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

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