The asymptotic behavior of simple eigenvalues of particle-in-well systems

Abstract

The particle in a well in dimension one is a classical problem in quantum mechanics. We study higher-dimensional analogues of the problem, where the well is a smooth domain in ${\mathbb{R}}^d$. We show that simple eigenvalues and eigenfunctions of the corresponding Schrödinger operator depend smoothly on the square root $h$ of the inverse depth of the well and provide an explicit first-order expansion of the eigenvalues at $h=0$. Our proof consists of two steps. In the first step, we construct $\mathcal{O}(h^{\infty })$ quasimodes (approximate eigenfunctions) on a resolution of $[0,1{)}_h\times {\mathbb{R}}^d$ which allows us to capture fine structure near the boundary of the well. The second step corrects these quasimodes to true eigenfunctions via a fixed point argument.