Minimizing Schr\"odinger eigenvalues for confining potentials

TL;DR

This paper investigates the problem of minimizing the lowest eigenvalue of the Schrödinger operator with a potential under a specific integral constraint, identifying the harmonic oscillator as the optimal potential.

Contribution

It proves that the harmonic oscillator minimizes the eigenvalue under the given constraint and provides a quantitative inequality related to this minimization.

Findings

Harmonic oscillator minimizes the lowest eigenvalue under the integral constraint.

Derived a quantitative version of the eigenvalue inequality.

Established the optimality of the harmonic oscillator in this setting.

Abstract

We consider the problem of minimizing the lowest eigenvalue of the Schr\"odinger operator $-\Delta+V$ 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.