On double eigenvalues of Hill's operator
type: Item
instance of: scholarly article
title: On double eigenvalues of Hill's operator
zbMATH Open document ID: 0697.47050
DOI: 10.1016/0022-1236(89)90067-0
author: Yves Colin de Verdière
author: T. Kappeler
published in: Journal of Functional Analysis
publication date: 1989-01-01T00:00:00Z
review text: The author considers Hill's equation \[ -y''+p(x)y=\lambda y \] with periodic boundary conditions \(y(x+2\pi)=y(x)\), where the potential p is in \(L^ 2(<0,\pi >)\), extended periodically to all of \({\mathbb{R}}\). Let \(\{\lambda_ k\}\), \(\lambda_ 0<\lambda_ 1\leq \lambda_ 2<\lambda_ 3\leq...\), be the periodic spectrum of p. \textit{S. P. Novikov} [Funct. Anal. Appl. 8(1974), 236-246 (1975; Zbl 0299.35017)] conjectured that the set of potentials p in \(L^ 2(<0,\pi >)\) with periodic spectrum having finitely many simple periodic eigenvalues is dense in the norm topology of \(L^ 2(<0,\pi >)\). This result has been proved by various authors, e.g. \textit{V. A. Marchenko} [Sturm Liouville Operators and Applications, Basel (1986; Zbl 0592.34011)], \textit{B. M. Levitan} [Math. USSR-Izv. 20, 55-87 (1983; Zbl 0517.34037)], \textit{J. Garnett} and \textit{E. Trubowitz} [Comment. Math. Helv. 62, 18-37 (1987; Zbl 0649.34034)]. All of them use the inverse spectral theory. In this paper a new proof is given without using the inverse spectral theory.
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 34G10
zbMATH DE Number: 4142848
zbMATH Keywords: eigenvalues of Hill's operator
zbMATH Keywords: Hill's equation
zbMATH Keywords: periodic boundary conditions
reviewed by: Svatoslav Staněk
MaRDI profile type: Publication
full work available at URL: 0022-1236(89)90067-0
OpenAlex ID: W2060758719
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description: scientific article; zbMATH DE number 4142848