Optimal relaxation for Witten Lindbladians

Abstract

We prove optimal trace-norm relaxation for the one-dimensional Lindbladian associated with the Witten differential $a=h{\partial }_x+V^{\prime }$ which annihilates the classical Gibbs density: if ${\lambda }_1(h)$ is the first positive eigenvalue of $H=a^{\ast }a$, then the relaxation rate is ${\gamma }_h={\lambda }_1(h)/(2h)$. It applies to initial operators whose Schwartz kernels, after a Gibbs conjugation, satisfy $L^2$ estimates for the restriction to the diagonal and for the normal derivative. An appendix by Chat GPT 6 presents a stronger result specialised to positive initial data.