Quantitative inequality for the eigenvalue of a Schr\"odinger operator in the ball

TL;DR

This paper establishes a sharp quantitative inequality for the first eigenvalue of a Schr"odinger operator in a ball, showing how the eigenvalue difference relates quadratically to potential deviations.

Contribution

It proves a new sharp growth rate inequality for the eigenvalue minimization problem involving potentials in a ball, using novel shape and parametric derivative techniques.

Findings

The eigenvalue difference is bounded below by a quadratic function of potential difference.

A new method for handling radial distributions in shape optimization is developed.

The result extends to all potentials via a dichotomy argument.

Abstract

The aim of this article is to prove a quantitative inequality for the first eigenvalue of a Schr\"odinger 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 $V^*$ is a minimizer, then $\lambda(V)-\lambda(V^*)\geq C ||V-V^*||_{L^1(\Omega)}^2$ 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.