Multidimensional Schrödinger operators whose spectrum features a half-line and a Cantor set

Damanik, David; Fillman, Jake; Gorodetski, Anton

doi:10.1016/j.jfa.2020.108911

Cited by 5 publications

(5 citation statements)

References 39 publications

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“…Proof. It was shown in [8] that Σ 1/2 λ is eventually (A, a, τ )-thick for suitable A, a, τ . Since x → x 2s is in TRV, the result follows immediately from Theorem 1.6.…”

Section: Definitions and Resultsmentioning

confidence: 99%

“…Clearly the latter implies the former, so we focus on proving Theorem 1.7. We follow the strategy from [8].…”

Section: Proofs Of Main Theoremsmentioning

confidence: 99%

“…In the present work, we will demonstrate classes of functions and sets for which the result of [8] holds. We give a class of examples for which one can observe a sharp phase transition between containing and not-containing a half-line.…”

mentioning

confidence: 97%

“…In the recent work [8], it was shown that if V j (x) denotes a suitable locally constant version of the Fibonacci potential, then σ(L V ) contains a half-line for each d ≥ 2. One of the key ingredients of the proof was an affirmative answer to the main question for the function g(x) = x 2 and Cantor sets sufficiently thick in the sense of Newhouse [25,26].…”

mentioning

confidence: 99%

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On Sums of Semibounded Cantor Sets

Fillman¹,

Tidwell²

2022

Preprint

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Motivated by questions arising in the study of the spectral theory of models of aperiodic order, we investigate sums of functions of semibounded closed subsets of the real line. We show that under suitable thickness assumptions on the sets and growth assumptions on the functions, the sums of such sets contain half-lines. We also give examples to show our criteria are sharp in suitable regimes. Contents 1. Introduction 1 2. Functions of Bounded Relative Variation 6 3. Proofs of Main Theorems 10 4. Examples Not Containing Half-Lines 13 References 16

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“…Proof. It was shown in [8] that Σ 1/2 λ is eventually (A, a, τ )-thick for suitable A, a, τ . Since x → x 2s is in TRV, the result follows immediately from Theorem 1.6.…”

Section: Definitions and Resultsmentioning

confidence: 99%

“…Clearly the latter implies the former, so we focus on proving Theorem 1.7. We follow the strategy from [8].…”

Section: Proofs Of Main Theoremsmentioning

confidence: 99%

mentioning

confidence: 97%

mentioning

confidence: 99%

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On Sums of Semibounded Cantor Sets

Fillman¹,

Tidwell²

2022

Preprint

Self Cite

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show abstract

“…Moreover, examples exist for which the spectrum has Lebesgue measure zero and even arbitrarily small but positive Hausdorff dimension (see e.g. [DFG21]). Despite the potentially wild behavior of the spectrum, our results show that, at high energy, the spectrum wants to be absolutely continuous.…”

Section: Introductionmentioning

confidence: 99%

Classical Wave methods and modern gauge transforms: Spectral Asymptotics in the one dimensional case

Galkowski¹,

Parnovski²,

Shterenberg³

2022

Preprint

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In this article, we consider the asymptotic behaviour of the spectral function of Schrödinger operators on the real line. Let H : L 2 (R) → L 2 (R) have the formwhere V is a formally self-adjoint first order differential operator with smooth coefficients, bounded with all derivatives. We show that the kernel of the spectral projector, 1 (−∞,ρ 2 ] (H), has a complete asymptotic expansion in powers of ρ. This settles the 1-dimensional case of a conjecture made by the last two authors.

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Classical wave methods and modern gauge transforms: spectral asymptotics in the one dimensional case

Galkowski,

Parnovski,

Shterenberg

2023

Geom. Funct. Anal.

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In this article, we consider the asymptotic behaviour of the spectral function of Schrödinger operators on the real line. Let $H: L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ have the form $$ H:=-\frac{d^{2}}{dx^{2}}+Q, $$ where Q is a formally self-adjoint first order differential operator with smooth coefficients, bounded with all derivatives. We show that the kernel of the spectral projector, ${1}_{(-\infty ,\rho ^{2}]}(H)$, has a complete asymptotic expansion in powers of ρ. This settles the 1-dimensional case of a conjecture made by the last two authors.

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Multidimensional Schrödinger operators whose spectrum features a half-line and a Cantor set

Cited by 5 publications

References 39 publications

On Sums of Semibounded Cantor Sets

On Sums of Semibounded Cantor Sets

Classical Wave methods and modern gauge transforms: Spectral Asymptotics in the one dimensional case

Classical wave methods and modern gauge transforms: spectral asymptotics in the one dimensional case

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