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arXiv 0903.0919math-phmath.APmath.FAmath.MPmath.SP
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非线性特征值问题的渐近展开

Asymptotic expansion for nonlinear eigenvalue problems

Fatima Aboud, Didier Robert

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摘要：

本文考虑一类对复参数二次依赖的算子族的广义特征值问题。我们的模型是$L(λ)=-\triangle +(P(x)-λ)^2$ in $L^2(\R^d)$，其中$P$是次数为$m\geq 2$的$\R^d$的正椭圆多项式。已知对于$d$甚至，或$d=1$，或$d=3$和$m\geq 6$，存在$λ\in\C$和$u\in L^2(\R^d)$，$u\neq 0$，使得$L(λ)u=0$。本文给出了一个证明方程$L(λ)u=0$非平凡解的存在性的方法，该方法在任意维都成立。这是对\cite{herowa}中一个猜想的部分回答。

英文摘要：

In this paper we consider generalized eigenvalue problems for a family of operators with a quadratic dependence on a complex parameter. Our model is $L(λ)=-\triangle +(P(x)-λ)^2$ 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 $λ\in\C$ and $u\in L^2(\R^d)$, $u\neq 0$, 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. This is a partial answer to a conjecture in \cite{herowa}.

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