Stability estimates for the lowest eigenvalue of a Schr\"odinger operator

TL;DR

This paper establishes stability estimates for the lowest eigenvalue of a Schr"odinger operator, quantifying how deviations from optimal potentials increase the eigenvalue, and also proves a new stability estimate for H"older's inequality.

Contribution

It provides effective bounds on eigenvalue increases for non-optimal potentials and introduces a novel stability estimate for H"older's inequality.

Findings

Effective estimates for eigenvalue increase when potential deviates from optimal

Identification of a family of potentials minimizing the eigenvalue under L^p constraint

New stability estimate for H"older's inequality

Abstract

There is a family of potentials that minimize the lowest eigenvalue of a Schr\"odinger eigenvalue under the constraint of a given L^p norm of the potential. We give effective estimates for the amount by which the eigenvalue increases when the potential is not one of these optimal potentials. Our results are analogous to those for the isoperimetric problem and the Sobolev inequality. We also prove a stability estimate for H\"older's inequality, which we believe to be new.