APPROXIMATION RESULTS FOR REFLECTIONLESS JACOBI MATRICES

ALEXEI POLTORATSKI AND CHRISTIAN REMLING

Abstract. We study spaces of reflectionless Jacobi matrices. The main theme is the following type of question: Given a reflectionless Jacobi matrix, is it possible to approximate it by other reflectionless and, typically, simpler Jacobi matrices of a special type? For example, can we approximate by periodic operators?

1. Introduction

We continue our study of reflectionless Jacobi matrices, which was begun in [10, 12]. In this paper, by a Jacobi matrix we mean a bounded, self-adjoint operator J on ℓ^{2}(ℤ) (always whole line!) which acts as follows:

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

Alternatively, one can represent J by the following tridiagonal matrix with respect to the standard basis of ℓ^{2}(ℤ):

J = (\begin{matrix}... & ... & ... \\ & a(−2) & b(−1) & a(−1) \\ & & a(−1) & b(0) & a(0) \\ & & & a(0) & b(1) & a(1) \\ & & & & ... & ... & ... \end{matrix})

Here, a(n) ≥ 0 and b(n) ∈ ℝ, and we also assume that a, b ∈ ℓ^{∞}(ℤ). (Usually, one insists that a(n) > 0, but for reasons of formal elegance, our convention seems preferable here.) The set of all such Jacobi matrices will be denoted by J.

Date: May 10, 2010.

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

2000 Mathematics Subject Classification. Primary 47B36 81Q10; Secondary 30E20.

AP’s work supported by NSF grant 0800300; CR’s work supported by NSF grant DMS 0758594.

A Jacobi matrix is called reflectionless on a bounded Borel set B ⊂ ℝ if for all n ∈ ℤ,

Re g_{n}(t) = 0 for almost every t ∈ B,

where g_{n}(z) = ⟨δ_{n}, (J − z)^{−1}δ_{n}⟩, and, as usual, δ_{n}(j) = 1 if j = n and = 0 if j ≠ n. It will be convenient to denote the set of reflectionless (on B) Jacobi matrices by R(B).

Reflectionless Jacobi matrices are of special interest because they provide the basic building blocks for arbitrary Jacobi matrices with non-empty absolutely continuous spectrum. See [12].

We will usually work with topological spaces of reflectionless Jacobi matrices rather than single operators. These spaces will always be contained in

J_{R} = {J ∈ J : ∥J∥ ≤ R}

for some R > 0. The topology we are interested in can be described as product topology on the coefficients; it is induced by the metric

d(J, J′) = ∑_{n=−∞}^{∞} 2^{−|n|} (|a(n) − a′(n)| + |b(n) − b′(n)|) .

This is also the topology that is induced on J_{R} by the weak or strong operator topology. This topology is by far the most useful one for the questions we are interested in here for many reasons, not the least of which are its smooth interaction with other natural topologies and the fact that it makes J_{R} a compact space. These two themes will play a prominent role throughout this paper.

A Jacobi matrix is called periodic if its coefficients have this property, that is, a(n + p) = a(n), b(n + p) = b(n) for all n ∈ ℤ and some period p ∈ ℕ. It is well known that periodic Jacobi matrices are reflectionless on their spectrum, which is a finite gap set (a union of finitely many compact intervals). Our first main result says that, conversely, any J ∈ R(B) can be approximated by periodic operators with spectrum almost equal to B. More precisely:

Theorem 1.1. J ∈ R(B) if and only if there are R > 0 and periodic Jacobi operators J_{n} ∈ J_{R} with spectra σ(J_{n}) = P_{n}, such that d(J_{n}, J) → 0 and |BΔP_{n}| → 0.

Here, BΔP denotes the symmetric difference of B and P (the set of all points that are in one set, but not in the other), and | · | refers to Lebesgue measure.

The main interest lies in the “only if” part; the converse statement is an immediate consequence of the fact that R(B) ∩ J_{R} is a compact set with respect to d; see Proposition 3.1(a) below for more details. We have stated this converse for completeness.

Our original motivation for Theorem 1.1 came from Corollary 1.3 below. Let us first formulate an abstract version of this statement. We define the action of the (left) shift S on J ∈ J in the obvious way: J′ = SJ has coefficients a′(n) = a(n + 1), b′(n) = b(n + 1).

Theorem 1.2. Let φ : J → J be a continuous (with respect to d, as always) map that preserves spectra and commutes with the shift: σ(φ(J)) = σ(J), φS = Sφ. Then φ(J) ∈ R(B) whenever J ∈ R(B).

The application we have in mind here is to Toda flows, as spelled out in Corollary 1.3 below. Let us quickly recall the basic setup here; for more background information on Toda flows, see, for example, [15]. For any (real valued) polynomial p, we can define an associated Toda flow, as follows. Let p_{a}(J) be the anti-symmetric part of p(J): write p(J) as a matrix with respect to the standard basis of ℓ^{2}(ℤ), change the signs in the lower triangular part and delete the diagonal to obtain p_{a}(J). Then the differential equation

J̇ = [p_{a}(J), J]

defines a global flow on J, which we call the Toda flow associated with p.

Corollary 1.3. If J(0) ∈ R(B), then J(t) ∈ R(B) for any Toda flow and all times t ∈ ℝ.

This follows because the Toda maps J = J(0) ↦ J(t) (for fixed t ∈ ℝ and polynomial p) are known to have all the properties required in Theorem 1.2. (The continuity with respect to d is perhaps not addressed explicitly in the existing literature, but this is easy to establish. It will also be discussed in a forthcoming publication [13], and, in any event, this discussion would take us too far afield here.)

In the second part of this paper, we will try to analyze the spaces

R_{0}(K) = {J ∈ R(K) : σ(J) ⊂ K},

for compact subsets K ⊂ ℝ. If K is essentially closed, then all J ∈ R_{0}(K) will in fact satisfy σ(J) = K.

For any compact K ⊂ ℝ, we have:

Proposition 1.4. R_{0}(K) is compact.

As usual, this statement refers to the topology that is induced by the metric d. Proposition 1.4 is not a new result; compare, for example, [5, 12]. However, the Proposition will also be a consequence of our Proposition 3.1 below, so we will briefly discuss its proof here.

It is also clear that R_{0}(K) is shift invariant, and, ideally, one would like to understand the dynamical system (R_{0}(K), S). This task has been accomplished for so-called homogeneous sets K by Sodin-Yuditskii [14]; the special case of a finite gap set K is classical and has been studied in very great detail. Beyond the Sodin-Yuditskii result, very little is known at present.

In this paper, we set ourselves the more modest task of analyzing the R_{0}(K) for compact K ⊂ ℝ as a collection of topological spaces rather than dynamical systems, but we would like to do this for rather general sets K. In this endeavor, the main difficulty comes from a possible singular part of spectral measures and related measures on K. If there can’t be any such singular part, the analysis becomes much easier. For example, the following statement holds.

Theorem 1.5. Suppose that ρ_{s}(K) = 0 for all ρ ∈ H(K). Then R_{0}(K) is homeomorphic to the N-dimensional torus 𝕋^{N}, where N ∈ ℕ_{0} ∪{∞} is the number of bounded components (“gaps”) of K^{c}.

Here, H(K) denotes the collection of measures that come from the socalled H functions of the J ∈ R_{0}(K). This H function is defined as the negative reciprocal of the Green function at n = 0: H(z) = −1/g_{0}(z). H is a Herglotz function and thus there is a unique associated measure. Please see Section 2 for a more detailed discussion of these definitions.

The hypothesis of Theorem 1.5 will be satisfied by many sets K. For example, Theorem 1.5 applies to all weakly homogeneous sets K [10, Corollary 2.3]; see also [14].

If N = ∞ in Theorem 1.5, then 𝕋^{∞} = S^{ℕ} is defined as the infinite Cartesian product of countably many copies of the circle S, and we use product topology on this space. If N = 0 (so K is an interval), then we just define 𝕋^{0} to be a single point. This of course is a well known special case: if K is a compact interval, then the only Jacobi matrix that is reflectionless on K and has K as its spectrum is the one with the appropriate constant coefficients.

While we haven’t seen it in print in this form, we don’t want to claim much credit for Theorem 1.5. There is a natural way of setting up a bijection between R_{0}(K) and 𝕋^{N}, which has been used by many authors [3, 5, 14, 15], and what we add here is the observation that this map is continuous with respect to the chosen topologies.

Moving on to the more original (we hope) parts of our discussion of the spaces R_{0}(K), we would now like to understand how R_{0}(K) changes with K. The spaces R_{0}(K) are compact subsets of J, so it seems natural to use Hausdorff distance to compare their coarse structure, what they look like if viewed from a distance. Recall that the Hausdorff distance between two compact, non-empty subsets A, B of a metric space is defined as

h(A, B) = max{sup_{x∈A} d(x, B), sup_{y∈B} d(y, A)}.

We would now like to know what conditions need to be imposed on K, K′ if we want R_{0}(K), R_{0}(K′) to be close in Hausdorff distance. For instance, is it always possible to approximate a potentially complicated space R_{0}(K) by a simpler space, say R_{0}(K′) for a finite gap set K′?

More formally, we can observe that K ↦ R_{0}(K) is an injective map that is defined on non-empty, compact sets K ⊂ ℝ, so we can pull back the Hausdorff metric and ask for a description of the metric on the space {K ⊂ ℝ : K compact, K ≠ ∅} that is obtained in this way. More precisely, we would like to be able to write down an equivalent metric (one that generates the same topology).

We will not be able to completely answer this question in this paper, but we can report on some progress. Based on what we do below, it in fact seems reasonable to conjecture that a possible choice for the sought metric is

(1.1)  δ(K, K′) = h(K, K′) + |KΔK′|.

It is easy to see that this indeed defines a metric δ on the non-empty compact subsets of ℝ; we will discuss this fact in Proposition 4.1 below.

Notice that δ generates a stronger topology on {K} than its first summand Hausdorff distance h. It is actually clear that {K ⊂ [−R, R]} with the correct metric, whatever it may be, is not a compact space, and this immediately rules out Hausdorff distance as the answer to our question. For example, one can show that if K_{n} = ⋃_{j=1}^{n}[j/n, j/n + 1/n^{2}], say, then with respect to Hausdorff distance,

lim_{n→∞} R_{0}(K_{n}) = {J : σ(J) ⊂ [0, 1]},

which is not equal to R_{0}(K) for any K.

Here’s one possible formulation of what we will actually prove.

Theorem 1.6. Let K, K_{n} ⊂ [−R, R] be non-empty, compact sets, and abbreviate Ω = R_{0}(K), Ω_{n} = R_{0}(K_{n}).

(a) If h(Ω_{n}, Ω) → 0, then δ(K_{n}, K) → 0.

(b) Suppose that ρ_{sc}(K) = 0 for all ρ ∈ H(K). Then δ(K_{n}, K) → 0 implies that h(Ω_{n}, Ω) → 0.

This is an incomplete answer to the question we posed because of the additional assumption in part (b) (which is similar to, but considerably weaker than the assumption we made in Theorem 1.5 above). We can be somewhat more specific here: Notice, first of all, that for two compact sets to be close in Hausdorff distance, we have to be able to come close to each point from one set by an element of the other set, and vice versa. However, as we’ll explain below, at the beginning of Section 5, the only real issue here is the question of whether we will be able to approximate any J ∈ Ω by J_{n} ∈ Ω_{n}. Indeed, in Theorem 1.7 below, the additional assumption is required only in part (b), when we try to approximate J ∈ Ω.

Theorem 1.7. Let K_{n}, K ⊂ [−R, R] be non-empty compact sets, and suppose that δ(K_{n}, K) → 0. Then: (a)

lim_{n→∞} sup_{J∈Ω_{n}} d(J, Ω) = 0

(b) If J ∈ Ω and the associated measure satisfies ρ_{sc}(K) = 0, then

lim_{n→∞} d(J, Ω_{n}) = 0.

As we’ll discuss in Section 5, Theorem 1.6(b) will be a quick consequence of Theorem 1.7. The crucial ingredient to the proof of Theorem 1.7, in turn, will be the following approximation result, which could be of some independent interest.

Theorem 1.8. Let J ∈ R_{0}(K) and suppose that the associated ρ ∈ H(K) satisfies ρ_{sc}(K) = 0. Then there are J_{n} ∈ R_{0}(K) with ρ_{s}^{(n)}(K) = 0, so that d(J_{n}, J) → 0.

We organize this paper as follows. Section 2 discusses basic material from inverse spectral theory, but in a version that is tailor made for the investigation of specifically reflectionless operators. We also prove Theorem 1.5 in this section. Theorems 1.1 and 1.2 are proved in Section 3. In Section 4, we discuss Theorem 1.6(a). Section 5 has the proofs of Theorems 1.6(b), 1.7, and 1.8.

In the main body of this paper, that is, in this introduction and Sections 2–5, we assume that the sets B and K are of positive Lebesgue measure. This is really the relevant case here since we want to understand aspects of the absolutely continuous spectrum. However, from a formal point of view, our results remain correct if |B| = 0 or |K| = 0. Note that if |B| = 0, then R(B) = J, and R_{0}(K) = {J : σ(J) ⊂ K} if |K| = 0. The arguments needed in these cases are easier than, but also different from those for the positive measure case. Therefore, we very briefly discuss them separately, in the final section. In Sections 1–5, we always assume that |B| > 0, |K| > 0.

In fact, we can strengthen Theorems 1.6, 1.7, and 1.8 if |K| = 0, and we do obtain complete answers in this case. For example, if |K| = 0, then h(Ω_{n}, Ω) → 0 if and only δ(K_{n}, K) → 0, which, in this case, happens if and only if h(K_{n}, K) → 0.

2. Spectral data for reflectionless Jacobi matrices

In this section, we review and develop further basic tools that will allow us to conveniently describe reflectionless Jacobi matrices in terms of carefully chosen spectral data. This material will be fundamental for everything we do in this paper. See [3, 5, 12, 15] for earlier applications of this basic method, and also [4] for related uses of the ξ function. Our presentation here follows [12, Sections 5, 6] very closely, with some additional material added. We only sketch most of the proofs here and refer the reader to this reference for full details.

This discussion will come in three parts. In the first part, we introduce convenient (for our purposes) spectral data for arbitrary bounded Jacobi matrices. We then modify these data to obtain a related parametrization of R(B), and, in the final part, we introduce still another variant of this, which will be particularly useful when we discuss the spaces R_{0}(K).

Let J ∈ J, and consider the Herglotz function

H(z) = −\frac{1}{g(z)},

where g(z) = ⟨δ_{0}, (J − z)^{−1}δ_{0}⟩. Let ξ be the Krein function of H, that is,

ξ(t) = \frac{1}{π} lim_{y→0+} Im ln H(t + iy).

Since H(z) is never zero on the upper half plane ℂ^{+}, we can take a holomorphic logarithm ln H(z). In fact, H(z) is a Herglotz function, so we can demand that the imaginary part of this logarithm lies in (0, π). The limit defining ξ exists for (Lebesgue) almost every t ∈ ℝ; we view ξ as an element of L^{∞}(ℝ), and in fact we have that 0 ≤ ξ(t) ≤ 1. The Krein function is an exceedingly useful tool here, mainly because of the following fact:

Proposition 2.1. If J ∈ R(B), then ξ = 1/2 almost everywhere on B.

Proof. This is obvious because if J is reflectionless on B, then, by definition, g and thus also H are purely imaginary on this set. □

We can recover H from its Krein function ξ; in fact, we have the explicit formula

(2.1)  H(z) = (z + R) exp (∫_{−R}^{R} \frac{ξ(t) dt}{t − z}) .

Here, R is chosen so large that ∥J∥ ≤ R; we have also made use of the asymptotic formula H(z) = z + O(1) as |z| → ∞ to determine an otherwise unknown constant.

However, we cannot, in general, recover the Jacobi matrix J from H or, equivalently, ξ. We must introduce additional spectral data. We proceed as follows. Write down the Herglotz representation for H:

(2.2)  H(z) = z + A + ∫_{(−R,R)} \frac{dρ(t)}{t − z}

Here, ρ is a finite and (obviously) compactly supported Borel measure. The constant A can be identified in terms of the previously used data as

A = R − ∫_{−R}^{R} ξ(t) dt.

Furthermore, we can use the half line spectral measures ρ_{±} to decompose ρ as follows:

ρ = a(0)^{2}ρ_{+} + a(−1)^{2}ρ_{−}

Please see [15, Chapter 2] for this information and precise definitions of ρ_{±}. (Warning: These measures are called ρ̃_{±} in [15].)

It will be convenient to introduce ν_{+} = a(0)^{2}ρ_{+}, ν_{−} = a(−1)^{2}ρ_{−} and rewrite this as

(2.3)  ρ = ν_{+} + ν_{−}.

We will also refer to these measures ν_{±} as half line spectral measures. Note that ν_{+} or ν_{−} or both of these can be the zero measure.

Every pair of positive finite measures (ρ_{+}, ρ_{−}) with ρ_{±}(ℝ) = 1 is admissible here as a pair of half line spectral measures; we thus obtain the following (almost) parametrization of (bounded) Jacobi matrices: With each J ∈ J, associate its ξ function and also its positive half line spectral measure ν_{+}. As just discussed, these data have the property that if ρ is defined via (2.1), (2.2), and then ν_{−} via (2.3), then ν_{−} is a positive (finite) measure (possibly the zero measure).

Conversely, suppose that such a pair (ξ, ν_{+}) is given. In other words, if again ρ is the measure associated with ξ and (2.3) is used to define ν_{−}, then ν_{−} is a positive measure. Then there exists a Jacobi matrix J that has (ξ, ν_{+}) as its spectral data; moreover, J will be unique if both ν_{+} and ν_{−} have infinite supports (in particular, neither is the zero measure). This follows from the usual inverse spectral theory for half line problems: If ν_{±}(ℝ) > 0, then we can of course recover ρ_{±} (and a(0), a(−1)) from ν_{±}, because these measures are probability measures. Then the measures ρ_{±} determine all the remaining coefficients except for b(0); here, we use the assumption that ρ_{±} are not finitely supported. Moreover, b(0) = −A, with A from (2.2), which is determined by ξ. See [15, Chapter 2] and [12, Section 5] for more background information on this procedure.

If ν_{+} or ν_{−} is finitely supported, then a(n) = 0 for some n and only the coefficients up to this index are determined by (ξ, ν_{+}). For instance, to give an extreme example, if ξ = χ_{(−R,b)}, then H(z) = z − b and ρ = ν_{+} = ν_{−} = 0, and we can only conclude that a(−1) = a(0) = 0, b(0) = b.

We are interested in approximation properties in this paper, so it is essential to use parametrizations with good continuity properties. Indeed, we have:

Proposition 2.2. Fix R > 0. The map that sends J ∈ J_{R} to (ξ, ν_{+}) is a continuous map between compact metric spaces. Here, we use the weak-∗ topology for both ξ dt and dν_{+} on the image.

More precisely, we make the image a (compact) metric space in the following way. Fix a metric D that induces the weak-∗ topology on the Borel measures μ on [−R, R] with ∥μ∥ ≤ C. Then let

(2.4)  d((ξ, ν_{+}), (ξ′, ν′_{+})) = D(ξ dt, ξ′ dt) + D(ν_{+}, ν′_{+}).

Sketch of proof. The spectral measure ν_{+} depends continuously on J with respect to the chosen topologies; this is a well known basic fact. See, for example, [12, Lemma 3.2]. Moreover, if J_{n} → J, then H_{n}(z) → H(z) locally uniformly on z ∈ ℂ^{+}. But then ln H_{n}(z) also converges, to ln H(z), and these new Herglotz functions have associated measures ξ_{n} dt and ξ dt, respectively, so we now obtain the asserted convergence ξ_{n} dt → ξ dt in weak-∗ sense. Compare [12, Theorem 2.1] for this last step. □

If J ∈ R(B) for some positive measure Borel set B ⊂ ℝ, then, as we saw above, ξ = 1/2 on B, and, moreover, ν_{+} also has to satisfy a related condition. We have that

(2.5)  dν_{+}(t) = f(t) dρ(t),

where f is a Borel function that satisfies 0 ≤ f ≤ 1. Of course, so far this is just a rephrasing of what (2.3) says about ν_{+}. However, J ∈ R(B) also implies that f(t) = 1/2 for Lebesgue almost every t ∈ B. See [12, Corollary 5.3]. These data (ξ, f) provide a complete parametrization of R(B). Here, f is thought of as an element of L^{1}(ℝ, dρ), where ρ is the measure associated with ξ.

So, if (ξ, f) as above is given (that is, ξ = f = 1/2 almost everywhere on B and 0 ≤ ξ, f ≤ 1), then there is a unique J ∈ R(B) whose spectral data are (ξ, f). This time, there are no strings attached: We know that χ_{B} dt is absolutely continuous with respect to dρ, and, since f = 1/2 Lebesgue almost everywhere on B, also with respect to dν_{±}, so these measures have infinite supports. See again [12, Corollary 5.3] for the details. The correspondence J ↔ (ξ, f) is continuous in both directions:

Proposition 2.3. Fix R > 0 and a Borel set B ⊂ (−R, R) with |B| > 0. The map that sends J ∈ R(B) ∩ J_{R} to (ξ, f) is a homeomorphism onto its image. On this image, we use the weak-∗ topology for the measures dν_{+} = f dρ; both spaces are compact metric spaces.

We could, of course, use the metric from (2.4) again (with dν_{+} = f dρ), but it is also possible, as indicated, to just use

d((ξ, f), (ξ′, f′)) = D(f dρ, f′ dρ′)

instead. This works because a reflectionless Jacobi matrix J ∈ R(B) is already determined by its half line restriction and thus also by dν_{+} = f dρ. Compare, for example, [12, Proposition 4.1]. So it indeed suffices to work with the half line spectral measures ν_{+} when defining d: if d((ξ, f), (ξ′, f′)) = 0, then ν_{+} = ν′_{+}, which, as just explained, implies that J = J′; in particular ξ = ξ′ and thus (ξ, f) = (ξ′, f′), as required.

Proof. We already know that this map is injective, and it is obviously continuous, being a restriction of the map from Proposition 2.2. Moreover, R(B) ∩ J_{R} is compact (see again [12, Proposition 4.1]), and a continuous bijection between compact metric spaces automatically has a continuous inverse. □

We now specialize further and seek a parametrization of R_{0}(K), for a compact set K ⊂ ℝ with |K| > 0. We will use the symbol H(K) to denote the collection of all measures ρ from (2.2) that correspond to some J ∈ R_{0}(K). This is the set that was referred to in the formulation of Theorems 1.5 and 1.6(b). Similarly, we let X(K) be the collection of the ξ functions of the J ∈ R_{0}(K).

By the spectral theorem, g has a representation of the type g(z) = ∫_{K} \frac{dμ(t)}{t−z}, with a probability measure μ. This implies that on each bounded component (a, b) ⊂ K^{c}, ξ is a step function that jumps from 0 to 1:

(2.6)  ξ(t) = χ_{(μ,b)}(t) (a < t < b)

for some μ ∈ [a, b]. These intervals (a, b) will also be referred to as gaps (of K). Notice that the parameters (μ_{j}) (one for each gap) determine ξ. Indeed, we always have that ξ = 1/2 on K and ξ = 1 to the left of K and ξ = 0 to the right of K, so it suffices to specify ξ on each gap to have a complete definition of a ξ ∈ X(K).

The correspondence between μ and ξ that is obtained in this way is again a homeomorphism with respect to the natural topologies:

Proposition 2.4. Fix a compact, non-empty set K ⊂ ℝ, with gaps (a_{j}, b_{j}). The map X(K) → ∏[a_{j}, b_{j}] that sends ξ to μ is a homeomorphism if we use the product topology on the second space.

On X(K), we of course use the weak-∗ topology for ξ dt, as before. Both spaces are in fact compact metric spaces, a possible choice for the metric on the second space is

d(μ, μ′) = ∑_{j=1}^{∞} |μ_{j} − μ′_{j}|.

Sketch of proof. It is clear from our discussion above that the map ξ ↦ μ is bijective, and continuity (in both directions) is easy to confirm. □

Every ρ ∈ H(K) is of the following form:

(2.7)  dρ = χ_{K}F dt + χ_{K} dρ_{s} + ∑ w_{j}δ_{μ_{j}}

Here, the sum is taken over all gaps for which a_{j} < μ_{j} < b_{j}. We can be sure that there will indeed be a point mass at all these μ_{j} (in other words, w_{j} > 0) because ξ jumps from 0 to 1 at these points, so we can refer to a basic criterion for the existence of point masses. See [6, pg. 201] or [10, Lemma 2.4]. Similarly, ρ can not have an additional singular part off K; in fact, H has a holomorphic continuation through every interval I ⊂ K^{c} with μ_{j} ∉ I. We would like to again remind the reader that F > 0 almost everywhere on K, because πF = Im (−1/g) = |1/g|, and this can not be zero on a positive measure set.

The function f from above, which determines ν_{+} through (2.5), now has to satisfy f = 1/2 Lebesgue almost everywhere on K and f(μ_{j}) = 0 or 1 for all j that contribute to the sum in (2.7). The first property was noted above for general J ∈ R(K), and the second property follows quickly from the additional requirement that σ(J) ⊂ K; see [12, Section 6] for a more detailed discussion.

In other words,

(2.8)  dν_{+} = \frac{1}{2}χ_{K}F dt + χ_{K}g dρ_{s} + ∑ σ_{j}w_{j}δ_{μ_{j}},

where g is a Borel function with 0 ≤ g ≤ 1 and σ_{j} ∈ {0, 1}. We will use (μ, σ, g), with μ = (μ_{j})_{j}, σ = (σ_{j})_{j}, and g from (2.8), as the spectral data for J ∈ R_{0}(K). As above, we view g as an element of L^{1}(K, dρ_{s}), that is, we identify g’s that agree almost everywhere on K with respect to ρ_{s}. Frequently, no ρ ∈ H(K) can have a singular part on K, and then we can discard g altogether, and we parametrize R_{0}(K) by just (μ, σ).

It is again true that, conversely, any set of parameters (μ, σ, g) of this type will correspond to a unique J ∈ R_{0}(K). Let us describe the corresponding procedure one more time: Given (μ, σ, g), we first of all define ξ by (2.6) and then, as usual, construct ρ from (2.1), (2.2). Then (2.8) gives us ν_{+}, and then ν_{−} is obtained from (2.3). From the half line spectral measures, we can recover a unique J if ν_{±} have infinite supports, and this is automatic here. This Jacobi matrix J will lie in R_{0}(K).

This parametrization of R_{0}(K) again has the desired continuity properties:

Proposition 2.5. Fix a compact, non-empty set K ⊂ ℝ of positive Lebesgue measure. The map that sends J ∈ R_{0}(K) to (μ, σ, g) is a homeomorphism from R_{0}(K) onto its image. Here, we use the metric from Proposition 2.3:

d((μ, σ, g), (μ′, σ′, g′)) = D(ν_{+}, ν′_{+}),

where ν_{+} = ν_{+}(μ, σ, g) is as in (2.8), and the metric D induces the weak-∗ topology on these measures.

Proof. As explained above, this map is a bijection onto its image, and it is clearly continuous because it can be thought of as a restriction of the map from Proposition 2.3. Since R_{0}(K) is compact, the image is compact, too, and continuity of the inverse map is automatic. □

Note that the individual parts of the decomposition (2.8) of ν_{+} do not necessarily depend continuously on J ∈ R_{0}(K). For example, Theorem 1.8 shows that there are situations where J_{n} → J and (ν_{+})_{s}(K) > 0, but (ν_{n,+})_{s}(K) = 0.

As our first application of the material discussed in this section, let us show that Theorem 1.5 is a direct consequence of the parametrization from Proposition 2.5.

Proof of Theorem 1.5. We now assume that ρ_{s}(K) = 0 for all ρ ∈ H(K), so Proposition 2.5 in fact provides a homeomorphism that sends J ∈ R_{0}(K) to (μ, σ). We want to map these data onto 𝕋^{N}, with N ∈ ℕ∪{∞} being the number of gaps. We will do this componentwise, by mapping (μ_{j}, σ_{j}) for fixed j to a copy of the unit circle S. Before we give the precise definition of this map, let us observe the following: Suppose that J_{n}, J ∈ R_{0}(K), J_{n} → J. Then μ_{j}^{(n)} → μ_{j} as n → ∞ for every fixed j, and if μ_{j} ≠ a_{j}, b_{j}, then also σ_{j}^{(n)} → σ_{j}. Recall in this context that the parameter σ_{j} is not used if μ_{j} = a_{j} or μ_{j} = b_{j}.

To prove this, we can argue as follows: The convergence μ_{j}^{(n)} → μ_{j} is an immediate consequence of Propositions 2.2, 2.4. We also know from the proof of Proposition 2.2 that ρ_{n} → ρ. In our current situation, with ρ_{s}(K) = 0, we can rewrite (2.8) as

(2.9)  dν_{+} = \frac{1}{2} dρ + \frac{1}{2} ∑(2σ_{j} − 1)w_{j}δ_{μ_{j}}.

Now notice that if μ_{j} ≠ a_{j}, b_{j}, then we must also have that

(2.10)  w_{j} = lim_{n→∞} w_{j}^{(n)}

here. Indeed, w_{j} = ∫ f dρ and w_{j}^{(n)} = ∫ f dρ_{n} for all large n for a suitable continuous test function f that is supported by the gap (a_{j}, b_{j}) and equal to 1 in a neighborhood of μ_{j}, so (2.10) follows from the weak- ∗ convergence ρ_{n} → ρ. We also have that ν_{n,+} → ν_{+}, so (2.9) implies that σ_{j}^{(n)} → σ_{j}, as claimed.

We are now ready to describe the sought map from our parameter space onto 𝕋^{N}. More precisely, we will give the inverse map. Let (z_{j})_{j} = (e^{iπx_{j}})_{j} ∈ 𝕋^{N}. For fixed j, let F_{j} be the following map:

F_{j}(e^{iπx_{j}}) = \begin{cases}(a_{j} + x_{j}(b_{j} − a_{j}), 1) & 0 < x_{j} < 1 \\ (a_{j} − x_{j}(b_{j} − a_{j}), 0) & −1 < x_{j} < 0; \end{cases}

we also send z_{j} = 1 to a_{j} and z_{j} = −1 to b_{j}, as suggested by these formulae. We then define F as the map that sends (z_{j})_{j} ∈ 𝕋^{N} to (F (z_{j}))_{j}.

Our preparatory discussion makes it clear that the induced map J ↦ z is continuous. Since we clearly have a bijection between the compact metric spaces R_{0}(K) and 𝕋^{N}, this map automatically has to be a homeomorphism. □

In this argument, we established the continuity of the map R_{0}(K) → 𝕋^{N} that we set up above. As usual, this was sufficient because we have a bijection between compact metric spaces. It is interesting to note that, contrary to what one would normally expect, an easy direct argument for the continuity of the inverse map does not seem available. Rather, we run into difficulties very similar to the ones that will occupy us in the main part of this paper: Could it happen that for z′ ∈ 𝕋^{N} arbitrarily close to a fixed z ∈ 𝕋^{N}, part (or all) of the measure χ_{K}ρ gets sucked out of K and moved into the gaps? The weak-∗ continuity of ρ does not prevent this, and if it happens, then (σ_{j})_{j} could be discontinuous (so it does not happen here, but this an indirect argument and in fact the one we just gave).

3. Approximation by periodic operators

Our main goal in this section is to prove Theorem 1.1. Before we embark on this assignment, we discuss how to derive Theorem 1.2 from Theorem 1.1. We will need the following observation. Actually, we will only need part (a) of Proposition 3.1 here, but part (b) will be needed later on, in the proof of Theorem 1.6, and this seems a good place to present it.

Proposition 3.1. Let J_{n}, J ∈ J_{R} and suppose that d(J_{n}, J) → 0.

(a) If J_{n} ∈ R(B_{n}) and |B \ B_{n}| → 0, then J ∈ R(B).

(b) If h(K_{n}, K) → 0 and σ(J_{n}) ⊂ K_{n}, then σ(J) ⊂ K.

In particular, if J_{n} ∈ R_{0}(K_{n}) and δ(K_{n}, K) → 0, then J ∈ R_{0}(K).

Proof. (a) Let ϵ > 0 be given. By passing to a subsequence, we can assume that |B \ B_{n}| < 2^{−n}ϵ. Let A = ⋂(B ∩ B_{n}). Then J_{n} ∈ R(A), and R(A) ∩ J_{R} is a compact set (see, for example [12, Proposition 4.1(d)]), hence J ∈ R(A) as well. Here, |B \ A| < ϵ, and ϵ > 0 was arbitrary, so from the definition of R(B) we now see that in fact J ∈ R(B), as claimed.

(b) Let x ∈ ℝ \ K. Since K is compact, we then have that (x − 2r, x + 2r) ⊂ K^{c} for suitable r > 0, and thus (x − r, x + r) ⊂ K_{n}^{c} for all sufficiently large n. As J_{n} → J in the strong operator topology, this implies that x ∉ σ(J); see [11, Theorem VIII.24(a)]. □

Note that considerations of this type also give Proposition 1.4; in fact, Proposition 1.4 could be viewed as a special case of the last part of Proposition 3.1, with K_{n} = K. Indeed, the Proposition says that R_{0}(K) is closed, so, since R_{0}(K) ⊂ J_{R} for large R > 0 and J_{R} is compact, it follows that R_{0}(K) is compact also.

Now return to our original topic. Assuming Theorem 1.1, we obtain Theorem 1.2 as follows.

Proof of Theorem 1.2. Let J ∈ R(B). Pick J_{n} ∈ R_{0}(P_{n}) as in Theorem 1.1. Periodicity means that S^{N_{n}}J_{n} = J_{n}, where N_{n} ∈ ℕ is the period, and thus the φ(J_{n}) are periodic as well. Moreover, they have the same spectra as the J_{n}, and since a periodic operator is reflectionless precisely on its spectrum, it follows that φ(J_{n}) ∈ R(P_{n}). By continuity, φ(J_{n}) → φ(J), and thus we can apply Proposition 3.1(a) to conclude the proof. □

Proof of Theorem 1.1. Let J ∈ R(B) ∩ J_{R}, with B ⊂ (−R, R). By removing a set of Lebesgue measure zero from B, we can assume that ρ_{s}(B) = 0; here, ρ again denotes the measure that is associated with the H function of J, as in (2.2). In fact, by passing to a subset of almost the same Lebesgue measure, we can also assume that B is compact and has no isolated points.

Given the material from Section 2, especially Proposition 2.2, we can describe our goal as follows: We would like to construct finite gap sets P_{n} (the periodicity will follow from a density argument; this issue can be ignored for now) and data (ξ_{n}, ν_{n,+}) (say), corresponding to J_{n} ∈ R_{0}(P_{n}), so that these data approach the corresponding data (ξ, ν_{+}) of J ∈ R(B), and |BΔP_{n}| → 0.

Let ϵ > 0 be given. The complement B^{c} ∩(−R, R) is a disjoint union of bounded open intervals, and by keeping only a finite large number of these gaps, we obtain a finite gap set

A_{0} = ⋃_{j=1}^{N} J_{j}, J_{j} = [a_{j}, b_{j}]

that satisfies A_{0} ⊃ B, |A_{0} \ B| < ϵ, ρ(A_{0} \ B) < ϵ. Notice that the endpoints of these intervals J_{j} belong to B. We chose B so that ρ_{s}(B) = 0. Therefore, ρ(∂A_{0}) = 0. As a consequence, we obtain the following:

Lemma 3.2. If ρ_{n} → ρ, then also χ_{A_{0}}ρ_{n} → χ_{A_{0}}ρ and χ_{A_{0}^{c}}ρ_{n} → χ_{A_{0}^{c}}ρ.

As usual, these limits refer to the weak-∗ topology.

We decompose

ρ = ρ_{1} + ρ_{2}, ρ_{1} = χ_{B}ρ = χ_{B}ρ_{ac}, ρ_{2} = χ_{B^{c}}ρ,

and we write accordingly

(3.1)  ν_{+} = \frac{1}{2}ρ_{1} + fρ_{2},

with a Borel function f that satisfies 0 ≤ f ≤ 1. We know that this density equals 1/2 on B because J ∈ R(B); compare the discussion that precedes Proposition 2.3. For future use, we record that

(3.2)  ∥χ_{A_{0}}ρ − ρ_{1}∥ = ∥χ_{A_{0}^{c}}ρ − ρ_{2}∥ = ρ(A_{0} \ B) < ϵ.

We now construct preliminary versions A_{n} of the sets P_{n} and associated Krein function ξ_{n} (which will correspond to certain J_{n} ∈ R_{0}(A_{n}), to be specified at a later stage). For each component I ⊂ B^{c} ∩(−R, R), we have either I ⊂ A_{0}^{c} or I ⊂ J_{j} ⊂ A_{0}. Let {I_{n}}_{n≥1} be an enumeration of those gaps of B that are contained in A_{0}, and set

Ã_{n} = A_{0} \ ⋃_{j=1}^{n} I_{j}.

This is still a finite gap set, and B ⊂ Ã_{n} ⊂ A_{0}. Obtain A_{n} from Ã_{n} by subdividing each component of Ã_{n}^{c} ∩ (−R, R) into n smaller gaps of equal length, which are separated by n − 1 very small intervals of size δ_{n} > 0 each; these intervals will be referred to as bands from now on; the band size δ_{n} will be chosen later.

On each of these new (small) gaps, let ξ_{n} jump from 0 to 1 in a such a way that the average has the correct value. More precisely, if (a, b) is a gap of A_{n}, then put ξ_{n} = χ_{(μ,b)} on (a, b), where μ = b − ∫_{a}^{b} ξ dx (so ∫_{a}^{b} ξ_{n} = ∫_{a}^{b} ξ). If this leads to μ = b and b is the left endpoint of a band, then we delete this band and set ξ_{n} = 0 on the slightly larger interval (a, b + δ_{n}). Do this for all such gaps (if any). Then use the analogous procedure for those gaps where μ = a and a is the right endpoint of a band. Finally, we set ξ_{n} = 1/2 on A_{n}.

This Krein function ξ_{n} lies in X(A_{n}); it corresponds to certain operators from R_{0}(A_{n}). As usual, it also defines a measure ρ_{n} via (2.1), (2.2). We claim that if the δ_{n} approach zero sufficiently rapidly, then we can achieve that

(3.3)  ρ_{n}(A_{n} \ A_{0}) → 0.

This will certainly follow if we can show that ρ_{n} gives little weight to the newly introduced bands of A_{n}. We need the following auxiliary calculation.

Lemma 3.3. Fix 0 < A, B < R and define, for sufficiently small δ > 0,

ξ_{δ}(x) = \begin{cases}0 & δ < x < B \\ 1 & −A < x < 0 \\ 1/2 & 0 < x < δ \end{cases} .

Then

lim_{δ→0+} sup ρ([0, δ]) = 0,

where the supremum is taken over all ρ whose Krein functions ξ ∈ L^{∞}(−R, R) agree with ξ_{δ} on (−A, B).

Proof. Since ξ = 1/2 on (0, δ), it follows that ρ is purely absolutely continuous on this interval, with density equal to (1/π)|H(x)|. From (2.1), we have that for 0 < x < δ,

|H(x)| = (x + R)h_{δ}(x) exp (∫_{(−R,R)\(−A,B)} \frac{ξ(t) dt}{t − x}) ,

where h_{δ}(x) = lim_{y→0+} exp (∫_{−A}^{B} \frac{(t−x)ξ_{δ}(t) dt}{(t−x)^{2}+y^{2}}). In fact, this can be evaluated:

h_{δ}(x) = \frac{\sqrt{x(δ − x)}}{A + x}

These formulae make it clear, first of all, that the arrangement that maximizes ρ([0, δ]) is the one where ξ = 0 on (−R, −A) and ξ = 1 on (B, R). It is then straightforward to estimate ρ([0, δ]) for this measure and confirm that this quantity approaches zero as δ → 0+. We leave the details to the reader. □

This Lemma indeed establishes (3.3) because we deleted those bands for which we don’t have the situation described in the Lemma (ξ_{n} = 1 to the left of the band and ξ_{n} = 0 to the right).

Moreover, by taking δ_{n} → 0 so fast that also |A_{n} \ B| → 0, we can make sure that ξ_{n} dx → ξ dx in weak-∗ sense. This follows now because ξ = 1/2 on B, so if a continuous f is given, we can split

(3.4)  ∫ fξ_{n} dx = \frac{1}{2} ∫_{A_{n}} f dx + ∫_{A_{n}^{c}} fξ_{n} dx.

The first integral on the right-hand side converges to ∫_{B} fξ dx as n → ∞. To deal with the last integral from (3.4), we approximate f on B^{c} ⊃ A_{n}^{c} uniformly by functions g_{n} that are constant on the gaps of A_{n}, and now our definition of ξ_{n} on these intervals guarantees that ∫_{I} g_{n}ξ_{n} dx = ∫_{I} g_{n}ξ dx for each gap I of A_{n}^{c}. Since |B^{c} \ A_{n}^{c}| → 0, this implies that ∫_{A_{n}^{c}} fξ_{n} → ∫_{B^{c}} fξ.

It follows from this that ρ_{n} → ρ. Indeed, the weak-∗ convergence ξ_{n} dx → ξ dx clearly implies that H_{n}(z) → H(z) locally uniformly on ℂ^{+}, and this in turn shows that ρ_{n} → ρ; compare again [12, Theorem 2.1].

Thus we also have that χ_{A_{0}}ρ_{n} → χ_{A_{0}}ρ and χ_{A_{0}^{c}}ρ_{n} → χ_{A_{0}^{c}}ρ, by Lemma 3.2.

Next, pick a g ∈ C_{0}^{∞}(ℝ) so that 0 ≤ g ≤ 1 and

(3.5)  ∥fρ_{2} − gρ_{2}∥ < ϵ,

where f is the density from (3.1). Since g is continuous, we then have that gχ_{A_{0}^{c}}ρ_{n} → gχ_{A_{0}^{c}}ρ, and, by (3.3), it is also true that

gχ_{A_{0}^{c}∩A_{n}^{c}}ρ_{n} → gχ_{A_{0}^{c}}ρ.

These approximating measures are pure point measures:

(3.6)  χ_{A_{0}^{c}∩A_{n}^{c}}ρ_{n} = ∑_{j=1}^{N_{n}} w_{j}^{(n)}δ_{x_{j}^{(n)}}

Our original goal was to approximate ν_{+} from (3.1) by a ν_{n,+} that is obtained by splitting ρ_{n} and thus corresponds to a J_{n} ∈ R_{0}(A_{n}). Recall from Section 2 how those plus measures were obtained: we split ρ = ν_{+} + ν_{−}, and if J ∈ R(B), then we have to put exactly one half of the absolutely continuous part of ρ on B into ν_{+}. For a general J ∈ R(B), there is no restriction on how to distribute the singular part between ν_{+} and ν_{−}. However, our goal is to construct approximations J_{n} ∈ R_{0}(A_{n}) (note the index 0!), and then either all or nothing of each point mass w_{j}^{(n)}δ_{x_{j}^{(n)}} has to go into ν_{n,+}. This is an unwelcome restriction because

ideally we would have liked to put the fraction g(x_{j}^{(n)})w_{j}^{(n)}δ_{x_{j}^{(n)}} into ν_{n,+}. We overcome this obstacle by splitting each point mass into two new point masses whose ratio is at our disposal.

Lemma 3.4 (The Splitting Lemma). Let 0 < A, B < R, and let ξ_{0} : (−R, R) → [0, 1] be a Borel function whose restriction to (−A, B) is χ_{(0,B)}. For sufficiently small δ > 0 and fixed 0 < g < 1, define

ξ_{δ}(x) = \begin{cases}1 & −gδ < x < 0 \\ 1/2 & 0 < x < δ^{2} \\ 0 & δ^{2} < x < (1 − g)δ \\ ξ_{0}(x) & otherwise \end{cases} .

Let ρ_{0}, ρ_{δ} be the measures that are associated with ξ_{0} and ξ_{δ}, respectively. Then, as δ → 0+, we have that ρ_{δ} → ρ_{0}, ρ_{δ}([0, δ^{2}]) → 0, and

ρ_{δ}({−gδ}) → gρ_{0}({0}), ρ_{δ}({(1 − g)δ}) → (1 − g)ρ_{0}({0}),

So a point mass inside a gap (at x = 0 here) can be split into two nearby point masses of approximately the same total weight, with a ratio between the two that can be specified in advance, by introducing an additional tiny band at the original point mass.

Proof. This is proved by an explicit calculation. First of all, we immediately obtain the weak-∗ convergence ρ_{δ} → ρ_{0} from ∥ξ_{δ} − ξ_{0}∥_{1} → 0. Next, we can control ρ_{δ}([0, δ^{2}]) by a calculation similar to the one that was used in the proof of Lemma 3.3, so we will explicitly discuss only the point masses of ρ_{δ} here. Observe also in this context that ρ_{δ} on (−A, B) is supported by {−gδ} ∪ {(1 − g)δ} ∪ (0, δ^{2}).

We will use the formulae

ρ_{δ}({−gδ}) = lim_{y→0+} y|H_{δ}(−gδ + iy)|,
ρ_{0}({0}) = lim_{y→0+} y|H_{0}(iy)|,

where, as usual, H_{δ}, H_{0} are the H functions of ξ_{δ} and ξ_{0}, respectively, as in (2.1). We rewrite (2.1) as

H_{δ}(z) = h(z) exp (∫_{−A}^{B} \frac{ξ_{δ}(t) dt}{t − z}) ,
H_{0}(z) = h(z) exp (∫_{−A}^{B} \frac{ξ_{0}(t) dt}{t − z});

here, h(z) is independent of δ and holomorphic in a neighborhood of (−A, B). Explicit calculation now shows that ρ_{0}({0}) = Bh(0) and

ρ_{δ}({−gδ}) = g^{1/2}(g − δ)^{1/2}(B + gδ)h(−gδ),

which obviously converges to gBh(0) as δ → 0+.

The calculation for ρ_{δ}({(1 − g)δ}) is of course analogous; alternatively, we could combine the previous calculations with the fact that ρ_{δ} → ρ_{0}. □

Apply the Splitting Lemma to all point masses from (3.6), with x_{j}^{(n)} taking the role of x = 0 in the Lemma, and with g = g_{j}^{(n)} = g(x_{j}^{(n)}) and the δ’s chosen so small that the statements below will be true. More precisely, we only do this if g_{j}^{(n)} ≠ 0, 1; there is of course no need to split the point mass if we already have g_{j}^{(n)} = 0 or 1.

For reasons that will become clear in a moment, we will also apply the Splitting Lemma to the point masses of ρ_{n} on A_{0}∩A_{n}^{c}, with g = 1/2. We obtain new sets from the A_{n}’s, with additional, very small bands added. We call these news sets P_{n}, and otherwise use tildes to refer to the new, modified data. Again, we have that

χ_{A_{0}^{c}∩P_{n}^{c}} ρ̃_{n} = Ñ_{n} ; ∑ ; j=1 w̃_{j}^{(n)}δ_{x_{j}^{(n)}};

the point is that we can now achieve that

(3.7)  ∑_{x_{j}^{(n)}∉A_{0}} σ̃_{j}^{(n)}w̃_{j}^{(n)}δ_{x_{j}^{(n)}} → gχ_{A_{0}^{c}}ρ

for suitably chosen σ̃_{j}^{(n)} ∈ {0, 1}. More precisely, for each fixed set of indices j, n, we split the old point mass at x_{j}^{(n)} into two new point masses. One of these has weight approximately equal to w̃ = g(x_{j}^{(n)})w_{j}^{(n)}, and we set the corresponding σ̃ = 1, and σ̃ = 0 for the other new point mass of this pair. This procedure makes sure that the point masses from (3.7) with σ̃ = 1 can be put in one-to-one correspondence with those of χ_{A_{0}^{c}∩A_{n}^{c}}ρ_{n}, and the two corresponding point masses x_{j}^{(n)}, x̃_{j}^{(n)} will get arbitrarily close to each other and the weights will satisfy w̃ ≈ g(x)w, up to an error that will approach zero as we take the δ’s from the Splitting Lemma closer and closer to zero. So we can indeed make sure that (3.7) holds.

Similarly and as already announced above, the Splitting Lemma with g = 1/2 can be used on A_{0} \ A_{n}, and we then obtain that (for suitable σ̃; as before we can take one σ̃ = 1 and the other equal to 0 for each split pair)

(3.8)  \frac{1}{2}ρ̃_{n,ac} + ∑_{x_{j}^{(n)}∈A_{0}} σ̃_{j}^{(n)}w̃_{j}^{(n)}δ_{x_{j}^{(n)}} → \frac{1}{2}χ_{A_{0}}ρ.

This follows because χ_{A_{0}}ρ_{n} → χ_{A_{0}}ρ, as we saw above, and the left-hand side of (3.8) is close to (1/2)χ_{A_{0}}ρ_{n}. Let us explain in more detail why this is true. In fact, it will only be true if the band sizes δ are chosen small enough when we apply Lemma 3.3 and the Splitting Lemma.

Notice that while ρ̃_{n,ac} is supported by P_{n}, only very little weight is given to Ã_{n}^{c} if the band sizes δ were chosen small enough. The set Ã_{n} was introduced at the beginning of this proof; it is equal to P_{n}, but with the bands removed. So ρ̃_{n,ac} is close (in fact, in norm) to its restriction to Ã_{n}, and this in turn is close (again, in norm) to χ_{A_{n}∩A_{0}}ρ_{n} (no tilde!), because the main part of this measure also sits on Ã_{n}, which is at some distance from the small set where we changed ξ when going from ρ_{n} to ρ̃_{n}. The part of (1/2)ρ_{n} on A_{0} ∩ A_{n}^{c}, on the other hand, is approximated by the sum from (3.8), by its construction.

By combining (3.8) with (3.7), we obtain that

\frac{1}{2}ρ̃_{n,ac} + ∑ σ̃_{j}^{(n)}w̃_{j}^{(n)}δ_{x_{j}^{(n)}} → \frac{1}{2}χ_{A_{0}}ρ + gχ_{A_{0}^{c}}ρ.

If we call the measure on the right-hand side ν̃_{+}, then (3.1), (3.2), (3.5) show that ∥ν̃_{+} −ν_{+}∥ < (5/2)ϵ. Moreover, the measure on the left-hand side is a measure of the type ν̃_{n,+}; it corresponds to some (unique) J_{n} ∈ R_{0}(P_{n}). We can also make sure that |P_{n}ΔB| < ϵ here. Indeed, P_{n} ⊃ B, and from the way P_{n} was constructed, it is clear that we can make |P_{n} \ B| arbitrarily small. Here it becomes again essential to add only very small bands during the construction. Recall also that we perhaps replaced B with a slightly smaller set at the very beginning of the proof, so the originally given set B is not necessarily a subset of P_{n}, and we can really only make a claim about P_{n}ΔB.

By the material from Section 2, especially Propositions 2.2, 2.3, we know that the Jacobi matrices J_{n} ∈ R_{0}(P_{n}) that correspond to the spectral data we have constructed will come as close to J ∈ R(B) as we wish, provided ϵ > 0 was taken sufficiently small and n is large. This follows because, by construction, the spectral data (ξ_{n}, ν_{n,+}) of J_{n} will come arbitrarily close to those of J. At the same time, |P_{n}ΔB| can also be made arbitrarily small.

These operators J_{n} ∈ R_{0}(P_{n}) are not necessarily periodic. However, it is known [1, 7, 16] that an arbitrary finite gap set can be transformed into a periodic set by arbitrarily small perturbations of the endpoints of its intervals. Here, we call a finite gap set P periodic if every (equivalently: one) J ∈ R_{0}(P) is periodic. Therefore, a final small adjustment of our sets P_{n} will give the full claim. We then need to know that for the new sets P̃_{n} we will be able to find new Jacobi matrices J̃_{n} ∈ R_{0}(P̃_{n}) that are close to the original operators J_{n} ∈ R_{0}(P_{n}). This issue will be discussed in great detail later in this paper; here we only need a small part of these later results. We can refer to Theorem 5.1 and the discussion preceding it to finish the present proof. This result applies here because for a finite gap set P, it is definitely true that ρ_{s}(P) = 0 for all ρ ∈ H(P). □

4. The distance δ

In this section, we prove Theorem 1.6(a). This will be an easier discussion than the proof of part (b) of this Theorem and will thus serve as a good warm-up. As a preliminary, we first confirm that δ is a metric.

Proposition 4.1. (1.1) defines a metric on the non-empty compact subsets of ℝ.

Proof. It is of course well known that h is a metric, and clearly δ is symmetric and non-negative, so it suffices to show that |KΔL| satisfies the triangle inequality. This, however, follows immediately from the observation that

AΔC ⊂ (AΔB) ∪ (BΔC)

for any three sets A, B, C. □

It will also be useful to keep in mind the following basic fact about Hausdorff distance: If A_{n}, A are non-empty, compact subsets of a compact metric space X and h(A_{n}, A) → 0, then

A = {x ∈ X : a_{j} → x for some sequence a_{j} ∈ A_{N(j)}, N(j) → ∞},

and also

A = {x ∈ X : a_{n} → x for some sequence a_{n} ∈ A_{n}}.

Compare, for example, [17, Lemma 1.11.2].

Proof of Theorem 1.6(a). We will actually show that if we just know that the collections of ξ functions X(K), X(K′) are close in Hausdorff distance, that already forces δ(K, K′) to be small as well. So we can completely avoid all issues related to the splitting ρ = ν_{+} + ν_{−} of the measures ρ.

Let K_{n}, K be non-empty compact subsets of [−R, R], so that h(Ω_{n}, Ω) → 0. Suppose, first of all, that it were not true that h(K_{n}, K) → 0, say h(K_{n}, K) ≥ 2r > 0 (on a subsequence, which, for simplicity, we assume to be the original sequence). We pass to another subsequence if necessary and then find ourselves in one of the following two situations. Either: (i) There are x_{n} ∈ K_{n} so that (x_{n} − 2r, x_{n} + 2r) ∩ K = ∅; or (ii) There are x_{n} ∈ K so that (x_{n} − 2r, x_{n} + 2r) ∩ K_{n} = ∅. In both cases, we can make the x_{n} converge to x ∈ ℝ by passing to still another subsequence.

Case (i) can then be ruled out as follows: Let ξ_{n} = 1 on all gaps (of K_{n}) to the left of x_{n} and ξ_{n} = 0 on all gaps to the right of x_{n}, and, of course, ξ_{n} = 1/2 on K_{n}. Then ξ_{n} ∈ X(K_{n}) and thus there are J_{n} ∈ Ω_{n} that have these functions as their ξ functions, but we claim that no accumulation point of the ξ_{n} lies in X(K). This is a contradiction because we can pass to a convergent subsequence so that J_{n} → J. Since h(Ω_{n}, Ω) → 0, the limit must satisfy J ∈ Ω. By Proposition 2.2, this implies that ξ_{n} dt → ξ dt, where ξ ∈ X(K) is the ξ function of J.

Since (x − r, x + r) ∩ K = ∅, an arbitrary ξ ∈ X(K) can only take the values 0 and 1 on this interval, and if both values occur, then ξ has to jump from 0 to 1 at some point μ ∈ [x − r, x + r]. Suppose that we had ξ_{n_{j}} dt → ξ dt in weak-∗ sense for such a ξ and a subsequence of the sequence ξ_{n} that was defined above. Now ξ_{n} ≥ 1/2 on (x − r, x − δ) for all large n for arbitrary δ > 0, so by testing against functions f ∈ C(ℝ) that are supported by [x − r, x], equal to 1 on some subinterval and take values between 0 and 1, we see that we must have μ = x − r, that is, ξ = 1 on [x − r, x + r]. This, however, leads to a contradiction when we test against similar functions that are supported by [x, x + r]. We have to admit that the accumulation points of the sequence ξ_{n} are not ξ functions of operators J ∈ Ω, but, as explained above, this contradicts our hypothesis that h(Ω_{n}, Ω) → 0.

Case (ii) is handled similarly. As observed earlier, we can in fact assume that there is an x ∈ K so that (x − r, x + r) ∩ K_{n} = ∅ for all large n. This time, we use the procedure from above to define a ξ ∈ X(K): Put ξ = 1 on all gaps of K to the left of x, ξ = 0 on the gaps to the right of x, and ξ = 1/2 on K. We can now argue as above to show that this ξ can not be the limit of a (sub-)sequence of ξ functions ξ_{n} ∈ X(K_{n}), basically, as before, because these functions can only jump from 0 to 1 on (x − r, x + r), not from 1 to 0. Again, it would follow that h(Ω_{n}, Ω) ↛ 0, which is a contradiction. So we have now shown that h(K_{n}, K) → 0. Note that this in particular implies that min K_{n} → min K, max K_{n} → max K.

It remains to show that |K_{n}ΔK| → 0 as well. We first observe that we must definitely have that |K_{n}| → |K|. Indeed, if this were false, say |K_{n}| ≤ |K| − ϵ on a subsequence, then the sequence of functions ξ_{n} ∈ X(K_{n}) that are equal to 0 on all gaps cannot have a limit point ξ ∈ X(K). If, on the other hand, we had that |K_{n}| ≥ |K| + ϵ, then the function ξ ∈ X(K) that is equal to 0 on all gaps cannot be reached as a limit of any sequence ξ_{n} ∈ X(K_{n}), so again we obtain a contradiction.

This argument also shows that, more generally,

|K_{n} ∩ I| → |K ∩ I|

for every fixed interval I ⊂ ℝ. To establish this version, just argue as above, but test against functions that are close to χ_{I}.

In particular, I can be any gap of K here, and it then follows that |K_{n} ∩ I| → 0. Since for any ϵ > 0, we can find finitely many gaps so that the total measure of the remaining gaps is < ϵ, this implies that

lim_{n→∞} |K_{n} \ K| = 0.

Since

|K \ K_{n}| = |K| − |K_{n}| + |K_{n} \ K|

and, as pointed out at the beginning of this argument, |K_{n}| → |K|, we also obtain that |K \ K_{n}| → 0. □

5. The spaces R_{0}(K)

We will first discuss how Theorem 1.6(b) follows from its variant Theorem 1.7 and then prove this statement. Suppose that what Theorem 1.6(b) asserts were not true. Then there are compact sets K_{n}, K with δ(K_{n}, K) → 0, but h(Ω_{n}, Ω) ≥ ϵ > 0. So, on a subsequence (which, for notational simplicity, we assume to be the original sequence), one of the following alternatives will hold: (i) There are J_{n} ∈ Ω_{n} so that d(J_{n}, Ω) ≥ ϵ; (ii) There are J_{n} ∈ Ω so that d(J_{n}, Ω_{n}) ≥ ϵ. By compactness, the J_{n} will approach a limit J on a subsequence. In case (i), Proposition 3.1 forces J ∈ Ω, an obvious contradiction. This rules out case (i), and we have also inadvertently established Theorem 1.7(a).

So it just remains to discuss case (ii), and we can actually restrict our attention to the slightly simpler scenario:

(ii’) There exists J ∈ Ω so that

(5.1)  lim sup_{n→∞} d(J, Ω_{n}) > 0.

Theorem 1.7(b) claims that this can not happen if ρ_{sc}(K) = 0, where, as usual, ρ denotes the measure of the H function of J, as in (2.2). So Theorem 1.6(b) will follow if we can prove Theorem 1.7(b).

We begin our discussion with the special case when ρ_{s}(K) = 0.

Theorem 5.1. Suppose that δ(K_{n}, K) → 0, and suppose further that J ∈ Ω with ρ_{s}(K) = 0. Then (5.1) cannot hold: d(J, Ω_{n}) → 0.

Proof. We know that

(5.2)  ρ = ρ_{ac} + ∑ w_{j}δ_{μ_{j}},
(5.3)  ν_{+} = \frac{1}{2} ρ_{ac} + ∑ σ_{j}w_{j}δ_{μ_{j}},

and our goal is to find J_{n} ∈ Ω_{n} so that ξ_{n} dx → ξ dx and ν_{n,+} → ν_{+} in the weak-∗ topology. The sums in (5.2), (5.3) are taken over those gaps (a_{j}, b_{j}) for which μ_{j} ≠ a_{j}, b_{j}; in other words, these sums give us the singular parts of ρ, ν_{+} on K^{c} (which is the complete singular part here, by assumption).

We start by labeling the gaps of K once and for all by integers j ≥ 1, for example in order of decreasing size. For each gap (a_{j}, b_{j}) ⊂ K^{c} of K and sufficiently large n, there has to be a corresponding gap (a_{j}^{(n)}, b_{j}^{(n)}) ⊂ K_{n}^{c} that converges to (a_{j}, b_{j}) in the sense that a_{j}^{(n)} → a_{j}, b_{j}^{(n)} → b_{j}. We use this fact to introduce integers N(j), for j ≥ 1, as follows: If μ_{j} ≠ a_{j}, b_{j}, then we define N(j) as the smallest index for which every K_{n} for n ≥ N(j) has a gap (A, B) that satisfies

|A − a_{j}| + |B − b_{j}| < \frac{1}{10} min\{b_{j} − μ_{j}, μ_{j} − a_{j}\},

say. If μ_{j} = a_{j} or b_{j}, we proceed similarly, but replace the minimum with b_{j} − a_{j} here.

Note that if n ≥ N(j), then the gap (A, B) of K_{n} that is close to (a_{j}, b_{j}) ⊂ K^{c} in this sense is unique, so we can label the gaps of K_{n} so that this gap (A, B) (where A, B depend on n) also gets the label j, for n ≥ N(j). We do not impose any conditions on the labels of the remaining gaps of K_{n}, other than the obvious requirement that no label can be used more than once for fixed n. It could actually happen here that when applying this procedure, we run out of labels j ≥ 1. More precisely, this happens if there is a finite set M ⊂ ℕ so that N(j) is bounded on ℕ \ M, but K_{n} for n ≥ sup_{j∉M} N(j) has more than |M| additional gaps. In this case, we just invent new labels; for example, we could use negative integers.

Each μ_{j} from (5.2) is in the interior of its gap, and we can put μ_{j}^{(n)} = μ_{j}, at least for n ≥ N(j). If n ≥ N(j) and μ_{j} = a_{j}, then we put μ_{j}^{(n)} = a_{j}^{(n)}; the analogous procedure is used if μ_{j} = b_{j} and n ≥ N(j). Finally, on the remaining gaps of K_{n}, we can put ξ_{n} = 0, say.

The assumption that δ(K_{n}, K) → 0 then makes sure that ξ_{n} differs from ξ only on a small set. In particular, ∥ξ_{n} − ξ∥_{1} → 0, and, as usual, this implies that ρ_{n} → ρ in weak-∗ sense. We rewrite (5.3) as

ν_{+} = \frac{1}{2}ρ + ∑ (σ_{j} − \frac{1}{2}) w_{j}δ_{μ_{j}},

and our goal is to find parameters σ_{j}^{(n)}, g_{n}(x) so that the corresponding measures

(5.4) ν_{n,+} = \frac{1}{2}ρ_{n} + (g_{n} − \frac{1}{2}) χ_{K_{n}}ρ_{n,s} + ∑ (σ_{j}^{(n)} − \frac{1}{2}) w_{j}^{(n)}δ_{μ_{j}^{(n)}}

approach ν_{+}. As will become clear from the argument we are about to give, the existence of a singular part of ρ_{n} on K_{n} would only make this task easier because the functions g_{n}, 0 ≤ g_{n} ≤ 1, are completely at our disposal. In fact, we could just set g_{n}(x) ≡ 1/2 here right away and then proceed as outlined below. We will therefore only discuss the case where ρ_{n,s}(K_{n}) = 0 explicitly.

We will establish the desired weak-∗ convergence ν_{n,+} → ν_{+} as follows: We will verify that given ϵ > 0 and finitely many intervals I_{1}, …, I_{N} ⊂ ℝ with ρ(∂I_{j}) = 0, there exists n_{0} ∈ ℕ so that for all n ≥ n_{0}, it is possible to assign values to the σ_{j}^{(n)} so that

(5.5)  |ν_{n,+}(I_{j}) − ν_{+}(I_{j})| < ϵ.

We can assume that the I_{j} are disjoint, and we can then focus on a single interval I because ν_{n,+}(I) obviously only depends on those σ_{j}^{(n)} for which μ_{j}^{(n)} ∈ I.

So fix such an interval I. We make the obvious first step in our attempts to choose the parameters σ_{j}^{(n)} appropriately: we put σ_{j}^{(n)} = σ_{j} if n ≥ N(j). To find suitable values for the remaining σ’s, we make the following preliminary observation: If N_{n} ∈ ℕ, N_{n} → ∞, then

(5.6)  lim_{n→∞} sup_{j∉{1,...,N_{n}}} w_{j}^{(n)} = 0.

To prove this, we argue by contradiction. Suppose (5.6) were wrong. Then there are μ_{j_{n}}^{(n)} with arbitrarily large n so that (on this subsequence, which we won’t make explicit in the notation, as always) eventually j_{n} ∉ {1, …, N} for every N ≥ 1 and

w_{j_{n}}^{(n)} = ρ_{n}({μ_{j_{n}}^{(n)}}) ≥ ϵ > 0.

Here, we may also assume that μ_{j_{n}}^{(n)} → x. But then the weak-∗ convergence ρ_{n} → ρ implies that ρ({x}) > 0. Since ρ_{s}(K) = 0 by hypothesis, this forces x to lie in some gap (a_{j}, b_{j}) ⊂ K^{c}, but then only μ_{j}^{(n)} can be close to x for large n, so in particular it is not possible to have indices j_{n} ∉ {1, …, j}.

We now split ν_{n,+} = ν_{n}^{(1)} + ν_{n}^{(2)}, where

ν_{n}^{(2)} = ∑_{j∉\{1,...,N_{n}\}} (σ_{j}^{(n)} − \frac{1}{2}) w_{j}^{(n)}δ_{μ_{j}^{(n)}}.

Here, we take cut-offs N_{n} ∈ ℕ_{0} that satisfy N_{n} → ∞ but increase so slowly that N(j) ≤ n if 1 ≤ j ≤ N_{n}, and

lim_{n→∞} ∑_{\begin{matrix}j=1,...,N_{n} \\ μ_{j}≠a_{j},b_{j} \end{matrix}} |w_{j}^{(n)} − w_{j}| = 0.

This is possible because w_{j}^{(n)} → w_{j} for fixed j, and this latter statement just follows from the convergence ρ_{n} → ρ together with the fact that if μ_{j} ≠ a_{j}, b_{j}, then ρ_{n} is supported by {μ_{j}} in a neighborhood of μ_{j} = μ_{j}^{(n)} for all large n.

Notice that

2 |ν_{n}^{(1)}(I) − ν_{+}(I)| ≤ |ρ_{n}(I) − ρ(I)| + ∑_{\begin{matrix}j=1,...,N_{n} \\ μ_{j}≠a_{j},b_{j} \end{matrix}} |w_{j}^{(n)} − w_{j}| + ∑_{j>N_{n}} w_{j},

and the right-hand side approaches zero here. So now our task is to show that for large n, the remaining σ_{j}^{(n)} can be chosen so that |ν_{n}^{(2)}(I)| becomes small. This, however, follows immediately from (5.6): We have ν_{n}^{(2)}(I) ≥ 0 if we take all the σ’s equal to 1 and ν_{n}^{(2)}(I) ≤ 0 if we set them all equal to 0, and (5.6) says that we can go from one extreme value to the other in very small steps by changing individual σ’s, so we will be able to come close to zero and thus obtain (5.5) for all large n. □

Given this, Theorem 1.8 will now indeed imply Theorem 1.7(b) because if J ∈ R_{0}(K) with ρ_{sc}(K) = 0 and ϵ > 0 are given, then we can first use Theorem 1.8 to find a J′ ∈ R_{0}(K) with ρ′_{s}(K) = 0 and d(J, J′) < ϵ and then Theorem 5.1 says that d(J′, Ω_{n}) → 0, so lim sup d(J, Ω_{n}) < ϵ.

Proof of Theorem 1.8. Our assumption says that

(5.7)  ρ = ρ_{ac} + ∑ v_{j}δ_{x_{j}} + ∑ w_{j}δ_{μ_{j}},
(5.8)  ν_{+} = \frac{1}{2} ρ_{ac} + ∑ g_{j}v_{j}δ_{x_{j}} + ∑ σ_{j}w_{j}δ_{μ_{j}},

where x_{j} ∈ K, μ_{j} ∈ (a_{j}, b_{j}) ⊂ K^{c}, and 0 ≤ g_{j} ≤ 1, σ_{j} = 0, 1. We will first show that given ϵ > 0, we can find J′ ∈ R_{0}(K) so that d(J, J′) < ϵ and ρ′_{s}(K) < ϵ. This will be done by removing sufficiently many of the point masses δ_{x_{j}} by modifying ξ on small neighborhoods of the x_{j}.

To keep the argument transparent and for notational convenience, we will start with the special case where we remove just one point mass, say v_{1}δ_{x_{1}}. We also assume that x_{1} = 0.

We will make use of Proposition 2.5. So we will try to construct new data ξ′, ν′_{+} so that ν′_{+} is close to ν_{+} in the weak-∗ topology, and ρ′_{s}(K ∩ (−r, r)) = 0 for some r > 0 (so the point mass at x_{1} = 0 has been removed). We will handle the weak-∗ topology in the same way as in the proof of Theorem 5.1: We assume that we are given ϵ > 0 and disjoint open intervals I_{1}, …, I_{N} whose endpoints are not point masses of ρ, and our task is to achieve that

(5.9)  |ν′_{+}(I) − ν_{+}(I)| < ϵ,

for these intervals and, of course, ρ′_{s}(K ∩ (−r, r)) = 0 for some r > 0. As will become clear later on, we will obtain (5.9) quite easily for those intervals that are at some distance from x = 0. So we’ll focus on the interval that contains 0; call this interval I.

With these preliminaries out of the way, we are now ready for the main part of the proof. We will first focus on the situation where gaps accumulate at 0 from both sides; equivalently, 0 ∈ K is not an endpoint of a gap. The easier alternative cases will be discussed later.

We choose A, B > 0 so small that (−A, B) ⊂ I,

(5.10)  ρ((−A, B) \ {0}) < ϵ,

and

(5.11)  ∫_{(−A,B)} \frac{|ξ(t) − χ_{(0,∞)}(t)|}{|t|} dt < ϵ.

This can be done because we know that the integral from (5.11), extended over (−1, 1), say, is finite. See again [6, pg. 201] or [10, Lemma 2.4]. It will also be convenient to choose −A as the left endpoint of a gap and, similarly, B as the right endpoint of a (different) gap.

Notice that since ξ = 1/2 on K, (5.11) in particular shows that

(5.12)  ∫_{(−A,B)∩K} \frac{dt}{|t|} < 2ϵ.

Now for 0 < d < A, B (typically d ≪ A, B), we define a new ξ function ξ_{d} ∈ X(K) as follows: Let ξ_{d} = ξ on (−A, B)^{c}. If (a, b) is a gap of K that is contained in (−d, d), jump from 0 to 1 at the center of this gap; in other words, μ = (a + b)/2. If d ∈ (a, b) for some gap (a, b) ⊂ K^{c}, proceed similarly, but now only the part of (a, b) inside (−d, d) counts: put μ = (a + d)/2. Use the same procedure at −d. Finally, put ξ_{d} = 1 on (d, B) \ K and ξ_{d} = 0 on (−A, −d) \ K and ξ_{d} = 1/2 on K.

Then ξ_{d} ∈ X(K), and, as usual, ξ_{d} defines a measure ρ_{d} via (2.1), (2.2). We claim that

(5.13)  ρ_{d,ac}((−A, B)) ≤ Cϵ^{1/2},

for some constant C > 0. Here and in the remainder of this proof, by a constant we mean a number that is independent of ϵ, d, A, B. It may depend on the set K and the measure ρ. We will also apply the usual convention that the value of a constant may change from one expression to the next, even though we use the same symbol C for these different constants.

The absolutely continuous part of ρ_{d} is supported by K, and since ξ_{d} = 1/2 on K, its density is given by (1/π)Im H_{d}(x) = (1/π)|H_{d}(x)|. By (2.1), this is equal to

\frac{x + R}{π} exp ( lim_{y→0+} ∫_{−R}^{R} \frac{t − x}{(t − x)^{2} + y^{2}} ξ_{d}(t) dt) .

For almost every x ∈ ℝ, the limit in the exponent (exists and) is equal to the Hilbert transform

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

Here, the integral is only extended over t ∈ (−R, R); we will use similar conventions throughout this paper whenever integrals of ξ functions are involved.

Observe that for almost every x ∈ K, (T ξ_{d})(x) goes up if we replace ξ_{d} by the constant function 1/2 on (−d, d). To show this, consider the effect of replacing ξ_{d} by the constant 1/2 on a single gap. It is then obvious from the definition of (T ξ_{d})(x) that this quantity goes up at all x not from this gap. To handle the limiting process that is involved here when we replace ξ_{d} by 1/2 on infinitely many gaps, it suffices to recall that T is a continuous (in fact, unitary) map on L^{2}(ℝ).

Similarly, for −d < x < d, the Hilbert transform will also go up if we replace ξ_{d} by 1 on (d, R) and by 0 on (−R, −d). For this modified function (call it ζ) and x ∈ (−d, d), we have that

(T ζ)(x) = lim_{y→0+} \frac{1}{2} ∫_{(−d,d)∩\{|t−x|>y\}} \frac{dt}{t − x} + ∫_{d}^{R} \frac{dt}{t − x},

and from this we obtain that the density of ρ_{d,ac} satisfies

\frac{dρ_{d}}{dx} ≤ \frac{R^{2} − x^{2}}{π\sqrt{(d − x)(x + d)}}

on x ∈ (−d, d). Hence

(5.14)  ρ_{d,ac}((−d, d)) ≤ C ∫_{K∩(−d,d)} \frac{dx}{\sqrt{d^{2} − x^{2}}} .

Now (5.12) clearly implies that |K ∩ (−d, d)| < 2dϵ, and the integrand from (5.14) becomes largest close to the endpoints x = ±d, so we can further estimate (5.14) as follows:

ρ_{d,ac}((−d, d)) ≤ C ∫_{(1−ϵ)d}^{d} \frac{dx}{\sqrt{d^{2} − x^{2}}} = C ∫_{1−ϵ}^{1} \frac{ds}{\sqrt{1 − s^{2}}} ≤ Cϵ^{1/2}.

This establishes part of (5.13). Let us now take a look at ρ_{d,ac}((d, B)). As before, we can replace ξ_{d} by 1/2 on (−d, d), set it equal to 0 to the left of −d and equal to 1 to the right of B, and the density of ρ_{d,ac} will only go up on (d, B) because the Hilbert transform has this monotonicity property. Call this modified Krein function ζ, and let μ be the associated measure. We will now estimate μ_{ac}((d, B)) by introducing a final modification. We subdivide I = (d, B) into intervals I_{n} = (q^{n}d, q^{n+1}d), where n = 0, 1, …, N and 2 ≤ q ≤ 4. Let s_{n} = |K ∩ I_{n}| and

ζ_{1}(x) = \begin{cases}1/2 & q^{n}d < x < q^{n}d + s_{n} \\ 1 & q^{n}d + s_{n} < x < q^{n+1}d \\ ζ(x) & x ∉ (d, B) \end{cases} .

Recall that ζ = 1 on (d, B) \ K and ζ = 1/2 on K. This new function ζ_{1} is a function of the same type, but we shifted the part of K inside I_{n} to the very left of this interval. We first claim that μ(I^{c}) can only decrease under this change. Again, this follows by comparing Hilbert transforms: it is clear that (T ζ_{1})(x) ≤ (T ζ)(x) if x ∉ I (and both limits exist). This implies that μ_{1,ac}(I^{c}) ≤ μ_{ac}(I^{c}), by comparing densities, as explained above. To obtain the same conclusion for the singular parts, we make use of the formula

dρ_{s} = lim_{λ→∞} \frac{π}{2} λχ_{\{|H|>λ\}}(x) dx.

See [9, Theorem 1] and also [2, Section 9.7] (for the disk version of this statement).

On the other hand, ∥ζ − ζ_{1}∥_{1} < B, so the following Lemma will make sure that |μ(ℝ) − μ_{1}(ℝ)| < ϵ, if B > 0 was chosen small enough initially.

Lemma 5.2. Fix R > 0. Then, for any ϵ > 0, there exists δ > 0 so that

|ρ_{1}(ℝ) − ρ_{2}(ℝ)| < ϵ

if ρ_{1}, ρ_{2} are the measures of two H functions, as in (2.1), (2.2), whose ξ functions satisfy ∥ξ_{1} − ξ_{2}∥_{L^{1}(−R,R)} < δ.

Proof of Lemma 5.2. As usual, the map ξ ↦ ρ is continuous if we use the weak-∗ topology for both the measures ξ dx and the ρ’s. The domain {ξ dx} is compact, so the map is in fact uniformly continuous, and clearly the L^{1} norm controls a distance that generates the weak-∗ topology. □

Let us summarize: μ_{1}(I^{c}) ≤ μ(I^{c}) and μ(ℝ) < μ_{1}(ℝ) + ϵ, hence μ_{ac}(I) = μ(I) < μ_{1}(I) + ϵ, and thus it suffices to estimate μ_{1}(I) = μ_{1,ac}(I). This is done by a calculation, which is similar to the one we used above. First of all, notice that

(5.15)  ∑_{n=0}^{N} \frac{s_{n}}{q^{n+1}d} ≤ ∫_{(d,B)∩K} \frac{dt}{t} < 2ϵ.

Let’s now look at μ_{1}(I_{n}) = μ_{1}((q^{n}d, q^{n}d + s_{n})) for fixed n ≥ 1. If we had ζ_{1} = 1 on all of I \ I_{n}, that would lead to a density of the form

\frac{R^{2} − x^{2}}{π\sqrt{x^{2} − d^{2}}} (\frac{x − q^{n}d}{q^{n}d + s_{n} − x})^{1/2}

on x ∈ (q^{n}d, q^{n}d + s_{n}). Clearly, this expression can be estimated by

(5.16)  \frac{C}{q^{n}d} (\frac{x − q^{n}d}{q^{n}d + s_{n} − x})^{1/2} .

This might get bigger by a factor of

(∏_{j=0}^{n−1} \frac{(q^{n} − q^{j})d}{(q^{n} − q^{j})d − s_{j}} )^{1/2} = (∏_{j=0}^{n−1} (1 + \frac{s_{j}}{(q^{n} − q^{j})d − s_{j}} ))^{1/2} ,

due to the presence of intervals where ζ_{1} = 1/2 to the left of I_{n}. However, q^{n} − q^{j} ≥ q^{j} for j < n, and s_{j} < q^{j}d/2 if ϵ < 1/16, by (5.15), so (5.15) now shows that there is a uniform bound on this factor, independent of n ∈ ℕ, B ≤ 1 and d ≤ d_{0}. So we may work with (5.16) after all. By integrating, it then follows that

μ_{1}(I_{n}) ≤ \frac{C}{q^{n}d} ∫_{0}^{s_{n}} (\frac{t}{s_{n} − t})^{1/2} dt = C \frac{s_{n}}{q^{n}d} (n ≥ 1).

For n = 0, similar reasoning applies and yields the (worse) bound μ_{1}(I_{0}) ≤ C(s_{0}/d)^{1/2}. We now see from (5.15) that μ_{1}(I) ≤ C_{1}ϵ^{1/2}+C_{2}ϵ, as desired. Since we can of course apply similar arguments to estimate ρ_{d,ac}((−A, −d)), we have now established (5.13).

Our next goal is to show the following: If x ∉ [−A, B] (and the limits defining H(x), H_{d}(x) exist), then

(5.17)  |H_{d}(x)| ≤ (1 + Cϵ)|H(x)|.

Since ln |H_{d}/H| = T (ξ_{d} − ξ), it suffices to compare the Hilbert transforms. Suppose that x > B, say. When going from ξ to ξ_{d}, we only changed ξ on (−A, B), and the Hilbert transform at x > B will only get smaller when we increase ξ on this interval. Only on (−A, d) could ξ_{d} perhaps be smaller than ξ. To estimate the possible effect of this change, let M = {x ∈ (−A, d) : ξ(x) ≠ 0}. Clearly, if x ∉ M, then ξ_{d}(x) ≥ ξ(x), so we can focus on M. Recall that ξ only takes the values 0, 1/2, 1, so (5.11) shows that |M| < 3ϵA, at least if we only consider d ≤ ϵA. On the other hand, if also d < B/2, say, then the Hilbert transform at an x > B will not increase by more than 2|M|/B when going from ξ to ξ_{d}. If now A ≤ 10B, say, then this is bounded by 60ϵ, and, as explained above, (5.17) follows.

If A > 10B, we introduce

m_{0} = |M ∩ (−B, d)|,
m_{n} = |M ∩ (−q^{n}B, −q^{n−1}B)|,  n = 1, …, N,

with 2 ≤ q ≤ 4, and then proceed similarly. Now (5.11) shows that

∑_{n=0}^{N} \frac{m_{n}}{q^{n}B} < Cϵ,

at least if we again insist that d is taken sufficiently small, say d ≤ ϵ. However, this sum also bounds the possible increase of the Hilbert transform at an x > B, so we obtain (5.17) in this case also.

We also saw earlier that (5.17) implies a corresponding (local) bound on the measures, by comparing separately the absolutely continuous and singular parts of ρ and ρ_{d}. So we have that ρ_{d}(S) ≤ (1 + Cϵ)ρ(S) for all Borel sets S ⊂ (−A, B)^{c}.

Let us summarize what we have achieved so far; for convenience, we adjust the constants here. Given ϵ > 0 and an open interval I with ρ(∂I) = 0 and 0 ∈ I, we have constructed a family of new Krein functions ξ_{d} that agree with ξ outside the interval (−A, B) ⊂ I. Here, A, B > 0 and 0 < d ≤ d_{0} can be chosen as small as we wish; in fact, it will usually be necessary to take these quantities small enough for the following statements to be true. Recall also that we arrive at these small values in a two step procedure: We first take A, B > 0 sufficiently small, in response to the value of ϵ > 0 that was given to us. Then, in a second step, we pick an appropriate value of d_{0} > 0, which will typically be much smaller still (at the very least d_{0} ≲ ϵ min{A, B}). This means that we’re not allowed to decrease A, B once a range for d has been specified, but it is permitted to make d_{0} smaller, according to our needs, while keeping A, B fixed.

Lemma 5.3. The associated measures ρ_{d} have no singular part on K ∩ (−A, B), and they have the following additional properties, for small enough A, B, d_{0} > 0 and for all 0 < d ≤ d_{0}:

(a) |ρ_{d}(I) − ρ(I)| < ϵ;

(b) ρ_{d,ac}((−A, B)) < ϵ;

(c) ρ_{d}(S) ≤ (1 + ϵ)ρ(S) if S ∩ (−A, B) = ∅

Proof. We will again compare ρ_{d} with the measure that is obtained when we replace ξ_{d} by the constant value 1/2 on (−d, d). Denote the corresponding H function by H_{1}. We claim that then

(5.18)  lim inf_{y→0+} \frac{|H_{1}(x + iy)|}{|H_{d}(x + iy)|} > 0

for all x ∈ K. This follows because the logarithm of this fraction differs from the corresponding truncated Hilbert transform T_{y}(ξ_{1} − ξ_{d})(x) by at most a fixed constant; here, T_{y} is defined as

(T_{y}g)(x) = ∫_{|t−x|>y} \frac{g(t) dt}{t − x}.

The Hilbert transform goes up when replace ξ_{d} by 1/2 on a gap. We used this important observation earlier, and the truncation does not seriously affect the argument; it introduces another constant.

Now we compare (5.18) with the result from [8] that if ρ_{1} = fρ_{d,s}+σ, where σ ⊥ ρ_{d,s}, then

lim_{y→0+} \frac{H_{1}(x + iy)}{H_{d}(x + iy)} = f(x)

for ρ_{d,s}-almost every x ∈ ℝ. So if we had ρ_{d,s}((−d, d) ∩ K) > 0, then (5.18) would force ρ_{1} to have a singular part on K also, but clearly ρ_{1} is purely absolutely continuous on (−d, d). Thus ρ_{d,s}((−d, d) ∩ K) = 0.

The Krein function avoids one of the values 0, 1 on (−A, −d) and (d, B), so ρ_{d,s} gives zero weight to these sets, and clearly ±d can not be point masses for the chosen arrangement ξ_{d}. Hence ρ_{d,s}((−A, B)∩K) = 0.

Part (a) follows from the fact that ∥ξ_{d} − ξ∥_{1} ≤ A + B, and this L^{1} norm controls the distance of ρ_{d} and ρ in a metric that generates the weak-∗ topology, so we just need to take A, B > 0 small enough. See also the proof of Lemma 5.2 for this argument. Part (b) is (5.13), and part (c) was discussed in the paragraphs preceding the formulation of the Lemma. □

Now return to (5.8) and our goal (5.9); ν′_{+} will be chosen as a measure ν_{d,+} for small d > 0 and suitable parameters σ_{j}(d), g_{j}(d). We can come close to ν_{+}(I) with finite sums in (5.8). More precisely, we pick N ∈ ℕ so that, after relabeling if necessary, the expression

(5.19) \frac{1}{2}ρ(I) + (g_{1} − \frac{1}{2}) v_{1} + ∑_{j=2}^{N} (g_{j} − \frac{1}{2}) v_{j} + ∑_{j=1}^{N} (σ_{j} − \frac{1}{2}) w_{j}

differs from ν_{+}(I) by not more than ϵ. Here, we of course assume that x_{j}, μ_{j} ∈ I for j ≤ N, and there may be (infinitely many) other point masses in I, but these have total mass less than ϵ. So now our task is to make sure that ν_{d,+}(I) is close to (5.19).

We first demand that A, B are so small that all these x_{j} (for 2 ≤ j ≤ N) and μ_{j} (for 1 ≤ j ≤ N) are well separated from (−A, B), say

A + B < \frac{1}{D} min\{|x_{j}|, |μ_{j}|\},

for a large constant D. Then the argument that was used to establish Lemma 5.3(c) also shows that if we take D large enough here (equivalently: A, B small enough), then

1 − \frac{ϵ}{N} ≤ \frac{v_{j}(d)}{v_{j}} ≤ 1 + \frac{ϵ}{N}

for j = 2, …, N. Here, v_{j}(d) = ρ_{d}({x_{j}}). Again, this follows because the Hilbert transform of ξ doesn’t change much at x = x_{j} when we pass to ξ_{d}. A similar two-sided estimate holds for w_{j}(d)/w_{j} for j = 1, …, N, where w_{j}(d) = ρ_{d}({μ_{j}}).

If we now take g_{j}(d) = g_{j} and σ_{j}(d) = σ_{j} for j ≤ N, then the corresponding contributions to ν_{d,+} will differ from the last two sums from (5.19) by at most ϵ, and this will be true uniformly in d ≤ d_{0}. Moreover, by Lemma 5.3(a), ρ_{d}(I) will be close to ρ(I) for all small d.

The other point masses of ρ_{d} on I \(−A, B) (if any) can be controlled with the help of Lemma 5.3(c). Indeed, we immediately obtain that

∑_{j>N}(v_{j}(d) + w_{j}(d)) < (1 + ϵ)ϵ < 2ϵ;

here, the sum is really taken only over those j for which the corresponding point mass (that is, x_{j} or μ_{j}) lies in I \ (−A, B).

So, to finish the proof of our claim that we can get close to (5.19) with ν_{d,+}(I), it just remains to show that d ∈ (0, d_{0}] and the σ_{j}(d) for |μ_{j}(d)| < d can be chosen so that

(5.20)  |(g_{1} − \frac{1}{2}) v_{1} − ∑_{|μ_{j}(d)|<d} (σ_{j}(d) − \frac{1}{2}) w_{j}(d)| < Cϵ.

Recall that the point masses from this latter sum make up the complete singular part of ρ_{d} on (−A, B). So, by Lemma 5.3(b),

(5.21)  ρ_{d}((−A, B)) − ϵ ≤ ∑_{|μ_{j}(d)|<d} w_{j}(d) ≤ ρ_{d}((−A, B)).

Moreover, by combining parts (a) and (c) of Lemma 5.3, we see that ρ_{d}((−A, B)) ≥ ρ((−A, B)) − Cϵ, and, in particular,

(5.22)  ρ_{d}((−A, B)) ≥ v_{1} − Cϵ.

We will now describe our choice of the parameters σ_{j}(d) ∈ {0, 1} from (5.20) by specifying the index set M on which σ_{j}(d) = 1. Then clearly

∑_{|μ_{j}(d)|<d} (σ_{j}(d) − \frac{1}{2}) w_{j}(d) = ∑_{j∈M} w_{j}(d) − \frac{1}{2} ∑_{|μ_{j}(d)|<d} w_{j}(d).

So, taking (5.21), (5.22) into account, we now see that it suffices to show that for arbitrary 0 ≤ α ≤ 1, there exists a choice of d > 0 and M that makes

|∑_{j∈M} w_{j}(d) − αρ_{d}((−A, B))| < 2ϵ.

Now for fixed d = d_{0}, we can clearly achieve that

(5.23)  ∑_{j∈M} w_{j}(d_{0}) ≥ αρ_{d_{0}}((−A, B)) − 2ϵ,

by just taking a large enough index set M. We can and will still insist that M be finite. Now decrease d. The w_{j}(d) are continuous functions, and each of the finitely many point masses μ_{j}(d) with j ∈ M will eventually merge into an endpoint of its gap. For example, if μ_{j}(d) > 0, then μ_{j}(d) = (a_{j} + b_{j})/2 is constant as long as d ≥ b_{j}, but then starts decreasing, and finally μ_{j}(a_{j}) = a_{j}. When this happens, we get w_{j} = 0. This follows because with the chosen configuration for ξ_{d}, these endpoints can not be point masses of ρ_{d}. So there is a smaller positive value d_{1} < d_{0} for which the left-hand side of (5.23) will become zero. Since ρ_{d}((−A, B)) also is a continuous function of d, some intermediate d ∈ [d_{1}, d_{0}] will work. This finishes the main part of the argument.

We now discuss the case where 0 is an endpoint of a gap. The case where there are two such gaps, say (−a, 0) and (0, b), is elementary. In this case, ξ = χ_{(0,b)} on (a, b). We can now use a (simplified) version of the Splitting Lemma 3.4 (where the small band in the middle has been shrunk to a point) to handle this case by splitting the point mass at x = 0 into two nearby point masses in the two gaps, with the ratio of their weights at our disposal.

Finally, if (−a, 0) is a gap, but no gap has 0 as its left endpoint, then we can run a one-sided version of the argument given above. Note that ξ = 0 on (−a, 0); otherwise, x = 0 couldn’t be a point mass of ρ. We now take ξ_{d}(x) = ξ(x) for x < 0 and modify ξ(x) only for x > 0.

This whole procedure can also be used to remove finitely many mass points x_{1}, …, x_{N} ∈ K. We just pick disjoint small neighborhoods of these points and then modify ξ and choose the parameters g_{j}, σ_{j} in the same way as above on each of these neighborhoods separately. Note that what we do on one such set will have a negligible effect on what happens on the other sets because these are separated and thus the Hilbert transform of ξ on one of the sets will not change much when we modify ξ on the other sets (here it is important that the sets are chosen to be small compared to their separations).

We have now shown the following: Given ϵ > 0 and J_{0} ∈ R_{0}(K) with ρ_{0,sc}(K) = 0, there exists J ∈ R_{0}(K) so that d(J_{0}, J) < ϵ and ρ_{s}(K) < ϵ. (We in fact also know that ρ_{sc} = 0, but this will not be used.)

To completely remove the singular part of ρ on K here, we proceed as follows. We know that, as usual,

ρ = ρ_{ac} + χ_{K}ρ_{s} + ∑ w_{j}δ_{μ_{j}},
ν_{+} = \frac{1}{2}ρ + (g − \frac{1}{2}) χ_{K}ρ_{s} + ∑ (σ_{j} − \frac{1}{2}) w_{j}δ_{μ_{j}},

and we would like to come close to ν_{+} with a new measure ν′_{+} whose corresponding ρ′ measure satisfies ρ′_{s}(K) = 0. The argument is quite similar to the ones given above.

We would like to define new Krein functions ξ_{n} by putting μ_{j}^{(n)} = b_{j} for j > n and μ_{j}^{(n)} = μ_{j} for 1 ≤ j ≤ n. However, this runs into minor technical problems, so the actual definition will be slightly different. We would like to achieve that ρ_{n,s}(K) = 0, and this is almost, but not quite, true with our preliminary definition of ξ_{n}.

Notice that ρ_{n,s} does give zero weight to every open set on which ξ_{n} avoids the value 1. So, since ξ_{n} = 0 on all but finitely many gaps, the singular part of ρ_{n} on K is supported by the set {a_{j_{1}(n)}, …, a_{j_{N} (n)}}, where this list contains all the left endpoints of those gaps on which ξ_{n} = 1 (if any). We now exclude the possibility of point masses of this type by preemptively setting μ = a + δ_{n}, with a very small δ_{n} > 0, if a = a_{j_{k}(n)} is one of these points. If the δ_{n} are chosen small enough here, then ∥ξ_{n} − ξ∥_{1} → 0 and thus also ρ_{n} → ρ in the weak-∗ topology.

Next, fix N_{0} ∈ ℕ so that

(5.24)  ∑_{j>N_{0}} w_{j} < ϵ.

By relabeling, if necessary, we may assume that w_{j} > 0 for j = 1, …, N_{0} (equivalently, the corresponding μ_{j}’s lie in the interiors of their gaps), and we then define σ_{j}^{(n)} = σ_{j} for j = 1, …, N_{0}. To compare ν_{+} and ν_{n,+} in the weak-∗ topology, we again compare the weights these measures give to fixed intervals I with ρ(∂I) = 0. If n ≥ N_{0}, then (5.25)

ν_{n,+}(I) = \frac{1}{2}ρ_{n}(I)+ ∑_{j≤N_{0};μ_{j}∈I} (σ_{j} − \frac{1}{2}) w_{j}^{(n)}+ ∑_{j>N_{0};μ_{j}^{(n)}∈I} (σ_{j}^{(n)} − \frac{1}{2}) w_{j}^{(n)},

and here ρ_{n}(I) → ρ(I), w_{j}^{(n)} → w_{j} as n → ∞ for j = 1, …, N_{0}. Since also ρ_{s}(K) < ϵ, this implies that the first two terms from the righthand side of (5.25) will differ from ν_{+}(I) by not more than ϵ for all large n. So we have to show that the last sum from (5.25) can be made small by choosing the corresponding σ_{j}^{(n)} appropriately. This follows as in the proof of Theorem 5.1 from the fact that

(5.26)  w_{j}^{(n)} < ϵ  (j > N_{0}, n ≥ n_{0});

here n_{0} must be taken sufficiently large. Indeed, (5.26) guarantees that the last sum from (5.25) can be changed from positive values (all σ’s equal to 1) to negative values in small steps of size < ϵ, so there is a configuration for which it will be within distance ϵ from 0, as desired.

To prove (5.26), assume the contrary. Then we find w_{j_{n}}^{(n)} ≥ ϵ with j_{n} > N_{0} for certain arbitrarily large values of n. We may assume that the corresponding μ_{j_{n}}^{(n)} converge, to x ∈ ℝ, say. Since ρ_{n} → ρ, this implies that ρ({x}) ≥ ϵ. This is a contradiction because ρ_{s}(K) < ϵ and (5.24) holds, so the only point masses of ρ whose weight could possibly be ≥ ϵ are μ_{1}, …, μ_{N_{0}}, but these are in the interiors of their gaps, so μ_{j_{n}}^{(n)} with j_{n} > N_{0} can certainly not converge to one of these. □

6. Sets of measure zero

On a zero measure set, the condition of being reflectionless becomes vacuous, and this changes the character of our results. Formally, they remain true, though, and we now discuss them one by one in this new situation. Since this discussion is somewhat removed from our main topic in this paper, we will be extremely sketchy here.

If |B| = 0, then Theorem 1.1 says that any J ∈ J can be approximated by periodic J_{n} ∈ J_{R} whose spectra satisfy |σ(J_{n})| → 0. We can do this directly, as follows: Let a(j), b(j) be the coefficients of J, and put a_{n}(j) = a(j), b_{n}(j) = b(j) for |j| ≤ n. This already guarantees that J_{n} → J, provided the remaining coefficients are chosen so that there is a uniform bound on their size. We now put a_{n}(n + 1) = 0, b_{n}(n+1) = 0 (the value of b is irrelevant here), and extend periodically. Then J_{n} is an infinite sum of copies of a finite-dimensional operator, so σ(J_{n}) is a finite set. This proves the trivial measure zero version of Theorem 1.1. If we want to, we can in fact avoid a zero coefficient a_{n}(j) here; we then just assign a very small positive value to a_{n}(n + 1).

Proposition 1.4 remains true for |K| = 0, with the same proof. Theorem 1.5 also continues to hold unless K consists of exactly two points, simply because its hypothesis is never satisfied if |K| = 0 and K has at least three points.

As for the next set of results, our treatment from above still applies if |K| = 0. In fact, as we will see, things become much easier now. If |K| = 0 and any J ∈ R_{0}(K) is given, then we can use the methods from Section 5 to produce, for any ϵ > 0, a J′ ∈ R_{0}(K) so that ρ′_{pp}(K) < ϵ and d(J, J′) < ϵ. Having done that, we can now run the construction from the last part of Section 5 to find a J′′ ∈ R_{0}(K) with ρ′′(K) = 0 and d(J′, J′′) < ϵ. The key new feature here is the fact that whether or not ρ′_{sc}(K) was small to start with, this measure can never be transformed into absolutely continuous measure. The corresponding weight can only (approximately) go into the point masses in the gaps if |K| = 0. Much of the analysis of Section 5 centered around this question, how do we prevent singular measure on K from becoming absolutely continuous measure on K under a small perturbation, and this problem has completely disappeared now.

A new technical issue arises here due to the possibility of finitely supported measures ρ, ν_{+}; in fact, K itself could be a finite set. Recall from Section 2 that the spectral data (ξ, ν_{+}) do not uniquely determine a Jacobi matrix J in this case, so it is not enough to construct such data that are close to the ones corresponding to the given Jacobi matrix. However, we can simply observe that a Jacobi matrix with finite spectrum splits into finite-dimensional blocks, and we can then discuss these blocks separately, using the techniques outlined above. We leave the matter at that and hope that this very brief sketch has given an impression of how the following result could be proved.

Theorem 6.1. Let K ⊂ ℝ be a non-empty compact set with |K| = 0. Recall that

R_{0}(K) = {J ∈ J : σ(J) ⊂ K}.

For every J ∈ R_{0}(K), there are J_{n} ∈ R_{0}(K) with ρ_{n}(K) = 0, so that d(J_{n}, J) → 0.

By following the pattern described at the beginning of Section 5, this improved version of Theorem 1.8 in a special case then leads to improved versions of Theorems 1.6, 1.7. In fact, the treatment again becomes much simpler, and one could also make more specific statements. For example, if a sequence K_{n} with δ(K_{n}, K) → 0 is given and |K| = 0, then one could pick finite subsets F_{n} ⊂ K_{n} so that still δ(F_{n}, K) → 0. This means that for any J with σ(J) ⊂ K, there exists a sequence of operators J_{n} whose spectra are finite subsets of the K_{n}, and J_{n} → J.

Finally, notice that if |K| = 0 and h(K_{n}, K) → 0, then |K_{n}| → 0, so, as noted in the introduction, h(K_{n}, K) → 0 implies that δ(K_{n}, K) → 0 in this case.

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

E-mail address: alexei@math.tamu.edu
URL: www.math.tamu.edu/∼alexei.poltoratski/

Mathematics Department, University of Oklahoma, Norman, OK 73019

E-mail address: cremling@math.ou.edu
URL: www.math.ou.edu/∼cremling