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 λ(V) of the operator Lv:=−Δ−V with Dirichlet boundary conditions with respect to the potential V, under L1 and L∞ 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 λ(V)−λ(V⁎)⩾C||V−V⁎||L1(Ω)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.