Half-range solutions of indefinite Sturm-Liouville problems
type: Item
instance of: scholarly article
title: Half-range solutions of indefinite Sturm-Liouville problems
zbMATH Open document ID: 0619.35062
DOI: 10.1016/0022-1236(87)90113-3
author: Martin Klaus
author: C. van der Mee
author: V. Protopopescu
published in: Journal of Functional Analysis
publication date: 1987-01-01T00:00:00Z
review text: We study diffusion equations of Sturm-Liouville type, i.e. boundary value problems of the form \[ w(\mu)(\partial \psi /\partial x)(x,\mu)=\partial /\partial \mu (p(\mu)(\partial \psi /\partial \mu)(x,\mu))-q(\mu)\psi (x,\mu)\quad (0<x<\infty,\quad \mu \in I) \] \[ \psi (0,\mu)=\phi +(\mu)\;if\;w(\mu)>0;\quad \int_{I}| \psi (x,\mu)|^ 2| w(\mu)d\mu =O(1)\quad (or\quad o(1)\quad or\quad O(x))\quad as\quad x\to \infty \] where \(\psi\) (x,\(\mu)\) satisfies self-adjoint boundary conditions of the differential operator \(h\to -(ph')'+qh\) whose spectrum is contained in the set \(\{\) \(0\}\cup [\epsilon,\infty)\) for some \(\epsilon >0\). I is a finite or infinite interval and w(\(\mu)\) is an indefinite weight function with finitely many sign changes. We develop two equivalent methods of constructing explicit representations for the solutions of the problem. The first method is based on an eigenfunction expansion whereas the second uses a Wiener-Hopf-type integral equation and factorization. Some illustrative examples are also worked out.
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35C05
Mathematics Subject Classification ID: 35P10
Mathematics Subject Classification ID: 35C15
zbMATH DE Number: 4004754
zbMATH Keywords: diffusion equations of Sturm-Liouville type
zbMATH Keywords: indefinite weight function
zbMATH Keywords: finitely many sign changes
zbMATH Keywords: explicit representations
zbMATH Keywords: eigenfunction expansion
zbMATH Keywords: Wiener-Hopf-type integral equation
MaRDI profile type: Publication
full work available at URL: 0022-1236(87)90113-3
OpenAlex ID: W2063271575
cites work: Discrete and continuous boundary problems
cites work: Sur une équation d'évolution changeant de type
cites work: Minimal factorization of matrix and operator functions
cites work: Factorizations of Transfer Functions
cites work: On an equation of mixed type from electron scattering theory
cites work: An abstract treatment of some forward-backward problems of transport and scattering
cites work: Erratum: Partial-range completeness and existence of solutions to two-way diffusion equations [J. Math. Phys. 2 2, 954 (1981)]
cites work: Indefinite Sturm-Liouville problems and half-range completeness
cites work: Half-range completeness for the Fokker-Planck equation. I
cites work: Existence and Uniqueness Theorems for the Neutron Transport Equation
cites work: Q4188958
cites work: Q5796557
cites work: On the Nature of the Spectrum of Singular Second Order Linear Differential Equations
cites work: Q3254057
cites work: Q3333353
cites work: Q3671491
cites work: Strong solutions of stationary equations in abstract kinetic theory
cites work: Generalized kinetic equations
cites work: Linear analysis and solution of neutron transport problems
cites work: Q3864214
cites work: Decompositions of a Hilbert Space and Factorization of a W–A Determinant
cites work: Q5570370
cites work: Full- and partial-range eigenfunction expansions for Sturm-Liouville problems with indefinite weights
cites work: 23.—The Linear Transport Equation. The Degenerate Case c = 1. II. Half-range Theory
cites work: Spectral analysis of the transport equation. I. Nondegenerate and multigroup case
cites work: Q3937851
cites work: Albedo operators and H-equations for generalized kinetic models
cites work: Q5562284
cites work: Second-Order Differential Equations with Fractional Transition Points
cites work: Q5588793
cites work: Q5604030
cites work: Q5563474
description: scientific article; zbMATH DE number 4004754