Asymptotics of instability zones of Hill operators with a two term potential
type: Item
instance of: scholarly article
title: Asymptotics of instability zones of Hill operators with a two term potential
zbMATH Open document ID: 1063.34081
author: Plamen Djakov
author: Boris Mityagin
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2004-11-23T00:00:00Z
review text: The authors consider the Hill operator \[ Ly:=-y''+[a\text{cos}2x+b\text{cos}4x]y, \] with \(a,b\in\mathbb{R}\), \(ab\neq 0\), and establish some estimates on the lengths \(\gamma_{n}\) corresponding to the intervals of instability zones or spectral gaps of the above operator. Using the parametrization \(a^{2}+8bt^{2}=0\), \(b=-2\alpha ^ {2}\) , \(a=-\alpha t\), they obtain the asymptotic behavior of the gaps \(\gamma_{n}(\alpha)\) both for \(\alpha \rightarrow 0\) and \(n\rightarrow\infty\).
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 34L20
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 34B30
zbMATH DE Number: 2118262
zbMATH Keywords: Hill operator
zbMATH Keywords: boundary value problems
zbMATH Keywords: instability zones
zbMATH Keywords: Mathieu operator
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: length of spectral gaps
zbMATH Keywords: eigenvalues of a operator
zbMATH Keywords: Lyapunov function
reviewed by: Antonio Linero Bas
MaRDI profile type: Publication
full work available at URL: j.crma.2004.06.019
OpenAlex ID: W2024939593
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DOI: 10.1016/J.CRMA.2004.06.019
description: scientific article; zbMATH DE number 2118262