Path Integration Via Summation of Perturbation Expansions and Applications to Totally Reflecting Boundaries, and Potential Steps

TL;DR

This paper develops a method to compute path integrals with perturbation series, summing them exactly for delta-function perturbations, and applies this to derive Green functions for reflecting boundaries and potential steps.

Contribution

It introduces a technique to sum perturbation expansions exactly for delta-function perturbations, enabling solutions for reflecting boundaries and potential steps.

Findings

Exact Green functions for totally reflecting boundaries derived

Path integral solutions for step and finite potential wells obtained

Method applicable to half-space quantum problems

Abstract

The path integral for the propagator is expanded into a perturbation series, which can be exactly summed in the case of $\delta$-function perturbations giving a closed expression for the (energy-dependent) Green function. Making the strength of the $\delta$-function perturbation infinite repulsive, produces a totally reflecting boundary, hence giving a path integral solution in half-spaces in terms of the corresponding Green function. The example of the Wood-Saxon potential serves by an appropriate limiting procedure to obtain the Green function for the step-potential and the finite potential-well in the half-space, respectively.