A second eigenvalue bound for the Dirichlet Schroedinger operator

TL;DR

This paper establishes bounds on the second eigenvalue of the Dirichlet Schrödinger operator using convexity and rearrangement techniques, extending classical inequalities and analyzing eigenvalue ratios for various potentials.

Contribution

It introduces new bounds for the second eigenvalue under convexity assumptions and explores eigenvalue ratios for spherically symmetric potentials, extending classical spectral inequalities.

Findings

The second eigenvalue is bounded above by that of a rearranged domain with the same first eigenvalue.

The ratio of the second to first eigenvalue decreases as the radius of the domain increases.

Results include bounds for eigenvalues with Gaussian and inverted Gaussian measures.

Abstract

Let $\lambda_i(\Omega,V)$ be the $i$th eigenvalue of the Schr\"odinger operator with Dirichlet boundary conditions on a bounded domain $\Omega \subset \R^n$ and with the positive potential $V$. Following the spirit of the Payne-P\'olya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential $V_\star$, we prove that $\lambda_2(\Omega,V) \le \lambda_2(S_1,V_\star)$. Here $S_1$ denotes the ball, centered at the origin, that satisfies the condition $\lambda_1(\Omega,V) = \lambda_1(S_1,V_\star)$. Further we prove under the same convexity assumptions on a spherically symmetric potential $V$, that $\lambda_2(B_R, V) / \lambda_1(B_R, V)$ decreases when the radius $R$ of the ball $B_R$ increases. We conclude with several results about the first two eigenvalues of the Laplace operator with respect to a measure of Gaussian or inverted Gaussian density.