THE KREIN SPECTRAL SHIFT AND RANK

ONE PERTURBATIONS OF SPECTRA

Alexei Poltoratski

Abstract. We use recent results on the boundary behavior of Cauchy integrals to study the Krein spectral shift of a rank one perturbation problem for self-adjoint operators. As an application, we prove that all self-adjoint rank one perturbations of a self-adjoint operator are pure point if and only if the spectrum of the operator is countable. We also study pairs of pure point operators unitarily equivalent up to a rank one perturbation and give various examples of rank one perturbations of singular spectra.

1. Introduction.

This paper discusses the spectral behavior of a self-adjoint operator under a rank one perturbation. We further study the well-known connections between one-dimensional perturbation theory and complex analysis. Our main tool is a recent result on the boundary behavior of Cauchy integrals (see Theorem 3.5 below).

We first apply this tool to the following problem of the perturbation theory of the singular spectrum: Given a pure point self-adjoint operator A, when is operator A + B pure point for all self-adjoint B belonging to a certain class? We answer this question for rank one operators B and show that analogous results for trace class and compact perturbations easily follow from the rank one result.

The Weyl–von Neumann Theorem [K] says that for any self-adjoint operator A there exists an operator B of arbitrarily small Hilbert–Schmidt norm such that A + B is pure point. I. e. any self-adjoint operator can be obtained from a pure point operator via an arbitrarily small Hilbert–Schmidt perturbation. The generalization of this result proved by Kuroda [Ku] allows to replace the Hilbert–Schmidt norm with any unitarily invariant cross norm except the trace norm. The statement does not hold for the trace norm simply because by the Kato–Rosenblum Theorem [K] trace class perturbations do not change absolutely continuous spectrum. But the Theorem proved by Carey and Pincus in [C-P] implies that any singular operator differs from a pure point operator by an operator of arbitrarily small trace norm. Finally, examples provided by Donoghue [D], Simon and Wolf [S-W] show that there exist self-adjoint rank one perturbations

A_{λ} = A + λ(·, ϕ)ϕ

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 47A55, 30E20.

Key words and phrases. The Krein spectral shift, rank one perturbations, self-adjoint operators, Clark’s measures.

Research at MSRI is supported in part by NSF grant DMS-9022140

for λ ∈ ℝ such that A is pure point but A_{λ} is singular continuous for any λ ≠ 0.

These facts demonstrate that, in general, the property of having no continuous part is not stable. Such a stability exists only under strong restrictions on operators A and B (for the perturbation A ↦ A + B). For instance, in [H] Howland proved that if A is pure point, and B is compact and small relatively to A in some special sense, then A + B is also pure point.

In this paper we show (Section 4) that even under such a small disturbance as a rank one perturbation, the pure point spectrum is stable only in degenerate situations.

The Weyl–von Neumann Theorem [K] on the stability of the essential spectrum under compact perturbations implies that if the spectrum σ(A) of A is countable, then for any compact B operator A + B is pure point. The Carey–Pincus Theorem [C-P] provides the following complement to this proposition: if σ(A) is uncountable, then there exists a trace class operator B such that A + B is not pure point. We refine this statement by proving that if σ(A) is uncountable then there exists a rank one operator B such that A + B has a non trivial continuous part. Moreover, B can be chosen as B = (·, ϕ)ϕ for some cyclic vector ϕ of A (see Theorem 4.1). This implies that all self-adjoint rank one perturbations of a self-adjoint operator are pure point if and only if the spectrum of the operator is countable (see Corollary 4.4). By the Weyl–von Neumann Theorem, “rank one” in the last statement can be replaced with “compact” or “trace class” (see Corollary 4.5). In particular, if a self-adjoint operator A has a compact or a trace class perturbation with a nontrivial continuous part, then it also must have a rank one perturbation with a nontrivial continuous part.

In Section 5 we analyze relative spectral properties of two self-adjoint pure point operators unitarily equivalent up to a rank one perturbation. The Weyl von–Neumann Theorem mentioned above states that two operators are unitarily equivalent up to a compact operator iff they have the same essential spectrum. The necessary and sufficient condition for two operators to be equivalent up to a trace-class operator, which is found in [C-P], is more delicate and involves some additional spectral invariants. An analogous condition for the rank one situation seems to be impossible to formulate in any reasonable terms. Instead, in Section 5 we present some sufficient conditions for the case of pure point operators. The restriction to the pure point case seems logical since the essential spectrum is invariant under rank one perturbations by the Weyl–von Neumann Theorem.

Let A be a cyclic self-adjoint operator, ϕ its cyclic vector and A_{λ} = A + λ(·, ϕ)ϕ for λ ∈ ℝ. Let us denote by Π the set of those λ for which A_{λ} has a nontrivial pure point part. Similarly, let Σ be the set of λ for which A_{λ} has a nontrivial singular continuous part. Then the set Π ∩ Σ will consist of those λ for which the spectrum is “mixed.”

The paper [D] of Donoghue gives examples in which Π = {0}, Σ = ℝ \ {0} and Σ = {0}, Π = ℝ\{0}. Therefore the set Π∩Σ can be empty when both Π and Σ are nonempty.

One of the natural questions which arise from the recent results on rank one perturbations (see [Si2] and [R-J-L-S]) is whether the set Π ∩ Σ can be empty (or almost empty) when the sets Π and Σ are big topologically or in measure.

In Section 6 we construct an example of “the absence of mixed spectra” which gives a partial answer to this question. We present a family of rank one perturbations such that the operators A_{λ} corresponding to the coupling constants λ ∈ [0; 1] are singular continuous and those corresponding to the coupling constants λ ∈ ℝ \ [0; 1] are pure point.

To prove our main results in Sections 4-6 we use the approach developed in Sections 2 and 3.

In [C] Clark introduced certain families of measures on the unit circle related to an inner function in the unit disk. It was shown that these measures are spectral measures of all possible rank one unitary perturbations of the model contraction corresponding to the given inner function. In Section 2 we discuss similar relations in the case of an arbitrary family of self-adjoint rank one perturbations of a cyclic self-adjoint operator.

Section 3 is devoted to the properties of the Krein spectral shift of a rank one perturbation problem. We study spectral shifts in terms of the boundary behavior of resolvent functions. Our goal in Section 3 is to extend the result of Martin and Putinar [M-P] (Lemma 3.8) from the case of point spectrum to the case of arbitrary singular spectrum. The main result of Section 3 is Lemma 3.7, which is extensively used in the remainder of the paper.

Acknowledgments. I would like to thank Nikolai Makarov for asking some of the questions discussed in this paper. I am deeply grateful to Aleksei Aleksandrov, Dmitri Jakubovich and Fedor Nazarov for useful discussions and comments, and to the Mathematical Sciences Research Institute for hospitality while this work was completed.

2. Families M_{φ} and rank one perturbations of self-adjoint operators.

We will denote by M(ℝ̂) the Banach space of Borel complex measures on ℝ̂ = ℝ ∪ {∞} such that the norm

||μ|| = \frac{1}{π} ∫_{−∞}^{∞} \frac{d|μ(t)|}{1 + t^{2}} + |μ|(\{∞\})

is finite. We will also use the notation M_{+}(ℝ̂) for the subset consisting of nonnegative measures.

Let φ be a non constant analytic function in the upper half plane ℂ_{+} such that |φ| ≤ 1. Then for any α from the unit circle 𝕋 function \frac{α+φ}{α−φ} has positive real part in ℂ_{+} . Thus there exists a measure μ_{α} ∈ M_{+}(ℝ̂) such that its Poisson integral satisfies

(2.1)  (Pμ_{α})(x + iy) = \frac{1}{π} ∫_{−∞}^{∞} \frac{ydμ_{α}(t)}{(x − t)^{2} + y^{2}} + yμ_{α}(\{∞\}) = Re \frac{α + φ}{α − φ}.

We will denote by M_{φ}(ℝ̂) the family of such measures {μ_{α}}_{α∈𝕋} corresponding to function φ.

If φ is defined in the unit disk one can replace the Poisson integral in (2.1) with the Poisson integral in the disk and consider the analogous family M_{φ}(𝕋) consisting of positive measures on the unit circle 𝕋.

Families M_{φ} possess many interesting properties, see [A1], [A2], [C], [P1], [P2] and [S]. As was shown by Clark in [C], if φ is an inner function in the unit disk 𝔻 then M_{φ}(𝕋) is the system of the spectral measures of all unitary rank one perturbations of the model contraction with the characteristic function φ. By adapting the argument of Clark to the case of the upper half-plane and arbitrary φ, |φ| ≤ 1, one can find a similar connection between rank one perturbations of self-adjoint operators and families M_{φ}(ℝ̂). Here we present a different way to establish this connection.

If A_{0} is a bounded cyclic self-adjoint operator and ϕ is its cyclic vector, then we can consider the family of rank one perturbations

A_{λ} = A_{0} + λ(·, ϕ)ϕ

for λ ∈ ℝ. Let ν_{λ} be the spectral measure of ϕ for A_{λ}:

ν_{λ}(B) = (ϕ, E_{B}(A_{λ})ϕ)

for any Borel B ∈ ℝ, where E_{B}(A) is the spectral projection of A. Since A_{λ} is bounded, supp ν_{λ} ⊂ ℝ is compact. Thus the Cauchy transform of ν_{λ}

(2.2)  Kν_{λ}(z) = \frac{1}{π} ∫_{−∞}^{∞} \frac{dν_{λ}(t)}{t − z} = \frac{1}{π} ((A_{λ} − z)^{−1}ϕ, ϕ)

converges for any z ∈ ℂ_{+}.

The relation for the resolvents

(A_{0} − z)^{−1} − (A_{λ} − z)^{−1} = [λ(·, ϕ) ((A_{λ} − z)^{−1}ϕ)] (A_{0} − z)^{−1},

for any z ∈ ℂ_{+}, gives us

(2.3)  Kν_{λ}(z) = \frac{Kν_{0}(z)}{1 + πλKν_{0}(z)},

see [A].

Since ν_{0} is a nonnegative Borel measure on ℝ, −iKν_{0} is an analytic function with a nonnegative real part in ℂ_{+}:

Re [−iKν_{0}] = Pν_{0}.

Thus

(2.4)  −iKν_{0} = \frac{1 + φ}{1 − φ}

for some bounded analytic function φ, |φ| ≤ 1 in ℂ_{+}.

Hence by (2.3)

(2.5)  (Pν_{λ})(x + iy) = Re −iKν_{λ}(x + iy) = Re \frac{\frac{1−φ}{1+φ}}{1 + iπλ\frac{1−φ}{1+φ}} = c Re \frac{β − φ}{β + φ}

where β = \frac{1+iπλ}{1−iπλ}, c = \frac{1}{1+π^{2}λ^{2}} . Thus

Pν_{λ} = cPμ_{β}

where {μ_{β}}_{β∈𝕋} = M_{φ}(ℝ̂). Therefore the family of spectral measures {ν_{λ}}_{λ∈ℝ} coincides up to constant multiples with the family M_{φ}(ℝ̂) for φ satisfying (2.4).

Conversely, if φ is an analytic function in ℂ_{+} with |φ| ≤ 1, we can consider the operator M_{μ} of multiplication by z in L^{2}(μ) where μ = μ_{1} ∈ M_{φ}(ℝ̂). Reversing the above argument we can show that for any α ∈ 𝕋 the measure μ_{α} ∈ M_{φ} is a spectral measure for the rank one perturbation M_{μ}+λ(·, 1)1 where λ = \frac{i}{π} \frac{1−α}{1+α}. Therefore, each family M_{φ}(ℝ̂) is a family of spectral measures of rank one perturbations of some self-adjoint operator.

Each measure from M_{+}(ℝ̂) belongs to a family M_{φ} for some φ (φ can be found from (2.1) uniquely up to a Möbius transform). One can also say that two given measures μ and ν from M(ℝ̂) belong to the same family M_{φ} iff there exist operator A and its rank one perturbation A_{λ} = A+λ(·, φ)φ such that μ and ν are the spectral measures of φ for A and A_{λ}. Indeed, as we proved above, such A and A_{λ} exist iff c_{1}μ, c_{2}ν ∈ M_{ψ} for some function ψ and positive constants c_{1} and c_{2}. Suppose c_{1}μ = μ_{α}, c_{2}ν = μ_{β} for some μ_{α}, μ_{β} ∈ M_{ψ}. Then μ, ν ∈ M_{φ} for φ = b ∘ ψ where b is the Möbius transform of the unit disk 𝔻 such that |b′(α)| = 1/c_{1}, |b′(β)| = 1/c_{2}, see [A2].

Further relations between the resolvent functions of the rank one perturbations of a self-adjoint operator are discussed in [A], [D], [M-P], [S-W], [Si1], [Si2] and [R-J-L-S].

Remark. The same argument works for the families of unitary operators (in particular for those studied by Clark [C]).

Let U_{1} be a unitary cyclic operator, v, ||v|| = 1 its cyclic vector. Then we can consider the family of unitary rank one perturbations of U_{1} :

U_{α} = U_{1} + (α − 1)(·, U_{1}^{−1}v)v,

α ∈ 𝕋. For the resolvents we have

(U_{1} − z)^{−1} − (U_{α} − z)^{−1} = (U_{α} − z)^{−1}[(α − 1)(·, U_{1}^{−1}v)v](U_{1} − z)^{−1}.

Denote by μ_{α} the spectral measure of v for U_{α}. Let Kμ_{α} and P μ_{α} be the standard Cauchy and Poisson integrals of measure μ_{α} in the unit disk 𝔻. Since

Kμ_{α} = ((U_{α} − z)^{−1}v, U_{α}^{−1}v)

for z ∈ 𝔻, we have that

(2.6)  Kμ_{α} = \frac{αKμ_{1}}{1 + (α − 1)Kμ_{1}} .

If we consider an analytic function θ in 𝔻 such that |θ| ≤ 1, θ(0) = 0 and

P μ_{1} = Re \frac{1 + θ}{1 − θ}

in 𝔻, then

(2.7)  Kμ_{1} = \frac{1}{1 − θ}

and by (2.6)

P μ_{α} = 2 Re Kμ_{α} − 1 = 2 Re [\frac{\frac{α}{1−θ}}{1 + \frac{α−1}{1−θ}} − \frac{1}{2}] = Re \frac{α + θ}{α − θ} .

Thus {μ_{α}}_{α∈𝕋} = M_{θ}(𝕋).

As we mentioned before, a similar result was obtained in [C] for rank one unitary perturbations of the model contraction T_{θ} = SP_{θ} , where θ is an inner function, S : f ↦ zf is a shift operator in the Hardy space H^{2} and P_{θ} is the orthogonal projector from H^{2} onto the model space θ^{∗}(H^{2}) = H^{2} ⊝ θH^{2}.

Let U be a singular cyclic unitary operator and μ be its spectral measure. As we discussed before, for any singular measure μ there exists analytic φ, |φ| ≤ 1 such that μ = μ_{1} ∈ M_{φ}. Since μ = μ_{1} is singular,

Re \frac{1 + φ}{1 − φ} = Pμ_{1} = 0

a. e. on 𝕋, which implies that φ is inner. Therefore, by Clark Theorem, U is unitarily equivalent to a rank one perturbation of the model contraction T_{φ}. I. e. any singular cyclic unitary operator is unitarily equivalent to a rank one perturbation of T_{φ} for some inner φ. Conversely, if the operator U_{1} from the last remark is singular, then U_{1} − (·, U_{1}^{−1}v)v is a C_{0} completely non unitary contraction with the characteristic function θ defined by (2.7). (Note that U_{α} are unitary if α ∈ 𝕋 and contractive if |α| < 1.)

3. The Krein spectral shift.

In this section we discuss the notion of the Krein spectral shift in the case of rank one perturbations. For basic results and definitions in this area we refer to [Kr], [A-D], [M-P] and [Si1].

Let A be a cyclic bounded self-adjoint operator, ϕ its cyclic vector. Let μ ∈ M_{+}(ℝ̂) be the spectral measure of ϕ for A. Since μ is a nonnegative measure, its Cauchy transform has nonnegative imaginary part. Therefore for each λ ∈ ℝ the function 1 + πλKμ admits the following representation:

(3.1)  1 + πλKμ = exp[Ku + c]

fore some real valued function u ∈ L^{∞}(ℝ), |u| ≤ π with compact support and some real c. If λ ≥ 0 then function u can be chosen to satisfy

(3.2)  u = arg(1 + πλKμ)

where arg stands for the principal branch of argument taking values in (−π; π] (we assume that Kμ is defined a. e. on ℝ by its angular boundary values). In this case we will have 0 ≤ u ≤ π. Similarly if λ < 0 then u can be chosen to satisfy (3.2) with arg taking values in [−π; π). Then −π ≤ u ≤ 0.

Definition. Such u is called the Krein spectral shift of the perturbation problem (A ↦ A + λ(·, ϕ)ϕ).

The Krein spectral shift is usually defined in more general settings of trace class perturbations (A ↦ A + K), see [Kr] and [M-P]. When K is one-dimensional the general definition can be reduced to the one given above, see [M-P].

Let ν be the spectral measure of ϕ for A + λ(·, ϕ)ϕ. Even though measure ν does not appear in (3.1), it is uniquely determined by μ and λ and satisfies a similar equation.

Lemma 3.1.

Measures μ, ν and the Krein spectral shift u satisfy

(3.3)  1 + πλKμ = exp[K(u) + c] = [1 − πλKν]^{−1}

for some real c. I. e.

(3.3’)  arg(1 + πλKμ) = u = −arg(1 − πλKν).

Proof. Apply formulas (3.1) and (2.3) with ν_{0} = μ and ν_{λ} = ν. ▲

Since in this paper we will mostly deal with the spectral measures of operators, it will be more convenient for us to call u the phase shift of the pair of measures (μ, ν) for the coupling constant λ. If λ = 1 we will call u the phase shift of the pair (μ; ν).

It is well known that the phase shift u determines μ and ν uniquely. This reflects the fact that among the spectral measures of A and A + λ(·, ϕ)ϕ we always choose those corresponding to the vector ϕ.

Any function u ∈ L^{∞}(ℝ), 0 ≤ u ≤ π with compact support is the Krein spectral shift of some perturbation problem (the phase shift of some pair of measures). Indeed, for any λ > 0 one can consider a nonnegative measure μ ∈ M(ℝ̂) with compact support satisfying (3.1). Then u is the Krein spectral shift of the perturbation problem (M_{μ} ↦ M_{μ} + λ(·, 1)1), where M_{μ} is the operator of multiplication by z in L^{2}(μ). Similarly, each u ∈ L^{∞}(ℝ), 0 ≥ u ≥ −π with compact support is also the Krein spectral shift of some perturbation problem (with λ < 0).

As we discussed in the previous Section, there exist operator A and its cyclic vector ϕ such that μ and ν are spectral measures of ϕ for A and A + (·, ϕ)ϕ respectively, iff both measures belong to the same family M_{φ} for some function φ analytic in ℂ_{+}. One can show that if we take μ_{α}, μ_{β} ∈ M_{φ} with α = e^{iθ_{1}}, β = e^{iθ_{2}}, 0 ≤ θ_{1} < θ_{2} < 2π, where φ is inner, then the phase shift u for the pair (μ_{α}, μ_{β}) can be defined as

u = πχ_{{x∈ℝ|φ(x)=e^{iθ},θ_{1}<θ<θ_{2}}}

where χ_{E} denotes the indicator function of the set E (once again we assume that φ is defined a. e. on 𝕋 by its angular boundary values).

As before, let u be the phase shift of a pair of measures (μ, ν) and φ be a function such that μ, ν ∈ M_{φ}. Then u has the following properties.

Lemma 3.2.

Let K ⊂ ℝ be a measurable set. Then the following conditions are equivalent

i) the restrictions of μ and ν on K are singular,

ii) |u| is equal to 0 or π a. e. on K,

iii) |φ| = 1 a. e. on K.

Proof. For i)⇔ii) see [M-P] or [Si1]; for i)⇔iii) see [C].▲

We will denote by Qu the conjugate Poisson integral of u:

Qu(x + iy) = − Re Ku(x + iy) = \frac{1}{π} ∫_{−∞}^{∞} \frac{x − t}{(x − t)^{2} + y^{2}} u(t)dt.

It is well-known that if v ∈ L^{∞}(ℝ) with a compact support then the difference

(3.4)  Qv(x + iy) −  ∫_{ℝ\(x−y;x+y)} \frac{v(t)dt}{x − t}

is O(1) as x + iy →_{≯} x_{0} for any x_{0} ∈ ℝ and o(1) as x + iy →_{≯} x_{0} for any Lebesgue point x_{0}

of v, see [G].

Let E ⊂ ℝ be a measurable set, μ ∈ M(ℝ). We will denote

p. v. ∫_{E} dμ(x) = lim_{ϵ→0} ∫_{E\(−ϵ;ϵ)} dμ(x).

We will also denote by μ^{s} the singular component of measure μ. The definition of the phase shift and the fact that difference (3.4) is O(1) as x + iy → x_{0} imply the following

Lemma 3.3.

i) For μ^{s}-a.e x

(3.5)  p. v. ∫_{ℝ} \frac{u(x + t)dt}{t} = ∞

and for ν^{s}-a.e x

(3.6)  p. v. ∫_{ℝ} \frac{u(x + t)dx}{t} = −∞.

ii) There exist α, β ∈ 𝕋 such that for μ^{s}-a.e x

φ(z) ⟶_{z⟶_{≯}x} α

and for ν^{s}-a.e x

φ(z) ⟶_{z⟶_{≯}x} β.

Proof. Follows from formulas (2.1) and (3.3). Also, for i) see [Si1]; for ii) see [C].▲ Example 3.4. Let x, y ∈ ℝ, x < y. Let u = πχ_{(x;y)}. Then by Lemma 3.2 ii) μ and ν are singular. By Lemma 3.3 i) μ is concentrated on the set {x} and ν is concentrated on the set {y}. Since u is non constant, μ and ν are nonzero. Thus μ = bδ_{x} and ν = cδ_{y} for some positive b and c.

To prove the next property of phase shifts we will need the following

Theorem 3.5([P1]). Let σ ∈ M(ℝ̂) be a singular measure with compact support, f ∈ L^{1}(|σ|). Define meromorphic function F in ℂ_{+} as

F = \frac{1 + K(fσ)}{1 + Kσ}.

Then F has non tangential boundary limits equal to f σ-a. e. on ℝ.

This Theorem is stated in [P1] for the case of the unit disk. The half-plane version presented above can be obtained from the disk version via a standard argument.

We will say that real function f defined in ℂ_{+} is less than ∞ (greater than −∞) at point x ∈ ℝ if

lim inf_{z⟶_{≯}x} f(z) < ∞ (lim sup > −∞).

We will write that

p. v. ∫_{E} f(x)dx < ∞ (> −∞)

if

lim inf_{ϵ→0} ∫_{E\(−ϵ;ϵ)} dμ(x) < ∞ (lim sup > −∞).

Corollary 3.6. Let σ, γ be singular positive measures on ℝ with compact supports. Define holomorphic function F in ℂ_{+} as

F = \frac{1 + Kσ}{1 + Kγ} .

Let K be a measurable subset of ℝ. Then |F | < ∞ σ-a. e. on K iff the restriction of σ on K is absolutely continuous with respect to γ.

Proof. Let σ = fγ + η where f ≥ 0, f ∈ L^{1}(γ) and η ⊥ γ. Then

(3.7)  F = \frac{1 + Kσ}{1 + Kγ} = \frac{1 + Kσ}{1 + K(σ + γ)} × \frac{1 + K(σ + γ)}{1 + Kγ}.

By Theorem 3.5 the first fraction has finite nonzero limits σ-a. e. Since

\frac{1 + K(σ + γ)}{1 + Kγ} = [\frac{1 + Kγ}{1 + K((1 + f)γ + η)}]^{−1} ,

by Theorem 3.5 the second fraction in (3.7) has finite limits γ-a. e. and tends to infinity η-a. e. Thus |F | < ∞ σ-a. e. on K iff η(K) = 0. ▲ Lemma 3.7. Let u_{1} and u_{2} be the phase shifts of the pairs of measures (μ_{1}, ν_{1}) and (μ_{2}, ν_{2}) respectively, i. e.

1 + πKμ_{1} = exp[K(u_{1}) + c_{1}] = [1 − πKν_{1}]^{−1}

and

1 + πKμ_{2} = exp[K(u_{2}) + c_{2}] = [1 − πKν_{2}]^{−1}

for some real c_{1} and c_{2}. Let K ⊂ ℝ be a measurable set. Put c = exp(c_{1} − c_{2}). Then i)

p. v. ∫_{ℝ} (u_{1}(t + x) − u_{2}(t + x))\frac{dt}{t} < ∞

for μ_{1}-a. e. x ∈ K iff the restriction of μ_{1}^{s} on K is absolutely continuous with respect to μ_{2};

p. v. ∫_{ℝ} (u_{1}(t + x) − u_{2}(t + x))\frac{dt}{t} > −∞

for ν_{1}-a. e. x ∈ K iff the restriction of ν_{1}^{s} on K is absolutely continuous with respect to ν_{2};

ii) if μ_{2}^{s}-a. e. x ∈ K is a Lebesgue point of u_{1} − u_{2} and

p. v. ∫_{ℝ} (u_{1}(t + x) − u_{2}(t + x))\frac{dt}{t} = f(x) < ∞

for μ_{2}^{s}-a. e. x ∈ K then the restriction of μ_{1}^{s} on K is equal to the restriction of ce^{f}μ_{2}^{s} on K; if ν_{2}^{s}-a. e. x ∈ K is a Lebesgue point of u_{1} − u_{2} and

p. v. ∫_{ℝ} (u_{1}(t + x) − u_{2}(t + x))\frac{dt}{t} = f(x) > −∞

for ν_{2}^{s}-a. e. x ∈ K then the restriction of ν_{1}^{s} on K is equal to the restriction of ce^{−f}ν_{2}^{s} on K.

Proof. Notice that

exp [K(u_{1} − u_{2})] = c\frac{1 + Kμ_{1}}{1 + Kμ_{2}} = c\frac{1 − Kν_{2}}{1 − Kν_{1}}

for some c > 0. Since difference (3.4) is O(1), Corollary 3.6 implies part i). Since difference (3.4) is o(1) at Lebesgue points of u_{1} − u_{2}, part ii) follows from part i) and Theorem 3.5.▲

Once again, let u be the phase shift of a pair of measures (μ, ν) and φ be a function such that μ, ν ∈ M_{φ}.

Lemma 3.8 [M-P]. Measure μ has a point mass at x iff

(3.8)  ∫_{x−1}^{x+1}(πχ_{(x;x+1)} − u(y))\frac{dy}{y − x} < ∞.

Measure ν has a point mass at x iff

(3.9)  ∫_{x−1}^{x+1}(πχ_{(x−1;x)} − u(y))\frac{dy}{x − y} < ∞.

Proof. As was shown in Example 3.5 measure μ_{0} corresponding to the phase shift u_{1} = πχ_{(x;x+1)} has a point mass at x. It is left to apply statement i) of Lemma 3.7 to u_{1}, u_{2} = u, and K = {x}.

The statement can also be proved directly, see [M-P]. ▲

Remark. In terms of the characteristic function φ, a point mass of μ_{α} ∈ M_{φ} can be described in the following way, see [C], [A-C] or [S]. For any x ∈ ℝ, μ_{α}({x}) > 0 iff φ(z) → α as z ⟶_{≯} x and φ has a finite nontangential derivative φ′(x) at x.

In terms of other measures from the family M_{φ}, μ_{1}({x}) > 0 iff Kμ_{−1}(z) → 0 as z ⟶_{≯} x

and (x − y)^{−1} ∈ L^{2}(μ_{−1}), see [C], [S-W] or [Si1].

4. On the stability of the absence of continuous spectra

Numerous examples given in [A-D], [D], [Si1], [Si2] and [R-J-L-S] show that for a general self-adjoint operator the property of having no continuous part is not stable under rank one perturbations. Here we prove that in order to have only pure point rank one perturbations the operator must have a very “thin” spectrum. We will denote the spectrum of A by σ(A). We will say that perturbation A + λ(·, ϕ)ϕ is cyclic if vector ϕ is cyclic for A.

Theorem 4.1. Let A be a cyclic self-adjoint operator. Then the following two conditions are equivalent:

1) All cyclic self-adjoint rank one perturbations of A are pure point,

2) σ(A) is countable.

Remark. We, of course, assume that A itself is included in the set of all its rank one perturbations.

In 1) “all” can not be replaced with “almost all” in any reasonable sense: it is not difficult to show that if μ is a standard singular Cantor measure on [0; 1] then M_{μ} +λ(·, ϕ)ϕ is pure point for a.e. λ ∈ ℝ for any cyclic vector ϕ.

The word ”cyclic” can be eliminated from the statement of Theorem 4.1, see Corollary 4.4 below.

To prove Theorem 4.1 we will need the following two Lemmas.

Lemma 4.2. Let F ⊂ ℝ, |F | = 0 be a closed set, F = ℝ \ ⋃_{n=1}^{∞} I_{n} where I_{n} = (x_{n}; y_{n}) are disjoint open intervals. Then one can choose two disjoint sets of positive integers L and M such that for any y ∈ F \ {x_{1}, x_{2}, x_{3}, ...}

(4.1)  ∫_{(y;y+1)∩ ⋃_{n∈L} I_{n}} \frac{dx}{x − y} =  ∫_{(y;y+1)∩ ⋃_{n∈M} I_{n}} \frac{dx}{x − y} = ∞

and for any y ∈ F \ {y_{1}, y_{2}, y_{3}, ...}

(4.2)  ∫_{(y−1;y)∩ ⋃_{n∈L} I_{n}} \frac{dx}{y − x} =  ∫_{(y−1;y)∩ ⋃_{n∈M} I_{n}} \frac{dx}{y − x} = ∞.

Proof. Since |F | = 0 we can choose intervals I_{n_{1}^{1}}, I_{n_{2}^{1}}, I_{n_{3}^{1}}, ... in such a way that

∫_{(y;y+1)∩⋃_{l} I_{n^{1}_{l}}} \frac{1}{x − y} dx > 1

for any y ∈ F \ {x_{1}, x_{2}, x_{3}, ...},

∫_{(y−1;y)∩ ⋃_{l=1}^{∞} I_{n^{1}_{l}}} \frac{1}{y − x}dx > 1

for any y ∈ F \ {y_{1}, y_{2}, y_{3}, ...} and the set of the endpoints of the intervals I_{n_{l}^{1}} has no cluster points except possibly ±∞. Indeed, for each n ∈ ℕ we can choose a finite number of intervals covering more than nine tenths (in measure) of the interval [n; n + 1] . Then we can put {I_{n_{l}^{1}}} to be the set of all chosen intervals for all n ∈ ℕ.

Since the set of the endpoints of I_{n_{l}^{1}} has no cluster points in ℝ, the set ℝ \ ⋃_{l=1}^{∞} I_{n_{l}^{1}} is a union of disjoint closed intervals. Therefore in a similar way we can choose intervals I_{n_{l}^{2}} ⊂ ℝ \ ⋃_{l} I_{n_{l}^{1}} such that

∫_{(y;y+1)∩⋃_{l} I_{n^{2}_{l}}} \frac{1}{x − y} dx > 1

for any y ∈ F \ {x_{1}, x_{2}, x_{3}, ...},

∫_{(y−1;y)∩⋃_{l} I_{n^{2}_{l}}} \frac{1}{y − x}dx > 1

for any y ∈ F \ {y_{1}, y_{2}, y_{3}, ...} and the set of the endpoints of the intervals I_{n_{l}^{2}} has no cluster points except possibly ±∞. Proceeding in this way we will obtain sets of intervals {I_{n_{l}^{k}}} for k = 1, 2, 3.... After that we can put L = ⋃_{k}{n_{l}^{2k−1}} and M = ⋃_{k}{n_{l}^{2k}}. ▲ Lemma 4.3. Let F ⊂ (0; 1) be a closed set with no isolated points, |F | = 0, F = (0; 1) \ ⋃ I_{n} where I_{n} = (x_{n}; y_{n}) are disjoint open intervals. Let {z_{n}}_{n>0} be a sequence of points such that z_{n} ∈ I_{n} and y_{n} − z_{n} < |I_{n}|^{2}. Let M be as in the previous Lemma. Then there exists N ⊂ M such that for all y ∈ F

i) if y = y_{n} for some n ∈ ℕ then

(4.3)  ∫_{(y;y+1)∩ ⋃_{k∈N} I_{k}} \frac{dx}{x − y} = +∞;

ii) if y ≠ y_{n} for any n and E = ⋃_{n∈N} I_{n} ∪ ⋃_{n∈ℕ\N}[z_{n}; y_{n}], then

(4.4)  p. v. ∫_{(−1;1)} χ_{E}(y + x)\frac{dx}{x} < +∞.

Proof. WLOG we can assume that there are infinitely many intervals I_{n} and that they are enumerated in such a way that |I_{1}| ≥ |I_{2}| ≥ ....

Let {δ_{k}}_{k>0} be small positive constants (the exact choice of δ_{k} will be made later). For each k ∈ ℕ choose a sequence of intervals {I_{n_{l}^{k}}}_{l>0}, such that for any l ∈ ℕ n_{l}^{k} ∈ M, I_{n_{l}^{k}} ⊂ (y_{k}; y_{k} + δ_{k}), y_{n_{1}^{k}} > x_{n_{1}^{k}} > y_{n_{2}^{k}} > x_{n_{2}^{k}} > ... and

∫_{(y_{k};y_{k}+1)∩ ⋃_{l>0} I_{n^{k}_{l}}} \frac{dx}{x − y_{k}} = ∞

(recall that F has no isolated points). Put N = {n_{l}^{k}| l, k ∈ ℕ}. Then condition i) is satisfied. We will show that if constants δ_{k} are small enough then N also satisfies ii).

Denote J(y_{k}) = ⋃_{l>0} I_{n_{l}^{k}}. Then J(y_{k}) ⊂ (y_{k}; y_{k} + δ_{k}). Choose constants δ_{k} > 0 to satisfy

both

(4.5)  δ_{k} < |I_{k}|^{2}

and

(4.6)  δ_{k} < \frac{1}{4}(y_{k} − z_{k}) exp (−\frac{4}{3} \frac{1}{dist(y_{k}; (y_{k}; ∞) ∩ ⋃_{m<k} J(y_{m}))})

for every k ∈ ℕ (notice that the distance in (4.6) is always nonzero because the only cluster points of the intervals from J(y_{m}), m < k are y_{m}, m < k). Let now y ∈ F, y ≠ y_{n} for any n > 0. Consider the sets of integers M′ = {k|x_{k} > y}, M′′ = {k| y_{k} + δ_{k} ≤ y} and M′′′ = {k| y_{k} < y, y_{k} + δ_{k} > y}. Then ℕ = M′ ∪ M′′ ∪ M′′′. Put

J′ = ⋃_{k∈M′} J(y_{k}), J′′ = ⋃_{k∈M′′} J(y_{k}) and J′′′ = ⋃_{k∈M′′′} J(y_{k}).

Then, since J(y_{k}) ⊂ (y_{k}; y_{k} + δ_{k}),

|∫_{J′} \frac{dx}{x − y} | < ∞

by (4.5). Also obviously

∫_{J′′} \frac{dx}{x − y} < 0 < ∞

since J′′ ⊂ (−∞; y).

Let {n_{m}}_{m∈ℕ} be the enumeration of M′′′ such that n_{1} < n_{2} < n_{3} < ... . Put ϵ_{m} = y − y_{n_{m}} + δ_{n_{m}} and consider

C_{m} = ∫_{(y−1;y+1)\(y−ϵ_{m};y+ϵ_{m})} χ_{E}(x)\frac{dx}{x − y}

(notice that 0 < ϵ_{m} ≤ 2δ_{n_{m}} → 0 as m → ∞). We shall prove that sequence {C_{m}} is bounded from above.

Note that (4.6) implies that for any m ∈ ℕ

δ_{n_{m}} < \frac{1}{4} dist(y_{n_{m}}; (y_{n_{m}}; ∞) ∩ ⋃_{k<m} J(y_{n_{k}}))

and therefore

dist(y; (y; ∞) ∩ ⋃_{k<m} J(y_{n_{k}})) > dist(y_{n_{m}}; (y_{n_{m}}; ∞) ∩ ⋃_{k<m} J(y_{n_{k}})) − δ_{n_{m}} >

> \frac{3}{4} dist(y_{n_{m}}; (y_{n_{m}}; ∞) ∩ ⋃_{k<m} J(y_{n_{k}})).

Since y − (y_{n_{m}} − δ_{n_{m}}) < 2δ_{n_{m}} by (4.6)

| ∫_{z_{nm}}^{y_{nm} −δ_{nm}} \frac{dx}{x − y} | = ln(y − z_{n_{m}}) − ln(y − (y_{n_{m}} − δ_{n_{m}})) >

> ln(y_{n_{m}} − z_{n_{m}}) − ln 2δ_{n_{m}} = ln \frac{y_{n_{m}} − z_{n_{m}}}{2δ_{n_{m}}} > \frac{4}{3} \frac{1}{dist(y_{n_{m}}; (y_{n_{m}}; ∞) ∩ ⋃_{k<m} J(y_{n_{k}}))} >

> \frac{1}{dist(y; (y; ∞) ∩ ⋃_{k<m} J(y_{n_{k}}))}.

Since |(y; ∞) ∩ ⋃_{k<m} J(y_{n_{k}})| ≤ | ⋃_{k<m} J(y_{n_{k}})| < 1

| ∫_{z_{nm}}^{y_{nm} −δ_{nm}} \frac{dx}{x − y} | > \frac{|(y; ∞) ∩ ⋃_{k<m} J(y_{n_{k}})|}{dist(y; (y; ∞) ∩ ⋃_{k<m} J(y_{n_{k}}))}.

Also

| ∫_{(y;∞)∩ ⋃_{k<m} J(y_{n_{k}})} \frac{dx}{x − y} | < \frac{|(y; ∞) ∩ ⋃_{k<m} J(y_{n_{k}})|}{dist(y; (y; ∞) ∩ ⋃_{k<m} J(y_{n_{k}}))}.

Thus

(4.7)  | ∫_{z_{nm}}^{y_{nm} −δ_{nm}} \frac{dx}{x − y} | > |  ∫_{(y;∞)∩ ⋃_{k<m} J(y_{n_{k}})} \frac{dx}{x − y} | .

Since y_{n} − z_{n} < |I_{n}|^{2},

∫_{(y;∞)∩ ⋃_{n∈ℕ}(z_{n};y_{n})} \frac{dx}{x − y} < C < +∞.

Thus by (4.7)

(4.8)  | ∫_{z_{nm}}^{y_{nm} −δ_{nm}} \frac{dx}{x − y} | > |  ∫_{(y;∞)∩[ ⋃_{n∈ℕ}(z_{n};y_{n})∪ ⋃_{k<m} J(y_{n_{k}})]} \frac{dx}{x − y} | − C.

Since |I_{n_{1}}| ≥ |I_{n_{2}}| ≥ ..., δ_{n_{m}} < \frac{1}{4}(y_{n_{m}} − z_{n_{m}}) < |I_{n_{m}}| and y ∈ (y_{n_{m}}; y_{n_{m}} + δ_{n_{m}}) , we have that I_{n_{m+1}} ⊂ (y_{n_{m}}; y) for all m. Indeed, if y_{n_{m+1}} < x_{n_{m}} and y_{n_{m+1}} < y ∈ (y_{n_{m+1}}; y_{n_{m+1}} + δ_{n_{m+1}}) then δ_{n_{m+1}} > |I_{n_{m}}|. Hence

y − ϵ_{m} = y_{n_{m}} − δ_{n_{m}} < y_{n_{m}} < y_{n_{m+1}}.

Also since

δ_{n_{m+1}} < |I_{n_{m+1}}| < y − y_{n_{m}} < δ_{n_{m}},

we have

y + ϵ_{m} = y + δ_{n_{m}} + (y − y_{n_{m}}) > y + δ_{n_{m}} > y + |I_{n_{m+1}}| > y_{n_{m+1}} + δ_{n_{m+1}}.

Therefore J(y_{n_{k}}) ⊂ (y − ϵ_{m}; y + ϵ_{m}) for all k > m. Thus

{x|χ_{E}(x)/(x − y) > 0} \ (y − ϵ_{m+1}; y + ϵ_{m+1}) = (y; ∞) ∩ [ ⋃_{n∈ℕ}(z_{n}; y_{n}) ∪ ⋃_{k<m} J(y_{n_{k}})] .

Since

\frac{χ_{E}(x)}{x − y} < 0

on (z_{n_{m}}; y_{n_{m}} − δ_{n_{m}}), (4.8) implies C_{m+1} < C for any m ∈ ℕ. Hence

lim inf_{ϵ→0} ∫_{(−1;1)\(−ϵ;ϵ)} χ_{E}(x)\frac{dx}{x} < +∞

and condition (ii) is satisfied. ▲

We will need the following

Definition. We will say that two disjoint sets of real numbers A and B are well-mixed if they satisfy the following conditions:

1) for any two points x, y ∈ A each of the sets (x; y) and ℝ \ [x; y] contains at least one point from B.

2) for any two points x, y ∈ B each of the sets (x; y) and ℝ \ [x; y] contains at least one point from A.

Proof of Theorem 4.1. 2) ⇒ 1). As follows from the Weyl–von Neumann Theorem on the stability of spectra, the essential spectrum is stable under rank one perturbations.

1) ⇒ 2). Suppose A is pure point and σ(A) is uncountable. We will show that there exists a self-adjoint rank one perturbation of A with nontrivial continuous part.

Denote by A the set of all eigenvalues of A. WLOG A ⊂ [0; 1]. Then there exists a closed uncountable set F ⊂ (0; 1), m(F) = 0 such that for any x ∈ F and any ϵ > 0 both sets (x − ϵ; x) ∩ A and (x; x + ϵ) ∩ A are nonempty. In fact, any uncountable subset F of the essential spectrum of A (which is uncountable) will satisfy the above conditions (recall, that A is cyclic). Also we can always choose F disjoint from A.

Let I_{1} = (x_{1}; y_{1}), I_{2} = (x_{2}; y_{2}), ... be disjoint open intervals such that F = (0; 1) \ ⋃ I_{n}. We always can assume that F contains no isolated points i.e. x_{i} ≠ y_{j} for any i, j.

First, using Lemmas 4.2 and 4.3 we will construct a phase shift u_{0} of a pair (μ_{0}; ν_{0}) such that μ_{0} = ∑ c_{n}δ_{a_{n}} for some a_{n} ∈ A, c_{n} > 0 and ν_{0} is continuous with supp ν_{0} ⊂ F.

By Lemma 4.2 we can choose two sets of integers L and M satisfying (4.1) and (4.2). For each n ∈ ℕ choose z_{n} ∈ I_{n} ∩A such that y_{n} −z_{n} < |I_{n}|^{2}. By Lemma 4.3 we can choose N ⊂ M satisfying (4.3) and (4.4). For each k ∈ ℕ \ N put a_{k} = z_{k}. For each k ∈ N choose a_{k} ∈ I_{k} ∩ A such that a_{k} − x_{k} < |I_{k}|^{2}. After that put u_{0} = π on ⋃_{k∈ℕ}(a_{k}; y_{k}) and u_{0} = 0 elsewhere. Then u_{0} has compact support.

Consider the pair of measures (μ_{0}; ν_{0}) such that u_{0} is its phase shift. We will prove that μ_{0} = ∑ c_{n}δ_{a_{n}}.

By Lemma 3.2 μ_{0} and ν_{0} are singular. Since supp μ_{0} ⊂ [F ∪ A], to prove that μ_{0} = ∑ c_{n}δ_{a_{n}} we need to show that μ_{0}(F) = 0.

To show that μ_{0}(F) = 0 let us notice that if y ∈ F then

0 ≥ ∫_{(y−1;y)∩ ⋃_{k∈N}(x_{k};a_{k})} \frac{dx}{x − y} > −C_{1} > −∞

because a_{k} − x_{k} < |I_{k}|^{2} for any k ∈ N. Thus (4.9)

∫_{\{u_{0}=π\}\(y−ϵ;y+ϵ)} \frac{dx}{x − y} =  ∫_{( ⋃_{k∈ℕ\N}(z_{k};y_{k})∪ ⋃_{k∈N} I_{k})\(y−ϵ;y+ϵ)} \frac{dx}{x − y} −  ∫_{⋃_{k∈N}(x_{k};a_{k})\(y−ϵ;y+ϵ)} \frac{dx}{x − y} <

<  ∫_{( ⋃_{k∈ℕ\N}(z_{k};y_{k})∪ ⋃_{k∈N} I_{k})\(y−ϵ;y+ϵ)} \frac{dx}{x − y} −  ∫_{(y−1;y−ϵ)∩ ⋃_{k∈N}(x_{k};a_{k})} \frac{dx}{x − y} <

< ∫_{( ⋃_{k∈ℕ\N}(z_{k};y_{k})∪ ⋃_{k∈N} I_{k})\(y−ϵ;y+ϵ)} \frac{dx}{x − y} + C_{1}.

If y ≠ y_{i} for any i ∈ ℕ then (4.9) and (4.4) (recall that N satisfies conditions of Lemma 4.3) imply that there exist positive ϵ_{1}, ϵ_{2}, ... such that ϵ_{n} → 0 as n → ∞ and

∫_{\{u_{0}=π\}\(y−ϵ_{n};y+ϵ_{n})} \frac{dx}{x − y} < C_{2} < ∞

for any n. Hence,

p. v. ∫_{ℝ} \frac{u_{0}(y + x)dx}{x} < ∞.

Therefore by Lemma 3.3 μ_{0}(F \ {y_{1}, y_{2}, ...}) = 0. Also, μ_{0}(y_{k}) = 0 for all k because u_{0}(x) → π as x → y_{k}− which means that condition (3.8) is not satisfied. Hence, μ_{0} = ∑ c_{n}δ_{a_{n}} for some positive c_{n}.

For ν_{0} we obviously have supp ν_{0} ⊂ F. Let y ∈ F, y = y_{i} for some i > 0. Then

∫_{(y;y+1)∩ ⋃_{k∈N} I_{k}} \frac{dx}{x − y} = ∞

because N satisfies (4.3). Since for k ∈ N we have a_{k} − x_{k} < (y_{k} − x_{k})^{2}, this implies that

∫_{(y;y+1)∩ ⋃_{k∈N}(a_{k};y_{k})} \frac{dx}{x − y} = ∞.

Indeed, if there are infinitely many k ∈ N such that (a_{k}; y_{k}) ⊂ (y; y+1) and a_{k}−y < y_{k}−a_{k} then the last integral is infinite. On the other hand, if there are finitely many of such k, then the integral over (y; y + 1) ∩ ⋃_{k∈N}(x_{k}; a_{k}) is finite (recall that a_{k} − x_{k} < |I_{k}|^{2} for k ∈ N) and the last equality follows from the previous one.

Since on (a_{k}; y_{k}) u_{0} = π, the last equality implies that condition (3.9) is not satisfied. Thus ν_{0}(y) = 0. If y ∈ F and y ≠ y_{i} for any i, then

∫_{(y−1;y)∩ ⋃_{k∈ℕ\N} I_{k}} \frac{dx}{y − x} = ∞

by (4.2) because L ⊂ ℕ \ N. Since for k ∈ ℕ \ N we have y_{k} − a_{k} < (y_{k} − x_{k})^{2}, by the same argument that we used above, the last equality implies that

∫_{(y−1;y)∩ ⋃_{k∈ℕ\N}(x_{k};a_{k})} \frac{dx}{y − x} = ∞.

Again, since u_{0} = 0 on (x_{k}; a_{k}), condition (3.9) does not hold and ν_{0}(y) = 0. Hence, ν_{0} is continuous.

Note that μ_{0} does not have point masses at some points of A, therefore it is not a spectral measure of A. So far we constructed a phase shift u_{0} corresponding to a continuous perturbation of a part of A. If we did not have to construct a cyclic perturbation, the proof would be essentially finished here.

Our next goal is to transform u_{0} into the phase shift u of a pair (μ; ν) such that μ is a spectral measure of A and ν has a nontrivial continuous part.

Let {b_{n}}_{n=1}^{∞} be some enumeration of the set A \ {a_{n}}_{n=1}^{∞}. For each b_{n} let us choose c_{n} > b_{n} such that if b_{n} ∈ I_{k} (recall that A ⊂ ℝ \ F) then c_{n} ∈ I_{k},

(4.10)  |b_{n} − c_{n}| < \frac{dist(b_{n}; (ℝ \ I_{k}) ∪ \{a_{n}, b_{1}, b_{2}, ..., b_{n−1}\})}{2^{n+1}}

and for each n ∈ ℕ the sets B_{n} = {a_{i}}_{i=1}^{∞} ∪ {b_{1}, b_{2}, ..., b_{n}} and C_{n} = {c_{1}, c_{2}, ..., c_{n}} ∪ F are well-mixed.

For each k ∈ ℕ define the function u_{k} in the following way:

1) u_{k} = 0 or π everywhere on ℝ;

2) u_{k} is continuous everywhere except B_{k} ∪ C_{k};

3) u_{k} jumps from 0 to π at each point of B_{k} and from π to 0 at each point of {c_{1}, c_{2}, ..., c_{k}}.

Let u_{k} be the phase shift of a pair (μ_{k}; ν_{k}). Then

μ_{k} = ∑_{n=1}^{∞} α_{n}^{k}δ_{a_{n}} + ∑_{n=1}^{k} β_{n}^{k}δ_{b_{n}}

for some positive constants α_{i}^{k} and β_{i}^{k} and

ν_{k} = ∑_{n=1}^{k} γ_{n}^{k}δ_{c_{n}} + f_{k}ν_{0}

for some positive constants γ_{1}^{k} and some positive function f_{k} ∈ L^{1}(ν_{0}). Since by condition (4.10) |b_{n} − c_{n}| < 1/2^{n+1} and {u_{k} ≠ u_{k−1}} ⊂ (b_{n}; c_{n}), sequence {u_{k}} converges in measure to some function u. Since supp u_{k} ⊂ [0; 1] for any k ∈ ℕ, u has compact support. Let u be the phase shift of a pair (μ; ν). Since u_{k} → u in measure, μ_{k} → μ and ν_{k} → ν in ∗-weak topology.

Since by (4.10)

(4.11)  |(Ku_{k})(y) − (Ku_{k+1})(y)| ≤ ∫_{ℝ} |\frac{u_{k}(x) − u_{k+1}(x)}{x − y}| dx <

< \frac{|b_{k+1} − c_{k+1}|}{dist(b_{n}; (ℝ \ I_{n}) ∪ \{a_{n}, b_{1}, b_{2}, ..., b_{n−1}\})} < \frac{1}{2^{k+1}}

for any y ∈ B_{k}, by part ii) of Lemma 3.7

μ_{k+1} = g_{k}μ_{k} + β_{k+1}^{k+1}δ_{b_{k+1}}

where g_{k} ∈ L^{∞}(μ_{k}),

(4.12)  1 − \frac{1}{2^{k}} < g_{k} < 1 + \frac{1}{2^{k}}

μ_{k}-a. e. Also by (4.11) and part ii) of Lemma 3.7

(4.13)  1 − \frac{1}{2^{k}} < \frac{f_{k}}{f_{k+1}} < 1 + \frac{1}{2^{k}}

ν_{0}-a. e.

Since μ_{k} → μ , ||μ_{k}|| → ||μ|| < ∞. Together with (4.12) this implies that

β_{k+1}^{k+1} ≤ \frac{||μ||}{2^{k−1}}

if k is big enough. Since by (4.12)

||μ_{k} − μ_{k+1}|| ≤ \frac{||μ_{k}||}{2^{k}} + β_{k+1}^{k+1},

this implies that μ_{k} → μ in norm. Since each μ_{k} is pure point, μ is pure point. By (4.12)

μ = ∑_{n=1}^{∞} α_{n}δ_{a_{n}} + ∑_{n=1}^{∞} β_{n}δ_{b_{n}}

where

0 < α_{i}^{1} ∏_{k∈ℕ} g_{k}(a_{i}) = α_{i} < ∞

and

0 < β_{i}^{i} ∏_{k>i} g_{k}(b_{i}) = β_{i} < ∞.

Thus A is unitarily equivalent to M_{μ}. Also, by (4.13) there exists f ∈ L^{1}(ν_{0}),

0 < f_{1}(x) ∏_{k>1} f_{k}(x) = f(x) < 3f_{1}(x)

for ν_{0}-a. e. x. Since ν is a ∗-weak limit of ν_{k} = ∑_{n=1}^{k} γ_{n}^{k}δ_{c_{n}} + f_{k}ν_{0}, where γ_{n}^{k} are always nonnegative, ν = η + σ for some nonnegative measure η and σ = fν_{0}. Since ν_{0} is continuous, ν is not discrete. Thus M_{μ} + (·, 1)1 has a nontrivial continuous part. ▲

As we mentioned before, operator A in Theorem 4.1 does not have to be cyclic:

Corollary 4.4. Let A be a self-adjoint operator. Then the following two conditions are equivalent:

1) All self-adjoint rank one perturbations of A are pure point,

2) σ(A) is countable.

Proof. 2)⇒ 1)

Follows from the Weyl–von Neumann Theorem.

1) ⇒ 2)

Suppose that 1) holds but σ(A) is uncountable. If A is not cyclic then we can always choose a cyclic subspace such that the restriction of A on this subspace has an uncountable spectrum. Indeed, A is unitarily equivalent to the multiplication by z in a direct integral of Hilbert spaces H_{ξ}

H = ∫ ⊕H_{ξ}dμ(ξ)

where μ is a scalar measure with uncountable support. If we choose a one-dimensional subspace l_{ξ} in each of these Hilbert spaces and consider the direct integral

L = ∫ ⊕l_{ξ}dμ(ξ)

then the restriction of multiplication by z on L will be cyclic with the spectral measure μ. Since supp μ is uncountable, the spectrum of such a restriction will be uncountable.

Notice that if some rank one perturbation of this restriction has a nontrivial continuous part then the corresponding rank one perturbation of the whole operator also has a nontrivial continuous part. Such a perturbation exists by Theorem 4.1. ▲ Corollary 4.5. Let A be a self-adjoint operator. Then the following conditions are equivalent

1) all self-adjoint compact perturbations of A are pure point;

2) all self-adjoint trace class perturbations of A are pure point;

3) all self-adjoint rank one perturbations of A are pure point;

4) σ(A) is countable.

Proof.

4)⇒1),4)⇒2) and 4)⇒3) follow from the Weyl–von Neumann Theorem on the stability of the essential spectrum under compact perturbations. 1)⇒4),2)⇒4) and 3)⇒4) follow from Theorem 4.1.

2)⇒4) also follows from the Theorem of Carey and Pincus [C-P] on the equivalence modulo the trace class. ▲

5. The problem of two spectra.

Definition. We shall say that two operators A and B are equivalent up to a rank one perturbation if there exists a unitary operator U such that rank(UAU^{∗} − B) = 1.

In this section we shall discuss the following question.

The problem of two spectra. Let μ and ν be two finite Borel mutually singular measures on ℝ. When do there exist two cyclic self-adjoint operators A and B equivalent up to a rank one perturbation such that μ and ν are spectral measures of A and B respectively?

Definition. If such A and B exist, we shall say that μ and ν solve the problem of two spectra (PTS).

We shall say that μ is equivalent to ν and write μ ∼ ν if there exists f ∈ L^{1}(μ) such that f > 0 μ-a. e. and fμ = ν. We shall say that pairs of measures (μ, ν) and (μ′, ν′) are equivalent, (μ, ν) ∼ (μ′, ν′), if μ ∼ μ′ and ν ∼ ν′.

In terms of the families M_{φ}, we can say that μ and ν solve PTS if and only if there exists φ such that (μ, ν) ∼ (μ′, ν′) for some μ′, ν′ ∈ M_{φ}. In terms of the phase shift, μ and ν solve PTS if and only if there is an equivalent pair possessing a phase shift i. e. satisfying (3.3′) for some real function u ∈ L^{∞}(ℝ), ||u||_{∞} ≤ π and λ ∈ ℝ.

This section discusses the case of pure point measures μ and ν.

We start with the situation of one point essential spectrum. We first consider the following well-known example.

Example 5.1. Let A = {a_{n}}_{n=1}^{∞} and B = {b_{n}}_{n=1}^{∞} be two disjoint sequences of real numbers, lim_{n→∞} a_{n} = lim_{n→∞} b_{n} = c ∈ ℝ . We will show that there exist unitarily equivalent up to a rank one perturbation cyclic self-adjoint operators A and B such that σ(A) = Clos A and σ(B) = Clos B if and only if the sets {a_{1}, a_{2}, ...} and {b_{1}, b_{2}, ...} are well-mixed. We will also show that if such A and B exist then they are unique up to a unitary equivalence.

To prove these statements we shall use the notion of the Krein spectral shift.

Define function u on ℝ to satisfy the following conditions:

1) u is continuous and equal to π or 0 everywhere on ℝ \ ({a_{n}} ∪ {b_{n}}), 2) at the points {a_{n}} u jumps from 0 to π,

3) at the points {b_{n}} u jumps from π to 0.

WLOG inf_{n∈ℕ} a_{n} ≤ inf_{n∈ℕ} b_{n} and sup_{n∈ℕ} a_{n} ≤ sup_{n∈ℕ} b_{n}. Then u has compact support. As we discussed before, there are unique measures μ, ν ∈ M_{+}(ℝ) such that u, μ and ν satisfy formula (3.3′) with λ = 1. Put A = M_{μ} and B = M_{ν}. Then B is unitarily equivalent to A + (·, 1)1. By Lemma 3.3 i) supp μ ⊂ Clos A and supp ν ⊂ Clos B. Also by Lemma 3.8 the measures μ and ν have point masses at points a_{i} and b_{i} respectively, therefore supp μ = Clos A and supp ν = Clos B. Thus the operators A and B satisfy all the conditions. Since the sequences are well-mixed and have only one cluster point, there exists a unique function u satisfying properties 1)-3). Therefore A and B are unique up to a unitary equivalence.

Remark. If μ = ∑ α_{n}δ_{a_{n}} and ν = ∑ β_{n}δ_{b_{n}}, where α_{i}, β_{i} > 0 and the sets A = {a_{1}, a_{2}, ...} and B = {b_{1}, b_{2}, ...} are disjoint, solve PTS then the sets A and B are well-mixed. To prove it one can, for instance, notice that if A and B are not well-mixed then the function u satisfying properties 1)-3) from the above Example does not exist.

Note that Example 5.1 provides us with operators such that μ and ν are only absolutely continuous with respect to spectral measures of A and A + (·, ϕ)ϕ but not necessarily equivalent to them. Therefore it does not imply that μ and ν, with A and B as in Example 5.1, solve PTS. As shown in Example 5.2 below, the condition that A and B are well-mixed is not sufficient for μ and ν to solve PTS.

Example 5.2. Put a_{n} = (−1)^{n+1}/2^{n} for n = 1, 2, 3, ..., b_{1} = 1, b_{n} = a_{n} + (−1)^{n}/4^{n} for n = 2, 3, .... Then the sets {a_{1}, a_{2}, ...} and {b_{1}, b_{2}, ...} are well-mixed.

Let μ and ν be defined as in the last remark. Suppose that μ and ν solve PTS. Then there exists a pair (μ′, ν′) ∼ (μ, ν) possessing a phase shift u. Such u must be continuous on ℝ \ ({a_{n}} ∪ {b_{n}}), jump from 0 to π at any a_{n} and jump from π to 0 at any b_{n}. Thus u = π on (−1/2^{2n}; −1/2^{2n} + 1/4^{2n+2}) and on (1/2^{2n+3}; 1/2^{2n+1} − 1/4^{2n+1}) for n = 1, 2, ... and u = π on the rest of ℝ.

But then Lemma 3.8 implies that μ(0) > 0. Since 0 ≠ a_{n} for any n ∈ ℕ we have a contradiction.

Hence in the conditions of Example 5.1 one can not prescribe arbitrarily whether or not spectral measures of A and B have a point mass at c.

If the sequences {a_{1}, a_{2}, ...} and {b_{1}, b_{2}, ...} are finite, disjoint and well-mixed then μ and ν obviously solve PTS. More interesting example is provided by the following theorem.

Theorem 5.3 ([A3]). Let A and B be two disjoint countable sets on the unit circle 𝕋. Suppose that Clos A = Clos B = 𝕋. Let {a_{n}}_{n=1}^{∞} and {b_{n}}_{n=1}^{∞} be some enumerations of A and B respectively. Then there exist sequences of positive real numbers {α_{n}} and {β_{n}} such that the measures ∑ α_{n}δ_{a_{n}} and ∑ β_{n}δ_{b_{n}} belong to the same family M_{φ}

An analogous result can be formulated for the case of the real line: if Clos A = Clos B = ℝ then the measures belong to the same M_{φ}(ℝ). In terms of operator theory this would mean that any two cyclic pure point self-adjoint operators whose spectrum is equal to ℝ are unitarily equivalent up to a rank one perturbation. Note that the condition Clos A = Clos B = 𝕋 automatically implies that A and B are well-mixed. However the following example shows that in general the condition Clos A = Clos B does not imply that the corresponding measures solve PTS even if A and B are well-mixed.

Example 5.4. Let C be the standard Cantor null set on the unit interval [0; 1], C = [0; 1] \ ⋃ I_{n} where I_{n} = (x_{n}; y_{n}) are disjoint open intervals. Let A and B be two disjoint well-mixed countable sets of points of C such that

(5.1)  0, x_{1}, x_{2}, .. ∈ B and 1 ∈ A.

Then Clos A = Clos B = C. Let {a_{n}}_{n=1}^{∞} and {b_{n}}_{n=1}^{∞} be some enumerations of A and B respectively. Define μ = ∑ \frac{1}{2^{n}} δ_{a_{n}} and ν = ∑ \frac{1}{2^{n}} δ_{b_{n}}. Suppose that μ and ν solve PTS. Then there exists the phase shift u of the pair (μ′; ν′) where (μ, ν) ∼ (μ′, ν′). Function u must be constant on each I_{n}. Condition (5.1) implies that u = 0 a. e. on ℝ. Then (3.3) implies that μ′ and ν′ are trivial and we obtain a contradiction.

It turns out that this example is, in a certain sense, typical: the following Theorem shows that if A and B do not contain the endpoints of complementary intervals (cf. (5.1)) then the corresponding measures solve PTS.

Theorem 5.5. Let K ⊂ ℝ be a closed set. Denote by I_{1} = (x_{1}; y_{1}), I_{2} = (x_{2}; y_{2}), ... disjoint open intervals such that K = ℝ \ ⋃ I_{n}. Let A and B be two self-adjoint cyclic pure point operators. Let A and B be the sets of all eigenvalues of A and B respectively. Suppose that σ(A) = σ(B) = K and A ∩ {x_{1}, y_{1}, x_{2}, y_{2}, ...} = B ∩ {x_{1}, y_{1}, x_{2}, y_{2}, ...} = ∅. Then A and B are equivalent up to a rank one perturbation.

Proof. WLOG K ⊂ (0; 1). Let {a_{n}}_{n=1}^{∞} and {b_{n}}_{n=1}^{∞} be some enumerations of A and B respectively.

Let us find reenumerations {a_{n_{k}}} and {b_{m_{k}}} of the sequences A = {a_{n}} and B = {b_{m}} such that for any k ∈ ℕ

1) the sets {a_{n_{1}}, ..., a_{n_{k}}} and {b_{m_{1}}, ..., b_{m_{k}}} are well-mixed and

2) the distance |a_{n_{k}} − b_{m_{k}}| is at least 2^{k+1} times less than the distance between a_{n_{k}} and any of the points a_{n_{1}}, ..., a_{n_{k−1}}, b_{m_{1}}, ..., b_{m_{k−1}}.

We can find such reenumerations in the following way.

WLOG a_{1} < b_{1}. Choose n_{1} = 1 and m_{1} = 1. After that for k = 2, 3, ... consequently do the following. Choose a point x in A ∪ B \ {a_{n_{1}}, ..., a_{n_{k−1}}, b_{m_{1}}, ..., b_{m_{k−1}}} with minimal index in the initial enumeration. Suppose x ∈ A. Then put a_{n_{k}} = x and choose b_{m_{k}} ∈ B to satisfy conditions 1) and 2) above. Similarly, if x ∈ B then put b_{m_{k}} = x and choose a_{n_{k}} ∈ A to satisfy conditions 1) and 2) above.

Note that it is always possible to choose b_{m_{k}} or a_{n_{k}} to satisfy 2) because a_{n_{k}}, b_{m_{k}} ∈ K \ {x_{1}, y_{1}, x_{2}, y_{2}, ...}.

Now for each k define function u_{k} on ℝ in such a way that u_{k}(x) ∈ {0, π} for any x ∈ ℝ, u_{k} is continuous everywhere except a_{n_{1}}, a_{n_{2}}, ...a_{n_{k}}, b_{n_{1}}, b_{n_{2}}, ..., b_{n_{k}}, u_{k} jumps from 0 to π at each a_{n_{i}}, 0 < i ≤ k and from π to 0 at each b_{n_{i}}, 0 < i ≤ k (this construction is possible because the sequences are well-mixed). Since a_{1} < b_{1}, u_{k} has compact support. Therefore u_{k} is the phase shift of a pair of measures (μ_{k}; ν_{k}). It is easy to see that μ_{k} = ∑_{i=1}^{k} α_{i}^{k}δ_{a_{n_{i}}} and ν_{k} = ∑_{i=1}^{k} β_{i}^{k}δ_{b_{m_{i}}} for some positive constants α_{1}^{k}, ..., α_{k}^{k}, β_{1}^{k}, ..., β_{k}^{k}.

Condition 2) implies that for any l > k |{u_{k} ≠ u_{l}}| ≤ ∑_{i=k}^{l} 1/2^{i}. Therefore the sequence {u_{k}} converges in measure to some function u. Let μ and ν be the measures such that u is the phase shift of the pair (μ; ν). By the same argument that we used in the proof of Theorem 4.1, the sequences of measures {μ_{k}} and {ν_{k}} converge in the ∗-week topology to measures μ and ν respectively. To complete the proof we need to show that μ = ∑ α_{i}δ_{a_{n_{i}}} and ν = ∑ β_{i}δ_{b_{n_{i}}} for some positive constants α_{i} and β_{i}.

Note that

(5.2)  ∑_{i=k+1}^{l} α_{i}^{l} ≤ | ||μ_{k}|| − ||μ_{l}|| | + ∑_{i=1}^{k} |α_{i}^{k} − α_{i}^{l}|.

But since u_{k} ≠ u_{k+1} only between a_{n_{k+1}} and b_{n_{k+1}}, condition 2) implies that

∫_{\{u_{k}≠u_{k+1}\}} \frac{dx}{|x − a_{n_{i}}|} < \frac{1}{2^{k}}

for any k and any i ≤ k. Hence, by part ii) of Lemma 3.7

(5.3)  |α_{i}^{k}/α_{i}^{k+1} − 1| < 1/2^{k−1}

for any k and any i ≤ k. Since supp u_{k+1} ⊂ (0; 1),

α_{i}^{k+1} ≤ ||μ_{k+1}|| = ∫_{ℝ} \frac{u_{k+1}(x)dx}{1 + x^{2}} ≤ 1.

Hence by (5.3)

(5.4)  |α_{i}^{k} − α_{i}^{k+1}| < 1/2^{k−1}

for any i and k ≥ i. Also since

||μ_{k}|| = Im exp [Ku_{k}(i)] → Im exp [Ku(i)] ,

we have that

||μ_{k}|| − ||μ_{l}|| → 0

as k, l → ∞.

Together with (5.2) and (5.4) this implies that

∑_{i=k+1}^{l} α_{i}^{l} ≤ | ||μ_{k}|| − ||μ_{l}|| | + \frac{k}{2^{k−1}} → 0

as k, l → ∞. The last inequality and (5.4) together imply

||μ_{k} − μ_{l}|| → 0

as k, l → ∞. Put α_{i} = lim_{k→∞} α_{i}^{k}, and μ′ = ∑_{i=1}^{∞} α_{i}δ_{a_{n_{i}}}. Since the sequence {μ_{k}} converges in norm, it must converge to μ′. But since μ is the ∗-weak limit of {μ_{k}}, μ = μ′ = ∑_{i=1}^{∞} α_{i}δ_{a_{n_{i}}}. It is left to notice that (5.3) implies that

α_{i} = α_{i}^{i} ∏_{l>i} \frac{α_{i}^{l+1}}{α_{i}^{l}} > 0

for each i. In the same way we can prove that ν = ∑ β_{i}δ_{b_{i}} for some positive β_{i}. ▲

Our next Theorem shows that even when the spectra contain the endpoints of complementary intervals, the situation of Example 5.4 is impossible unless K is very “porous” (like the Cantor set in Example 5.4).

Theorem 5.6. Let K ⊂ ℝ be a closed set with no isolated points. Denote by I_{1}, I_{2}, ... the disjoint open intervals I_{k} = (x_{k}; y_{k}) such that K = ℝ \ ⋃ I_{n}.

Then the following two conditions are equivalent:

i)Any two cyclic self-adjoint pure point operators A and B such that the sets of their eigenvalues are well-mixed and σ(A) = σ(B) = K are unitarily equivalent up to a rank one perturbation.

ii) If y ∈ K \ {x_{1}, y_{1}, x_{2}, y_{2}, ...} then

(5.5)  ∫_{(y−1;y+1)\K} \frac{dx}{|y − x|} < ∞;

if y = x_{k} or y = y_{k} for some k ∈ ℕ then

(5.6)  ∫_{(y−1;y+1)\(K∪I_{k})} \frac{dx}{|y − x|} < ∞.

Example 5.7. Condition ii) fails for K = C for any standard Cantor set C (even if |C| > 0). Condition ii) obviously holds if ∂K is a finite set. We will show that there exist more complicated examples of such K, namely that for any nowhere dense set L ⊂ ℝ there exists a set K such that K satisfies ii) and ∂K ⊃ L.

Denote by J_{i} = (x_{i}; y_{i}) the disjoint open intervals such that Clos L = ℝ \ ⋃ J_{n}. On each J_{i} choose a sequence of intervals {Δ_{i}^{k}}_{k=−∞}^{∞}, Δ_{i}^{k} = (x_{i}^{k}; y_{i}^{k}) such that x_{i}^{k} → x_{i} as k → −∞, x_{i}^{k} → y_{i} as k → ∞ and

∫_{∪_{k}Δ_{i}^{k}} \frac{dx}{|x − y|} < \frac{1}{2^{i}}

for all y ∈ ℝ \ J_{i}. Then we can put K = ℝ \ ⋃_{i,k} Δ_{i}^{k}.

Similarly, one can show that for any closed L ⊂ ℝ there exists a closed K ⊃ L satisfying ii) and such that ∂K ⊃ ∂L.

Proof of Theorem 5.6. ii)⇒i)

Let K satisfy ii). WLOG K ⊂ (0; 1). Let operators A and B be such as in i). Denote by A and B the sets of eigenvalues of A and B respectively. Put N = {n|∂I_{n} ∩ [A ∪ B] ≠ ∅}. Consider disjoint countable sets A′ = A ∪ C and B′ = B ∪ D where C = {c_{n}} ⊂ [∪_{n∈N}I_{n}] ∩ (0; 1) and D = {d_{n}} ⊂ [∪_{n∈N}I_{n}] ∩ (0; 1) satisfy:

1) C and D are dense in [∪_{n∈N}I_{n}] ∩ (0; 1) and

2) if for some n ∈ N one of the endpoints x of the interval I_{n} belongs to A (B) but the other endpoint y of I_{n} is not in A ∪ B then y ∈ A′ (∈ B′).

Recall that K does not have isolated points. Hence x_{k} ≠ y_{l} for any k, l ∈ ℕ which makes it possible to choose C and D satisfying 2).

Then the set A′ ∪ B′ contains all the endpoints of the intervals I_{n}, n ∈ N.

Put K′ = K ∪ ⋃_{n∈N} I_{n} and let A′ and B′ be pure point self-adjoint cyclic operators such that A′ and B′ are the sets of eigenvalues of A′ and B′ respectively. Then K′, A′ and B′ satisfy the conditions of Theorem 5.5. Therefore we can construct a phase shift u′ corresponding to spectral measures μ′ = ∑ α_{i}δ_{a_{i}} + ∑ σ_{i}δ_{c_{i}} and ν′ = ∑ β_{i}δ_{b_{i}} + ∑ η_{i}δ_{d_{i}} for some positive α_{i}, σ_{i}, β_{i}, η_{i}. Define N_{π} = {n|x_{n} ∈ A or y_{n} ∈ B} and N_{0} = {n|x_{n} ∈ B or y_{n} ∈ A}. Then N_{π} ∩ N_{0} = ∅, because the sets A and B are well-mixed, and N = N_{π} ∪ N_{0}. Let function u be such that u = u′ outside ∪_{n∈N}I_{n}, u = π on each I_{n}, n ∈ N_{π} and u = 0 on each I_{n}, n ∈ N_{0}. Let μ and ν be the measures such that u is the phase shift of the pair (μ; ν). Then ii) together with Lemma 3.7 imply that the restriction of μ on K \ ∪_{n∈N}∂I_{n} is absolutely continuous with respect to μ′. Also, obviously μ(∪_{n∈N}I_{n}) = 0 because u is constant on each I_{n}, n ∈ N. It is left to check the endpoints of the intervals I_{n}, n ∈ N.

If for some n ∈ N_{π} x_{n} ∈ A, then μ′ has a point mass at x_{n}. Thus phase shift u′ satisfies the condition (3.8) at the point y = x_{n}. Since u = π on I_{n} and u′ satisfies (3.8), by (5.6) we have

∫_{\{u≠u′\}} \frac{dx}{|x − x_{n}|} < ∞.

Hence by (3.8) μ also has a point mass at x_{n}. Similarly if x_{n} ∈ B then ν has a point mass at x_{n}.

If for some n ∈ N_{π} x_{n} belongs to neither A nor B then, by the choice of N_{π} and D, x_{n} ∈ D. Therefore ν′ has a point mass at x_{n}. Thus phase shift u′ satisfies (3.9) at y = x_{n}. Since u = π on I_{n} and u′ satisfies (3.9), by (5.6) we have that

∫_{\{u≠π\}∩(0;1)} \frac{dx}{|x − x_{n}|} < ∞.

Thus u does not satisfy (3.8) or (3.9) at y = x_{n} and therefore μ(x_{n}) = ν(x_{n}) = 0. Other endpoints of I_{n}, n ∈ N can be checked in the same way. Thus μ = ∑ α_{n}δ_{a_{n}} for some positive α_{n} and ν = ∑ β_{n}δ_{b_{n}} for some positive β_{n}. i)⇒ii) Suppose condition ii) is not satisfied at some y ∈ K. Let us first consider the case when

y ≠ y_{n}, x_{n} for any n ∈ ℕ. Since (5.5) is not satisfied, WLOG we can assume that

(5.7)  ∫_{(y;y−1)\K} \frac{dx}{|y − x|} = ∞.

Let A = {a_{n}} and B = {b_{n}} be disjoint dense in K well-mixed sequences such that y ∈ B and y_{i} ∈ A for all i such that y_{i} < y. Suppose that some pair of measures (μ; ν), such that μ = ∑ α_{n}δ_{a_{n}} and ν = ∑ β_{n}δ_{b_{n}} for some positive β_{n} and α_{n}, possesses a phase shift u. Then u must be equal to 0 on ∪_{{n|y_{n}<y}}I_{n}. By (5.7), condition (3.9) is not satisfied at y. Therefore ν({y}) = 0 and we have a contradiction.

If y = y_{n} for some n we must choose A = {a_{n}} and B = {b_{n}} to be disjoint dense in K well-mixed sequences such that y ∈ B and x_{i} ∈ A for all i such that x_{i} > y. Since (5.6) is not satisfied, in the same way one can obtain a contradiction with condition (3.9). The case y = x_{n} can be proven similarly. ▲

In the rest of this Section we are going to discuss the following question.

Remark. Let A be a singular self-adjoint cyclic operator, ϕ and ψ its cyclic vectors and μ, ν are spectral measures corresponding to ϕ and ψ. We will say that ϕ is equivalent to ψ (or write φ ∼ ψ) if μ = ν.

Is it true that if A^{ϕ} = A + (·, ϕ)ϕ and A^{ψ} = A + (·, ψ)ψ are unitarily equivalent, then ϕ ∼ ψ?

If A is a finite rank operator, then the answer is positive. Indeed, denote by μ and ν the spectral measures of ϕ for A and A^{ϕ} respectively; denote by μ′ and ν′ the spectral measures of ψ for A and A^{ψ} respectively. If μ and ν are linear combinations of point masses at points a_{1}, a_{2}, .., a_{n} and b_{1}, b_{2}, .., b_{n} respectively, a_{1} < b_{1} < a_{2} < b_{2} < ... < a_{n} < b_{n}, then the phase shift u of the pair (μ; ν) depends only on the sequences {a_{n}} and {b_{n}}: u = 0 a. e. on (−∞; a_{1}) ∪ (b_{1}; a_{2}) ∪ ... ∪ (b_{n−1}; a_{n}) ∪ (b_{n}; ∞) and u = π elsewhere. If A^{ϕ} is unitarily equivalent to A^{ψ} then ν ∼ ν′. Also we have that μ ∼ μ′. Thus measures μ′ and ν′ have point masses at the same points a_{1}, a_{2}, ..., a_{n}, b_{1}, b_{2}, ..., b_{n}. Hence the phase shift u′ of the pair (μ′, ν′) must be equal to u a. e. on ℝ. Since the phase shift determines the pair of measures uniquely, we have μ = μ′ and ν = ν′. Therefore, ϕ ∼ ψ.

In the same way one can prove that if all eigenvalues of A are isolated, then any two different (corresponding to nonequivalent vectors) rank one perturbations of A can not be unitarily equivalent.

However if σ(A) = σ_{s}(A) is more complicated there may exist two different rank one perturbations of A which are unitarily equivalent. To show this, it is enough to give an example of two equivalent pairs of singular measures (μ, ν) and (μ′, ν′) with different phase shifts. Indeed, if such pairs exist then M_{μ} + (·, 1)1 and M_{μ} + (·, \sqrt{f})\sqrt{f}, where f ∈ L^{1}(μ), f ≥ 0, μ′ = fμ, are unitarily equivalent and |f|^{2} ≠ const.

To construct such an example one can notice, for instance, that function u in the proof of Theorem 5.5 is not unique. Indeed, the reenumerations of the sequences A and B can be found in many different ways. It can be shown that some of the enumerations will give us different phase shifts u at the end of our construction. For instance, let n_{k}, m_{k} ∈ ℕ, k = 1, 2, ... be such as in the proof of Theorem 5.5. Then we can choose n′_{k}, m′_{k} ∈ ℕ to satisfy the conditions analogous to 1) and 2) and such that n′_{1} = 1, m′_{1} = 1 and |a_{n′_{k}} − b_{m′_{k}}| < |a_{n_{k}} − b_{m_{k}}|/2 for k = 2, 3, .... Proceeding in the same way as in the proof of Theorem 5.5, we can define functions u′_{k} and consider the limit u′ of the sequence {u′_{k}}. Then

|{χ_{(a_{1};b_{1})} ≠ u′}| = |{u′_{1} ≠ u′}| < |{u_{1} ≠ u}| /2 = |{χ_{(a_{1};b_{1})} ≠ u}| .

Therefore |{u ≠ u′}| > 0. At the same time we again can prove that u′ is the phase shift of the pair (∑ α′_{i}δ_{a_{i}}, ∑ β′_{i}δ_{b_{i}}) for some positive α′_{i}, β′_{i}. Therefore u and u′ are different phase shifts of equivalent pairs of measures.

In terms of the families M_{φ}, we can say that if (μ, ν), μ, ν ∈ M_{φ} and (μ′, ν′), μ′, ν′ ∈ M_{φ} are equivalent pairs of measures and φ is a finite Blaschke product, then φ is equal to b ∘ φ for some Möbius transform b. However, if φ is not a finite Blaschke product then φ and φ can be completely different functions.

In fact, modifying the constructions from the proofs of Theorems 5.5 and 5.6 one can show that there exist infinitely many different phase shifts such that the corresponding pairs of measures are equivalent to (∑ α_{i}δ_{a_{i}}, ∑ β_{i}δ_{b_{i}}). It means that if A and B are two pure point operators whose sets of eigenvalues are such as in Theorems 5.5 or 5.6 then there exist infinitely many different (corresponding to nonequivalent vectors) rank one perturbations of A unitarily equivalent to B. However, it is absolutely unclear if such situation is possible in the case when A or B has a nontrivial singular continuous part.

Added in proof. Some of the results from this Section allow generalizations from the case of pure point operators to the case of arbitrary singular operators. These results are to be published elsewhere.

6. One example of the absence of the mixed spectrum

In this Section we will show that there exists a self-adjoint cyclic operator A such that for some cyclic vector ϕ the operators A + λ(·, ϕ)ϕ are singular continuous on [0; 1] for all 0 ≤ λ ≤ 1 and pure point for all other λ ∈ ℝ. As we discussed in the Introduction, this example shows that even when the sets of coupling constants λ corresponding to pure point operators A_{λ} and to singular continuous operators A_{λ} are big, we still may have no “mixed” spectrum for any λ.

Example 6.1.

We shall first construct the Krein spectral shift u.

Let {a_{n}} be a sequence of real numbers 0 < a_{n} < 1 monotonically decreasing to 0 and such that

(6.1)  ∏_{n=1}^{∞}(1 − a_{n}) = c > 0

and

(6.2)  1 − ∏_{k=n}^{∞}(1 − a_{k}) ≥ \frac{1}{n}.

Consider the Cantor set C corresponding to the sequence {a_{n}}: let

C_{0} = I_{0}^{0} = [0; 1], C_{1} = I_{1}^{1} ∪ I_{2}^{1}, ..., C_{n} = I_{1}^{n} ∪ ... ∪ I_{2^{n}}^{n}, ...

where

I_{2k}^{n} ∪ I_{2k−1}^{n} = I_{k}^{n−1} \ Δ_{k}^{n−1}

and Δ_{k}^{n} is the open interval placed in the center of the interval I_{k}^{n} such that |Δ_{k}^{n}| = a_{n}|I_{k}^{n}|. Put C = ⋂_{n=0}^{∞} C_{n}.

The Cantor set C has the following properties:

(6.3)  |C| = c

and

(6.4)  1 − \frac{|I_{k}^{n} ∩ C|}{|I_{k}^{n}|} = 1 − ∏_{n}^{∞}(1 − a_{k}) ≥ \frac{1}{n} ≥ \frac{ln 2}{− ln |I_{k}^{n}|}

for any n, k ∈ ℕ.

Define u = π on C and u = 0 elsewhere on ℝ. Denote U(z) = Ku(z) for z ∈ ℂ_{+}.

Claim. U has a finite nontangential derivative U′(x) = lim_{z⟶_{≯}x} \frac{U(x)−U(z)}{x−z} at x ∈ ℝ if and

only if x ∉ C.

Proof. Since u is locally constant on ℝ \ C, U′ obviously exists there.

Let intervals I_{k}^{n} be the same as in the construction of C above. The derivative U′ obviously does not exist on the set where the nontangential limit of U does not exist. Let x ∈ C be such that the nontangential limit U(x) of U exists. Let {I_{k_{n}}^{n}}_{n=1}^{∞} be the sequence of intervals containing x.

We will first prove that Pu(x) = π. Denote by x_{n} the middle of the interval I_{k_{n}}^{n} and put y_{n} = ϵ|I_{k_{n}}^{n}|. Then (6.4) imply that for any δ > 0 we can choose ϵ such that

lim sup_{n→∞} |π − Pu(x_{n} + iy_{n})| < δ

(recall that 1 − ∏_{n}^{∞}(1 − a_{n}) → 0). Since points x_{n} + iy_{n} approach x nontangentially and the limit Pu(x) exists, it has to be equal to π.

Now let us notice, that for y_{n} = |I_{k_{n}}^{n}| we have

|π − Pu(x_{n} + iy_{n})| > cπ (1 − \frac{|I_{k}^{n} ∩ C|}{|I_{k}^{n}|}) > \frac{d}{| ln y_{n}|}

for some c, d > 0 by (6.4). Thus U(x) − U(z) ≠ O(x − z) as z →_{≯} x and U′(x) does not

exist.▲

Let u be the phase shift of a pair (ν_{0}, ν). Let A = M_{ν_{0}}, where M_{ν_{0}} is the operator of multiplication by z in L^{2}(ν_{0}), and A_{λ} = A + λ(·, 1)1. Let ν_{λ} be the spectral measure of 1 for A_{λ} (then ν = ν_{1}).

By Lemma 3.2 ν_{0} is singular. Therefore all ν_{λ} are singular. Hence

|Kν_{λ}(z)| ⟶_{z⟶_{≯}x} ∞

for ν_{λ}-a. e. x. By formula (2.3) this means that

1 + πKν_{0} = exp U(z) ⟶_{z⟶_{≯}x} 1 − \frac{1}{λ}

for ν_{λ}-a. e. x. Thus by the definition of the phase shift for all λ ∈ (0; 1)

arg(1 + πKν_{0}(z)) = Pu(z) ⟶_{z⟶_{≯}x} π

for ν_{λ}-a. e. x. Thus all ν_{λ}, λ ∈ (0; 1) are concentrated on the set C. Since U′ does not exist on C and

π (Kν_{0})′ = (exp U − 1)′ = U′ exp U

ν_{λ}-a. e., (Kν_{0})′ does not exist on C (recall that

exp U(z) ⟶_{z⟶_{≯}x} 1 − 1/λ ≠ 0

ν_{λ}-a. e.). Hence the characteristic function φ defined in ℂ_{+} by formula (2.4) does not have a nontangential derivative on C. Thus by the Remark after Lemma 3.8 all ν_{λ} for λ ∈ (0; 1) are singular continuous. Similarly all ν_{λ} for λ ∈ ℝ \ [0; 1] are concentrated on ℝ \ C. Since U′ exists everywhere on ℝ \ C, μ_{λ} for λ ∈ ℝ \ [0; 1] are pure point.

To prove that ν_{0} is continuous, let us notice that since by Lemma 3.3

p. v. ∫_{ℝ} \frac{u(t + x)dt}{t} = ∞

for ν_{0}-a. e. x, ν_{0} is concentrated on C.

Let y ∈ C and let I_{n_{k}}^{k} be the sequence of intervals from the construction of C containing y. Then (6.4) implies that condition (3.8) is not satisfied at y. Indeed, if y is the right endpoint of a complementary interval, then y is the right endpoint of every I_{n_{k}}^{k} starting with some k. Therefore (6.4) implies that

∫_{(y;y+1)} \frac{(π − u(x))dx}{x − y} = ∞

and the integral in (3.8) is infinite. If y is not the right endpoint of a complementary interval, then there are infinitely many I_{n_{k}}^{k} such that y is in the right half of I_{n_{k}}^{k}. Then (3.8) can not be satisfied because

\frac{|I_{n_{k}}^{k} ∩ C|}{|I_{n_{k}}^{k}|} → 1

as k → ∞ and u = π on C. Thus ν_{0} can not have a point mass at y. Hence ν_{0} is continuous. Similarly, ν_{1} is continuous.

Remark. Let again Π be the set of those λ for which A_{λ} has nontrivial pure point part on [0; 1] and let Σ be the set of λ for which A_{λ} has nontrivial singular continuous part on [0; 1].

It is still unclear if there exist examples such that the sets Π and Σ are big (topologically or in measure) but Π ∩ Σ = ∅ when σ(A) contains the interval [0; 1]. Modifying the above example, we can obtain a dense in [0; 1] set C′ by inserting smaller Cantor sets into each complimentary interval I_{n}^{k}, then inserting smaller Cantor sets into each new complimentary interval and so on. If the size of these Cantor sets decreases to 0 fast enough then replacing C in the above example with C′ we will obtain an example of A and ϕ such that σ(A) = [0; 1], A + λ(·, ϕ)ϕ is continuous on [0; 1] for all λ ∈ [0; 1] and pure point for almost all λ ∈ ℝ \ [0; 1].

The case σ(A) ⊃ [0; 1] became especially interesting after it was shown in [Go] and [R-M-S] that σ(A) ⊃ [0; 1] implies Σ \ Π is a dense G_{δ}.

References

[A] N. Aronszajn, On a problem of Weyl in the theory of singular Sturm- Liouville equations, Amer. J. Math. 79 (1957), 597-610.

[A-C] P. Ahern and D. Clark, Radial limits and invariant subspaces, Amer. J. Math. 92 (1970), 332-

342.

[A-D] N. Aronszajn and W. Donoghue, On exponential representations of analytic functions in the

upper half-plane with positive imaginary part, J. Analyse Math. 5 (1956-1957), 321-388.

[A1] A. B. Aleksandrov, Multiplicity of boundary values of inner functions, Izv. Acad. Nauk. Arm. SSR, Matematica 22 5 (1987), 490-503. (Russian)

[A2] A. B. Aleksandrov, Inner functions and related spaces of pseudo continuable functions, Proceedings of LOMI seminars 170 (1989), 7-33. (Russian)

[A3] A. B. Aleksandrov, Private communications.

[C] D. Clark, One dimensional perturbations of restricted shifts, J. anal. math. 25 (1972), 169-91. [C-P] R. W. Carey and J. D. Pincus, Unitary equivalence modulo the trace class for self-adjoint

operators, Amer J. Math. 98 (1976), 481-514.

[D] W. Donoghue, On the perturbation of spectra, Comm. Pure Appl. Math. 18 (1965), 559-576.

[G] J. B. Garnett, Bounded analytic functions, Academic Press, New York, 1981.

[Go] A. Gordon, Pure point spectrum under 1-parameter perturbations and instability of Anderson localization, to appear in Commun. Math. Phys..

[H] J. S. Howland, Perturbation theory of dense point spectra, J. Func. An. 74 (1987), 52-80.

[K] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1966.

[Kr] M. G. Krein, Perturbation determinants and a formula for the traces of unitary and self-adjoint operators, Sov. Math. Dokl. 3 (1962), 707-710.

[Ku] S. T. Kuroda, On a Theorem of Weyl-von Neumann, Proc. Japan Acad. 34 (1958), 11-15. [M-P] M. Martin and M. Putinar, Lectures on Hyponormal operators, vol. 39, Operator Theory: Ad-

vances and Applications, 1989.

[P1] A. Poltoratski, On the boundary behavior of pseudocontinuable functions, St. Petersburg Math. J. 5 (1994), 389-406.

[P2] A. Poltoratski, On the distributions of boundary values of Cauchy integrals, Proc. Amer. Math. Soc. 124 No 8 (1996), 2455–2463.

[R-J-L-S] R. del Rio, S. Jitomirskaya, Y. Last and B. Simon, Operators with singular continuous spectrum, 4. Hausdorff dimension and rank one perturbations, to appear.

[R-M-S] R. del Rio, N. Makarov and B. Simon, Operators with singular continuous spectrum, II. Rank

one operators, to appear in Commun. Math. Phys..

[S] D. Sarason, Sub-Hardy Hilbert Spaces in the unit disk, The University of Arkansas lecture notes in the mathematical sciences; v. 10, J. Wiley and Sons, New York, 1994.

[Si1] B. Simon, Spectral analysis of rank one perturbations and applications, Proc. 1993 Vancouver Summer School in Mathematical Physics.

[Si2] B. Simon, Singular spectrum: recent results and open questions., to appear. [S-W] B. Simon, T. Wolff, Singular continuous spectrum under rank one perturbations and localization

for random Hamiltonians, Comm. Pure Appl. Math. 39 (1986), 75-90.

Department of Mathematics, MIT, 77 Massachusetts Ave., Cambridge, MA 02139 E-mail address: agp@math.mit.edu