\alpha-Continuity Properties of Stable Processes

Let $D$ be a domain of finite Lebesgue measure in ${\text{bR}}^d$ and let $X_t^D$ be the symmetric $\alpha$-stable process killed upon exiting $D$. Each element of the set $\{{\lambda }_i^{\alpha }{\}}_{i=1}^{\mathrm{\infty }}$ of eigenvalues associated to $X_t^D$, 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. We also prove that if $D$ is a domain of finite Lebesgue measure, then for all $0<\alpha <\beta \le 2$ and $i\ge 1$, $${\lambda }_i^{\alpha }\le [{\lambda }_i^{\beta }{]}^{\alpha \mathrm{/}\beta }\mathrm{.}$$ Previously, this bound had been known only for $\beta =2$ and $\alpha$ rational.