Path integrals of systems on the half-line

TL;DR

This paper explores how path integrals for quantum systems on the half-line are affected by potentials, especially those with inverse square dependence, and how reflection contributions are generalized.

Contribution

It generalizes the reflection sign in path integrals for systems with inverse square potentials on the half-line, extending the known free particle case.

Findings

Reflection sign depends on the potential parameter.

Path integral formulation is extended to inverse square potentials.

Provides a unified framework for path integrals with boundary conditions.

Abstract

It is wellknown that the Feynman kernel for the free particle on the half-line can be expressed as a sum over classical paths if we take the contribution from the reflected path into account. The minus sign for the reflected path needs to be generalized depending on the parameter of the potential if we add a potential proportional to $1/x^{2}$.