First and second order differentiation operators with weight functions of variable sign
type: Item
instance of: scholarly article
title: First and second order differentiation operators with weight functions of variable sign
zbMATH Open document ID: 0835.34093
DOI: 10.1007/BF02110552
published in: Mathematical Notes
publication date: 1996-01-08T00:00:00Z
review text: Considerable difficulty arises in the study of convergence of expansions in eigen- and associated-functions (e.a.f.) by the method of contour integration in the case of exponential growth of the resolvent for large \(|\lambda |\). At present the problem has been well-studied for differential operators with irregularly decomposing boundary conditions. The function being expanded must have some properties of the type of analyticity and this is precisely what allows one to overcome the difficulties mentioned. In the present paper we show that one also observes exponential growth of the resolvent in the case of first and second order differentiation operators when weight functions of variable sign and can overcome the difficulties caused by it for a first order differentiation operator using some functional equations which the function being expanded must satisfy, and for a second order differentiation operator using its analyticity in addition. The complexity of the arguments makes it necessary to restrict oneself for now to just these operators and moreover with just one sign change. Regular boundary problems with more complicated weight functions have been studied by many authors. Different results have been obtained by \textit{M. H. Stone} [Trans. Am. Math. Soc. 26, 335-355 (1924)] for the nonregular case with \(n = 1\).
Mathematics Subject Classification ID: 34L10
Mathematics Subject Classification ID: 47E05
zbMATH DE Number: 830683
zbMATH Keywords: convergence of expansions in eigen- and associated-functions
zbMATH Keywords: method of contour integration
zbMATH Keywords: exponential growth of the resolvent
zbMATH Keywords: weight functions of variable sign
zbMATH Keywords: first order differentiation operator
zbMATH Keywords: second order differentiation operator
author: Aleksandr Petrovich Gurevich
author: A. P. Khromov
MaRDI profile type: Publication
cites work: Die Entwicklungen nach Eigenfunktionen irregulärer Eigenwertprobleme mit zerfalleneden Randbedingungen
cites work: Expansion theorems for a class of regular indefinite eigenvalue problems
cites work: Q5332797
description: scientific article; zbMATH DE number 830683