Effects of confinement for single-well potentials

TL;DR

This paper analyzes how confinement affects bound states in single-well potentials, deriving asymptotic formulas for eigenvalue shifts due to boundary conditions in the semiclassical limit.

Contribution

It provides explicit asymptotic expansions for eigenvalue shifts caused by confinement, including applications to harmonic oscillator and Coulomb potentials.

Findings

Eigenvalues differ from unconfined case by exponentially small terms

Explicit formulas for eigenvalue shifts are derived

Applications to harmonic oscillator and Coulomb potentials are demonstrated

Abstract

We study bound states generated by a unique potential minimum in the situation where the system is strongly confined to a bounded region containing the minimum (by imposing Dirichlet boundary conditions). In this case the eigenvalues of the confined system differ from those of the unconfined system by an exponentially small quantity in the semiclassical limit. An asymptotic expansion for this shift is established. The formulas are evaluated explicitly for the harmonic oscillator and an application to the Coulomb potential at a fixed angular momentum is given.