The log-Sobolev inequality for the ground state of a Schrödinger operator on bounded convex domains

Abstract

We consider the ground state ${\varphi }_0$ of the Schrödinger operator $L=-\Delta +V$ on the bounded convex domain $\Omega \subset {\mathbb{R}}^n$, satisfying the Dirichlet boundary condition. Assume that $V\in C^1(\Omega )$ and it admits an even function $\tilde{V}\in C^1([-D/2,D/2])$ as its modulus of convexity, where $D$ is the diameter of $\Omega$. If the first Dirichlet eigenvalue ${\tilde{\lambda }}_0$ of $-\frac{\d^2}{\d t^2}+\tilde V$ on the interval $[-D/2,D/2]$ satisfies ${\tilde{\lambda }}_0>\tilde{V}(0)$, then the measure $\dμ=ϕ_0 \d x$ satisfies the log-Sobolev inequality on $\Omega$ with the constant ${\tilde{\lambda }}_0-\tilde{V}(0)$. In particular, if $V$ is convex, then the constant is explicitly given by $\frac{{\pi }^2}{D^2}$.

This paper has been withdrawn since we put it in another paper as a section