Abstract

In this work we study the polynomial family of operators L(z) = H0 + z H1 + z²,
where the coefficients H0,H1 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 greater or equal to 2 (The complex number z0 is called an eigenvalue of L(z), if there exists u0 in H, u0 different from 0 such that
L(z0)u0=0).
We consider here a quadratic family (m=2) and in particular we are interested in the case L(z)=- \Delta_x+(P(x)-z)^2, which is defined on the Hilbert space L^2(R^n), where P is an elliptic positive polynomial of degree M greater or equal to 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 greater or equal to 2, the spectrum of L is non empty.
We prouve this conjecture for the following cases :
1) n=1,3, for every polynomial P of degree M greater or equal to 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)=x^2+y^4, x in R^{n1}, y in
R^{n2}, n1+n2=n, and n is even.
We prove this results by computing the coefficients of a semi-
classical trace formula and by using the theorem of Lidskii.