Differentiability of spectral functions for nonsymmetric diffusion processes

Abstract

Let $L=(1/2)\mathrm{\nabla }\cdot a\mathrm{\nabla }+b\cdot \mathrm{\nabla }+W$ be a critical elliptic operator on ${\mathbb{R}}^d$. For a certain class of potential functions, we consider the generalized principal eigenvalues $\lambda \left(t\right)$ of $L_t=L+tV$. We show that it is differentiable if and only if $L_0$ is null critical.