Path Integral Treatment of Singular Problems and Bound States: Quantum Mechanics

TL;DR

This paper introduces a path-integral method for calculating quantum propagators and Green's functions, effectively handling singular potentials like the inverse square potential, and discusses their connection to quantum field theory limits.

Contribution

It presents a novel path-integral approach specifically designed for singular quantum problems, including inverse square and delta-function interactions, with insights into their field theory origins.

Findings

Effective computation of propagators for singular potentials

Regularization techniques for singular quantum problems

Connection between singular potentials and quantum field theory limits

Abstract

A path-integral approach for the computation of quantum-mechanical propagators and energy Green's functions is presented. Its effectiveness is demonstrated through its application to singular interactions, with particular emphasis on the inverse square potential--possibly combined with a delta-function interaction. The emergence of these singular potentials as low-energy nonrelativistic limits of quantum field theory is highlighted. Not surprisingly, the analogue of ultraviolet regularization is required for the interpretation of these singular problems.