Zeroes of the spectral density of the Schrödinger operator with the slowly decaying Wigner–von Neumann potential

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Zeroes of the spectral density of the Schrödinger operator with the slowly decaying Wigner–von Neumann potential. / Simonov, Sergey.

In: Mathematische Zeitschrift, Vol. 284, No. 1-2, 01.10.2016, p. 335-411.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{450d7b77a4764d339c3839fe6c66c9ea,

title = "Zeroes of the spectral density of the Schr{\"o}dinger operator with the slowly decaying Wigner–von Neumann potential",

abstract = "We consider the Schr{\"o}dinger operator Lα on the half-line with a periodic background potential and a perturbation which consists of two parts: a summable potential and the slowly decaying Wigner–von Neumann potential csin(2ωx+δ)xγ, where γ∈(12,1). The continuous spectrum of this operator has the same band-gap structure as the continuous spectrum of the unperturbed periodic operator. In every band there exist two points, called critical, where the eigenfunction equation has square summable solutions. Every critical point νc r is an eigenvalue of the operator Lα for some value of the boundary parameter α= αc r, specific to that particular point. We prove that for α≠ αc r the spectral density of the operator Lα has a zero of the exponential type at νc r.",

keywords = "Multiple scale method, Schr{\"o}dinger operator, Spectral density, Titchmarsh–Weyl theory, Wigner–von Neumann potential",

author = "Sergey Simonov",

year = "2016",

month = oct,

day = "1",

doi = "10.1007/s00209-016-1659-0",

language = "English",

volume = "284",

pages = "335--411",

journal = "Mathematische Zeitschrift",

issn = "0025-5874",

publisher = "Springer Nature",

number = "1-2",

}

RIS

TY - JOUR

T1 - Zeroes of the spectral density of the Schrödinger operator with the slowly decaying Wigner–von Neumann potential

AU - Simonov, Sergey

PY - 2016/10/1

Y1 - 2016/10/1

N2 - We consider the Schrödinger operator Lα on the half-line with a periodic background potential and a perturbation which consists of two parts: a summable potential and the slowly decaying Wigner–von Neumann potential csin(2ωx+δ)xγ, where γ∈(12,1). The continuous spectrum of this operator has the same band-gap structure as the continuous spectrum of the unperturbed periodic operator. In every band there exist two points, called critical, where the eigenfunction equation has square summable solutions. Every critical point νc r is an eigenvalue of the operator Lα for some value of the boundary parameter α= αc r, specific to that particular point. We prove that for α≠ αc r the spectral density of the operator Lα has a zero of the exponential type at νc r.

AB - We consider the Schrödinger operator Lα on the half-line with a periodic background potential and a perturbation which consists of two parts: a summable potential and the slowly decaying Wigner–von Neumann potential csin(2ωx+δ)xγ, where γ∈(12,1). The continuous spectrum of this operator has the same band-gap structure as the continuous spectrum of the unperturbed periodic operator. In every band there exist two points, called critical, where the eigenfunction equation has square summable solutions. Every critical point νc r is an eigenvalue of the operator Lα for some value of the boundary parameter α= αc r, specific to that particular point. We prove that for α≠ αc r the spectral density of the operator Lα has a zero of the exponential type at νc r.

KW - Multiple scale method

KW - Schrödinger operator

KW - Spectral density

KW - Titchmarsh–Weyl theory

KW - Wigner–von Neumann potential

UR - http://www.scopus.com/inward/record.url?scp=84964048369&partnerID=8YFLogxK

U2 - 10.1007/s00209-016-1659-0

DO - 10.1007/s00209-016-1659-0

M3 - Article

AN - SCOPUS:84964048369

VL - 284

SP - 335

EP - 411

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 1-2

ER -

ID: 9365999