Quantitative inequality for the eigenvalue of a Schrödinger operator in the ball

Abstract

The aim of this article is to prove a quantitative inequality for the first eigenvalue of a Schrödinger operator in the ball. More precisely, we optimize the first eigenvalue $\lambda (V)$ of the operator ${\mathcal{L}}_v:=-\Delta -V$ with Dirichlet boundary conditions with respect to the potential $V$, under $L^1$ and $L^{\infty }$ constraints on $V$. The solution has been known to be the characteristic function of a centered ball, but this article aims at proving a sharp growth rate of the following form: if $V^{\ast }$ is a minimizer, then $\lambda (V)-\lambda (V^{\ast })\ge C||V-V^{\ast }|{|}_{L^1(\Omega )}^2$ for some $C>0$. The proof relies on two notions of derivatives for shape optimization: parametric derivatives and shape derivatives. We use parametric derivatives to handle radial competitors, and shape derivatives to deal with normal deformation of the ball. A dichotomy is then established to extend the result to all other potentials. We develop a new method to handle radial distributions and a comparison principle to handle second order shape derivatives at the ball. Finally, we add some remarks regarding the coercivity norm of the second order shape derivative in this context.