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

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.