$\alpha $-continuity properties of the symmetric $\alpha $-stable process

Abstract

Let $D$ be a domain of finite Lebesgue measure in $\mathbb {R}^d$ and let $X^D_t$ be the symmetric $\alpha$-stable process killed upon exiting $D$. Each element of the set $\{ \lambda _i^\alpha \}_{i=1}^\infty$ of eigenvalues associated to $X^D_t$, regarded as a function of $\alpha \in (0,2)$, is right continuous. In addition, if $D$ is Lipschitz and bounded, then each $\lambda _i^\alpha$ is continuous in $\alpha$ and the set of associated eigenfunctions is precompact.