} \end{matrix})  (7.6)

In particular, for n ≥ 1,

(\begin{matrix}p_{n+1}(x, ω) \\ a_{n+1}p_{n}(x, ω) \end{matrix}) = T (n, −1; x, ω) (\begin{matrix}1 \\ 0 \end{matrix})  (7.7)

The ergodic and subadditive ergodic theorems produce the following well-known facts:

Theorem 7.1. There exists Ω_{0} ⊂ Ω of full σ measure so that for ω ∈ Ω_{0},

(a) σ(J̃(ω)) = E, a fixed perfect subset of ℝ independent of ω (in Ω_{0}). Moreover, for any ω ∈ Ω_{0}, each J_{k}^{±} obeys

σ_{ess}(J_{k}^{±}(ω)) = E  (7.8)

(b) There is a measure dν_{∞} with

supp(dν_{∞}) = E  (7.9)

If dν_{n}^{k,±,ω} is the zero counting measure for J_{k}^{±}(ω), then for any ω ∈ Ω_{0}, as n → ∞,

dν_{n}^{k,±,ω} ⟶^{ω} dν_{∞}  (7.10)

(c) Define the Lyapunov exponent γ(z) for z by

γ(z) = lim_{n→∞} 𝔼(\frac{1}{n} log ∥T (n − 1, −1; z, ω)∥)  (7.11)

where

𝔼(f) = ∫ f(ω) dσ(ω)  (7.12)

and (7.11) includes that the limit exists. Moreover, for any k ∈ ℤ,

γ(z) = lim_{n→∞} 𝔼(\frac{1}{n} log ∥T (n + k, k; z, ω)∥)  (7.13)
γ(z) = lim_{n→∞} 𝔼(\frac{1}{n} log ∥T (k − n, k; z, ω)∥)  (7.14)

(d) For any ω ∈ Ω_{0} and z ∉ E and k fixed,

lim_{n→∞} \frac{1}{n} log ∥T (n + k, k; z, ω)∥ = γ(z)  (7.15)
lim_{n→∞} \frac{1}{n} log ∥T (k − n, k; z, ω)∥ = γ(z)  (7.16)

(e) For any z ∈ E and σ-a.e. ω ∈ Ω_{0}, (7.15) and (7.16) hold.

(f) For ω ∈ Ω_{0}, lim_{n→∞}(a_{1} … a_{n})^{1/n} = A exists and is ω-independent, and one has the Thouless formula,

γ(z) = log(A^{−1}) + ∫ log(|z − x|) dν_{∞}(x)  (7.17)

Moreover, for all z,

γ(z) ≥ 0  (7.18)

Remarks. 1. For proofs, see [21, 26, 87, 99]. (7.17) is due (in the physics literature) to Herbert–Jones [51] and Thouless [110]. It is, of course, just (1.4) for z ∉ cvh(Σ). Almost everything else here is a simple consequence of the Birkhoff ergodic theorem/the Kingman subadditive ergodic theorem and translation invariance which implies, for example, that the expectation in (7.13) is k-independent for each n.

2. There are two subtleties to OP readers. First, (7.14) comes from ∥A^{−1}∥ = ∥A∥ for 2 × 2 matrices A with det(A) = 1. It implies that the Lyapunov exponent is the same in both directions. det(T) = 1 also implies (7.18).

3. The second subtlety concerns equality in (7.17) for all z, including those in Σ. This was first proven by Avron–Simon [12]; the simplest proof is due to Craig–Simon [25] who were motivated by work of Herman [52]. The point is that, in general, lim sup \frac{1}{n} log ∥T (n + k, k; z, ω)∥ (and lim sup \frac{1}{n} log|p_{n}(z, ω)|) may not be upper semicontinuous but 𝔼(\frac{1}{n} log ∥T (n+k, k; z, ω)∥) is because of translation invariance, Hölder’s inequality, and

T (n + ℓ + k, k; z, ω) = T (n + ℓ + k, ℓ + k; z, T^{ℓ}ω)T (ℓ + k, k; z, ω) (7.19)

This implies that the expectation is subadditive so the limit is an inf.

Two main examples are the Anderson model and almost periodic functions. For the former, (a_{n}(ω), b_{n}(ω)) are independent (0, ∞) × ℝvalued (bounded with a_{n}^{−1} also bounded) identically distributed random variables. In the almost periodic case, Ω is a finite- or infinitedimensional torus with dσ Haar measure and Ã, B̃ continuous functions. A key observation (of Avron–Simon [12]) is that in this almost periodic case, the density of states exists for all, not only a.e., ω ∈ Ω so we can then take Ω_{0} = Ω in Theorem 7.1.

Here is the first consequence of potential theory ideas in this setting:

Theorem 7.2. E has positive capacity; indeed,

C(E) ≥ A  (7.20)

Moreover, E is always potentially perfect (as defined in Appendix A). Each dμ_{ω} (ω ∈ Ω_{0}) is regular if and only if equality holds in (7.20).

Proof. Use γ ≥ 0 in (7.17), integrating dν_{∞}, to see that

E(ν_{∞}) ≤ log(A^{−1})  (7.21)

so E(ν_{∞}) < ∞, implying C(E) > 0. By (7.21), we get (7.20).

By (7.21), ν_{∞} has finite energy and so, by Proposition A.6, ν_{∞} gives zero weight to any set of capacity zero. It follows that if x ∈ supp(dν_{∞}), then C((x − δ, x + δ)) > 0 for all δ. By (7.9), E is potentially perfect.

By definition of A, regularity for all ω ∈ Ω_{0} is equivalent to C(E) = A. □

Proof of Theorem 1.15. We will prove that (1.37) holds. By (7.21), ν_{∞} has finite Coulomb energy, so ν_{∞} gives zero weight to sets of zero capacity. Since equality holds in (A.23), q.e. on E, we conclude that

log(C(E)^{−1}) = ∫ dν_{∞}(x) Φ_{ρ_{E}}(x)
= ∫ dρ_{E}(x) Φ_{ν_{∞}}(x)  (by (A.2))
= ∫ dρ_{E}(x) [log(A^{−1}) − γ(x)] (by (7.17))

This is (1.37).

By (1.37), we have (1.35) ⇔ ∫ γ(x) dρ_{E}(x) = 0 which, given that γ(x) ≥ 0, holds if and only if γ(x) = 0 for ρ_{E}-a.e. x.

If (1.35) holds, then each dμ_{ω} is regular, so by Theorem 1.7, dν_{∞} = dρ_{E}. The converse part follows from Theorem 1.9. □

Note: Remling remarked to me that Theorem 1.15 has a deterministic analog with essentially the same proof.

Kotani theory says something about when γ(x) = 0 but we have not succeeded in making a tight connection, so we will postpone the precise details until we discuss conjectures in the next section. As a final topic, we want to prove Theorem 1.16 and a related result.

Proof of Theorem 1.16. By (4.14) for dμ_{ω}-a.e. x, we have

lim sup|p_{n}(x)|^{1/n} ≤ 1  (7.22)

On the other hand, by the upper envelope theorem and (1.4) for q.e. x,

lim |p_{n}(x)|^{1/n} = A^{−1} exp(−Φ_{ν_{∞}}(x))  (7.23)
= exp(γ(x))  (7.24)

by (7.17). Let Q_{ω} be the capacity zero set where (7.24) fails.

On S, exp(γ(x)) > 1, so since (7.22) holds for a.e. x, we have dμ_{ω}(S \ Q_{ω}) = 0 as claimed. □

Remark. All we used was that dν_{∞} is the limit of dν_{n}, so this holds for all ω ∈ Ω_{0}. In particular, in the almost periodic case, it holds for all ω in the hull.

One is also interested in the whole-line operator.

Theorem 7.3. Let J̃(ω) be the whole-line Jacobi matrix associated with {a_{n}(ω), b_{n}(ω)}_{n=−∞}^{∞} and dμ_{ω,k} its spectral measures. Let S ⊂ ℝ be the Borel set of x with γ(x) > 0. Then for each ω ∈ Ω_{0}, there exists a set Q̃_{ω} of capacity zero so that

μ_{ω,k}(S \ Q̃_{ω}) = 0  (7.25)

for all k.

Proof. By (7.7), the transfer matrix T (n, −1; x, ω) has matrix elements given by p_{n+1}, p_{n} and the second kind polynomials q_{n+1}, q_{n}. As in the last proof, there is a set Q̃_{ω}^{(1)} of capacity zero so for x ∉ Q_{ω}^{(1)},

lim |p_{n}(x)|^{1/n} = exp(γ(x))  (7.26)

and (zeros of p_{n} and q_{n} interlace, so the zero counting measure for q_{n} also converges to dν_{∞})

lim |q_{n}(x)|^{1/n} = exp(γ(x))  (7.27)

In particular, for x ∉ Q̃_{ω}^{(1)}, we have

lim_{n→∞} \frac{1}{n} log ∥T (n, −1; x, ω)∥ = γ(x)  (7.28)

By the Ruelle–Osceledec theorem (see, e.g., [99, Sect. 10.5]), for any w ≠ 0 ∈ ℂ^{2}, either

∥T (n, −1; x, ω)w∥^{1/n} → e^{γ(x)}  (7.29)

or

∥T (n, −1; x, ω)w∥^{1/n} → e^{−γ(x)}  (7.30)

Similarly, there is a set Q̃_{ω}^{(2)} of capacity zero with similar behavior as n → ∞.

This says that every solution of (7.3) for x ∉ Q̃_{ω}^{(1)} ∪ Q̃_{ω}^{(2)} either grows exponentially at ±∞ or decays exponentially. Thus, polynomial boundedness implies ℓ^{2} solutions. If Q̃_{ω}^{(3)} is the set of eigenvalues of J̃(ω) which is countable and so of capacity zero, and if Q̃_{ω} = Q̃_{ω}^{(1)} ∪ Q̃_{ω}^{(2)} ∪ Q̃_{ω}^{(3)}, then

J̃(ω)u = xu with u polynomially bounded ⇒ x ∈ Q̃_{ω}

By the BGK expansion discussed in Section 4, this implies the spectral measures of J̃(ω) are supported on Q̃_{ω}, that is, (7.25) holds. □

Remarks. 1. The reader will recognize this proof as a slight variant of the Pastur–Ishii argument [86, 53] that proves absence of a.c. spectrum on S.

2. As above, in the almost periodic case, this holds for all ω in the hull.

3. This is the first result on zero Hausdorff dimension in this generality. But for suitable analytic quasi-periodic Jacobi matrices, the result is known; see Jitomirskaya–Last [59] and Jitomirskaya [56].

8. Examples, Open Problems, and Conjectures

Here we consider a number of illustrative examples and raise some open questions and conjectures. The conjectures are sometimes mere guesses and could be wrong. Indeed, when I started writing this paper, I had intended to make a conjecture for which a counterexample appears below as Example 8.12. So the reader should regard the conjectures as an attempt to stimulate work with my own guesses. I will try to explain my guesses, but they are not always compelling.

Example 8.1 (Random and Decaying Random OPRL). Let Ω = ×_{n=1}^{∞}[(0, ∞) × ℝ] with dσ({a_{n}, b_{n}}) = ⊗dη(a_{n}, b_{n}), where η is a measure of compact support on (0, ∞) × ℝ. For each ω ∈ Ω, there is an associated Jacobi matrix, and we want results on J(ω) that hold for σa.e. ω. The traditional Anderson model is the case where a_{n} ≡ 1 and b_{n} is uniformly distributed on [α, β], that is, dη(a, b) = δ_{a1} \frac{1}{β−α}χ_{(α,β)}(b) db.

The decaying random model has two extra parameters, λ ∈ (0, ∞) and γ ∈ (0, 1), takes ã_{n}(ω) ≡ 1, b̃_{n}(ω) the Anderson model with β = −α = 1, and takes

b_{n}(ω) = λn^{−γ}b̃_{n}(ω)  (8.1)
a_{n}(ω) = 1  (8.2)

The Anderson model is ergodic; the decaying random model is not. The Anderson model goes back to his famous work [2] with the first mathematical results by Kunz–Souillard [72] and the decaying model to Simon [95] (see also [67]).

For the Anderson model, it is known for a.e. ω, σ_{ess}(J(ω)) = [−2 + α, 2 + β], while for the decaying random model, σ_{ess}(J(ω)) = [−2, 2] by Weyl’s theorem (i.e., J(ω) is in Nevai class). Clearly, (a_{1} … a_{n})^{1/n} = 1. For the Anderson model,

C(σ_{ess}(J(ω))) = \frac{1}{4} (4 + (β − α)) > 1  (8.3)

while for the decaying Anderson model,

C(σ_{ess}(J(ω))) = 1  (8.4)

so the former is not regular, while the latter is.

Of course, for the regular model, the density of zeros is the equilibrium measure where ρ_{E}(x) = \frac{dρ}{dx} = \frac{1}{π}(4 − x^{2})^{−1/2} by (A.34). For the Anderson model, on the other hand, \frac{dν}{dx} is very different. It is C^{∞} even at the endpoints (by [103]) and decays exponentially fast to zero at the ends of the spectrum (Lifshitz tails; see [66]).

The Anderson model is known to have dense pure point spectrum and so is the decaying model if γ < \frac{1}{2}. It is known for the Anderson model (see [31]) that for some ω-dependent labeling of the eigenvalues,

dμ_{ω} = ∑ w_{n}(ω)δ_{e_{n}(ω)}  (8.5)

where for some c > 0,

|w_{n}(ω)| ≤ e^{−c|n|}  (8.6)

The same methods should allow one to prove for the decaying model on each [−A, A] ⊂ (−2, 2) that there is a labeling so that

|w_{n}(ω)| ≤ e^{−c|n|^{1−2γ}}  (8.7)

One expects that there are lower bounds of the same form and that the labels are such that the e_{n}(ω) are quasi-uniformly distributed (i.e., for n ≫ m, the first n e_{j}(ω) are at least within \frac{1}{m} of each point away from the edge of the spectrum). If these expectations are met, this example nicely illustrates Theorems 1.13 and 1.14.

In the not regular Anderson case, one expects μ_{ω}([\frac{j}{m}, \frac{j+1}{m}]) ∼ e^{−cm} for fixed c, while in the regular decaying random model, one expects μ_{ω}([\frac{j}{m}, \frac{j+1}{m}]) ∼ e^{−cm^{1−2γ}} > C_{η}e^{−ηm} for any η. □

Example 8.2 (Generic Regular Measures). Fix a_{n} ≡ 1, 0 < γ < \frac{1}{2}, and let B = {{b_{n}} | lim n^{γ}b_{n} → 0} normed by |||b||| = sup_{n}|n^{γ}b_{n}|. It is known ([97]; see also [77, 22]) that for a dense G_{δ} in B, the associated Jacobi matrix has singular continuous measure. We believe there is some suitable sense in which a generic regular measure is singular continuous. □

Example 8.3 (Almost Mathieu Equation). Perhaps the most studied model in spectral theory is the whole-line Jacobi matrix with a_{n} ≡ 1 and

b_{n} = λ cos(nα + θ)  (8.8)

where λ, α, θ are parameters with \frac{α}{π} irrational. (See [56] for a review on the state of knowledge.) We will use some of the most refined results and comment on whether they are needed for the main potential theoretic conclusions. We fix α, λ. θ ∈ Ω = [0, 2π) labels the hull of an almost periodic family.

It is known since Avron–Simon [12] that for |λ| > 2, there is no a.c. spectrum for almost all θ (and by Kotani [71] and Last–Simon [75], for all θ) and by [12] for α which are Liouville numbers (irrational but very well approximated by rationals) only singular continuous spectrum for all θ. Jitomirskaya [55] proved that for α’s with good Diophantine properties and |λ| > 2, there is dense pure point spectrum for a.e. θ (and there is also singular continuous spectrum for a dense set of θ’s [60]). On the other hand, Last [74] proved that for |λ| < 2 and all irrational α that the spectrum is a.c. for almost all θ (now known for all θ [71, 75]). It is now known the spectrum in this region is purely a.c. (see [9, 8, 6]).

At the special point λ = 2, it is known that for all irrational α, the spectrum has measure zero [74, 10], and therefore for all irrational α and a.e. θ, the spectrum is purely singular continuous [49].

An important special feature for our purposes is Aubry duality (found by Aubry [5]; proven by Avron–Simon [12]) that relates the Lyapunov exponent γ(z) and integrated density of states, k(E) = ∫_{−∞}^{E} dν_{∞}(x), for α fixed (they are θ-independent) at λ and \frac{4}{λ}. Making the λ-dependence explicit,

k(E, \frac{4}{λ}) = k(\frac{2E}{λ} , λ) γ(z, \frac{4}{λ}) = γ(\frac{2z}{λ} , λ) + log(\frac{λ}{2}) (8.9)

Kotani theory implies in the a.c. region (i.e., λ < 2) that γ(E) = 0 for a.e. E ∈ spec(J), and Bourgain–Jitomirskaya [19] proved continuity of γ. So using (8.9),

E ∈ spec(J) ⇒ γ(E) = max(0, log(\frac{λ}{2}))  (8.10)

and, in particular, γ(E) = 0 on the spectrum if |λ| ≤ 2. □

We thus have:

Theorem 8.4. The density of zeros for the almost Mathieu equation is the equilibrium measure for its spectrum. For λ ≤ 2, the measures are regular (for all ω). For λ > 2, they are not regular since

lim(a_{1} … a_{n})^{1/n} = 1 < C(spec(J_{0}(ω))) = log(\frac{λ}{2})  (8.11)

Proof. By Theorem 1.14, the measures are regular if λ ≤ 2 since γ(E) = 0 on the spectrum. That the measure is the equilibrium measure even if λ > 2 follows from (8.9) as does (8.11). □

Remarks. 1. Thus we see an example where the density of zeros is the equilibrium measure even though dμ_{ω} is not regular. Consistently with Theorem 2.5, dμ_{ω} lives on a set of capacity 0 by Theorem 7.3.

2. If we knew a priori that dρ_{σ_{ess}(J_{ω})} were absolutely continuous, Kotani theory then would suffice for Theorem 8.4. But as it is, we need the continuity result of [19].

3. It should be an exceptional situation that J(ω) has some singular spectrum but the density of states is still dρ_{E}. In particular, if there are separate regions in σ(J) of positive capacity where γ(x) = 0 and where γ(x) > 0, the density of states cannot be dρ_{E} since, for it, γ(x) is constant on supp(dρ_{E}). For examples with such coexistent spectrum (some only worked for the continuum case), see [17, 18, 36, 37].

Example 8.5 (Rotation Invariant Anderson Model OPUC). Let dσ_{0} be a rotation invariant measure on the disk, 𝔻 (i.e., on \overline{𝔻} with σ_{0}(∂𝔻) = 0). Let σ on ×_{j=0}^{∞} 𝔻 be ⊗_{j=0}^{∞}dσ_{0}(z_{j}). The ergodic OPUC with Verblunsky coefficients α_{j} distributed by σ is called the rotation invariant Anderson model, and it is discussed in [99, Sect. 12.6] and earlier in Teplyaev [109] and Golinskii–Nevai [47].

If

∫ − log(1 − |z|) dσ_{0}(z) < ∞  (8.12)

then

lim_{n→∞} (ρ_{0} … ρ_{n−1})^{1/n} = exp(∫ log(1 − |z|^{2}) dσ_{0}(z)) > 0 (8.13)

If also

∫ − log|z| dσ_{0}(z) < ∞  (8.14)

then, by a use of the ergodic theorem,

lim_{n→∞} |α_{n}|^{1/n} = 1

with probability 1. By a theorem of Mhaskar–Saff [82] (see [98, Thm. 8.1.1]), any limit point of the zero counting measure lives on ∂𝔻 so, by the ergodic theorem, ν_{n} has a limit ν_{∞} on ∂𝔻.

By the rotation invariance of σ_{0}, the distribution of {α_{j}} is invariant under α_{j} → e^{i(j+1)θ}α_{j}. So the collection of measures is rotation invariant and thus, by ergodicity, dν_{∞} is rotation invariant, that is, it is \frac{dθ}{2π}. By the Thouless formula and (8.13),

γ(e^{iθ}) = − ∫ log(1 − |z|^{2}) dσ_{0}(z) > 0

so long as σ_{0} ≠ δ_{z=0}. This is constant on ∂𝔻.

Thus, this family of measures is not regular, but the density of zeros is the equilibrium measure for supp(dμ_{ω}) = ∂𝔻. This is the simplest example of a nonregular measure for which the density of zeros is the equilibrium measure. As is proven in Theorem 12.6.1 of [99], the measure is a pure point measure, so dμ_{w} is for a.e. ω supported on a countable set, so of zero capacity, consistent with Theorem 2.5. □

Example 8.6 (Subshifts). This is a rich class of ergodic Jacobi matrices (with a_{n} ≡ 1), reviewed in [28] (see also [99, Sect. 12]). For many of them, it is known that E ≡ σ(J) is a set of Lebesgue measure zero on which γ(x) is everywhere 0. By Theorem 1.15, C(E) = 1 and a.e. ω has regular dμ_{ω}, so, in particular, dν_{∞} = dρ_{E}. □

Notice that, by Craig’s argument (see Theorem A.13), if dμ is any probability measure whose support, E, has measure zero, then G(z) = ∫ \frac{dμ(y)}{y−z} has the form

G(z) = −\frac{1}{\sqrt{(z − a)(z − b)}} ∏_{j=1}^{∞} \frac{(z − λ_{j})}{\sqrt{(z − ℓ_{j})(z − u_{j})}}  (8.15)

where the gaps in E are (ℓ_{j}, u_{j}) and a = inf ℓ_{j}, b = sup μ_{j}. This is so regular that we wildly make the following:

Conjecture 8.7. Any ergodic matrix that has a spectrum of measure zero has vanishing Lyapunov exponent on the spectrum; equivalently, γ(x) > 0 for some x ∈ Σ implies |Σ| > 0. Such zero Lyapunov exponent examples would thus be regular.

We note that for analytic functions on the circle with irrational rotation, this result is known to be true [57], following from combining results from Bourgain [16] and Bourgain–Jitomirskaya [19]. Of two experts I consulted, one thought it was false and the other, “likely true but too little support to make it a conjecture.” Fools rush in where experts fear to tread.

Open Question 8.8 (The Classical Cantor Set). Of course, one of the simplest of measure zero sets is the classical Cantor set. It would be a good first step to understand its “isospectral tori.” Which whole-line Jacobi matrices have ⟨δ_{0}, (J_{0} − z)^{−1}δ_{0}⟩ = (8.15)? Are they regular? As suggested by Deift–Simon [30], are they mainly mutually singular? Are any or all almost periodic?

Conjecture 8.9 (Last’s Conjecture). A little more afield from potential theory, but worth mentioning, is the conjecture of Last that any ergodic Jacobi matrix (whole- or half-line) with some a.c. spectrum is almost periodic. Does it help to consider the case where the spectrum is purely a.c.? We note that a result of Kotani [70] implies Last’s conjecture if a_{n}, b_{n} take only finitely many values.

And it links up to the next question:

Open Question 8.10 (Denisov–Rakhmanov Theorem). Let E be an essentially perfect set, that is, for every x ∈ E and δ > 0, |(x − δ, x + δ) ∩ E| > 0. In [29], E was called a DR set if any half-line Jacobi matrix with σ_{ess}(J) = Σ_{ac}(J) = E has a set of right limit points which is uniformly compact (and so the limits are all almost periodic). A classical theorem of Rakhmanov, as extended by Denisov (see [99, Ch. 9]), says that [−2, 2] is a DR set. Damanik–Killip–Simon [29] proved a number of E’s, including those associated with periodic problems, are DR sets. Remling [90] recently proved any finite union of closed intervals is a DR set, and he remarks that it is possible to combine his methods with those of Sodin–Yuditskii [104] to prove that any homogeneous set in the sense of Carleson (see [104] for a definition) is a DR set.

Following this section’s trend to make (foolhardy?) conjectures:

Conjecture 8.11. Any essentially perfect compact subset of ℝ is a DR set.

A counterexample would also be very interesting. This is relevant to this paper because, as we have explained, Widom’s theorem (Theorem 1.12) is a kind of poor man’s DR condition.

Related to this: it would be interesting to find a proof of the almost periodicity of every reflectionless two-sided Jacobi matrix with spectrum a finite union of intervals that did not rely on the theory of meromorphic functions on a Riemann surface.

Example 8.12. Remling [90] has some interesting Jacobi matrices that are regular on [−2, 2], not in Nevai class, and have Σ_{ac} = [0, 2]. Related examples for Schrödinger operators appeared earlier in Molchanov [83].

□

9. Continuum Schrödinger Operators

The theory presented earlier was developed by the OP community dealing with discrete (i.e., difference) equations. The spectral theory community knows there are usually close analogies between difference and differential equations, so it is natural to ask about regularity ideas for continuum Schrödinger operators—a subject that does not seem to have been addressed before. We begin this exploration here. This is more a description of a research project than a final report. We will be discursive without proofs.

The first problem that one needs to address is that there is no natural potential theory for infinite unbounded sets. log|x − y|^{−1} is unbounded above and below so Coulomb energies can go to −∞. Moreover, the natural measures are no longer probability measures. There is no reasonable notion of capacity, even of renormalized capacity. But at least sometimes there is a natural notion of equilibrium measure and equilibrium potential.

Consider E = [0, ∞). We may not know the precise right question but we know the right answer: For V = 0, the solutions of −u′′ +V u = λu with u(0) = 0 are u(x) = C sinh(x\sqrt{−λ}), and so

lim_{x→∞} \frac{log|u(x)|}{x} = \sqrt{−λ}  (9.1)

which must be the correct analog of the potential theorist Green’s function. And there is a huge literature on continuum density of states, which for this case is

dρ(λ) = χ_{[0,∞)}(λ)(λ)^{−1/2}(2π)^{−1} dλ  (9.2)

This comes from noting the eigenvalues on [0, 1] with u(0) = u(L) = 0 boundary conditions are (\frac{πn}{L})^{2}, n = 1, 2, . . .. Here is a first attempt to find the right question.

It is the derivative of ∫ log|x−y|^{−1} dμ(x) that is a Herglotz function, so we make

Definition. We say dν is an equilibrium measure associated to a set E ⊂ [a, ∞) for some a, if and only if there is a Herglotz function, F_{E}(z), on ℂ so that

(i) Im F (λ + i0) is supported on λ ∈ E. (ii) Re F (λ + i0) = 0 for a.e. λ ∈ E. (iii) F (λ) → 0 as λ → −∞.

(iv) π^{−1}F (λ + iε) dλ ⟶^{w} dν(λ)

(v) For any bounded connected component (a, b) of ℝ \ E, we have

∫_{a}^{b} F (λ) dλ = 0  (9.3)

We will say dν is normalized if

F (λ) ∼ \frac{1}{2} (−λ)^{−1/2}(1 + o(1))  (9.4)

near −∞.

The reason for choosing (9.3) and (9.4) will be made clear shortly. Once we have F, we define the equilibrium potential of E by

Φ_{E}(z) = Re(∫_{x_{0}}^{z} F (ω) dω)  (9.5)

where x_{0} ∈ E and the integral is in a path in ℂ \ [a, ∞) with a = inf{y ∈ ℝ | y ∈ E}. That Re F = 0 on E and that (9.3) holds show Φ_{E} is independent of x_{0}. (9.3) also implies Φ_{E}(z) = 0 on E. For this reason, we need to take E = σ_{ess}(−\frac{d^{2}}{dx^{2}} + V), not σ(−\frac{d^{2}}{dx^{2}} + V).

With (9.4), we have

Φ_{E}(z) = Re(\sqrt{−z})(1 + o(1))  (9.6)

near −∞. We can explain why we normalize as we do. For regular situations, we expect that the absolute value of the eigenfunction, ψ_{z}(x), analogous to OPs (see below) are asymptotic to

exp(xΦ_{E}(z))

as x → ∞. This, in turn, is related to integrals of the negative of the real part of

m(z, x) = \frac{η′_{z}(x)}{η_{z}(x)}  (9.7)

where η is the solution of L^{2} at infinity.

It is a result of Atkinson [4] (see also [45]) that in great generality that as |z| → ∞, −\frac{d^{2}}{dx^{2}} + V is bounded from below in sectors about (−∞, a) and, in general, in sectors |arg z| ∈ (c, π − ε),

m(z, λ) = −\sqrt{−z} + o(1)  (9.8)

ψ should grow as the inverse of η, so Φ ∼ −m as z → −∞.

This is stronger than (9.6) (if one can interchange limits x → ∞ and z → ∞) since the error in (9.6) is o(1)\sqrt{−z}, while in (9.7) it is o(1). The lack of a constant term is an issue to be understood.

If we take E = [0, ∞) since F′ > 0 on (−∞, 0), we have F > 0 on (−∞, 0), and so log F (x + i0) has boundary values 0 on (−∞, 0) and \frac{1}{2} on (0, ∞). This plus log F (z) = o(z) at −∞ uniquely determine log F, and so F, up to an overall constant which is fixed by the normalization yielding

F (z) = \frac{1}{2\sqrt{−z}}  (9.9)

so there is a unique “potential” for [0, ∞) that gives the right Φ(z) = \sqrt{−z}.

Similarly, for a finite number of gaps removed from [0, ∞), one gets a unique F. Craig’s argument yields F up to positions of zeros in the gap, which are then fixed by (9.1).

Open Project 9.1. Develop a formal theory of equilibrium measures and equilibrium potentials for unbounded sets that are “close” to [0, ∞) (e.g., one might require that E \ [0, ∞) has finite Lebesgue measure). Can one understand the o(1) in (9.8) from this theory?

With potentials in hand, we can define regularity. We recall first that given any V on [0, ∞) which is locally in L^{1}, one can define the regular solution, ψ(x, z), obeying

−ψ′′(x, z) + V (x)ψ(x, z) = zψ(x, z)  (9.10)
ψ(0, z) = 0 ψ′(0, z) = 1  (9.11)

Here ψ is C^{1} (and so, locally bounded), its second distributional derivative is L^{1}, and obeys (9.10) as a distribution. For fixed x, ψ is an entire function of x of order \frac{1}{2}. If ψ is not L^{2} at infinity for (one and hence all) z ∈ ℂ_{+}, V is called limit point at infinity and then there is a unique selfadjoint operator H which is formally −\frac{d^{2}}{dx^{2}} + V (x) with u(0) = 0 boundary conditions. We only want to consider the case where H is bounded below (which never happens if V is not limit point). η_{z}(x) is then the solution L^{2} at ∞ determined up to a constant, so

m(x, z) = \frac{η′_{z}(x)}{η_{z}(x)}  (9.12)

is determined by V.

Definition. Let E = σ_{ess}(H). We say H is regular if and only if for all z ∉ σ(H),

lim sup_{x→∞} \frac{1}{x} log|ψ(x, z)| = Φ_{E}(z)  (9.13)

Of course, for this to make sense, E has to be a set for which there is a potential. This will eliminate a case like V (x) = x^{2} where σ_{ess}(H) is empty). We expect the following should be easy to prove:

Metatheorem 9.2. (a) If H is regular, for z ∉ σ(H), lim sup in

(9.13) can be replaced by lim.

(b) H is regular if and only if (9.13) holds q.e. on E (where Φ_{E}(z) is q.e. = 0).

(c) If H is regular, the density of states exists and equals the equilibrium measure for E.

(d) Conversely, if the density of states exists and equals the equilibrium measure for E_{0}, either H is regular or else the spectral measure for H is supported on a set of capacity zero.

(e) If H is regular and

lim_{n→∞} ∫_{n}^{n+1}|(δV)(x)| dx = 0

then H + δV is also regular.

Remarks. 1. Here capacity zero and q.e. are defined in the usual way, that is, any probability measure of compact support contained in E has infinite Coulomb energy.

2. By density of states, we mean the following (see [11, 13, 54, 61, 65, 84, 85]). Take H_{L} to be the operator −\frac{d^{2}}{dx^{2}} +V with u(0) = 0 boundary conditions on L^{2}([0, L], dx). This has infinite but discrete spectrum E_{1,L} < E_{2,L} < E_{3,L} < … (the solutions of ψ(L, z) = 0). Let dν_{L} be the infinite measure that gives weight \frac{1}{L} to each E_{j,L}. If w-lim dν_{L} (as functions on continuous functions of compact support) exist, we say the density of states exists and the limit is called the density of states.

3. (e) should follow from a standard use of an iterated DuHamel’s formula.

Open Project 9.3. Verify Metatheorem 9.2 and explore, in particular, analogs of (a) Widom’s theorem, Theorem 1.12

(b) The Stahl–Totik criterion, Theorem 1.13

(c) For ergodic continuum Schrödinger operators, the analog of Theorem 1.16.

Appendix A: A Child’s Garden of Potential Theory in the Complex Plane

We summarize the elements of potential theory relevant to this paper. For lucid accounts of the elementary parts of the theory, see the appendix of Stahl–Totik [105], Martinez-Finkelshtein [81], and especially Ransford [88]. More comprehensive are Helms [50], Tsuji [113], and especially Landkof [73]. We will try to sketch some of the most important notions in remarks but refer to the texts, especially for the more technical aspects.

The two-dimensional Coulomb potential is log|x−y|^{−1} which has two lacks compared to the more familiar |x−y|^{−1} of three dimensions: It is neither positive nor positive definite. We will deal with lack of positivity by only considering measures of compact support, and conditional positive definitiveness can replace positive definitiveness in some situations.

If μ is a positive measure of compact support on ℂ, its potential is defined by

Φ_{μ}(x) = ∫ log|x − y|^{−1} dμ(y)  (A.1)

Because μ has compact support, log|x − y|^{−1} is bounded below for x fixed, so if we allow the value +∞, Φ_{μ} is always well defined and Fubini’s theorem is applicable and implies that for another positive measure, ν, also of compact support, we have

∫ Φ_{μ}(x) dν(x) = ∫ Φ_{ν}(x) dμ(x)  (A.2)

Sometimes it is useful to fix M > 0 and define the cutoff

Φ_{μ}^{M}(x) = ∫ log[min(M, |x − y|^{−1})] dμ(y)  (A.3)

Φ_{μ}^{M} is continuous and Φ_{μ}^{M} is an increasing sequence in M, so

Proposition A.1. Φ_{μ}(x) is harmonic on ℂ \ supp(dμ), lower semicontinuous on ℂ, and superharmonic there.

One might naively think that Φ_{μ}(x) only fails to be continuous because it can go to infinity and that it is continuous in the extended sense—but that is wrong!

Example A.2. Let x_{n} = −n^{−1} and let

dμ = ∑_{n=1}^{∞} n^{−2}δ_{x_{n}}  (A.4)

Then Φ_{μ}(x_{n}) = ∞ and x_{n} → 0, but

Φ_{μ}(0) = ∑_{n=1}^{∞} n^{−2} log n < ∞  (A.5)

Notice that this is consistent with lower semicontinuity, that is, Φ_{μ}(lim x_{n}) ≤ lim inf Φ_{μ}(x_{n}). Also notice, given Hydrogen atom spectra, that this example is relevant to spectral theory.

Lest you think this kind of behavior is only consistent with unbounded Φ_{μ}, one can replace δ_{x_{n}} by a smeared out probability measure, η_{n} (using equilibrium measures on a small interval, I_{n}, about x_{n}), so Φ_{η_{n}} = λn^{2} on I_{n} and have with μ = ∑ n^{−2}η_{n}, then Φ_{μ} is bounded, Φ_{μ}(x_{n}) ≥ λ while Φ_{μ}(0) ≤ 2 ∑_{n=1}^{∞} n^{−2} log n. Hence one loses continuity for λ large. □

The following is sometimes useful:

Proposition A.3. If Φ_{μ}(x) restricted to supp(μ) is continuous, then Φ_{μ} is continuous on ℂ.

Remarks. 1. The general case can be found in [73, Theorem 1.7]. Here we will sketch the case where supp(μ) ⊂ ℝ which is most relevant to OPRL.

2. By lower semicontinuity, if Φ_{μ} fails to be continuous on ℂ, there exists z_{n} → z_{∞}, so Φ_{μ}(z_{n}) → a > Φ_{μ}(z_{∞}). Continuity off supp(μ) is easy, so we must have z_{∞} ∈ ℝ (since we are supposing supp(μ) ⊂ ℝ).

3. If w, x, y ∈ ℝ, then |w − x − iy|^{−1} ≤ |w − x|^{−1}, so

Φ_{μ}(x + iy) ≤ Φ_{μ}(x)

and thus lim inf Φ_{μ}(Re z_{n}) ≥ a > Φ_{μ}(Re z_{∞}), so without loss, we can suppose z_{n} are real.

4. If (α, β) ⊂ ℝ \ supp(μ) with α, β ∈ supp(μ), it is easy to see that Φ_{μ}(x) is continuous when restricted to [α, β] (using monotone convergence at the endpoints) and convex on [α, β] since log|x|^{−1} is convex. Thus, max_{[α,β]} Φ_{μ}(x) = max(Φ_{μ}(α), Φ_{μ}(β)). From this, it is easy to see that if such a z_{n} ∈ ℝ exists, one can take z_{n} ∈ supp(μ) and so get a contradiction to the assumed continuity of Φ_{μ} restricted to supp(μ).

The energy or Coulomb energy of μ is defined by

E(μ) = ∫ Φ_{μ}(x) dμ(x) = ∫ log|x − y|^{−1} dμ(x) dμ(y)  (A.6)

where, again, the value +∞ is allowed. If E ⊂ ℂ is compact, we say it has capacity zero if E(μ) = ∞ for all μ ∈ M_{+,1}(E), the probability measures on E. If E does not have capacity zero, then the capacity, C(E), of E is defined by

C(E) = exp(− min(E(ρ) | ρ ∈ M_{+,1}(E)))  (A.7)

One indication that this strange-looking definition is sensible is seen by, as we will show below (see Example A.17),

C([a, b]) = \frac{1}{4} (b − a)  (A.8)

It is useful to define the capacity of any Borel set. For bounded open sets, U,

C(U) = sup(C(K) | K ⊂ U, K compact)  (A.9)

and then for arbitrary bounded Borel X,

C(X) = inf(C(U) | X ⊂ U, U open)  (A.10)

It can then be proven (see [73, Thm. 2.8]) that

C(X) = sup(C(K) | K ⊂ X, K compact)  (A.11)

for any Borel sets and that (A.10) holds for compact X. In particular, C(X) = 0 if and only if E(μ) = ∞ for any measure μ with supp(μ) ⊂ X.

The key technical fact behind Theorem 1.16 is the following:

Proposition A.4. If C(X) > 0 for a Borel set X, there exists a probability measure, μ, supported in X so that Φ_{μ}(x) is continuous on ℂ.

Remarks. 1. Let μ have finite energy so ∫ Φ_{μ}(x) dμ(x) < ∞. By Lusin’s theorem (see, e.g., the remark after Theorem 6 of Appendix A of Lax [76] for the truly simple proof), we can find compact sets K ⊂ supp(dμ) so μ(K) > 0 and Φ_{μ} ↾ K is continuous.

2. Let ν = μ ↾ K, that is, ν(S) = μ(S ∩ K). Since μ(K) > 0, ν is a nonzero measure. By general principles, both Φ_{ν} and Φ_{μ−ν} are lower semicontinuous on K, so since Φ_{μ} is continuous,

Φ_{ν} = Φ_{μ} − Φ_{μ−ν}  (A.12)

is upper semicontinuous on K. Thus, Φ_{ν} is continuous on K (since Φ_{μ} is continuous on K, it is bounded there, so Φ_{ν} and Φ_{μ−ν} are both bounded there, so there are no ∞–∞ cancellations in (A.12)).

3. By Proposition A.3, Φ_{ν} is continuous on ℂ.

Now suppose μ is an arbitrary measure of compact support and that C({x | Φ_{μ}(x) = ∞}) > 0. Then, by the above proposition, there is an η supported on that set with Φ_{η} continuous and so bounded above on supp(dμ). Thus,

∫ Φ_{η}(x) dμ(x) < ∞  (A.13)

On the other hand, Φ_{μ}(x) = ∞ on supp(dη), so

∫ Φ_{μ}(x) dη(x) = ∞  (A.14)

This contradicts (A.2). We thus see that the last proposition implies:

Corollary A.5. For any measure of compact support, μ, {x | Φ_{μ}(x) = ∞} has capacity zero.

A main reason for defining capacity for any Borel set is that it lets us single out sets of capacity zero (also called polar sets), which are very thin sets (e.g., of Hausdorff dimension zero; see Theorem A.20). We say an event (i.e., a Borel set) occurs quasi-everywhere (q.e.) if and only if it fails on a set of capacity zero. “Nearly everywhere” is also used. A countable union of capacity zero sets is capacity zero. Note that if μ is any measure of compact support, with E(μ) < ∞, then E(μ ↾ E) < ∞ for any compact E (because log|x − y|^{−1} is bounded below) and thus, μ(E) = 0 if C(E) = 0. It follows (using (A.11)) that

Proposition A.6. If E(μ) < ∞, then μ(X) = 0 for any X with C(X) = 0.

Here is an important result showing the importance of sets of zero capacity. It is the key to Van Assche’s proof in Section 4 and the proof of our new Theorem 1.16 in Section 7.

Theorem A.7. Let ν_{n}, ν be measures with supports contained in a fixed compact set K and sup_{n} ν_{n}(K) < ∞. If ν_{n} → ν weakly, then

lim inf_{n→∞} Φ_{ν_{n}}(x) ≥ Φ_{ν}(x)  (A.15)

for all x ∈ ℂ and equality holds q.e.

Remarks. 1. (A.15) is called the “Principle of Descent” and the equality q.e. is the “Upper Envelope Theorem.”

2. Suppose ν_{n} has a point mass of weight \frac{1}{2^{n}} at \{\frac{j}{2^{n}} \}_{j=0}^{2^{n}−1}. Then dν_{n} → dx ≡ dν, Lebesgue measure. Φ_{ν_{n}}(\frac{j}{2^{n}}) = ∞ so lim inf Φ_{ν_{n}}(x) = ∞ at any dyadic rational, while Φ_{ν}(x) < ∞ for all x. This shows equality may not hold everywhere. This example is very relevant to spectral theory. For the Anderson model, we expect lim sup|p_{n}(x)|^{1/n} = e^{γ(x)} for almost all x and lim sup|p_{n}(x)|^{1/n} = e^{−γ(x)} at the eigenvalues. Thus, with ν_{n} the zero counting measure for p_{n}, so Φ_{ν_{n}}(x) = − log|p_{n}(x)|^{1/n}, we have lim inf Φ_{ν_{n}}(x) = −γ(x) for almost all x and γ(x) at the eigenvalue consistent with (A.15), and with (A.15) failing on a capacity zero set, including the countable set of eigenvalues.

3. (A.15) is easy. For Φ_{ν}^{M} is the convolution with a continuous function so lim_{n→∞} Φ_{ν_{n}}^{M}(x) = Φ_{ν}^{M}(x). Since Φ_{ν_{n}}(x) ≥ Φ_{ν_{n}}^{M}(x), we see lim inf_{n→∞} Φ_{ν_{n}}(x) ≥ Φ_{ν}^{M}(x). Taking M → ∞ yields (A.15).

4. Let X be the set of x for which the inequality in (A.15) is strict. Suppose C(X) > 0. Then, by Proposition A.2, there is η supported on X with Φ_{η}(x) continuous so

lim_{n→∞} ∫ Φ_{η}(x) dν_{n} = ∫ Φ_{η}(x) dν  (A.16)

By (A.2) and Fatou’s lemma (Φ_{ν_{n}}(x) is uniformly bounded below),

lim_{n→∞} ∫ Φ_{η}(x) dν_{n} = lim ∫ Φ_{ν_{n}}(x) dη
≥ ∫ lim inf Φ_{ν_{n}}(x) dη
> ∫ Φ_{ν}(x) dη  (A.17)
= ∫ Φ_{η}(x) dν

where (A.17) comes from the assumptions supp(dη) ⊂ X and (A.15) is strict on X. This contradiction to (A.16) shows C(X) = 0, that is, equality holds in (A.15) q.e.

If E_{M}(μ) = ∫ Φ_{μ}^{M} dμ(x), then it is easy to prove E_{M} is weakly continuous and conditionally positive definite in that

μ(ℂ) = ν(ℂ) ⇒ E_{M}(μ − ν) ≥ 0  (A.18)

where boundedness of log(min(|x − y|^{−1}, M)) implies E_{M} makes sense for any signed measure. By taking M to infinity, one obtains

Theorem A.8. The map μ ↦ E(μ) is weakly lower semicontinuous on M_{+,1}(E) for any compact E ⊂ ℂ. Moreover, it is conditionally positive definite in the sense that for μ, ν ∈ M_{+,1}(E), E(μ) < ∞ and E(ν) < ∞ imply

∫ Φ_{ν}(x) dμ(x) ≤ \frac{1}{2} E(μ) + \frac{1}{2} E(ν)  (A.19)

with strict inequality if μ ≠ ν.

Remark. The strict inequality requires an extra argument. One can prove that if μ, ν ∈ M_{+,1}(E) with finite energy, then μ̂(k) − ν̂(k) is analytic in k vanishing at k = 0 and

E(μ) + E(ν) − 2 ∫ Φ_{ν}(x) dμ(x) = \frac{1}{2π} ∫ |\frac{μ̂(k) − ν̂(k)}{k}|^{2} d^{2}k (A.20)

Since the inequality in (A.19) is strict and

E(\frac{1}{2} μ + \frac{1}{2} ν) = \frac{1}{4} E(μ) + \frac{1}{4} E(ν) + \frac{1}{2} ∫ Φ_{ν}(x) dμ(x)  (A.21)

we see that E(μ) is strictly convex on M_{+,1}(E), and thus

Theorem A.9. Let E be a compact subset of ℂ with C(E) > 0. Then there exists a unique probability measure, dρ_{E}, called the equilibrium measure for E, that has

E(ρ_{E}) = log(C(E)^{−1})  (A.22)

The properties of ρ_{E} are summarized in

Theorem A.10. Let E ⊂ ℂ be compact. Let Ω be the unbounded component of ℂ \ E and Ω̃ = ℂ \ (Ω ∪ E) the union of the bounded components of ℂ \ E. Suppose C(E) > 0 and dρ_{E} is its equilibrium measure. Then

(a) For all x ∈ ℂ,

Φ_{ρ_{E}}(x) ≤ log(C(E)^{−1})  (A.23)

(b) Equality holds in (A.23) q.e. on E and on Ω̃.

(c) Strict inequality holds in (A.23) on Ω.

(d) ρ_{E} is supported on ∂Ω, the boundary viewed as a set in ℂ.

(e) Φ_{ρ_{E}} is continuous on ℂ if and only if it is continuous when restricted to supp(dρ_{E}) if and only if equality holds in (A.23) on supp(dρ_{E}).

(f) If I ⊂ E ⊂ ℝ with I = (a, b), then dρ_{E} ↾ I is absolutely continuous with respect to Lebesgue measure, \frac{dρ_{E}}{dx} ↾ I is real analytic, and equality holds in (A.23) on I.

Remarks. 1. For example, if E = ∂𝔻, Ω = ℂ \ \overline{𝔻} and Ω̃ = 𝔻. 2. See [73, 88] for complete proofs.

3. (a)+(b) is called Frostman’s theorem.

4. Equality in (A.23) may not hold everywhere on E; for example, if E = [−1, 1] ∪ {2}, the equilibrium measure gives zero weight to {2}, so is the same as the equilibrium measure for [−1, 1] and that dρ_{E} has inequality on ℂ \ [−1, 1] by (c).

5. If f is supported on supp(dρ_{E}) and f bounded and Borel, and ∫ f dρ_{E} = 0, then (1+εf)dρ_{E} is a probability measure for ε small with E((1+εf) dρ_{E}) < ∞. Since \frac{d}{dε}E((1+εf)dρ_{E}) = 2 ∫ f(x)Φ_{ρ_{E}}(x) dρ_{E}(x), we see Φ_{ρ_{E}}(x) is a constant for dρ_{E}-a.e. x. Since E(ρ_{E}) = ∫ dρ_{E} Φ_{ρ_{E}}(x), the constant must be E(ρ_{E}) = log(C(E)^{−1}). By lower semicontinuity, (A.23) holds on supp(dρ_{E}). Since Φ_{ρ_{E}} is harmonic on ℂ \ supp(dρ_{E}) and goes to −∞ as |x| → ∞, (A.23) holds by the maximum principle.

6. Let η be a probability measure on E with E(η) < ∞. Then

E((1−t)dρ_{E} +tdη) = E(dρ_{E})+t(∫ Φ_{ρ_{E}}(x)[dη−dρ_{E}])+O(t^{2}) (A.24)

Since ∫ dρ_{E}Φ_{ρ_{E}}(x) = E(dρ_{E}) = log(C(E)^{−1}), if η is supported on a set where strict inequality holds in (A.23), E((1−t)dρ_{E} +tdη) < E(dρ_{E}) for small t, violating minimality. Thus the set where (A.23) has inequality cannot support a measure of finite energy, that is, it has zero capacity, proving (b).

7. Since Φ_{ρ_{E}} is harmonic on Ω and goes to −∞ at ∞, the maximum principle implies Φ_{ρ_{E}}(x) cannot take its maximum (which is log(C(E)^{−1})) on Ω. (e) follows from Proposition A.3. (d) is left to the references; see [73, 88].

8. If I ⊂ E ⊂ ℝ, one first shows equality holds in (A.23) on I and that Φ is continuous there. (This uses the theory of “barriers”; see [73, 88]. One can also prove this using periodic Jacobi matrices and approximations; see [102]). Then one can apply the reflection principle to see that Φ_{ρ_{E}} has a harmonic continuation across I. Indeed, Φ_{ρ_{E}} is then the real part of a function analytic on I with zero derivative there. That derivative for Im z > 0 is the real part of

F (z) = ∫ \frac{dρ_{E}(x)}{x − z}  (A.25)

so, by the standard theory of boundary values of Herglotz functions (see [98, Sect. 1.3]), we have that dρ_{E} ↾ I is absolutely continuous and

\frac{dρ_{E}}{dx} = \frac{1}{π} Im F (x + i0)  (A.26)

proving real analyticity of this derivative.

9. The same argument as in Remark 8 applies if I is replaced by an analytic arc with a neighborhood N obeying N ∩ E = I. In particular, if I is an “interval” in ∂𝔻 and I ⊂ E ⊂ ∂𝔻, we have absolute continuity and analyticity on I.

Here is an interesting consequence of (A.2):

Theorem A.11. Let ν be a measure of compact support, E, so that C(E) > 0. Then

Φ_{ν}(x) < ∞  for dρ_{E} a.e. x  (A.27)

Remarks. 1. This can happen even if E(ν) = ∞ so ∫ Φ_{ν}(x) dν(x) = ∞. 2. For (A.23) implies

∫ Φ_{ρ_{E}}(x) dν ≤ log(C(E)^{−1})ν(E) < ∞

so (A.2) implies

∫ Φ_{ν}(x) dρ_{E}(x) < ∞  (A.28)

The following illustrates the connection between potential theory and polynomials:

Theorem A.12 (Bernstein–Walsh Lemma). Let E be a compact set in ℂ with C(E) > 0 and let Ω be the unbounded component of ℂ \ E. Let p_{n} be a polynomial of degree n and let

∥p_{n}∥_{E} = sup_{z∈E} |p_{n}(z)|  (A.29)

Then for all z ∈ Ω,

|p_{n}(z)| ≤ C(E)^{−n}∥p_{n}∥_{E}[exp(−nΦ_{ρ_{E}}(z))]  (A.30)

Remarks. 1. This is named after Bernstein and Walsh [121], although the result appears essentially in Szegő [107].

2. Let {z_{j}}_{j=1}^{n} be the zeros of p_{n}. Define

g(z) = log|p_{n}(z)| + nΦ_{ρ_{E}}(z) + n log(C(E))  (A.31)

on Ω ∪ {∞} \ {z_{j}}_{j=1}^{n} = Ω′. g is harmonic on Ω′ including at ∞ since both log|p_{n}(z)| and −nΦ_{ρ_{E}}(z) are n log|z| plus harmonic near ∞. Since g_{n}(z) → −∞ at the z_{j} ∈ Ω, we see

sup_{z∈Ω′} |g(z)| ≤ lim_{δ↓0} [ sup_{\begin{matrix}dist(w,E)=δ \\ w∈Ω \end{matrix}}|g(w)|]  (A.32)

But, by (A.23), g(z) ≤ log|p_{n}(z)|, so

g(z) ≤ log ∥p_{n}∥_{E}

which is (A.30) on Ω′\{∞}. (A.30) holds trivially at the z_{j}, completing the proof.

Following ideas of Craig [24], one can say much more about \frac{dρ_{E}}{dx} when E contains an isolated closed interval:

Theorem A.13. Let E ⊂ ℝ be compact and a < c < d < b so E ∩ (a, b) = [c, d]. Then there exists g real, real analytic, and strictly positive on [c, d] so that

dρ_{E} ↾ [c, d] = g(x)[(d − x)(x − c)]^{−1/2} dx  (A.33)

If E = ∪_{j=1}^{ℓ+1}[a_{j}, b_{j}] with a_{1} < b_{1} < a_{2} < ⋯ < b_{ℓ+1}, there are x_{j} ∈ (b_{j}, a_{j+1}) for j = 1, 2, …, ℓ so that

dρ_{E}(x) = \frac{1}{π} [ ∏_{j=1}^{ℓ} \frac{x − x_{j}}{\sqrt{(x − b_{j})(x − a_{j+1})}}]\frac{1}{\sqrt{(x − a_{1})(b_{ℓ+1} − x)}} dx

(A.34)

Remarks. 1. (A.34) is from Craig [24].

2. The idea behind the proof is simple. One lets F (z) = ∫ \frac{dρ_{E}(x)}{x−z}. By the arguments above, F is pure imaginary on [c, d] as the derivative of Φ_{ρ_{E}}(x). Thus, arg F (x + i0) is \frac{π}{2} on [c, d], and by a simple argument, 0 on [c−δ, c) and π on (d, d+δ]. A Herglotz representation for log F (x+ i0) yields (A.33) and (A.34).

3. The x_{j}’s are uniquely determined by

∫_{b_{j}}^{a_{j+1}} F (x) dx = 0  (A.35)

Recall a set S is called perfect if it is closed and has no isolated points. A standard argument shows that any compact E has a unique decomposition into disjoint sets, D ∪ S where D is a countable set and S is perfect (similarly, any compact E ⊂ ℝ can be written Z ∪ F where Z has Lebesgue measure zero and F is essentially perfect, that is, |F ∩ (x − δ, x + δ)| > 0 for any x ∈ F and δ > 0).

Similarly, we call a set P potentially perfect (the terminology is new) if P is closed and C(P ∩{x | |x−x_{0}| < δ}) > 0 for all x_{0} ∈ P and δ > 0. It is easy to see that any compact E ⊂ ℂ can be uniquely written as a disjoint union E = Q∪P where C(Q) = 0 and P is potentially perfect.

These notions are related to equilibrium measures. If cap(E) > 0 and E = Q ∪ P is this decomposition, then

P = supp(dρ_{E})  (A.36)

In particular, supp(dρ_{E}) = E if and only if E is potentially perfect.

Just as one writes σ(dμ) = σ_{disc}(dμ) ∪ σ_{ess}(dμ), we single out the potentially perfect part of σ(dμ) and call it σ_{cap}(dμ).

Next, we want to state a kind of converse to Frostman’s theorem.

Theorem A.14. Let E ⊂ ℂ be compact. Suppose E is potentially perfect. Let η ∈ M_{+,1}(E) be a probability measure on E with supp(dη) ⊆ E so that for some constant, α,

Φ_{η}(x) = α  dρ_{E}-a.e. x  (A.37)

Then η = ρ_{E}, the equilibrium measure, and α = log(C(E)^{−1}).

Remark. By lower semicontinuity, Φ_{η}(x) ≤ α on supp(dρ_{E}) = E by hypothesis. Thus,

E(η) = ∫ Φ_{η}(x) dη(x) ≤ α < ∞  (A.38)

so η must give zero weight to zero capacity sets. Thus, Φ_{ρ_{E}}(x) = log(C(E)^{−1}) for dη-a.e. x and thus,

∫ Φ_{ρ_{E}}(x) dη(x) = log(C(E)^{−1})  (A.39)

By (A.2) and (A.37),

LHS of (A.39) = ∫ Φ_{η}(x) dρ_{E}(x) = α

Thus, α = log(C(E)^{−1}), and by (A.38) and uniqueness of minimizers, η = ρ_{E}.

Next, we note that the Green’s function for a compact E ⊂ ℂ is defined by

G_{E}(z) = −Φ_{ρ_{E}}(z) + log(C(E)^{−1})  (A.40)

It is harmonic on ℂ \ E, G_{E}(z) − log|z| is harmonic at infinity, and G_{E}(z) has zero boundary values q.e. on E. Notice that G_{E}(z) ≥ 0 on ℂ. If

lim_{z→E} G_{E}(z) = 0  (A.41)

in the sense that

lim_{δ↓0} sup_{dist(z,E)<δ} G_{E}(z) = 0  (A.42)

we say E is regular for the Dirichlet problem (just called regular). By Theorem A.10(c), this is true if and only if Φ_{ρ_{E}}(x) = log(C(E)^{−1}) for all x ∈ E. By Theorem A.13, this is true for finite unions of disjoint closed intervals.

Notice that the Bernstein–Walsh lemma (A.30) can be rewritten

|p_{n}(z)| ≤ ∥p_{n}∥_{E} exp(nG_{E}(z))  (A.43)

Closely related are comparison theorems and limit theorems. We will state them for subsets of ℝ:

Theorem A.15. Let E_{1} ⊂ E_{2} ⊂ ℝ be compact sets. Then

(i) C(E_{1}) ≤ C(E_{2}) (ii)

G_{E_{2}}(z) ≤ G_{E_{1}}(z)  (A.44)
for all z ∈ ℂ

(iii)

dρ_{E_{2}} ↾ E_{1} ≤ dρ_{E_{1}}  (A.45)

(iv) If I = (a, b) ⊂ E_{1}, then on I,

\frac{dρ_{E_{2}}}{dx} ≤ \frac{dρ_{E_{1}}}{dx}  (A.46)

for all x ∈ I.

Theorem A.16. Let E_{1} ⊃ E_{2} ⊃ … be compact subsets of ℝ. Let E_{∞} = ∩_{j=1}^{∞}E_{j}. Then (i)

lim_{n→∞} C(E_{n}) = C(E_{∞})  (A.47)

(ii) ρ_{E_{n}} → ρ_{E_{∞}} weakly (iii) For z ∈ ℂ \ E_{∞},

lim_{n→∞} G_{E_{n}}(z) = G_{E_{∞}}(z)  (A.48)

and (A.48) holds q.e. on E_{∞}. (iv) If I = (a, b) ⊂ E_{∞}, then uniformly on compact subsets of I,

lim_{n→∞} \frac{dρ_{E_{n}}}{dx} = \frac{dρ_{E_{∞}}}{dx}  (A.49)

Remarks. 1. (A.45) has the pleasing physical interpretation that if one conductor is connected to another, charge leaks out in a way that there is less charge everywhere in the original conductor.

2. Part (i) of each theorem is easy. For Theorem A.15, it follows from the minimum energy definition. For Theorem A.16(i), we note that if U is open with E_{∞} ⊂ U, then eventually E_{n} ⊂ U, so (A.10) implies (A.47).

3. One proves (ii)–(iv) of Theorem A.15 first for E, a finite union of closed intervals, then proves Theorem A.16, and then for general compact E_{∞} ⊂ ℝ defines E_{n} = \{x | dist(x, E_{∞}) ≤ \frac{1}{n}\} and proves ∩_{n}E_{n} = E_{∞} and each E_{n} is a finite union of closed intervals. Theorem A.16 then yields Theorem A.15 for general E’s (see [102]).

4. For E_{1}, E_{2} finite union of closed intervals and z ∉ E_{2}, one gets (A.44) by noting the difference G_{E_{1}}(z) − G_{E_{2}}(z) is harmonic on ℂ \ E_{2}, zero on E_{1}, and positive on E_{2} \ E_{1}, where G_{E_{1}} > 0 and G_{E_{2}} = 0. The inequality for z ∈ E_{2} then follows from the fact that any subharmonic function h obeys

h(z_{0}) = lim_{r↓0} \frac{1}{2πr} ∫_{0}^{2π} h(z_{0} + re^{iθ}) dθ  (A.50)

5. For this case, one gets (A.45)/(A.46) by noting that (A.26) can be rewritten

\frac{dρ_{E}(x_{0})}{dx} = \frac{1}{π} lim_{ε↓0} ε^{−1}G_{E}(x_{0} + iε)  (A.51)

for x_{0} ∈ E and using (A.44) for x ∈ E_{1}.

6. If {E_{n}}_{n=1}^{∞}, E_{∞} are as in Theorem A.16 and dη is a weak limit point of dρ_{E_{n}}, then η is supported on E_{∞}, and by lower semicontinuity of the Coulomb energy E,

E(η) ≤ lim E(ρ_{E_{n}})
= lim log(C(E_{n})^{−1})
= log(C(E_{∞})^{−1})

by (A.47), so η = ρ_{E_{∞}}, that is, ρ_{E_{n}} → ρ_{E_{∞}} weakly. (A.48) then follows for z ∉ E_{∞} from (A.47) and continuity of Φ_{ν}(z) in ν for z ∉ supp(dν). (A.50) implies convergence for z ∈ E_{∞}.

7. (A.49) follows from ρ_{E_{n}} → ρ_{E_{∞}} and uniform bounds on derivatives of \frac{dρ}{dx} on I, which in turn follow from the proof of (A.33).

Example A.17. Harmonic functions are conformally invariant, which means (since Green’s functions are normalized by G_{E}(z) = log|z| + O(1) near infinity and boundary values of 0 on E), if Q is an analytic bijection of ℂ\\overline{𝔻}∪\{∞\} to Ω∪\{∞\} with Q(z) = Cz+O(1) near infinity, then, since log|z| is the Green’s function for E = ∂𝔻, log|Q^{−1}(z)| is the Green’s function for E and C its capacity. In particular, with Q(z) = z + \frac{1}{z}, we see

C([−2, 2]) = 1  (A.52)

and consistent with (A.34)

dρ_{[−2,2]}(x) = \frac{1}{π} \frac{dx}{\sqrt{4 − x^{2}}}  (A.53)

Notice that, by scaling, if λ > 0 and λE = {λz | z ∈ E} and μ ∈ M_{+,1}(E) is mapped to μ_{λ} in M_{+,1}(λE) by scaling, then

E(μ_{λ}) = − log(λ) + E(μ)  (A.54)

so

C(λE) = λC(E)  (A.55)

This plus translation invariance shows

C([a, b]) = \frac{1}{4} (b − a)

Now, let E ⊂ ℝ, E_{−} = E ∩ (−∞, 0), E_{+} = E \ E_{−}, and for a > 0, let E^{(a)} = E_{−} ∪ (E_{+} + a). Let dρ_{E} be equilibrium measure for E. Since log|x + a − y|^{−1} < log|x − y|^{−1} for x > 0, y < 0, a > 0, we see

E(ρ ↾ E_{−} + T_{a}(ρ ↾ E_{+})) < E(ρ)  (A.56)

(where T_{a}μ is the translate of μ). Thus

C(E^{(a)}) ≥ C(E)

This is an expression of the repulsive nature of the Coulomb force! Thus, by joining together all the pieces of E (via a limiting argument), one sees that if |E| is the Lebesgue measure of E ⊂ ℝ, then

C(E) ≥ \frac{1}{4} |E|  (A.57)

and sets of capacity zero have Lebesgue measure zero. □

Example A.18. Let dμ be the conventional Cantor measure on [0, 1] which can be thought of as writing x = ∑ \frac{a_{n}(x)}{3^{n}} with a_{n} = 0, 1 or 2 and taking dμ as the infinite product of measures given weight \frac{1}{2} to a_{n} = 0 or 2. Looking at a_{1}, we get the usual two pieces of mass \frac{1}{2} with minimum distance \frac{1}{3} between them. Look at a_{1}, …, a_{k} and we have 2^{k} pieces of mass 2^{−k} and minimum distance 3^{−k}. Given x, y in the Cantor set, dist|x − y| < 3^{−k} if and only if they are in the same pieces, that is,

μ({x | |x − y| < 3^{−k}}) = 2^{−k}

Thus

∫ log|x − y|^{−1} dμ(x) dμ(y) ≤ ∑_{k} (k + 1)(log 3) μ({x, y | |x − y| < 3^{−k}})

= ∑_{k} (k + 1)(log 3) 2^{−k} < ∞

This shows the Cantor set has positive capacity. Generalizing, we get sets of any Hausdorff dimension α > 0 with positive capacity. In fact, as we will see shortly, any set of positive Hausdorff dimension has positive capacity. □

Example A.19. Fix a > 0 and let E = (−\frac{a}{2} − Δ, −\frac{a}{2}) ∪ (\frac{a}{2}, \frac{a}{2} + Δ) where Δ = \frac{4}{a}. When a is very large, the equilibrium measure is very close to the average of the equilibrium measure for the two individual intervals. This measure has energy approximately

2 \frac{1}{4} log(\frac{4}{Δ}) + 2 \frac{1}{4} log(a) = 0

so the asymptotic capacity is 1. This phenomenon of distant pieces of individually small capacity having total capacity bounded away from zero is a two-dimensional phenomenon. □

Sets of capacity zero not only have zero Lebesgue measure, but they also have zero α-dimensional Hausdorff measure for any α > 0:

Theorem A.20. Any compact set E of capacity zero has zero Hausdorff dimension.

Remarks. 1. We will sketch a proof where E ⊂ ℝ. What one needs to do, for any α > 0, ε > 0, is to find a cover of E by intervals I_{1}, …, I_{n} … of length |I_{j}| so that

∑ |I_{j}|^{α} < ε  (A.58)

2. We begin by noting that there is a measure μ (not necessarily supported by E) so that Φ_{μ}(x) = ∞ for all x ∈ E (we do not care that E is exactly the set where Φ_{μ} = ∞ but note that by combining the ideas here with Corollary A.5, one can show E is the set where some potential is infinite if and only if E is a G_{δ}-set of zero capacity). Here is how to construct μ. Let E_{m} = \{x ∈ ℝ | dist(x, E) ≤ \frac{1}{m}\}. E_{m} is a finite union of closed intervals and, by (A.10), C(E_{m}) ↓ 0. Pass to a subsequence Ẽ_{m}, so C(Ẽ_{m}) ≤ exp(−\frac{1}{m^{2}} ) so Φ_{ρẼm}(x) ≥ m^{2} on Ẽ_{m} and so on E. Let μ = ∑_{m} m^{−2}ρẼ_{m}. μ is a finite measure with Φ_{μ} = ∞ on E.

3. Let x ∈ E. Suppose for some α > 0 and c > 0, we have with I_{r}^{x} = (x − r, x + r),

μ(I_{r}^{x}) ≤ c(2r)^{α}  (A.59)

Then picking r = 2^{−n}, we see (with n_{0} large and negative so supp(dμ) ⊂ I_{2^{−n}}^{x})

∫ log|y − x|^{−1} dμ(y) ≤ ∑_{n=n_{0}}^{∞}[(n + 1) log 2]μ(I_{2^{−n}}^{x})
< ∞

We thus conclude (A.59) always fails, that is, for any x ∈ E and any α,

lim sup_{r↓0} (2r)^{−α}μ(I_{r}^{x}) = ∞  (A.60)

4. Given α > 0, δ > 0 fixed, by (A.60), we can find for each x ∈ E, so I_{r_{x}}^{x} with

μ(I_{r_{x}}^{x}) ≥ δ^{−1}(2r_{x})^{α}  (A.61)

5. There is standard covering lemma used in the proof of the Hardy– Littlewood maximal theorem (see [64], p. 74, the proof of the lemma) that we can find a suitable sequence x_{j} with

I_{r_{x_{j}}}^{x_{j}} ∩ I_{r_{x_{k}}}^{x_{k}} = ∅  (A.62)

and

⋃_{x} I_{r_{x}}^{x} ⊂ ⋃_{j=1}^{∞} I_{4r_{x_{j}}}^{x_{j}}  (A.63)

6. Thus, {I_{4r_{x_{j}}}^{x_{j}} } cover E and, by (A.61),

∑_{j} |I_{4r_{x_{j}}}^{x_{j}} |^{α} ≤ 4^{α} ∑_{j} |I_{r_{x_{j}}}^{x_{j}} |^{α}

≤ 4^{α}δ ∑_{j} μ(I_{r_{x_{j}}}^{x_{j}})

≤ 4^{α}δμ(ℝ)

by (A.62). Since δ is arbitrary, we have the required covers to see dim(E) = 0.

A final comparison result will be needed in Appendix B:

Theorem A.21. Let μ, ν be two probability measures on ℝ so that for all z near infinity,

Φ_{μ}(z) ≥ Φ_{ν}(z)  (A.64)

Then μ = ν. In particular, if either Φ_{μ}(z) ≥ Φ_{ρ_{E}}(z) or Φ_{μ}(z) ≤ Φ_{ρ_{E}}(z) for all z near infinity, then μ = ρ_{E}.

Remark.

Φ_{μ}(z) + log|z| = − Re ∫ log(1 − \frac{w}{z}) dμ(w)

= Re[ ∑_{n=1}^{∞} z^{−n} ∫ w^{n} dμ(w)]

Thus Φ̃_{μ}(z) ≡ Φ_{μ}(z)+log|z| is harmonic near infinity with Φ̃_{μ}(∞) = 0. Thus, Φ_{μ} − Φ_{ν} = Φ̃_{μ} − Φ̃_{ν} is harmonic and vanishing at ∞. The only way it can have a definite sign near infinity is if it is identically 0. By harmonicity off ℝ, Φ_{μ} = Φ_{ν} on ℂ \ ℝ and then, by (A.50), on ℝ. Thus, Φ_{μ} = Φ_{ν} as distributions. Since −ΔΦ_{μ} = 2πμ, we see μ = ν.

Appendix B: Chebyshev Polynomials, Fekete Sets, and Capacity

For further discussion of the issues in this appendix, see Andrievskii– Blatt [3], Goluzin [48], and Saff–Totik [91] whose discussion overlaps ours here. Let E ⊂ ℂ be compact and infinite. The Chebyshev polynomials, T_{n}(x), are defined as those monic polynomials of degree n which minimize

∥T_{n}∥_{E} = sup_{z∈E} |T_{n}(z)|  (B.1)

By T_{n}^{R}, the restricted Chebyshev polynomials, we mean the monic polynomials, all of whose zeros lie in E, which minimize ∥·∥_{E} among all such polynomials. They can be distinct: for example, if E = ∂𝔻, T_{n}(z) = z^{n} while T_{n}^{R}(z) = 1 + z^{n} (not unique). It can be proven (see [113, Thm. III.23]) that Chebyshev (but not restricted Chebyshev) polynomials are unique.

Clearly,

∥T_{n}∥_{E} ≤ ∥T_{n}^{R}∥_{E}  (B.2)

and since T_{n}T_{m} is a monic polynomial of degree n + m,

∥T_{n+m}∥_{E} ≤ ∥T_{n}∥_{E}∥T_{m}∥_{E}  (B.3)

so lim_{n→∞} ∥T_{n}∥_{E}^{1/n} exists, and similarly, so does lim_{n→∞} ∥T_{n}^{R}∥_{E}^{1/n}. An n point Fekete set is a set {z_{j}}_{j=1}^{n} ⊂ E that maximizes

q_{n}(z_{1}, …, z_{n}) = ∏_{i≠j}|z_{i} − z_{j}|  (B.4)

There are n(n − 1) terms in the product and the Fekete constant is defined by

ζ_{n}(E) = q_{n}(z_{1}, …, z_{n})^{1/n(n−1)}  (B.5)

for the maximizing set. The set may not be unique: for example, if E = ∂𝔻 and ω_{n} is an nth root of unity, {z_{k} = z_{0}ω_{n}^{k}} is a minimizer for any z_{0} ∈ ∂𝔻.

Let z_{1}, …, z_{n+1} be an n + 1-point Fekete set. For each j,

∏_{\begin{matrix}k,ℓ≠j \\ ℓ≠k \end{matrix}} |z_{k} − z_{ℓ}| ≤ ζ_{n}^{n(n−1)}  (B.6)

Thus, taking the product over the n + 1 values of j and noting that each z_{k} − z_{ℓ} occurs n − 1 times,

[ζ_{n+1}^{(n+1)n}]^{n−1} ≤ [ζ_{n}^{n(n−1)}]^{n+1}  (B.7)

so ζ_{n} is monotone decreasing. Thus ζ_{n} has a limit, called the transfinite diameter. The main theorem relating these notions and capacity is Theorem B.1. For any compact set E ⊂ ℂ, we have

C(E) ≤ ∥T_{n}∥_{E}^{1/n} ≤ ∥T_{n}^{R}∥_{E}^{1/n} ≤ ζ_{n+1}(E)  (B.8)

Moreover,

lim_{n→∞} ζ_{n}(E) ≤ C(E)  (B.9)

(so all limits equal C(E)). Finally, if C(E) > 0,

(i) The normalized density of Fekete sets converges to dρ_{E}, the equilibrium measure for E.

(ii) If E ⊂ ℝ, the normalized zero counting measure for T_{n} and for T_{n}^{R} converges to dρ_{E}.

Remarks. 1. Normalized densities and zero counting measure are the point measures that give weight k/n to a point in the set of multiplicity k (for Fekete sets, k = 1, but for polynomials there can be zeros of multiplicity k > 1).

2. If E = ∂𝔻, T_{n}(z) = z^{n}, so (ii) fails for T_{n}. If E = 𝔻, T_{n}^{R}(z) = z^{n} and (ii) fails for T_{n}^{R} also. It can be shown that if E ⊂ ∂𝔻, E ≠ ∂𝔻, (ii) also holds.

3. Fekete sets have the interpretation of sets minimizing the point Coulomb energy ∑_{j≠k} log|z_{j} − z_{k}|^{−1}. Parts of this theorem can be interpreted as saying the point minimizer and associated energy without self-energies converge to the minimizing continuum distribution and energy, which is physically pleasing!

4. The equality of lim ζ_{n} and lim ∥T_{n}∥^{1/n} is due to Fekete [38]. The rest is due to Szegő [107], whose proof we partly follow.

5. Stieltjes [106] considered what we call Fekete sets for E = [−1, 1], proving that, in that case, the set is unique and consists of 1, −1, and the n − 2 zeros of a suitable Jacobi polynomial (see [108]). The general set up is due to Fekete [38].

6. When E ⊂ ∂𝔻, there are two other sets of polynomials related to minimizing ∥P_{n}∥_{∞,E}. We can restrict to either

(a) “Quasi-real” monic polynomials, that is, degree n polynomials, so for some φ, e^{−iφ}e^{−inθ/2}P_{n}(e^{iθ}) is real for θ real (these are exactly polynomials for which P_{n}^{∗}(z) = e^{−2iφ}P_{n}(z) where ^{∗} is the Szegő dual). Equivalently, zeros are symmetric about ∂𝔻.

(b) Monic P_{n} all of whose zeros lie on ∂𝔻. These Chebyshev-like polynomials are used in [100].

Since there are classes of polynomials between all monic and monic with zeros on E, the nth roots of the norms also converge to C(E).

∥T_{n}∥_{E}^{1/n} ≤ ∥T_{n}^{R}∥_{E}^{1/n} is (B.2). Here is the last inequality in (B.8):

Proposition B.2.

∥T_{n}^{R}∥_{E} ≤ ζ_{n+1}(E)^{n}  (B.10)

Proof. Let {z_{j}}_{j=1}^{n+1} be an (n + 1)-Fekete set. Let

P_{k}(z) = ∏_{j≠k}(z − z_{j})  (B.11)

called a Fekete polynomial. (Note: There is a different set of polynomials occurring in a different context also called Fekete polynomials.) By the maximizing property of Fekete sets,

∥P_{k}∥_{E} = ∏_{j≠k} |z_{k} − z_{j}|  (B.12)

since if z′_{j} = z_{j} (j ≠ k), z′_{k} = z, then ∏_{ℓ≠k}|z′_{ℓ} − z′_{k}| ≤ ∏_{ℓ≠k}|z_{ℓ} − z_{k}|. Since ∥T_{n}^{R}∥_{E} ≤ ∥P_{k}∥_{E} (by the minimizing property of ∥T_{n}^{R}∥_{E}), taking the n + 1 choices of k,

∥T_{n}^{R}∥_{E}^{n+1} ≤ ∏_{k=1}^{n+1} ∥P_{k}∥_{E} = ∏_{all j≠k}|z_{k} − z_{j}| = ζ_{n+1}^{n(n+1)}

which is (B.10). □

The following completes the proof of (B.8):

Proposition B.3. For any monic polynomial P_{n}(z),

∥P_{n}∥_{E} ≥ C(E)^{n}  (B.13)

Proof. There is nothing to prove if C(E) = 0, so suppose C(E) > 0. By the Bernstein–Walsh lemma (A.30),

|P_{n}(z)| ≤ ∥P_{n}∥_{E}C(E)^{−n} exp(−nΦ_{ρ_{E}}(z))  (B.14)

Divide by |z|^{n} and take z → ∞. The left side of (B.14) goes to 1. Since Φ_{ρ_{E}}(z) = − log|z| + o(1), the right side goes to ∥P_{n}∥_{E}C(E)^{−n}. □

Next we turn to the convergence of Fekete set counting measures to dρ_{E}.

Proposition B.4. Let dν_{n} be finite point probability measures supported at {z_{j}^{(n)}}_{j=1}^{N_{n}} with weight ν_{n,j} = ν({z_{j}^{(n)}}). Suppose dν_{n} → dη weakly for some measure η. Suppose there is a compact K ⊂ ℂ containing all the {z_{j}^{(n)}} and that as n → ∞,

∑_{j} ν_{n,j}^{2} → 0  (B.15)

Then

lim sup_{n→∞} ∏_{j≠k}|z_{j}^{(n)} − z_{k}^{(n)}|^{ν_{n,j}ν_{n,k}} ≤ exp(∫ dη(z)dη(w) log|z − w|) (B.16)

Remark. Since ∑_{j} ν_{n,j} = 1,

(max_{j} ν_{n,j})^{2} ≤ ∑_{j} ν_{n,j}^{2} ≤ max ν_{n,j}  (B.17)

(B.15) is equivalent to

max_{j} ν_{n,j} → 0

Proof. Fix m ≥ 0 and let

g_{m}(z, w) ≡ log(max(m, |z − w|))  (B.18)

Then

m^{∑_{j} ν_{n,j}^{2}} ∏_{j≠k}|z_{j}^{(n)} − z_{k}^{(n)}|^{ν_{n,j}ν_{n,k}} ≤ exp(∫ dν_{n}(z)dν_{n}(w)g_{m}(z, w))

(B.19)

Now take n → ∞. By (B.15), m^{∑ ν_{n,j}^{2}} → 1, and by continuity of g_{m}(z, w) and the weak convergence,

∫ dν_{n}(z)dν_{n}(w)g_{m}(z, w) → ∫ dη(z)dη(w)g_{m}(z, w)  (B.20)

we have

LHS of (B.16) ≤ exp(∫ dη(x)dη(y)g_{m}(z, w))  (B.21)

Now take m → 0 using monotone convergence to get (B.16). □

Lemma B.5. Let E ⊂ ℝ. Let (a, b) ∩ E = ∅. Then T_{n}(z) has at most one zero in (a, b) which is simple. If (a, b) ∩ cvh(E) = ∅, T_{n} has no zero in (a, b) (where cvh(E) is the convex hull of E). In particular, if dη is a limit point of the normalized zero counting measure for T_{n}, then supp(dη) ⊂ E.

Proof. Suppose x_{1}, x_{2} are two zeros in (a, b) with x_{1} < x_{2}. Then

(z − (x_{1} − δ))(z − (x_{2} + δ)) = (z − x_{1})(z − x_{2}) − δ(x_{2} − x_{1}) − δ^{2}

so uniformly on E where (z − x_{1})(z − x_{2}) > 0,

|(z − (x_{1} − δ))(z − (x_{2} + δ))| < |(z − x_{1})(z − x_{2})|

for δ small. Thus, ∥T_{n}(z)∥_{E} is decreased by changing those two zeros. Similarly, if x is a zero below cvh(E), T_{n} is decreased by moving the zero up slightly. If x_{j} is a complex zero, ∥T_{n}(z)∥_{E} is decreased by replacing x_{j} by Re x_{j}.

The final statement is immediate if we note that if f is a continuous function supported in (a, b), then ∫ f dη = 0. □

Proof of Theorem B.1. We have already proved (B.8). The Fekete points are distinct, so ν_{n,j} = 1/n, in the language of Proposition B.4. So if we pass to a subsequence for which dν_{n(j)} has a weak limit η, we see (using lim ζ_{n} exists)

lim ζ_{n} = lim_{j→∞} ζ_{n(j)}^{(n(j)−1)/n(j)} ≤ exp(−E(dη))
≤ exp(− inf_{all dρ} E(dρ))  (B.22)
= C(E)

By (B.8), lim ζ_{n} ≥ C(E), so we have equality in (B.22) and dη = dρ_{E}. Thus, any limit point is dρ_{E}. By compactness, we have (i).

That leaves the proof of (ii). By the Berstein–Walsh lemma (A.30), for z ∈ ℂ \ E,

\frac{1}{n} log|T_{n}(z)| ≤ log(\frac{∥T_{n}∥_{E}^{1/n}}{C(E)} ) − Φ_{ρ_{E}}(z)  (B.23)

and similarly for T_{n}^{R}.

Now let dη be a limit point of the normalized density of zeros of T_{n}(z). By the last lemma, dη is supported on E, so (B.23) plus lim ∥T_{n}∥_{∞}^{1/n} = C(E) implies

Φ_{η}(z) ≥ Φ_{ρ_{E}}(z)  (B.24)

for z ∈ ℂ \ E. By Theorem A.21, this implies dη = dρ_{E}. Thus, dρ_{E} is the only limit point of the zeros, and so the limit is dρ_{E}. □

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