The spectral asymptotics of a quadratic pencil of differential operators
type: Item
instance of: scholarly article
title: The spectral asymptotics of a quadratic pencil of differential operators
zbMATH Open document ID: 1079.47047
author: Ya. T. Sultanaev
author: E. V. Silova
published in: Differential Equations
publication date: 2005-06-17T00:00:00Z
review text: The authors study the distribution function of eigenvalues of the pencil \(L(\lambda)=A+\lambda B-\lambda^2 I\), where \(A\) is the self-adjoint differential operator induced in \(H=L_2(-\infty,+\infty)\) by the differential expression \(y^{(4)}+q(x)y\) with a positive function \(q(x)\), \(B\) is the self-adjoint operator induced in \(H\) by the expression \((i/2)((p(x)y'')' +(p(x)y')'')\), where \(p(x)\) is a twice continuously differentiable positive function. In this case, \(L(\lambda)\) is not a Keldysh pencil. In Section 1 of the paper, the asymptotic behavior of the Green function \(G(x,\xi,\lambda)\) of the pencil \(L(\lambda)\) for \(\lambda=i\mu\) as \(\mu\to+\infty\) is investigated. In Section 2, the Keldysh Tauberian theorem is used to derive an asymptotic formula for the distribution of eigenvalues of \(L(\lambda)\) from the representation of \(G(x,\xi,\lambda)\).
reviewed by: Victor S. Rykhlov
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 34B07
Mathematics Subject Classification ID: 34L20
Mathematics Subject Classification ID: 34B40
zbMATH DE Number: 2176520
zbMATH Keywords: quadratic pencil of differential operators
zbMATH Keywords: spectral asymptotics
zbMATH Keywords: distribution function of eigenvalues
MaRDI profile type: Publication
full work available at URL: b:dieq.0000011285.73769.ef
OpenAlex ID: W1992874436
DOI: 10.1023/B:DIEQ.0000011285.73769.EF
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description: scientific article; zbMATH DE number 2176520