REFLECTIONLESS HERGLOTZ FUNCTIONS AND JACOBI MATRICES

ALEXEI POLTORATSKI AND CHRISTIAN REMLING

Abstract. We study several related aspects of reflectionless Jacobi matrices. First, we discuss the singular part of the corresponding spectral measures. We then show how to identify sets on which measures are reflectionless by looking at the logarithmic potentials of these measures.

1. Introduction

We study several aspects of reflectionless Jacobi matrices and Herglotz functions in this paper. This is part of a larger program; the (perhaps too ambitious) goal is to reach a systematic understanding of the absolutely continuous spectrum of Jacobi operators J on ℓ_{2}(ℤ_{+}),

(Ju)(n) = a(n)u(n + 1) + a(n − 1)u(n − 1) + b(n)u(n).

We will always assume that the coefficients a, b satisfy bounds of the form

(C + 1)^{−1} ≤ a(n) ≤ C + 1, |b(n)| ≤ C,

for some C > 0. Note that if J has some absolutely continuous spectrum, then, by the decoupling argument of Dombrowski and Simon-Spencer [3, 17], it actually suffices to assume that a(n) is bounded above; the other two inequalities follow automatically.

Let us recall some definitions. A Herglotz function is a holomorphic mapping of ℂ^{+} = {z ∈ ℂ : Im z > 0} to itself. We denote the set of Herglotz functions by H. If F ∈ H, then F (t) ≡ lim_{y→0+} F (t + iy) exists for (Lebesgue) almost every t ∈ ℝ. We call F reflectionless (on E ⊂ ℝ) if

(1.1) Re F (t) = 0 for almost every t ∈ E.

Date: August 8, 2008.

Key words and phrases. Jacobi matrix, Herglotz function, reflectionless measure.

The first author is supported in part by N.S.F. grant DMS 0800300.

2000 Mathematics Subject Classification. Primary 31A20 34L40 47B39 81Q10; Secondary 30C85 31A15.

We will also use the notation

N (E) = {F ∈ H : F reflectionless on E}.

Herglotz functions have unique representations of the form

F (z) = F_{μ}(z) = a + bz + ∫_{−∞}^{∞} (\frac{1}{t − z} − \frac{t}{t^{2} + 1}) dμ(t), (1.2)

with a ∈ ℝ, b ≥ 0, and a (positive) Borel measure μ on ℝ, ∫_{ℝ} \frac{dμ(t)}{t^{2}+1} < ∞. We will call such a measure μ reflectionless (on E) if F_{μ} ∈ N (E) for some choice of a ∈ ℝ, b ≥ 0; for easier reference, it will also be convenient to introduce the notation

R(E) = {μ : μ reflectionless on E}.

We emphasize again that in particular ∫_{ℝ} \frac{dμ(t)}{t^{2}+1} < ∞ for all μ ∈ R(E). Also, if μ ∈ R(E), then F_{μ} will refer to the unique Herglotz function F_{μ} ∈ N (E) that is associated with μ as in (1.2).

There are several reasons for being interested in the class N (E); here, our main motivation is provided by the following fact: Call a (whole line) Jacobi matrix J reflectionless (on E) if g_{n} ∈ N (E) for all n ∈ ℤ, where

g_{n}(z) = ⟨δ_{n}, (J − z)^{−1}δ_{n}⟩

is the nth diagonal element of the resolvent of J (also known as the Green function). Then [13, Theorem 1.4] says that all ω limit points of a Jacobi matrix J with some absolutely continuous spectrum are reflectionless on E = Σ_{ac}; here, Σ_{ac} denotes an essential support of the absolutely continuous part of the spectral measure ρ of J. This is defined up to sets of Lebesgue measure zero; we can obtain a representative as Σ_{ac} = {t : dρ/dt > 0}. Please see [13] for the details.

If μ ∈ R(E), then χ_{E} dt ≪ dμ_{ac}. Indeed, this follows immediately from (1.1) because dμ_{ac}(t) = (1/π)Im F (t) dt and the boundary value of a Herglotz function can not be zero on a set of positive measure. However, it is not so clear in general if μ can also have a singular part on E. See [4, 5, 6] for earlier work on this question. We have the following criterion. We say that a (positive) measure ν is supported by a (measurable) set S if ν(S^{c}) = 0; supports are not assumed to be closed in this paper.

Theorem 1.1. Let μ ∈ R(E). Then:

(a) μ_{s}, the singular part of μ, is supported by

\{x ∈ ℝ : lim_{h→0+} \frac{|E ∩ (x − h, x + h)|}{2h} = 0\} .

(b) Let θ ∈ L_{∞}(E) be an arbitrary bounded measurable function. Then μ_{s} is also supported by

{x ∈ ℝ : (H̃_{E}θ)(x) exists}.

Here, we define H̃_{E}f as

(H̃_{E}f)(x) = lim_{y→0+} ∫_{E} (\frac{t − x}{(t − x)^{2} + y^{2}} − \frac{t}{t^{2} + 1}) f(t) dt,

if the limit exists. This is closely related to the Hilbert transform

(Hf)(x) = lim_{y→0+} ∫_{|t−x|>y} \frac{f(t)}{t − x} dt.

For instance, if f ∈ L_{1}(ℝ), then both H̃_{E}f and Hχ_{E}f exist (Lebesgue) almost everywhere and define the same function (almost everywhere), up to an additive constant. Here, we are interested in the singular part of μ, so sets of Lebesgue measure zero do matter, and we distinguish between the two transforms. However, for a bounded, integrable function, the difference between the integrals defining H̃_{E} and H stays bounded, so we also obtain the following variant of Theorem 1.1(b):

(c) Let θ ∈ L_{∞}(E). Then μ_{s} is supported by

\{x ∈ ℝ : sup_{0<y≤1} |∫_{y<|t−x|≤1} \frac{θ(t)}{t − x} χ_{E}(t) dt| < ∞\} .

It is tempting to compare Theorem 1.1(a) with the results of Sodin-Yuditskii [18] on reflectionless Jacobi matrices with homogeneous spectrum. Sodin and Yuditskii prove that these operators are almost periodic and have purely absolutely continuous spectra. We can obtain this latter conclusion for the much larger class of weakly homogeneous spectra (to be defined in Section 2) directly from Theorem 1.1(a); see Definition 2.1 and Corollary 2.3 below. However, this will not give any additional information on the structure of the associated Jacobi matrices. Perhaps Theorem 1.1 can actually be used as the starting point for such a refined analysis, but at present, this is only a hope for the future.

It is easier to keep track of the pure point part of a reflectionless measure, and in fact Theorem 1.1(c) gives a sharp criterion in this case.

Theorem 1.2. Let E ⊂ ℝ be a Borel set, and fix x ∈ ℝ. Then the following are equivalent:

(a) μ({x}) = 0 for all μ ∈ R(E).

(b)

∫_{x−1}^{x+1} \frac{χ_{E}(t)}{|t − x|} dt = ∞

Our final result on the singular part of reflectionless measures is of a conditional nature. It says that if μ is also non-zero outside E, then this will only make it more difficult to produce a singular part on E. To be able to formulate this concisely, we introduce

R_{0}(E) = {μ ∈ R(E) : μ(E^{c}) = 0} .

Theorem 1.3. Let E ⊂ ℝ be a closed set. Suppose that μ_{s}(E) = 0 for μ ∈ R_{0}(E). Then ν_{s}(E) = 0 for all ν ∈ R(E).

As our next topic, we would like to address the following question: Given a set E, how can we produce examples of measures that are reflectionless on E? Two quick answers are immediately available: As already mentioned above, [13, Theorem 1.4] says that if we start out with a Jacobi matrix with Σ_{ac} ⊃ E and then take ω limit points, then we can be sure that these will be reflectionless on E. A different answer to our question was obtained in [9, Theorem 5.4] (see also [10]): the potential theoretic equilibrium measure is reflectionless on its support.

These two results are not totally unrelated. The equilibrium measure frequently arises as the density of states measure of Jacobi matrices with some absolutely continuous spectrum. See, for example, [15]; the book [19] has a systematic exposition of similar methods in a slightly different setting. We will prove a general result here that puts these facts in perspective.

Let ν be a finite Borel measure on ℝ of compact support. We introduce the following two functions:

γ(z) = ∫_{ℝ} ln |t − z| dν(t) (z ∈ ℂ),
g(z) = ∫_{ℝ} \frac{dν(t)}{t − z} (z ∈ ℂ^{+})

The integral defining the logarithmic potential γ(z) could diverge to −∞; however, for simplicity, we assume here that ν is such that γ(z) > −∞ for all z ∈ ℂ. This assumption could be relaxed, but it holds in all cases of interest, so we do not bother.

We will need the following definition.

Definition 1.1. Let f : ℝ → ℝ be a (Lebesgue) measurable function. We say that f is approximately differentiable at x ∈ ℝ if there exists d ∈ ℝ so that

lim_{h→0+} \frac{1}{2h} |\{y ∈ (−h, h) : |\frac{f(x + y) − f(x)}{y} − d| ≥ ϵ\}| = 0

for all ϵ > 0. In this case, we call d the approximative derivative of f at x, and we write (D_{ap}f)(x) = d.

Please see [1, 14, 20] for much more on this and related topics.

Theorem 1.4. For almost all x ∈ ℝ, we have that

(D_{ap}γ)(x) = −Re g(x). (1.3)

In particular, γ is approximately differentiable almost everywhere.

This is a development of [9, Theorem 5.4]. See also [10] for subsequent work inspired by the same result. Theorem 1.4 may be viewed as a result on interchanging limits. Indeed, Re g(x + iy) = −∂_{x}γ(x + iy) for y > 0, so, for almost every x ∈ ℝ, Re g(x) = − lim_{y→0+} ∂_{x}γ(x+iy). This raises the question of whether it is possible to perform these operations in the opposite order; in other words, can we first take the boundary value of γ to obtain γ(x) and then take the derivative? Theorem 1.4 provides an affirmative answer if the derivative is taken in the approximate sense.

However, for us here, Theorem 1.4 is significant mainly because it identifies sets on which g is reflectionless; in particular, this set will contain the points of constancy of γ. More precisely, we obtain the following:

Corollary 1.5. Let

K = {c ∈ ℝ : |γ^{−1}({c})| > 0}; C = γ^{−1}(K).

Then g ∈ N (C).

Proof. K is countable and thus C is an at most countable union of sets C_{j} of the form C_{j} = γ^{−1}({c_{j}}). Almost every point of C_{j} is a point of density, and at such points, clearly D_{ap}γ = 0. Theorem 1.4 now gives the Corollary.

So, first of all, we recover the result from [9, 10] that the equilibrium measure of a compact set K ⊂ ℝ is reflectionless on K (see also [12] for information on potential theory). More importantly, Theorem 1.4 and Corollary 1.5 may be used to identify additional examples of reflectionless measures, and they unify these results. We will not pursue this theme in detail here, but just make a few quick remarks on how to proceed. Introduce density of states measures ν as follows:

dν(t) = lim_{j→∞} \frac{1}{N_{j}} ∑_{n=1}^{N_{j}} d∥E(t)δ_{n}∥^{2}

Here, E denotes the spectral resolution of the Jacobi matrix J, and the limit is taken in the weak-∗ sense and on a suitable subsequence N_{j} → ∞. By the Banach-Alaoglu Theorem, such limits always exist; of course, ν could depend on the choice of the sequence N_{j}. We also remark that there are other ways to obtain dν: one can use the eigenvalues of truncated problems or Christoffel-Darboux kernels; see [16, Theorem 1.5] for further discussion of this.

By (a generalized version of) the Thouless formula, the logarithmic potential γ will equal the (generalized) Lyapunov exponent, up to a constant. Therefore, Corollary 1.5 now says that ν will be reflectionless on sets of constancy of the Lyapunov exponent; from this in turn, it also follows that

ν ∈ R(Σ_{ac}), g ∈ N (Σ_{ac}),

where Σ_{ac} again denotes an essential support of the absolutely continuous part of the spectral measure of J.

The proofs of the results discussed in this introduction will be given in the following two sections, and we split the material in the obvious way: We will prove Theorems 1.1, 1.2, and 1.3 in Section 2, and Section 3 has the proof of Theorem 1.4.

2. The singular part of reflectionless measures

Please recall the notation F_{μ} introduced in (1.2). Also, if f ≥ 0 is a Borel function, then, as expected, fμ will denote the measure (fμ)(A) = ∫_{A} f dμ. The following result from [11] will be our main tool in this section.

Theorem 2.1 ([11]).

lim_{y→0+} \frac{F_{fμ}(x + iy)}{F_{μ}(x + iy)} = f(x)

for μ_{s}-almost every x ∈ ℝ.

A clarifying comment is in order: Given ν, the function F_{ν} is of course not completely determined yet (we don’t know a, b). This, however, is not an issue here; the statement from the Theorem holds for all such functions. This follows because |F_{μ}(x + iy)| → ∞ as y → 0+ for μ_{s}- almost every x ∈ ℝ. In Theorem 2.1, we of course implicitly assume that 1/(t^{2} + 1) is integrable for all measures involved here. See also [2, 7] for further discussion of this theorem.

We will also use the following consequence of Theorem 2.1.

Proposition 2.2. Suppose that ρ = ρ_{s} and σ ⊥ ρ. Then

lim_{y→0+} \frac{F_{σ}(x + iy)}{F_{ρ}(x + iy)} = 0

for ρ-almost every x ∈ ℝ.

Proof. Pick a Borel set T ⊂ ℝ with ρ(T^{c}) = σ(T) = 0, and abbreviate ρ + σ = μ. Then

\frac{F_{σ}}{F_{ρ}} = \frac{F_{χ_{T^{c}}μ}}{F_{χ_{T} μ}} = \frac{F_{χ_{T^{c}}μ}}{F_{μ}} \frac{F_{μ}}{F_{χ_{T} μ}} ,

and F_{χ_{T^{c}}μ}/F_{μ} → χ_{T^{c}} μ_{s}-almost everywhere by Theorem 2.1. In particular, this ratio goes to zero ρ-almost everywhere. Similarly, F_{χ_{T} μ}/F_{μ} → 1 ρ-almost everywhere, so the Proposition follows.

Proof of Theorem 1.1. Let μ ∈ R(E). Write F_{μ} for the associated Herglotz function F_{μ} ∈ N (E), as in (1.2), and let ξ be the Krein function of F_{μ}, that is,

ξ(x) = \frac{1}{π} lim_{y→0+} Im ln F_{μ}(x + iy),

where we take the logarithm with 0 < Im ln w < π for w ∈ ℂ^{+}. Since ln F_{μ} is a Herglotz function, the limit defining ξ exists almost everywhere and 0 ≤ ξ(x) ≤ 1.

If, conversely, a measurable function ζ with values in [0, 1] is given, then ζ is the Krein function of some Herglotz function G. We can in fact recover ln G, up to an additive real constant, from ζ, using the Herglotz representation of ln G. Here, we make use of the fact that since ln G has bounded imaginary part, the associated measure is purely absolutely continuous.

The condition that F_{μ} ∈ N (E) means that ξ = 1/2 (almost everywhere) on E. Given an arbitrary function θ ∈ L_{∞}(ℝ), with −1 ≤ θ ≤ 1 and θ = 0 on E^{c}, we can therefore introduce two new Krein functions ξ_{±}, as follows:

ξ_{±}(x) = ξ(x) ± \frac{1}{2}θ(x)

As just explained, this also defines two new Herglotz functions F_{±}, up to multiplicative constants. We fix these constants by demanding that |F_{±}(i)| = |F_{μ}(i)|. Call the measures associated with these functions μ_{+} and μ_{−}, respectively. Since ξ = (ξ_{+} + ξ_{−})/2, we then have that

F_{μ} = \sqrt{F_{μ_{+}}F_{μ_{−}}}. (2.1)

Our first aim is to show that

μ_{s} ≪ μ_{±}. (2.2)

Suppose this were wrong and write

μ_{s} = gμ_{+,s} + ν,

with ν ⊥ μ_{+,s}, ν ≠ 0. We can then find a Borel set T so that ν(T) > 0, μ_{+,s}(T) = ν(T^{c}) = |T | = 0. Theorem 2.1 now shows that

\frac{F_{ν}(x + iy)}{F_{μ_{s}}(x + iy)} = \frac{F_{χ_{T} μ_{s}}(x + iy)}{F_{μ_{s}}(x + iy)} → 1

for μ_{s}-almost every x ∈ T, and, similarly,

\frac{F_{ν}(x + iy)}{F_{μ_{+}+ν}(x + iy)} = \frac{F_{χ_{T} (μ_{+}+ν)}(x + iy)}{F_{μ_{+}+ν}(x + iy)} → 1

for (μ_{+,s} + ν)-almost every x ∈ T and thus also for μ_{s}-almost every x ∈ T. Put differently, this means that

\frac{F_{μ_{+}}(x + iy)}{F_{ν}(x + iy)} → 0

for μ_{s}-almost every x ∈ T. So on a set of positive μ_{s}-measure,

\frac{F_{μ_{+}}(x + iy)}{F_{μ_{s}}(x + iy)} = \frac{F_{μ_{+}}}{F_{ν}} \frac{F_{ν}}{F_{μ_{s}}} → 0. (2.3)

We also have that for μ_{s}-almost every x ∈ ℝ,

sup_{0<y≤1} |\frac{F_{μ_{−}}(x + iy)}{F_{μ_{s}}(x + iy)}| < ∞. (2.4)

This follows quickly from Proposition 2.2 with ρ = μ_{s} if we write μ_{−} = hμ_{s} + σ, with σ ⊥ μ_{s}. Indeed, F_{μ_{−}}/F_{μ_{s}} → h at μ_{s}-almost every point by the Proposition and Theorem 2.1, and h < ∞ μ_{s}-almost everywhere.

Finally, Theorem 2.1 also implies that

lim_{y→0+} \frac{F_{μ_{s}}(x + iy)}{F_{μ}(x + iy)} = 1

for μ_{s}-almost every x ∈ ℝ, and if this is combined with (2.3), (2.4), we obtain that

\frac{\sqrt{F_{μ_{+}}F_{μ_{−}}}}{F_{μ}} → 0

on a set of positive μ_{s}-measure, but by (2.1), this ratio is identically equal to one, so we reach a contradiction if (2.2) fails.

Thus we can write

μ_{s} = f_{±}μ_{±} = f_{±}μ_{±,s},

with f_{±} ≥ 0 and in fact 0 < f_{±} < ∞ at μ_{s}-almost all points. By Theorem 2.1,

lim_{y→0+} \frac{F_{μ_{s}}(x + iy)}{F_{μ_{±}}(x + iy)} → f_{±}(x + iy)

for μ_{±,s}-almost every x ∈ ℝ and thus also μ_{s}-almost everywhere. It follows that for μ_{s}-almost every x,

lim_{y→0+} \frac{F_{μ_{+}}(x + iy)}{F_{μ_{−}}(x + iy)} exists and is positive;

(2.5)

in fact, this limit is equal to f_{−}(x)/f_{+}(x) μ_{s}-almost everywhere. By definition of ξ_{±}, we have that ξ_{+} − ξ_{−} = θ, so, if we introduce

L(z) = ∫_{E} (\frac{1}{t − z} − \frac{t}{t^{2} + 1}) θ(t) dt,

then F_{μ_{+}}/F_{μ_{−}} = e^{L}. Since −1 ≤ θ ≤ 1, we have that Im L(z) ∈ (−π, π) on ℂ^{+}, and thus (2.5) implies that for μ_{s}-almost every x ∈ ℝ,

L(x) ≡ lim_{y→0+} L(x + iy) exists, Im L(x) = 0.

In particular, if we take θ = χ_{E}, then

Im L(x + iy) = ∫_{E} \frac{y}{(t − x)^{2} + y^{2}} dt ≥ \frac{1}{2y} |E ∩ (x − y, x + y)| ,

so part (a) of the Theorem follows. Part (b) is now also immediate, from the fact that Re L(x+iy) approaches a finite limit as y → 0+.

The set from part (a) of Theorem 1.1 contains \overline{E}^{c}, so this result really addresses the question of whether μ can have a singular part on \overline{E}. In particular, it says that no such singular part can be present for closed sets of the following type.

Definition 2.1. Call a Borel set E ⊂ ℝ weakly homogeneous if

lim sup_{h→0+} \frac{1}{2h} |E ∩ (x − h, x + h)| > 0

for all x ∈ E.

This condition is much weaker than the following, which is used to define homogeneous sets:

inf_{x∈E} inf_{0<h≤1} \frac{1}{2h} |E ∩ (x − h, x + h)| > 0.

From the work of Sodin-Yuditskii [18] it was previously known that if E is a compact (strongly) homogeneous set and μ ∈ R_{0}(E), then μ_{s} = 0. By using Theorem 1.1, we can go considerably beyond this:

Corollary 2.3. Suppose that E is a weakly homogeneous set. If μ ∈ R(E), then μ_{s}(E) = 0.

This is more general in two respects: E is only assumed to be weakly homogeneous (rather than homogeneous), and we can treat measures from R(E), not just from R_{0}(E). This latter improvement, of course, can also be obtained from the general principle that we formulated as Theorem 1.3.

We now move on to proving Theorem 1.2. This will follow quickly from the following known characterization of the point part of μ in terms of the Krein function ξ of F_{μ}. See, for example, [8, pg. 201]. We include a proof for the reader’s convenience.

Lemma 2.4. μ({x}) > 0 if and only if

∫_{x−1}^{x+1} \frac{|ξ(t) − χ_{(x,∞)}(t)|}{|t − x|} dt < ∞. (2.6)

Proof. First of all, we can recover the point part as

μ({x}) = −i lim_{y→0+} yF_{μ}(x + iy);

this is well known and follows quickly from the dominated convergence theorem. So μ({x}) > 0 if and only if

(2.7)  lim sup_{y→0+} (Re ln F_{μ}(x + iy) + ln y) > −∞.

To slightly simplify the notation, we will now assume that x = 0. In terms of the Krein function ξ, the expression from (2.7) equals

∫_{−1}^{1} \frac{t}{t^{2} + y^{2}} ξ(t) dt − ∫_{y}^{1} \frac{dt}{t} + O(1) (y → 0+).

By monotone convergence,

∫_{−1}^{0} \frac{t}{t^{2} + y^{2}} ξ(t) dt → ∫_{−1}^{0} \frac{ξ(t)}{t} dt

(and, of course, this limit could equal −∞). Also,

∫_{0}^{1} \frac{t}{t^{2} + y^{2}} ξ(t) dt − ∫_{y}^{1} \frac{dt}{t} = ∫_{y}^{1} \frac{t(ξ(t) − 1)}{t^{2} + y^{2}} dt − ∫_{y}^{1} \frac{y^{2}}{t(t^{2} + y^{2})} dt

+ ∫_{0}^{y} \frac{tξ(t)}{t^{2} + y^{2}} dt

= ∫_{y}^{1} \frac{t(ξ(t) − 1)}{t^{2} + y^{2}} dt + O(1),

and, by monotone convergence again,

∫_{y}^{1} \frac{t(ξ(t) − 1)}{t^{2} + y^{2}} dt → ∫_{0}^{1} \frac{ξ(t) − 1}{t} dt ≥ −∞.

These calculations have shown that (2.7), for x = 0, holds if and only if

∫_{−1}^{0} \frac{ξ(t)}{|t|} dt + ∫_{0}^{1} \frac{1 − ξ(t)}{t} dt < ∞,

as asserted by the Lemma.

Proof of Theorem 1.2. Suppose that condition (b) from Theorem 1.2 fails. Put

ξ(t) = \frac{1}{2}χ_{E}(t) + χ_{E^{c}∩(x,∞)}(t).

Let F ∈ H be the corresponding Herglotz function. Since ξ = 1/2 on E, we have that F ∈ N (E), but it is also clear that (2.6) holds, so the corresponding measure has a point mass at x.

The converse is an immediate consequence of Theorem 1.1(c), with θ(t) = sgn(t − x). Furthermore, we can also obtain this statement conveniently from Lemma 2.4, as follows: If F_{μ} ∈ N (E), then ξ = 1/2 almost everywhere on E, so the integrand from (2.6) equals 1/(2|t−x|) on E ∩ (x − 1, x + 1) and thus (2.6) can not hold if we have condition (b) from Theorem 1.2.

Proof of Theorem 1.3. Let ν ∈ R(E). We claim that if μ ∈ R_{0}(E), then we must have that

lim_{y→0+} \frac{F_{μ}(x + iy)}{F_{ν}(x + iy)} = 0 (2.8)

for ν_{s}-almost every x ∈ ℝ. Indeed, μ_{s} = 0, μ(E^{c}) = 0 by assumption, and, as discussed above, the condition that ν ∈ R(E) forces the absolutely continuous part of ν to be equivalent to χ_{E} dt on E. So μ ≪ ν, μ_{s} = 0, and thus (2.8) follows immediately from Theorem 2.1.

Starting from ν, we will now construct a measure μ ∈ R_{0}(E) for which (2.8) cannot hold at any point x ∈ E. This will prove that ν_{s}(E) = 0, as claimed.

We will again work with the Krein functions; the following simple monotonicity property is at the heart of the matter.

Lemma 2.5. For ξ ∈ L_{∞}(a, b), 0 ≤ ξ ≤ 1, and x ∉ [a, b], define

I_{x}(ξ) = ∫_{a}^{b} \frac{ξ(t) dt}{t − x}.

Let c = ∫_{a}^{b} ξ(t) dt. Then

I_{x}(ξ) ≤ I_{x} (χ_{(a,a+c)}) for all x ∉ [a, b].

Proof. It suffices to prove this for step functions ξ because these are dense in L_{1}. So assume that ξ = ∑_{j=1}^{N} s_{j}χ_{I_{j}}, with disjoint intervals I_{j}. If (c, c + h) is such an interval of constancy of ξ and ξ = s on (c, c + h), with 0 < s < 1, then, as an elementary argument shows, I_{x}(ξ) will go up if we redefine ξ on (c, c + h) as χ_{(c,c+sh)}. Use this procedure on all intervals of constancy. Since I_{x}(ξ) clearly also increases if we pass to the non-increasing rearrangement of ξ, we obtain the Lemma.

Let ξ be the Krein function of F_{ν}, and, motivated by Lemma 2.5, define ξ_{0} as follows: ξ_{0} = 1/2 on E, and if (a, b) is one of the bounded components of the open set E^{c}, set ξ_{0} = χ_{(a,a+c)} on (a, b), where c = ∫_{a}^{b} ξ dt, as in the Lemma. If E^{c} has unbounded components, put ξ_{0} = 1 (say) on these. Notice that ξ_{0} is the Krein function of a Herglotz function F_{μ} whose associated measure satisfies μ ∈ R_{0}(E). Indeed, μ is reflectionless on E because this property is equivalent to ξ_{0} = 1/2 on E, and μ(E^{c}) = 0 because F_{μ}(x) ≡ lim_{y→0+} F_{μ}(x + iy) exists and is real at all points of E^{c}, except possibly at the jumps of ξ_{0}. However, these can’t be discrete points of μ either because in order for this to happen, ξ_{0} would have to jump from 0 to 1, not the other way around, by Lemma 2.4.

Now fix x ∈ E and look at ln |F_{μ}/F_{ν}|. As y → 0+,

ln |F_{μ}(x + iy)| − ln |F_{ν}(x + iy)| = ∫_{y<|t−x|≤1} \frac{ξ_{0}(t) − ξ(t)}{t − x} dt + O(1). (2.9)

Since ξ = ξ_{0} = 1/2 on E, this set doesn’t contribute to the integral. Moreover, by Lemma 2.5 and construction of ξ_{0}, those components of E^{c} that are contained in the region of integration make non-negative contributions. This more or less finishes the proof except that there might also be up to four truncated components of E^{c} contributing to the integral. Suppose for example that (a, b) is such a component and x ≤ a < x + y < b ≤ x + 1. Suppose also, for simplicity, that x = 0. We claim that then

∫_{y}^{b} \frac{ξ_{0}(t) − ξ(t)}{t} dt ≥ −1.

This follows because Lemma 2.5 says that this integral will only become smaller if we replace the actual ξ on (y, b) by χ_{(y,y+h)}, where again h is chosen so that the integral of ξ over (y, b) is left unchanged. A similar process was used to construct ξ_{0}, so after this replacement, ξ_{0} and ξ are both characteristic functions of an interval, and the interval of ξ_{0} is not smaller than the one corresponding to ξ, so the difference ξ_{0} − ξ is zero, except perhaps on an interval of at most the size of the truncated piece (a, y), and this is obviously ≤ y.

Similar discussions of course apply to the other cases, so (2.9) is bounded below as y → 0+ and (2.8) cannot hold.

3. Proof of Theorem 1.4

For a more streamlined presentation, we first of all isolate the following simple (but key) calculation.

Lemma 3.1. If 0 < |t| ≤ 2y, then

∫_{0}^{y} |ln |1 − \frac{h}{t}|| dh ≤ 12y ln (1 + \frac{y}{|t|}) .

Proof. We will treat explicitly only the case 0 < t ≤ 2y here; the case where t < 0 is similar, but easier. In the former case,

∫_{0}^{y} |ln |1 − \frac{h}{t}| | dh = t ∫_{1−y/t}^{1} |ln |s|| ds. (3.1)

If y/t ≤ 2, then

(3.1) ≤ 2t ∫_{0}^{1} |ln s| ds = 2t ≤ 4y.

Similarly, if y/t > 2, then

(3.1) = 2t + t ∫_{1}^{y/t−1} ln s ds ≤ 2t + y ln(1 + y/t) ≤ 12y ln(1 + y/t).

Proof of Theorem 1.4. We will prove here that for almost every x ∈ ℝ, the one-sided right approximate derivative (D_{ap}^{+}γ) (x) exists and (1.3) holds. The same argument can then be used to establish the corresponding statement about the left derivative, and these two statements together will give the full claim. Here, one-sided derivatives are defined in the obvious way; for example, we say that (D_{ap}^{+}f)(x) = d if for all ϵ > 0,

lim_{h→0+} \frac{1}{h} |\{y ∈ (0, h) : |\frac{f(x + y) − f(x)}{y} − d| ≥ ϵ\}| = 0.

Our basic strategy is modelled on the proof of [9, Theorem 5.4]. The following statements hold at (Lebesgue) almost every point x ∈ ℝ; here, we write again ν_{s} for the singular part of ν, and we also use the notation ν(t) = ν((−∞, t)).

• x is a Lebesgue point of ν′(t);

• lim_{y→0+} g(x + iy) exists;

• lim_{y→0+} ν_{s}([x − y, x + y])/y = 0;

• lim_{y→0+} Im g_{s}(x + iy) = 0, where g_{s}(z) = ∫_{ℝ} \frac{dν_{s}(t)}{t−z}. We will now show that if x has all these properties, then (D_{ap}^{+}γ)(x) exists and (1.3) holds.

So fix such an x. To simplify the notation, we will again assume that x = 0. The basic idea is to look at averages of

F (y) ≡ Re g(iy) + \frac{γ(y) − γ(0)}{y}.

By the definitions of g and γ, we have that F (y) = ∫_{ℝ} ϕ_{y}(t) dν(t), where

(3.2)  ϕ(t) = \frac{t}{t^{2} + 1} + ln |1 − \frac{1}{t}| ,
ϕ_{y}(t) = \frac{1}{y}ϕ (\frac{t}{y} ) .

Recall that we assumed that ν has a finite logarithmic potential γ(z) everywhere, so ϕ_{y} in L_{1}(dν). Moreover, since ϕ(t) = O(t^{−2}) for large |t|, we have that ϕ, ϕ_{y} ∈ L_{1}(ℝ). For later use, we also observe that

∫_{−∞}^{∞} ϕ(t) dt = 0. (3.3)

To prove this, look at ∫_{−R}^{R} ϕ. Clearly, the first term from (3.2) is odd and thus doesn’t contribute to this integral, and

∫_{−R}^{R} ln |1 − \frac{1}{t}| dt = ∫_{−R}^{R} ln |\frac{t − 1}{t}| dt

= ∫_{−R−1}^{−R} ln |t| dt − ∫_{R−1}^{R} ln |t| dt → 0 (R → ∞),

so we obtain (3.3).

Suppose now that B_{y} is a family of Borel sets with the following properties:

B_{y} ⊂ [δy, y], |B_{y}| ≥ δy, (3.4)

for some fixed (but arbitrary) 0 < δ < 1/2. Define

ψ_{y}(t) = \frac{1}{|B_{y}|} ∫_{B_{y}} ϕ_{h}(t) dh.

We now claim that

|ψ_{y}(t)| ≲ \begin{cases}y^{−1} ln(1 + y/|t|) & 0 < |t| ≤ 2y \\ y/t^{2} & |t| > 2y \end{cases} . (3.5)

The constant implicit in (3.5) only depends on δ. Indeed, for |t| ≤ 2y this follows immediately from Lemma 3.1 and the obvious bound |t|/(t^{2} + y^{2}) ≤ 2/y (if |t| ≤ 2y). If, on the other hand, h ≤ y < |t|/2, then Taylor’s theorem shows that

|ϕ_{h}(t)| = \frac{h^{2}}{|t|(t^{2} + h^{2})} + O(h/t^{2}) ≲ \frac{y}{t^{2}} ,

and the second bound from (3.5) follows.

Next, (3.3) and the Fubini-Tonelli Theorem imply that

∫_{−∞}^{∞} ψ_{y}(t) dt = 0. (3.6)

Our next goal is to show that

lim_{y→0+} ∫_{−∞}^{∞} ψ_{y}(t) dν(t) = 0.

We rewrite this as

|∫_{ℝ} ψ_{y}(t) dν(t)| ≤ ∫_{ℝ} |ψ_{y}(t)| dν_{s}(t) + |∫_{ℝ} ψ_{y}(t)ν′(t) dt| . (3.7)

Our first step will be to show that the first integral on the right-hand side of (3.7) goes to zero as y → 0. Start by considering the contributions coming from |t| > 2y: By (3.5),

∫_{|t|>2y} |ψ_{y}(t)| dν_{s}(t) ≲ ∫_{ℝ} \frac{y}{t^{2} + y^{2}} dν_{s}(t) = Im g_{s}(iy) → 0,

by our choice of x (= 0). Next, if ϵ > 0 is given, we can find η > 0 so that if h ≤ η, then ν_{s}([−h, h]) ≤ ϵh. If 2y < η, then this, (3.5), and the monotone convergence theorem imply that

∫_{|t|≤2y} |ψ_{y}(t)| dν_{s}(t) = ∑_{n=0}^{∞} ∫_{2^{−n}y<|t|≤2^{−n+1}y} |ψ_{y}(t)| dν_{s}(t)

≲ \frac{1}{y} ∑_{n=0}^{∞} ν_{s} ([−2^{−n+1}y, 2^{−n+1}y]) ln (1 + 2^{n})

≲ ϵ ∑_{n=0}^{∞}(n + 1)2^{−n} = Cϵ.

So, putting things together, we see that lim_{y→0+} ∫_{ℝ} |ψ_{y}| dν_{s} = 0, as desired.

As for the last integral from (3.7), we recall (3.6) to estimate this as follows:

|∫_{ℝ} ψ_{y}(t)ν′(t) dt| ≤ ∫_{ℝ} |ψ_{y}(t)| |ν′(t) − ν′(0)| dt

Now a very similar argument works, so we will just give a sketch of this. First of all, for |t| > 2y, ψ_{y}(t) is dominated by the Poisson kernel y/(t^{2} + y^{2}), so this part goes to zero because x = 0 is a Lebesgue point of ν′. For small |t|, on the other hand, we again have that the contributions coming from |t| ≈ 2^{−n}y will be ≲ ϵn2^{−n}, and the sum over n is still ≲ ϵ.

Let us summarize: We have shown that

lim_{y→0} ∫ ψ_{y}(t) dν(t) = 0.

By unwrapping the definitions, we see that this means that

lim_{y→0} (\frac{1}{|B_{y}|} ∫_{B_{y}} \frac{γ(h) − γ(0)}{h} dh + \frac{1}{|B_{y}|} ∫_{B_{y}} Re g(ih) dh) = 0.

Since g(ih) converges, to g(0), by the choice of x = 0 again, the second term converges to Re g(0), so we can also say that

lim_{y→0} \frac{1}{|B_{y}|} ∫_{B_{y}} (\frac{γ(h) − γ(0)}{h} + Re g(0)) dh = 0, (3.8)

and this holds for any choice of sets B_{y} as in (3.4). This implies that the (right) approximate derivative of γ at x = 0 exists and (1.3) holds. Indeed, if this were not true, then we could find δ, ϵ > 0 and a sequence of sets A_{n} ⊂ [0, y_{n}], with y_{n} → 0, such that |A_{n}| ≥ 3δy_{n} and

|\frac{γ(h) − γ(0)}{h} + Re g(0)| ≥ ϵ

for all h ∈ A_{n}. But then we can also construct sets B_{n} ⊂ [δy_{n}, y_{n}], |B_{n}| ≥ δy_{n}, so that either

\frac{γ(h) − γ(0)}{h} + Re g(0) ≥ ϵ

for all h ∈ B_{n} or … ≤ −ϵ for all h ∈ B_{n}. However, then (3.8) with B_{y_{n}} = B_{n} leads to a contradiction, so we have to admit that the (onesided) approximate derivative exists and (1.3) holds, as claimed.

References

[1] A.M. Bruckner, Differentiation of Real Functions, Lecture Notes in Mathematics, 659, Springer, Berlin, 1978.

[2] J.A. Cima, A.L. Matheson, and W.T. Ross, The Cauchy Transform, Mathematical Surveys and Monographs, 125, American Mathematical Society, Providence, RI, 2006.

[3] J. Dombrowski, Quasitriangular matrices, Proc. Amer. Math. Soc. 69 (1978), 95–96.

[4] F. Gesztesy, M. Krishna, and G. Teschl, On isospectral sets of Jacobi operators, Comm. Math. Phys. 181 (1996), 631–645.

[5] F. Gesztesy and P. Yuditskii, Spectral properties of a class of reflectionless Schrödinger operators, J. Funct. Anal. 241 (2006), 486–527.

[6] F. Gesztesy and M. Zinchenko, Local spectral properties of reflectionless Jacobi, CMV, and Schrödinger operators, preprint,

http://arxiv.org/abs/0803.3177

[7] V. Jaksic and Y. Last, A new proof of Poltoratski’s theorem, J. Funct. Anal. 215 (2004), 103–110.

[8] M. Martin and M. Putinar, Lectures on Hyponormal Operators, Operator Theory: Advances and Applications, 39, Birkhäuser Verlag, Basel, 1989.

[9] M. Melnikov, A. Poltoratski, and A. Volberg, Uniqueness theorems for Cauchy

integrals, preprint, http://arxiv.org/abs/0704.0621

[10] F. Nazarov, A. Volberg, and P. Yuditskii, Reflectionless measures with a point mass and singular continuous component, preprint,

http://arxiv.org/abs/0711.0948

[11] A. Poltoratski, Boundary behavior of pseudocontinuable functions, St. Petersburg Math. J. 5 (1994), 389–406.

[12] T. Ransford, Potential Theory in the Complex Plane, London Mathematical Society Student Texts, 28, Cambridge University Press, Cambridge, 1995.

[13] C. Remling, The absolutely continuous spectrum of Jacobi matrices, preprint,

http://arxiv.org/abs/0706.1101

[14] S. Saks, Theory of the Integral, second revised edition, Dover Publications, New York, 1964.

[15] B. Simon, Equilibrium measures and capacities in spectral theory, Inverse Probl. Imaging 1 (2007), 713–772.

[16] B. Simon, Weak convergence of CD kernels and applications, to appear in Duke Math. J.

[17] B. Simon and T. Spencer, Trace class perturbations and the absence of absolutely continuous spectra, Comm. Math. Phys. 125 (1989), 113–125.

[18] M. Sodin and P. Yuditskii, Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions, J. Geom. Anal. 7 (1997), 387–435.

[19] H. Stahl and V. Totik, General Orthogonal Polynomials, Encyclopedia of Mathematics and its Applications, 43, Cambridge University Press, Cambridge, 1992.

[20] B.S. Thomson, Real Functions, Lecture Notes in Mathematics, 1170, Springer-Verlag, Berlin, 1985.

Mathematics Department, Texas A&M University, College Station, TX 77843

E-mail address: alexei@math.tamu.edu

Mathematics Department, University of Oklahoma, Norman, OK 73019

E-mail address: cremling@math.ou.edu