Estimates for the eigenvalues of Hill's equation and applications for the eigenvalues of the Laplacian on toroidal surfaces
type: Item
instance of: scholarly article
title: Estimates for the eigenvalues of Hill's equation and applications for the eigenvalues of the Laplacian on toroidal surfaces
zbMATH Open document ID: 0733.34080
DOI: 10.1007/BF02568765
author: Brigitte Beekmann
author: Hermann Lökes
published in: Manuscripta Mathematica
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155528
review text: Suppose p,q,s are real, continuous, periodic functions with period \(L>0\); in addition, assume that p,s are positive and \(p'\) is continuous. Hill's equation and associated periodic eigenvalue problem is: \[ (p(x)y'(x))'+(\lambda s(x)-q(x))y(x)=0;\quad y(0)=y(L);\quad y'(0)=y'(L). \] It is well known that the eigenvalues \(\{\lambda_ n\}\) satisfy \(\lambda_ 0<\lambda_ 1\leq \lambda_ 2<\lambda_ 3\leq \lambda_ 4<...<\lambda_{2n-1}\leq \lambda_{2n}<\lambda_{2n+1}...\); [see \textit{M. S. P. Eastham}, The spectral theory of periodic differential equations (1973; Zbl 0287.34016)]. The authors give bounds for these eigenvalues based on the Sturm comparison theorem and a result of \textit{O. Haupt} [Math. Ann. 76, 67-104 (1914)]. More specifically, they show that \[ \min_{x\in [0,L]}B_ n(x)\leq \lambda_{2n-1}\leq \lambda_{2n}\leq \max_{x\in [0,L]}B_ n(x), \] where \(B_ n\) is the estimating function defined by \[ B_ n(x)=\frac{p(x)}{q(x)}((\frac{2\pi}{L})^ 2 n^ 2+\frac{q(x)}{p(x)}-\frac{S(P^*)}{2}), \] \(P^*\) is any primitive of p and where \(S(P^*)\) is the Schwarz derivative of \(P^*\). They follow this result with some refinements and apply their results to estimate the eigenvalues of the Laplacian on a surface of revolution homeomorphic to the torus. Many authors have given either upper bounds or lower bounds for the eigenvalues; for example, under certain restrictions on q, Eastham gives the upper bound \[ \lambda_{2n}\leq \frac{\sup (p(x))}{\inf (s(x))}(\frac{2n\pi}{L})^ 2. \] It is interesting and impressive that these authors have been able to give both global upper and lower bounds for these eigenvalues. It appears that their estimates are very good for certain Laplacian problems. This reviewer would like to see further work and a follow-up paper by the authors in order to see further comparisons made between their estimates and known numerical techniques to estimate eigenvalues (i.e. from the NAG library) of various periodic boundary value problems associated…
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34B30
zbMATH DE Number: 4212545
zbMATH Keywords: toroidal surfaces
zbMATH Keywords: Hill's equation
zbMATH Keywords: periodic eigenvalue problem
zbMATH Keywords: Sturm comparison theorem
zbMATH Keywords: estimating function
zbMATH Keywords: Schwarz derivative
zbMATH Keywords: eigenvalues of the Laplacian
zbMATH Keywords: global upper and lower bounds
describes a project that uses: NAG
MaRDI profile type: Publication
Wikidata QID: Q125371070
cites work: Q3787077
cites work: Eigenfunctions and eigenvalues on surfaces of revolution
cites work: Le spectre d'une variété riemannienne. (The spectrum of a Riemannian manifold)
cites work: Constructing Isospectral Manifolds
cites work: Spektrale Starrheit gewisser Drehflächen
cites work: Q3344779
cites work: Eigenfunctions and nodal sets
cites work: Q5848416
cites work: Q4774804
cites work: Asymptotic estimates for the Sturm-Liouville spectrum
cites work: Q5540675
cites work: Q3342966
cites work: Q3962217
Recommended article: Q4202107
Recommended article: Eigenvalue estimates for hypersurfaces in \(\mathbb H^{m} \times \mathbb R\) and applications
Recommended article: Approximate formulas for eigenvalues of the Laplace operator on a torus arising in linear problems with oscillating coefficients
Recommended article: Equidistribution of toral eigenfunctions along hypersurfaces
Recommended article: A note on eigenvalue asymptotics for Hill's equation
Recommended article: Eigenvalue estimates and trace formulas for the matrix Hill's equation
Recommended article: Q5453782
Recommended article: On eigenvalue problem of the Laplace operators for ellipsoidal areas
Recommended article: Eigenvalue estimates with applications to minimal surfaces
Recommended article: Eigenvalue estimates for some natural elliptic operators on hypersurfaces
description: scientific article; zbMATH DE number 4212545