Boundary Conditions from Path Integrals

A particle moving in two dimensions in a repulsive $r^{-2\mathrm{}}$ potential, and in the presence of a magnetic flux line, is examined as an example of a system for which the Schrödinger Hamiltonian is not essentially self-adjoint. The path-integral propagator is, nevertheless, shown to exist, and to define a unique self-adjoint extension of the Hamiltonian, corresponding to specific boundary conditions at the origin. Physical implications are discussed.