Convergence and completeness for square-well Stark resonant state expansions

TL;DR

This paper studies the completeness and convergence properties of Stark resonant eigenstates in a square-well potential, showing convergence inside the well independent of potential depth and relating smoothness to convergence rate.

Contribution

It provides new theoretical insights into the convergence behavior of resonant state expansions for square-well potentials, including conditions for convergence and its dependence on function smoothness.

Findings

Resonant state expansions converge inside the well regardless of potential depth.

Convergence rate depends on the smoothness of target functions.

The paper establishes a relation between function smoothness and expansion convergence.

Abstract

In this paper we investigate the completeness of the Stark resonant eigenstates for a particle in a square-well potential. We find that the resonant state expansions for target functions converge inside the potential well and that the existence of this convergence does not depend on the depth of the potential well. By analyzing the asymptotic form of the terms in these expansions we prove some results on the relation between smoothness of target functions and the rate of convergence of the corresponding resonant state expansion.