Destruction of Absolutely Continuous Spectrum by Perturbation Potentials of Bounded Variation

Last, Yoram

doi:10.1007/s00220-007-0264-9

Cited by 13 publications

(18 citation statements)

References 23 publications

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“…A notable result in this direction has been obtained by Kupin [22] who showed that the essential support of the a.c. spectrum of is still preserved if a decaying potential of a square-summable variation is added to it under an additional restriction that this perturbation lies in m for some m ∈ N. But for a perturbation of a bounded variation of a general Jacobi matrix, a guess even weaker than an analog of Conjecture 1 would be wrong. Indeed, one of us have recently constructed [24] an example of a Jacobi matrix J(a,b +b) with a = 1, lim n→∞bn = lim n→∞bn = 0, so that {b n } ∞ n=1 is of bounded variation, both J(a,b) and J(a,b) have purely a.c. spectrum on (−2, 2) with essential support (−2, 2), but J(a,b +b) has empty absolutely continuous spectrum. In particular, adding a decaying perturbation of bounded variation to a Jacobi matrix can fully "destroy" its absolutely continuous spectrum.…”

Section: Conjecturementioning

confidence: 98%

Preservation of a.c. spectrum for random decaying perturbations of square-summable high-order variation

Kaluzhny

Last

2011

Journal of Functional Analysis

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We consider random self-adjoint Jacobi matrices of the form (J ω u)(n) = a n (ω)u(n + 1) + b n (ω)u(n) + a n−1 (ω)u(n − 1) on 2 (N), where {a n (ω) > 0} and {b n (ω) ∈ R} are sequences of random variables on a probability space (Ω, dP (ω)) such that there exists q ∈ N, such that for any l ∈ N,are independent random variables of zero mean satisfying ∞ n=1 Ω β 2 n (ω) dP (ω) < ∞.Let J p be the deterministic periodic (of period q) Jacobi matrix whose coefficients are the mean values of the corresponding entries in J ω . We prove that for a.e. ω, the a.c. spectrum of the operator J ω equals to and fills the spectrum of J p . If, moreover,then for a.e. ω, the spectrum of J ω is purely absolutely continuous on the interior of the bands that make up the spectrum of J p .

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Section: Conjecturementioning

confidence: 98%

Preservation of a.c. spectrum for random decaying perturbations of square-summable high-order variation

Kaluzhny

Last

2011

Journal of Functional Analysis

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“…Both Stolz's paper and ours give stronger results than some of the results for diagonal perturbations of DSO, e.g., those in [3,4,23]. For instance, Golinskii and Nevai [4] (see also [17 |β n+T − β n | < ∞ and β n → 0 for some T ∈ N, and the unperturbed operator is a T -periodic DSO.…”

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confidence: 44%

Slowly oscillating perturbations of periodic Jacobi operators in l²(N)

Moszyński

2009

Studia Math.

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Abstract. We prove that the absolutely continuous part of the periodic Jacobi operator does not change (modulo unitary equivalence) under additive perturbations by compact Jacobi operators with weights and diagonals defined in terms of the Stolz classes of slowly oscillating sequences. This result substantially generalizes many previous results, e.g., the one which can be obtained directly by the abstract trace class perturbation theorem of Kato-Rosenblum. It also generalizes several results concerning perturbations of the discrete (free or periodic) Schrödinger operator. The paper concerns "one-sided" Jacobi operators (i.e. in l 2 (N)) and is based on the method of subordinacy. We provide some spectral results for the unperturbed, periodic case, and also an appendix containing some subordination theory tools. 0. Introduction. Searching for perturbations of a specific "unperturbed" operator which preserve a certain spectral property of that operator is a popular kind of problem in spectral analysis. In this paper, we consider periodic Jacobi operators in l 2 (N) as unperturbed operators, and we perturb them by adding other Jacobi operators. The main spectral properties studied relate to the absolutely continuous part of the operator.Similar problems concerning perturbations of the free or periodic discrete Schrödinger operator (DSO) in l 2 (N) or l 2 (Z) by diagonal operators have been studied in many papers (see e.g. [2-4, 13-17, 21-23]; see also the introduction and references to [17]).A famous abstract result on stability of the absolutely continuous part of an operator under perturbations is the Kato-Rosenblum theorem (see e.g. [20]). It states that the absolutely continuous parts of a self-adjoint operator and of its perturbation by an arbitrary trace class operator are unitarily equivalent. Obviously, this theorem can be directly applied to perturbations by any Jacobi operators with weights α = {α n } n≥1 and diagonals β = 2000 Mathematics Subject Classification: 47B36, 47B39, 47B25, 47A55, 47A10, 39A11.

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“…To prove this theorem, we will rely on a method from [16]. The sequence b n will be constructed out of two parts,…”

Section: Proof Of Theorem 15mentioning

confidence: 99%

“…We will denote the set of right limits of J by R. We are interested in the following conjecture from [1]. A narrower version of this conjecture, for a n ≡ 1 and b n → 0, was previously made by Last [16] and proven by Denisov [6]. Further work of Kaluzhny-Shamis [11] proved (1.3) in the case where the sequences {a n }, {b n } are asymptotically periodic (so there is, up to shifts, only one right limit).…”

Section: Introductionmentioning

confidence: 99%

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ℓ 2 Bounded Variation and Absolutely Continuous Spectrum of Jacobi Matrices

Lukić

2017

Commun. Math. Phys.

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Destruction of Absolutely Continuous Spectrum by Perturbation Potentials of Bounded Variation

Abstract: We show that absolutely continuous spectrum of one-dimensional Schrödinger operators may be destroyed by adding to them decaying perturbation potentials of bounded variation.

Cited by 13 publications

References 23 publications

Preservation of a.c. spectrum for random decaying perturbations of square-summable high-order variation

Preservation of a.c. spectrum for random decaying perturbations of square-summable high-order variation

Slowly oscillating perturbations of periodic Jacobi operators in l²(N)

ℓ 2 Bounded Variation and Absolutely Continuous Spectrum of Jacobi Matrices

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Destruction of Absolutely Continuous Spectrum by Perturbation Potentials of Bounded Variation

Abstract: We show that absolutely continuous spectrum of one-dimensional Schrödinger operators may be destroyed by adding to them decaying perturbation potentials of bounded variation.

Cited by 13 publications

References 23 publications

Preservation of a.c. spectrum for random decaying perturbations of square-summable high-order variation

Preservation of a.c. spectrum for random decaying perturbations of square-summable high-order variation

Slowly oscillating perturbations of periodic Jacobi operators in l2(N)

ℓ 2 Bounded Variation and Absolutely Continuous Spectrum of Jacobi Matrices

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Slowly oscillating perturbations of periodic Jacobi operators in l²(N)