A Comparison Theorem of the Eigenvalue Gap for One-Dimensional Barrier potentials

Abstract

The fundamental gap between the lowest two Dirichlet eigenvalues for a Schrodinger operator $H_R = -frac{d^2}{dx^2} + V(x)$ on $L^2([-R,R])$ is compared with the gap for a same operator $H_S$ with a different domain $[-S,S]$ and the difference is exponentially small when the potential has a large barrier.