Path integral treatment of two- and three-dimensional delta-function potentials and application to spin-1/2 Aharonov-Bohm problem

TL;DR

This paper develops a path integral approach to analyze delta-function potentials in 2D and 3D quantum mechanics, including the spin-1/2 Aharonov-Bohm problem, revealing a unique self-adjoint extension parameter that yields a well-defined propagator.

Contribution

It introduces a novel path integral method incorporating self-adjoint extensions for delta-function potentials and explicitly computes Green functions for these systems.

Findings

Identifies a unique self-adjoint extension parameter for well-defined propagators.

Calculates explicit energy-dependent Green functions for delta potentials.

Applies the method to the spin-1/2 Aharonov-Bohm problem with new insights.

Abstract

Delta-function potentials in two- and three-dimensional quantum mechanics are analyzed by the incorporation of the self-adjoint extension method to the path integral formalism. The energy-dependent Green functions for free particle plus delta-function potential systems are explicitly calculated. Also the energy-dependent Green function for the spin-1/2 Aharonov-Bohm problem is evaluated. It is found that the only one special value of the self-adjoint extension parameter gives a well-defined and non-trivial time-dependent propagator. This special value corresponds to the viewpoint of the spin-1/2 Aharonov-Bohm problem when the delta-function is treated as a limit of the infinitesimal radius.