The asymptotics of eigenvalues and eigenfunctions and a trace formula for a potential with Delta functions
type: Item
instance of: scholarly article
title: The asymptotics of eigenvalues and eigenfunctions and a trace formula for a potential with Delta functions
zbMATH Open document ID: 1043.34092
DOI: 10.1023/A:1020302110566
author: V. A. Vinokurov
author: V. A. Sadovnichy
published in: Differential Equations
publication date: 2003-05-25T00:00:00Z
review text: Het, the authors extend two results on the boundary problem \[ z(x)=x-\int_{0}^{l}{\nu(x-t)z(t)}\,d(\sigma(t)+t\lambda), \quad x\in{[0,l]}, \quad z(l)=0, \] obtained by them in two previous papers [\textit{V. A.Vinokurov}, Dokl. Math., 57, No. 1, 43--46 (1998; Zbl 0989.34016) and Dokl. Math., 59, No. 2, 220--222 (1999; Zbl 0967.34076)]. In these papers, they considered the case of an absolutely continuous function \(\sigma\). In the paper under review, functions \(\sigma\) having \(\delta\)-type discontinuities are allowed. Namely, one assumes that \(\sigma\) is in the space \(BV_{c}[0,l]\) of functions with bounded variation, right continuous on \([0,l]\) and continuous at \(0\) and \(l\). By approximating \(\sigma\) with smooth functions and passing to the limit one proves that the second-order approximate formulas for the spectral parameter \(\lambda_{n}\) and for the eigenfunctions which were obtained in the first paper cited above are still valid. The regularized trace formula from the second paper cited above is also extended to the case of functions from \(BV_{c}[0,l]\).
reviewed by: Mihai Pascu
Mathematics Subject Classification ID: 34L20
Mathematics Subject Classification ID: 34B05
zbMATH DE Number: 1916101
zbMATH Keywords: first boundary value problem
zbMATH Keywords: asymptotics of eigenvalues and eigenfunctions
zbMATH Keywords: trace formula
zbMATH Keywords: potential with delta functions
MaRDI profile type: Publication
full work available at URL: a:1020302110566
OpenAlex ID: W275417501
description: scientific article; zbMATH DE number 1916101