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Problèmes aux valeurs propres non-linéaires

Abstract : In this work we study the polynomial family of operators : L(z)=H_0+z H_1+...+ zm-1Hm-1+zm , where the coefficients H0,H1,...,Hm-1 are operators defined on the Hilbert space H and z is a complex parameter. We are interested to study the spectrum of the family L(z). The problem L(z)u(x)=0, is called a non-linear eigenvalue problem for m≥2 (The complex number z is called an eigenvalue of L(z), if there exists u in H, u≠0 such that L(z)u=0). We consider here a quadratic family (m=2) and in particular we are interested in the case LP(z)=-∆x+(P(x)-z)2which is defined on the Hilbert space L2(Rn), where P is an elliptic positive polynomial of degree M≥2. For this example results for existence of eigenvalues are known for n=1 and n is even. The main goal of our work is to check the following conjecture, stated by Helffer-Robert-Wang : “ For every dimension n, for every M≥2, the spectrum of LP is non empty.” We prouve this conjecture for the following cases : (1) n=1,3, for every polynomial P of degree M≥2. (2) n=5, for every convex polynomial P satisfying some technical conditions. (3)n=7, for every convex polynomial P. This result extends to the case of quasi-homogeneous polynomial and quasi-elliptic, for example P(x,y)=x2+y4, x in Rn1, y in Rn2, n1+n2=n, where n is even. We prove this results by computing the coefficients of a semi-classical trace formula and by using the theorem of Lidskii.
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Submitted on : Thursday, August 20, 2009 - 4:54:37 PM
Last modification on : Monday, March 25, 2019 - 4:52:05 PM
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Fatima Aboud. Problèmes aux valeurs propres non-linéaires. Mathématiques [math]. Université de Nantes, 2009. Français. ⟨tel-00410455⟩

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