Perturbations of periodic Sturm–Liouville operators

Jussi Behrndt; Philipp Schmitz; Gerald Teschl; Carsten Trunk

doi:10.1016/j.aim.2023.109022

Perturbations of periodic Sturm–Liouville operators

Jussi Behrndt^*, Philipp Schmitz, Gerald Teschl, Carsten Trunk

^*Corresponding author for this work

Institute of Applied Mathematics (5040)

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

Abstract

We study perturbations of the self-adjoint periodic Sturm–Liouville operator A₀=[Formula presented](−[Formula presented]p₀[Formula presented]+q₀) and conclude under L¹-assumptions on the differences of the coefficients that the essential spectrum and absolutely continuous spectrum remain the same. If a finite first moment condition holds for the differences of the coefficients, then at most finitely many eigenvalues appear in the spectral gaps. This observation extends a seminal result by Rofe-Beketov from the 1960s. Finally, imposing a second moment condition we show that the band edges are no eigenvalues of the perturbed operator.

Original language	English
Article number	109022
Journal	Advances in Mathematics
Volume	422
DOIs	https://doi.org/10.1016/j.aim.2023.109022
Publication status	Published - 1 Jun 2023

Keywords

Absolutely continuous spectrum
Discrete eigenvalues
Essential spectrum
Periodic Sturm–Liouville operators
Perturbations
Spectral gaps

ASJC Scopus subject areas

Mathematics(all)

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10.1016/j.aim.2023.109022Licence: CC BY 4.0

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title = "Perturbations of periodic Sturm–Liouville operators",

abstract = "We study perturbations of the self-adjoint periodic Sturm–Liouville operator A0=[Formula presented](−[Formula presented]p0[Formula presented]+q0) and conclude under L1-assumptions on the differences of the coefficients that the essential spectrum and absolutely continuous spectrum remain the same. If a finite first moment condition holds for the differences of the coefficients, then at most finitely many eigenvalues appear in the spectral gaps. This observation extends a seminal result by Rofe-Beketov from the 1960s. Finally, imposing a second moment condition we show that the band edges are no eigenvalues of the perturbed operator.",

keywords = "Absolutely continuous spectrum, Discrete eigenvalues, Essential spectrum, Periodic Sturm–Liouville operators, Perturbations, Spectral gaps",

author = "Jussi Behrndt and Philipp Schmitz and Gerald Teschl and Carsten Trunk",

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T1 - Perturbations of periodic Sturm–Liouville operators

AU - Behrndt, Jussi

AU - Schmitz, Philipp

AU - Teschl, Gerald

AU - Trunk, Carsten

PY - 2023/6/1

Y1 - 2023/6/1

N2 - We study perturbations of the self-adjoint periodic Sturm–Liouville operator A0=[Formula presented](−[Formula presented]p0[Formula presented]+q0) and conclude under L1-assumptions on the differences of the coefficients that the essential spectrum and absolutely continuous spectrum remain the same. If a finite first moment condition holds for the differences of the coefficients, then at most finitely many eigenvalues appear in the spectral gaps. This observation extends a seminal result by Rofe-Beketov from the 1960s. Finally, imposing a second moment condition we show that the band edges are no eigenvalues of the perturbed operator.

AB - We study perturbations of the self-adjoint periodic Sturm–Liouville operator A0=[Formula presented](−[Formula presented]p0[Formula presented]+q0) and conclude under L1-assumptions on the differences of the coefficients that the essential spectrum and absolutely continuous spectrum remain the same. If a finite first moment condition holds for the differences of the coefficients, then at most finitely many eigenvalues appear in the spectral gaps. This observation extends a seminal result by Rofe-Beketov from the 1960s. Finally, imposing a second moment condition we show that the band edges are no eigenvalues of the perturbed operator.

KW - Absolutely continuous spectrum

KW - Discrete eigenvalues

KW - Essential spectrum

KW - Periodic Sturm–Liouville operators

KW - Perturbations

KW - Spectral gaps

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DO - 10.1016/j.aim.2023.109022

M3 - Article

AN - SCOPUS:85153604901

SN - 0001-8708

VL - 422

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 109022

ER -

Perturbations of periodic Sturm–Liouville operators

Abstract

Keywords

ASJC Scopus subject areas

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