Spectral properties of a limit-periodic Schrödinger operator in dimension two

Karpeshina, Yulia; Lee, Young-Ran

doi:10.1007/s11854-013-0014-1

Cited by 11 publications

(45 citation statements)

References 24 publications

(91 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…2.1.2. Description of methods: To prove the above results in [25], the authors considered the sequence of operators:…”

Section: Prior Resultsmentioning

confidence: 99%

“…The sets G n,λ and G ∞,λ are also bounded. It is shown in [25] that the measure of the symmetric difference of the two sets G ∞,λ and G n,λ converges to zero as n → ∞, uniformly in λ in every bounded interval: lim n→∞ |G ∞,λ ∆ G n,λ | = 0.…”

Section: )mentioning

confidence: 99%

“…Next, we consider the sequence of operators T n (G ∞,λ ) which are given by (2.18) and act from L 2 (R 2 ) to L 2 (G ∞,λ ). It is proven in [25] that the sequence T n (G ∞,λ ) has a strong limit T ∞ (G ∞,λ ). The operator T ∞ (G ∞,λ ) satisfies T ∞ ≤ 1 and can be described by the formula (T ∞ F )( k) = 1 2π F, Ψ ∞ ( k) for any F ∈ C c (R 2 ).…”

Section: )mentioning

confidence: 99%

“…It is proven in [25] that the sequence of operators S n (G ∞,λ ) has a strong limit S ∞ (G ∞,λ ). It follows T * ∞ (G ∞,λ ) = S ∞ (G ∞,λ ).…”

Section: )mentioning

confidence: 99%

See 3 more Smart Citations

Ballistic Transport for the Schrödinger Operator with Limit-Periodic or Quasi-Periodic Potential in Dimension Two

Karpeshina

Lee

Shterenberg

et al. 2017

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

Abstract. We prove the existence of ballistic transport for the Schrödinger operator with limit-periodic or quasi-periodic potential in dimension two. This is done under certain regularity assumptions on the potential which have been used in prior work to establish the existence of an absolutely continuous component and other spectral properties. The latter include detailed information on the structure of generalized eigenvalues and eigenfunctions. These allow to establish the crucial ballistic lower bound through integration by parts on an appropriate extension of a Cantor set in momentum space, as well as through stationary phase arguments.

show abstract

“…2.1.2. Description of methods: To prove the above results in [25], the authors considered the sequence of operators:…”

Section: Prior Resultsmentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

“…It is proven in [25] that the sequence of operators S n (G ∞,λ ) has a strong limit S ∞ (G ∞,λ ). It follows T * ∞ (G ∞,λ ) = S ∞ (G ∞,λ ).…”

Section: )mentioning

confidence: 99%

See 2 more Smart Citations

Ballistic Transport for the Schrödinger Operator with Limit-Periodic or Quasi-Periodic Potential in Dimension Two

Karpeshina

Lee

Shterenberg

et al. 2017

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

show abstract

“…For the Schrödinger operators with a periodic potential this conjecture was proved for various dimensions and under various assumptions for the potential in works [1]- [6]. For the magnetic Schrödinger operator this conjecture was established in papers [7], [8]. In works [9]- [11], a polyharmonic operator was considered with various perturbations.…”

Section: Introductionmentioning

confidence: 99%

On spectral gaps of a Laplacian in a strip with a bounded periodic perturbation

Борисов¹

2018

Ufimskii Matematicheskii Zhurnal

View full text Add to dashboard Cite

Abstract. In the work we consider the Laplacian subject to the Dirichlet condition in an infinite planar strip perturbed by a periodic operator. The perturbation is introduced as an arbitrary bounded periodic operator in 2 on the periodicity cell; then this operator is extended periodically on the entire strip.We study the band spectrum of such operator. The main obtained result is the absence of the spectral gaps in the lower part of the spectrum for a sufficiently small potential. The upper bound for the period ensuring such result is written explicitly as a certain number. It also involves a certain characteristics of the perturbing operator, which can be nonrigorously described as "the maximal oscillation of the perturbation". We also explicitly write out the length of the part of the spectrum, in which the absence of the gaps is guaranteed. Such result can be regarded as a partial proof of the strong Bethe-Sommerfeld conjecture on absence of internal gaps in the band spectra of periodic operators for sufficiently small periods.

show abstract

Thin spectra and singular continuous spectral measures for limit‐periodic Jacobi matrices

Damanik

Fillman

Wang

2023

Mathematische Nachrichten

View full text Add to dashboard Cite

This paper investigates the spectral properties of Jacobi matrices with limitperiodic coefficients. We show that generically the spectrum is a Cantor set of zero Lebesgue measure, and the spectral measures are purely singular continuous. For a dense set of limit-periodic Jacobi matrices, we show that the spectrum is a Cantor set of zero lower box counting dimension while still retaining the singular continuity of the spectral type. We also show how results of this nature can be established by fixing the off-diagonal coefficients and varying only the diagonal coefficients, and, in a more restricted version, by fixing the diagonal coefficients to be zero and varying only the off-diagonal coefficients. We apply these results to produce examples of weighted Laplacians on the multidimensional integer lattice having purely singular continuous spectral type and zero-dimensional spectrum.

show abstract

Spectral properties of a limit-periodic Schrödinger operator in dimension two

Cited by 11 publications

References 24 publications

Ballistic Transport for the Schrödinger Operator with Limit-Periodic or Quasi-Periodic Potential in Dimension Two

Ballistic Transport for the Schrödinger Operator with Limit-Periodic or Quasi-Periodic Potential in Dimension Two

On spectral gaps of a Laplacian in a strip with a bounded periodic perturbation

Thin spectra and singular continuous spectral measures for limit‐periodic Jacobi matrices

Contact Info

Product

Resources

About