A note on eigenvalue asymptotics for Hill's equation
type: Item
instance of: scholarly article
title: A note on eigenvalue asymptotics for Hill's equation
zbMATH Open document ID: 1205.34119
author: J. Martínez
published in: Applied Mathematics Letters
publication date: 2010-08-18T00:00:00Z
review text: The nonlinear eigenvalue problem for a Hill's type equation is considered with respect to periodic and nonperiodic boundary conditions. Asymptotic approximations of the periodic and antiperiodic eigenvalues are given up to an order of \(o(n^{-1})\). The theorem of \textit{V. A. Amburzamyan} [Z. f. Physik 53, 690--695 (1929; JFM 55.0868.01)] concerning the reconstruction of the potential from the spectrum of a Sturm-Liouville problem with Neumann boundary conditions on a finite interval can be considered as the starting point for inverse spectral theory. Later, the theorem was extended to more general boundary conditions. In 1997, \textit{H. H. Chern} and \textit{C. L. Shen} [Inverse Probl. 13, No.~1, 15--18 (1997; Zbl 0866.34020)] presented a proof for a vectorial version of the theorem. A short proof of the Ambazumyan theorem for periodic and anti-periodic Sturm-Liouville problems is given here by applying the obtained asymptotic estimates.
reviewed by: Ioana Dragomirescu
Mathematics Subject Classification ID: 34L25
Mathematics Subject Classification ID: 34L20
Mathematics Subject Classification ID: 34B09
zbMATH DE Number: 5772529
zbMATH Keywords: Hill's equation
zbMATH Keywords: periodic and anti-periodic eigenvalues
zbMATH Keywords: eigenvalue asymptotics
zbMATH Keywords: Ambarzumyan theorem
MaRDI profile type: Publication
full work available at URL: j.aml.2010.04.028
OpenAlex ID: W1987199528
cites work: On the Nature of the Spectrum of Singular Second Order Linear Differential Equations
cites work: Q5540675
cites work: Estimates for the periodic and semi-periodic eigenvalues of Hill's equation
cites work: Ambarzumyan's theorems for vectorial Sturm-Liouville systems with coupled boundary conditions
cites work: On the n -dimensional Ambarzumyan's theorem
cites work: On the determination of a Hill's equation from its spectrum
cites work: On the well-posedness of the inverse nodal problem
cites work: Method of spectral mappings in the inverse problem theory
DOI: 10.1016/J.AML.2010.04.028
description: scientific article; zbMATH DE number 5772529