Indefinite elliptic spectral problems
type: Item
instance of: scholarly article
title: Indefinite elliptic spectral problems
zbMATH Open document ID: 0943.34067
DOI: 10.1007/BF02677520
published in: Siberian Mathematical Journal
publication date: 2000-04-13T00:00:00Z
review text: The author considers the spectral problem of the form \[ L u = \lambda B u, \qquad x \in G \subset\mathbb{R}^n, \tag{1} \] \[ B_j u |_{\Gamma} = 0, \quad j= \overline{1,m}, \tag{2} \] where \(L\) is a selfadjoint uniformly elliptic operator of order \(2m\); \(\{B_j\}\) is a normal system of boundary operators covering \(L\) on \(\Gamma\); \(Bu\) is an operator \(g(x)u\); and \(g(x)\) is a real-valued measurable function assuming positive and negative values on \(G\). Let the sets \(G^+=\{x\in G:g(x) >0\}\) and \(G^-=\{x\in G:g(x) <0\}\) be open and let \(L_{2,g}(G^+ \cup G^-)\) be the space of measurable functions \(u(x)\) on \(G^+ \cup G^-\) such that \(u |g|^{1/2} \in L_{2}(G^+ \cup G^-)\). The author exhibits conditions which ensure that the eigenfunctions and associated functions to problem (1), (2) are dense in the space \(L_{2,g}(G^+ \cup G^-)\) and constitute an unconditional basis for this space. Similar conditions in terms of interpolation equalities were stated in the earlier article by the author [Sib. Adv. Math. 4, No. 2, 87-121 (1994; Zbl 0851.35100)]. Here, sufficient conditions on the function \(g(x) \) and the domain \(G\) are given ensuring the basis property.
Mathematics Subject Classification ID: 34L10
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 35P05
zbMATH DE Number: 1431677
zbMATH Keywords: indefinite spectral problem
zbMATH Keywords: basis property for the eigenelements and associated functions
zbMATH Keywords: sufficient conditions
author: S. G. Pyatkov
MaRDI profile type: Publication
cites work: Q5817707
cites work: Asymptotic analysis of a singular Sturm-Liouville boundary value problem
cites work: Eigenvalues of Elliptic Boundary Value Problems With an Indefinite Weight Function
cites work: Remainder estimates for the asymptotics of elliptic eigenvalue problems with indefinite weights
cites work: Q5185439
cites work: Partial-range completeness and existence of solutions to two-way diffusion equations
cites work: Half-range completeness for the Fokker-Planck equation. I
cites work: Q3967870
cites work: Full- and partial-range eigenfunction expansions for Sturm-Liouville problems with indefinite weights
cites work: Spectral methods in linear transport theory
cites work: Half-range solutions of indefinite Sturm-Liouville problems
cites work: On the relative completeness of the generalized eigenvectors of elliptic eigenvalue problems with indefinite weight functions
cites work: Elliptic Problems Involving an Indefinite Weight
cites work: Non-selfadjoint elliptic problems involving an ndeefinite weight
cites work: A Krein space approach to symmetric ordinary differential operators with an indefinite weigth function
cites work: Q3475718
cites work: Q4895320
cites work: Some properties of eigenfunctions of linear pencils
cites work: Q4005035
cites work: RIESZ COMPLETENESS OF THE EIGENELEMENTS AND ASSOCIATED ELEMENTS OF LINEAR SELFADJOINT PENCILS
cites work: Q3743707
cites work: Elliptic eigenvalue problems with an indefinite weight function
cites work: On the regularity of the critical point infinity of definitizable operators
cites work: A Krein space approach to elliptic eigenvalue problems with indefinite weights
cites work: Q3810395
cites work: Q3833085
cites work: Q3470741
cites work: Q4774569
full work available at URL: https://doi.org/10.1007/bf02677520
OpenAlex ID: W1988411794
description: scientific article; zbMATH DE number 1431677