Estimates for the Hill operator. I
type: Item
instance of: scholarly article
title: Estimates for the Hill operator. I
zbMATH Open document ID: 0954.34073
published in: Journal of Differential Equations
publication date: 2001-01-23T00:00:00Z
review text: The author studies one-dimensional Schrödinger operators \(-d^2/dx^2 + q(x)\) with periodic potentials \(q(x)=q(x+1)\), normalized by \(\int_0^1 q(x) dx = 0\). He proves various estimates involving the \(L_2(0,1)\) norm of the potential and some spectral data. Typical of his results is the following: Let \(\gamma = (\gamma_n)\) be a sequence of the gap lengths, and let \(\|\gamma\|\) be the \(\ell_2\) norm of this sequence. Then the inequalities \(\|\gamma\|\leq 6 \|q\|(1+\|q\|^{1/3})\) and \(\|q\|\leq 4 \|\gamma\|(1+ \|\gamma \|^{1/3})\) hold.
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 34B30
zbMATH DE Number: 1437337
zbMATH Keywords: periodic Schrödinger operator
zbMATH Keywords: Hill operator
author: E. Korotyaev
reviewed by: Christian Remling
MaRDI profile type: Publication
full work available at URL: jdeq.1999.3684
OpenAlex ID: W4210437089
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cites work: Sturm-Liouville operators and applications. Transl. from the Russian by A. Iacob
cites work: A CHARACTERIZATION OF THE SPECTRUM OF HILL'S OPERATOR
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DOI: 10.1006/JDEQ.1999.3684
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description: scientific article; zbMATH DE number 1437337