On the instability intervals of the Mathieu--Hill operator
type: Item
instance of: scholarly article
title: On the instability intervals of the Mathieu--Hill operator
zbMATH Open document ID: 1173.47028
published in: Letters in Mathematical Physics
publication date: 2009-04-20T00:00:00Z
review text: The author obtains the precise asymptotic formulas for the widths \(\gamma _{k}\) of the instability intervals of the operator \(L\) considered as the Mathieu--Hill operator defined by \(Ly=-y''+(2h\cos 2x)y,\) \(-\infty <x<+\infty \) in \(L_{2}(\mathbb R)\), where \(h\in\mathbb R\setminus\{0\}\). The formula states the isolated terms of arbitrary number in the asymptotics of the sequence \(\gamma_{k}\) for large \(k\) and verifies the results of \textit{E.\,M.\thinspace Harrell} [in: Contributions to analysis and geometry, Suppl.\ Am.\ J.\ Math., 139--150 (1981; Zbl 0544.34014)] and \textit{J.\,Avron} and \textit{B.\,Simon} [Ann.\ Phys.\ 134, 76--84 (1981; Zbl 0464.34020)].
reviewed by: Bilender P. Allahverdiev
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 34L20
Mathematics Subject Classification ID: 34B30
Mathematics Subject Classification ID: 34L40
zbMATH DE Number: 5544972
zbMATH Keywords: asymptotic formulas
zbMATH Keywords: Mathieu-Hill operator
zbMATH Keywords: instability interval
author: Alp Arslan Kiraç
MaRDI profile type: Publication
full work available at URL: s11005-007-0215-6
OpenAlex ID: W2079139720
cites work: Q5732789
cites work: The asymptotics of the gap in the Mathieu equation
cites work: Q5848416
cites work: On the Riesz basisness of the root functions of the nonself-adjoint Sturm-Liouville operator
cites work: Q4100368
cites work: Q3333352
cites work: Asymptotic estimates for the Sturm-Liouville spectrum
cites work: Estimates on the Stability Intervals for Hill's Equation
cites work: On the Width of the Instability Intervals of the Mathieu Equation
cites work: Q5540675
cites work: Q4185363
cites work: The inverse problem for periodic potentials
cites work: The spectral expansion for a nonself-adjoint Hill operator with a locally integrable potential
cites work: On the nonself-adjoint differential operators with the quasiperiodic boundary conditions
DOI: 10.1007/S11005-007-0215-6
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description: scientific article; zbMATH DE number 5544972