Asymptotic expansion for nonlinear eigenvalue problems

2009/03/05 by Fatima Aboud, Didier Robert, Aboud, Fatima +1
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.AP #math.FA #math.MP #math.SP

paper · pdf · doi:10.48550/arxiv.0903.0919

arxiv created 2009/03/05 · arxiv updated 2009/12/01

Abstract

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(λ)=-\triangle +(P(x)-λ)² in L²(\R^d) where P is a positive elliptic polynomial in \R^d of degree m≥ 2. It is known that for d even, or d=1, or d=3 and m≥ 6, there exist λ∈\C and u∈ L²(\R^d), u≠ 0, such that L(λ)u=0. In this paper, we give a method to prove existence of non trivial solutions for the equation L(λ)u=0, valid in every dimension. This is a partial answer to a conjecture in \citeherowa.

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