Perturbative and non-perturbative studies with the delta function potential

TL;DR

This paper demonstrates how the delta function potential can be used with perturbation theory to evaluate infinite series and analyze energy shifts in quantum systems, highlighting the role of the continuum in bound states.

Contribution

It introduces a method combining delta function potentials with perturbation theory to solve infinite series and study energy shifts in quantum mechanics.

Findings

Second-order energy shift expressed as an infinite sum.

Analytical determination of energy shifts using delta potential.

Importance of the continuum in bound state energy shifts.

Abstract

We show that the delta function potential can be exploited along with perturbation theory to yield the result of certain infinite series. The idea is that any exactly soluble potential if coupled with a delta function potential remains exactly soluble. We use the strength of the delta function as an expansion parameter and express the second-order energy shift as an infinite sum in perturbation theory. The analytical solution is used to determine the second-order energy shift and hence the sum of an infinite series. By an appropriate choice of the unperturbed system, we can show the importance of the continuum in the energy shift of bound states.