Asymptotic expansion for nonlinear eigenvalue problems

Fatima Aboud , Didier Robert (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

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 L2(Rd) where P is a positive elliptic polynomial in Rd 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(Rd), 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 [12]. key words: semiclassical analysis, nonlinear eigenvalue problems, nonselfadjoint operators, trace formula.

Domaines: Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]

Fatima Aboud, Didier Robert. Asymptotic expansion for nonlinear eigenvalue problems. Journal de Mathématiques Pures et Appliquées, 2010, 93 (2), pp.149-162. ⟨hal-01015315⟩