The log-Sobolev inequality for the ground state of a Schr\"odinger operator on bounded convex domains

We consider the ground state $\phi_0$ of the Schr\"odinger operator $L=-\Delta+V$ on the bounded convex domain $\Omega\subset\R^n$, 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 $\d\mu=\phi_0 \d x$ 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 $\frac{\pi^2}{D^2}$.