Asymptotic expansion for nonlinear eigenvalue problems

In this paper we consider generalized eigenvalue problems for a family of operators with a quadratic dependence on a complex parameter. Our model is \(L(\lambda)=-\triangle +(P(x)-\lambda)^2\) in \(L^2(\R^d)\) where \(P\) is a positive elliptic polynomial in \(\R^d\) of degree \(m\geq 2\). It is kno...

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Bibliographic Details
Published inarXiv.org
Main Authors Aboud, Fatima, Didier, Robert
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 05.03.2009
Subjects
Asymptotic series
Dependence
Eigenvalues
Polynomials
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Summary:In this paper we consider generalized eigenvalue problems for a family of operators with a quadratic dependence on a complex parameter. Our model is \(L(\lambda)=-\triangle +(P(x)-\lambda)^2\) in \(L^2(\R^d)\) where \(P\) is a positive elliptic polynomial in \(\R^d\) of degree \(m\geq 2\). It is known that for \(d\) even, or \(d=1\), or \(d=3\) and \(m\geq 6\), there exist \(\lambda\in\C\) and \(u\in L^2(\R^d)\), \(u\neq 0\), such that \(L(\lambda)u=0\). In this paper, we give a method to prove existence of non trivial solutions for the equation \(L(\lambda)u=0\), valid in every dimension. This is a partial answer to a conjecture in \cite{herowa}.
ISSN:2331-8422