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

Abstract

We consider the ground state $ϕ_0$ of the Schrödinger operator $L=-Δ+V$ on the bounded convex domain $Ω\subset\R^n$, satisfying the Dirichlet boundary condition. Assume that $V\in C^1(Ω)$ 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 $Ω$. If the first Dirichlet eigenvalue $\tildeλ_0$ of $-\frac{\d^2}{\d t^2}+\tilde V$ on the interval $[-D/2,D/2]$ satisfies $\tildeλ_0>\tilde V(0)$, then the measure $\dμ=ϕ_0 \d x$ satisfies the log-Sobolev inequality on $Ω$ with the constant $\tildeλ_0-\tilde V(0)$. In particular, if $V$ is convex, then the constant is explicitly given by $\frac{π^2}{D^2}$.