Equilibrium measures and capacities in spectral theory

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 Schrodinger 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-Szego theory.


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 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, and dν n its zero counting measure, that is, the point measure with nν n ({w}) = multiplicity of w as a root of P n , then |P n (z)| 1/n = exp(−Φ νn (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 X n (z) X m (z) dµ(z) = X n 2 L 2 δ nm (1.6) and x n (z) = 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 Φ n (1/z) 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 This has an immediate corollary (it appears with this argument in Widom [122] and may well be earlier): (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 1/n L 2 (dµ) = 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 (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. 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 1.6. Let a n ≡ 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 = 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 = 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) ≤ 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.
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 1/n L 2 (dµ) = C(E). (ii) For all z in C, uniformly on compacts, 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  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 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 ). 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), 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, Remarks. 1. We will prove Theorem 1.13 in Section 5.
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 n (ω)) 1/n = C( Remarks. 1. We will prove that for a.e. ω, 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]

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.
The Jacobi parameters a n obey (recall a n > 0) Proof. In a spectral representation, J is multiplication by In particular, p n (z) = 0 for all n and lim inf|p n (z, dµ)| 1/n ≥ 1 + d D Remark. Of course, it is well known that p n has all its zeros on H.
Remark. Thus, we have proven Corollary 1.2 again, Theorem 1.7, and one part of Theorem 1.10.
(ii) ⇒ (vi). Without loss, we can redefine Q n so Q n L 2 (dµ) = 1. Then 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 subsubsequenceñ(j), 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 On the other hand (see (4.14) below), for µ-a.e. x, we have (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.
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, Suppose Γ n < c − ε and let D = 1 2 diam(cvh(supp(µ))) ≥ a n by (2.2). Then (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 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 2 2ℓ → dρ [−1,3] and dν N 2 2ℓ+1 → 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.

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:

. (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 Maximizing over ω yields (3.1).
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 Remark. This is essentially "half" the CD formula; operator theoretic approaches to the CD formula are discussed in [101].
Proof. For any ϕ, 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 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 , In particular, x n (z 0 ) = 0 for all n and Remark. Again, it is well known (a theorem of Fejér) that zeros of x n lie in H.

Proof.
Define 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 and that (3.4) says (3.11) originally holds for a.e. s, t in supp(dµ), but since both sides are polynomials in s andt, for all s, t. Setting 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 where It is the unique measure on ∂D with by (3.14), we have 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 In particular, 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.

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) 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, Then for dµ-a.e. x, By (4.1) and B > A, we see Since B is arbitrary, we have (4.2).
Proof of Theorem 1.12. Let A be a limit point of P n(j)

1/n(j)
L 2 (dµ) . 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), By (1.4) for such x, By the upper envelope theorem (Theorem A.7) for q.e. x ∈ C, Thus, there exist sets S 1 and S 2 so that µ(S 1 ) = 0 and C(S 2 ) = 0, so that for We can now repeat the argument that led to (2.19). By hypothesis, Therefore, by (A.2), 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.
Suppose dρ E is a.c. with respect to dµ, and for some n(1) < n(2) < . . . , we have P n(j) for the monic P n (x, dµ). Let dν n(j) be the corresponding zero counting measure. Then dν n(j) 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.

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 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 for all z, Thus, using (4.10), we see 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 Berezanskiȋ [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) and thus, for dµ-a.e. x, 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 w(x) > 0 for a.e. x in E, and thus which, by constancy of the Wronskian, implies 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 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 Remark. We note that (4.17) holds a.e. on the a.c. spectrum and by the above arguments, a.e. on that spectrum, 1 n log T n (x) → 0, a deterministic analog of the Pastur-Ishii theorem.

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 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 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 iθn ; dµ)| = Φ n ∂D , the sup norm, and |θ − θ n | ≤ 1 2n , then so the sup norm obeys Φ n ∂D ≥ 1 (5.6) and so (5.4) implies 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 or (Markov's inequality) 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 √ 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( 1 n ) at best. They get |p n (x)| ≥ 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: 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].

(This can be proven by showing the worst case occurs when
and cosh(ε) = 1 + ε 2 2 + O(ε 4 ), we have 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.
Fix δ 1 and let 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 δ 1 4nℓ , but the total size of those is at most δ 1 2 , so we can find disjoint intervals I Now define for any δ 2 > 0 and m, If x lies in the set on the left side of (5.20), let since x ∈ L (n) (δ 1 ) and (5.18) holds. By x ∈ J(Mn, δ 2 ), 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 = δ 1 M . Then take δ 1 ↓ 0 and get lim inf P n 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 Then µ n has total mass at most n(|E| + ℓ) where ℓ is the number of intervals. Let which is easily seen to be equivalent to µ.
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 an ℓ 1 sequence of positive numbers. Let , which kills the contributions of the pure points at {x j } n j=1 , so 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 The following illuminates (1.33). For 2 n ≤ k < 2 n+1 , let x k = k−2 n 2 n and let 0 < y < 1. Define The x k are not distinct, but that does not change the bound (5.31). Thus ≥ 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 picked so the OPRL for the restriction are multiples of the Chebyshev where dη n is concentrated uniformly at the dyadic rationals of the form k/2 n not previously "captured," that is, The a n 's are carefully picked as follows. Define N j inductively by and a n = Our goal will be to prove that Intuitively, for m = 2 N 2 2k+1 , 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 2 2k , 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 2 m 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 = 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 is much smaller than the right side of (5. so there is an interval of size 4/m 2 where P m (x) ≥ 1, that is, so it is bounded from below by a power of 2 −N 2 2k+1 . 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 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) .

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.
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 To understand why the next theorem is so restrictive compared to Theorem 6.2, consider 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) 1/n L 2 (dµ) → C(supp(dµ)). 2. The analog of the sets I in [105] must have nonempty twodimensional interior. Our I obviously has empty two-dimensional inte- 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.
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.

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.
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 1 n log T (n + k, k; z, ω) (and lim sup 1 n log|p n (z, ω)|) may not be upper semicontinuous but E( 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 −1 n also bounded) identically distributed random variables. In the almost periodic case, Ω is a finite-or infinitedimensional torus with dσ Haar measure andÃ,B 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, 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).
By definition of A, regularity for all ω ∈ Ω 0 is equivalent to C(E) = A.
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.
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. 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 setQ (1) ω of capacity zero so for x / ∈ Q ω , 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 / ∈Q Similarly, there is a setQ (2) ω of capacity zero with similar behavior as n → ∞.
This says that every solution of (7.3) for x / ∈Q ω either grows exponentially at ±∞ or decays exponentially. Thus, polynomial boundedness implies ℓ 2 solutions. IfQ (3) ω is the set of eigenvalues of J(ω) which is countable and so of capacity zero, and ifQ ω =Q By the BGK expansion discussed in Section 4, this implies the spectral measures ofJ(ω) are supported onQ 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].

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. }) = ⊗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 The decaying random model has two extra parameters, λ ∈ (0, ∞) and γ ∈ (0, 1), takesã n (ω) ≡ 1,b n (ω) the Anderson model with β = −α = 1, and takes b n (ω) = λn −γb 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, while for the decaying Anderson model, 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) = dρ dx = 1 π (4 − x 2 ) −1/2 by (A. 34). For the Anderson model, on the other hand, 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 < 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 en(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 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 ([ j m , j+1 m ]) ∼ e −cm for fixed c, while in the regular decaying random model, one expects µ ω ([ j m , 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 < 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. 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 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.

Example 8.3 (Almost Mathieu
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].
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 σess(Jω) 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].
. 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 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 dθ 2π . By the Thouless formula and (8.13), 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) = dµ(y) y−z has the form 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.

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 √ −λ), and so 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 This comes from noting the eigenvalues on [0, 1] with u(0) = u(L) = 0 boundary conditions are ( 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, We will say dν is normalized if 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 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 (− d 2 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 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| → ∞, − d 2 dx 2 + V is bounded from below in sectors about (−∞, a) and, in general, in sectors |arg z| ∈ (c, π − ε), 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) √ −z, while in (9.7) it is o(1). The lack of a constant term is an issue to be understood.
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 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 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 − 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 is determined by V.
Definition. Let E = σ ess (H). We say H is regular if and only if for all z / ∈ σ(H), 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: 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 − 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 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.

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 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 Sometimes it is useful to fix M > 0 and define the cutoff Φ M µ is continuous and Φ M µ 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! Then Φ µ (x n ) = ∞ and x n → 0, but Notice that this is consistent with lower semicontinuity, that is, . 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 xn by a smeared out probability measure, η n (using equilibrium measures on a small interval, I n , about x n ), so Φ ηn = λn 2 on I n and have with µ = n −2 η n , then Φ µ is bounded, Hence one loses continuity for λ large.
The following is sometimes useful: 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.
and thus lim inf Φ µ (Re z n ) ≥ a > Φ µ (Re z ∞ ), so without loss, we can suppose z n are real.
The energy or Coulomb energy of µ is defined by 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 One indication that this strange-looking definition is sensible is seen by, as we will show below (see Example A.17), 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, It can then be proven (see [73,Thm. 2.8]) that 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.

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, On the other hand, Φ µ (x) = ∞ on supp(dη), so This contradicts (A.2). We thus see that the last proposition implies: 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 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 1 2 n at { j 2 n } 2 n −1 j=0 . Then dν n → dx ≡ dν, Lebesgue measure. Φ νn ( j 2 n ) = ∞ so lim inf Φ νn (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 νn (x) = − log|p n (x)| 1/n , we have lim inf Φ νn (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 M ν is the convolution with a continuous function so lim 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 By (A.2) and Fatou's lemma (Φ νn (x) is uniformly bounded below), 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 M µ dµ(x), then it is easy to prove E M is weakly continuous and conditionally positive definite in that 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 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 µ(k) − ν(k) is analytic in k vanishing at k = 0 and Since the inequality in (A. 19) is strict and 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 The properties of ρ E are summarized in 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 , 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 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 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 Remarks. 1. This can happen even if The following illustrates the connection between potential theory and polynomials: Remarks. 1. This is named after Bernstein and Walsh [121], although the result appears essentially in Szegő [107]. 2. Let {z j } n j=1 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 } n j=1 = 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 Following ideas of Craig [24], one can say much more about dρ E dx when E contains an isolated closed interval: Remarks. 3. The x j 's are uniquely determined by 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 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, Then η = ρ E , the equilibrium measure, and α = log(C(E) −1 ).
Next, we note that the Green's function for a compact E ⊂ C is defined by Closely related are comparison theorems and limit theorems. We will state them for subsets of R: for all x ∈ I.
Theorem A. 16. 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 ∞ ) ≤ 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 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 En , then η is supported on E ∞ , and by lower semicontinuity of the Coulomb energy E, by (A.47), so η = ρ E∞ , that is, ρ En → 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 En → ρ E∞ and uniform bounds on derivatives of 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\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 + 1 z , we see C([−2, 2]) = 1 (A.52) and consistent with (A.34) 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 This plus translation invariance shows 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 where T a µ is the translate of µ). Thus with a n = 0, 1 or 2 and taking dµ as the infinite product of measures given weight 1 2 to a n = 0 or 2. Looking at a 1 , we get the usual two pieces of mass 1 2 with minimum distance 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, 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 = (− a 2 − ∆, − a 2 ) ∪ ( a 2 , a 2 + ∆) where ∆ = 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 1 4 log 4 ∆ + 2 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) ≤ 1 m }. E m is a finite union of closed intervals and, by (A.10), C(E m ) ↓ 0. Pass to a subsequenceẼ m , so C(Ẽ m ) ≤ exp(− 1 m 2 ) so Φ ρẼ m (x) ≥ m 2 onẼ 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 x r = (x − r, x + r), µ(I x r ) ≤ c(2r) α (A.59) Then picking r = 2 −n , we see (with n 0 large and negative so supp(dµ) ⊂ I x 2 −n ) log|y − x| −1 dµ(y) ≤ 4. Given α > 0, δ > 0 fixed, by (A.60), we can find for each x ∈ E, so I x rx with µ(I x rx ) ≥ 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 by (A.62). Since δ is arbitrary, we have the required covers to see dim(E) = 0.

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 R n , 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 R n (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 R n E (B.2) and since T n T m is a monic polynomial of degree n + m, 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 k n } is a minimizer for any z 0 ∈ ∂D.
Let z 1 , . . . , z n+1 be an n + 1-point Fekete set. For each j, 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 (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 R n 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 R n (z) = z n and (ii) fails for T R n 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). 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 R n E ≤ P k E (by the minimizing property of T R n E ), taking the n + 1 choices of k, which is (B.10).
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 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 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, and similarly for T R n . 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 .
Note added in proof. Since completion of this manuscript, I have found a result (to appear in "Regularity and the Cesàro-Nevai class", in prep.) relevant to the subject of the current review. In the simplest case, it states that if a measure has [−2, 2] as its essential support and is regular, then (1.25) holds.