Asymptotic expansion for nonlinear eigenvalue problems

TL;DR

This paper develops a method to prove the existence of non-trivial solutions for a class of nonlinear eigenvalue problems involving quadratic operator dependence, applicable in all dimensions, advancing understanding of such spectral problems.

Contribution

It introduces a general method to establish solutions for nonlinear eigenvalue problems with quadratic operator dependence across all dimensions.

Findings

Existence of non-trivial solutions for the nonlinear eigenvalue problem in all dimensions.

Applicable to operators with quadratic dependence on complex parameters.

Partially confirms a conjecture in the literature.

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