Transformation operators and asymptotic formulas for the eigenvalues of a polynomial pencil of Sturm--Liouville operators
type: Item
instance of: scholarly article
title: Transformation operators and asymptotic formulas for the eigenvalues of a polynomial pencil of Sturm--Liouville operators
zbMATH Open document ID: 0962.34063
author: R. Smith
published in: Siberian Mathematical Journal
publication date: 2001-02-04T00:00:00Z
full work available at URL: https://eudml.org/doc/120978
review text: The authors consider the equation \[ -y'' +\bigl(q_0(x)+\lambda q_1(x)+\dots + \lambda^{n-1}q_{n-1}(x)\bigr)y =\lambda^{2n} y,\;x\in (0,a), \tag{1} \] with \(q_0\in C([0,a])\), \(q_i\in C^1([0,a])\) for \(i=1,\dots,n-1\), and \(\lambda\) a spectral parameter. By an eigenfunction the authors mean a solution to (1) satisfying the Dirichlet boundary conditions, i.e. the conditions \(y(0)=y(a)=0\). The first part of the article is devoted to constructing the transformation operators, i.e. operators taking the solution \(\exp\bigl((-1)^{j+1}i\lambda^n x\bigr)\) to the equation \(y''+\lambda^{2n}y=0\) into the solution to equation (1) satisfying the initial conditions \(y_j(0)=1\) and \(y_j'(0)=(-1)^{j+1}\lambda^n\). This operator allows the author to write out asymptotic representations for linear independent solutions to equation (1). In the last part of the article, the authors present asymptotic formulas for the eigenvalues.
Mathematics Subject Classification ID: 34L20
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 1560597
zbMATH Keywords: asymptotics of eigenvalues
zbMATH Keywords: transformation operator
zbMATH Keywords: polynomial pencil
zbMATH Keywords: asymptotics of linear independent solutions
zbMATH Keywords: Sturm-Liouville operators
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1560597