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 ( λ ) = − △ + ( P ( x ) − λ ) 2 in L 2 ( 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 =...

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Bibliographic Details
Published in:	Journal de mathématiques pures et appliquées Vol. 93; no. 2; pp. 149 - 162
Main Authors:	Aboud, Fatima, Robert, Didier
Format:	Journal Article
Language:	English
Published:	Kidlington Elsevier SAS 01.02.2010 Elsevier
Subjects:	Analysis of PDEs Exact sciences and technology Functional Analysis General mathematics General, history and biography Mathematical Physics Mathematics Non-selfadjoint operators Nonlinear eigenvalue problems Physics Sciences and techniques of general use Semiclassical analysis Spectral Theory Trace formula Semiclassical analysis Non-selfadjoint operators Nonlinear eigenvalue problems Trace formula Polynomial Asymptotic expansion Nonexistence of solution Mathematics Nonlinear problems Eigenvalue problem Self adjoint operator nonlinear eigenvalue partial differential equation non-seladjoint operator trace formula semiclassical analysis
ISSN:	0021-7824
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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 ( λ ) = − △ + ( P ( x ) − λ ) 2 in L 2 ( 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 ∈ L 2 ( 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 d ⩾ 1 . This is a partial answer to a conjecture in Helffer, Robert and Xue Ping Wang (2004) [13]. Dans cet article nous considérons un problème aux valeurs propres généralisé pour une famille d'opérateurs dépendant quadratiquement d'un paramètre complexe. Le modèle étudié concerne la famille L ( λ ) = − △ + ( P ( x ) − λ ) 2 dans L 2 ( R d ) où P un polynôme elliptique dans R d de degré m ⩾ 2 . Si d est paire ou si d = 1 ou d = 3 et m ⩾ 6 , on sait alors qu'il existe λ ∈ C et u ∈ L 2 ( R d ) , u ≠ 0 , tels que L ( λ ) u = 0 . L'objet principal de cet article est de donner une méthode pour démontrer l'existence de solutions non triviales pour l'équation L ( λ ) u = 0 pour toute dimension d ⩾ 1 . On répond ainsi partiellement à une conjecture formulée dans Helffer, Robert et Xue Ping Wang (2004) [13].
ISSN:	0021-7824
DOI:	10.1016/j.matpur.2009.08.009