Minimizing Schrödinger eigenvalues for confining potentials

We consider the problem of minimizing the lowest eigenvalue of the Schrödinger operator $-\mathrm{\Delta }+V$ in $L^2({\mathbb{R}}^d)$ when the integral $\int e^{-tV}dx$ is given for some $t>0$. We show that the eigenvalue is minimal for the harmonic oscillator and derive a quantitative version of the corresponding inequality.