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

TL;DR

This paper investigates how simple eigenvalues of quantum particle-in-well systems in higher dimensions behave asymptotically as the well depth increases, providing explicit expansions and a detailed construction of approximate eigenfunctions.

Contribution

It introduces a method to explicitly expand eigenvalues of Schrödinger operators in higher-dimensional wells as the inverse well depth parameter approaches zero.

Findings

Eigenvalues depend smoothly on the square root of inverse well depth.

Explicit first-order eigenvalue expansion at zero inverse depth.

Construction of high-precision quasimodes capturing boundary effects.

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 Schr\"odinger 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.