A free nonrelativistic particle moving in two dimensions on a half-plane can be described by self-adjoint Hamiltonians characterized by boundary conditions imposed on the systems. The most general boundary condition is parameterized in terms of the elements of an infinite-dimensional matrix. We construct the Brownian functional integral for each of these self-adjoint Hamiltonians. Nonlocal boundary conditions are implemented by allowing the paths striking the boundary to jump to other locations on the boundary. Analytic continuation in time results in the Green’s functions of the Schrödinger equation satisfying the boundary condition characterizing the self-adjoint Hamiltonian.
(D.O.E.) under contract #DE-AC02-76ER03069, and by the Fonds pour la Formation de Chercheurs et l’Aide à la Recherche. A free non-relativistic particle moving in two dimensions on a half-plane can be described by self-adjoint Hamiltonians characterized by boundary conditions imposed on the systems. The most general boundary condition is parameterized in terms of the elements of an infinite-dimensional matrix. We construct the Brownian functional integral for each of these self-adjoint Hamiltonians. Non-local boundary conditions are implemented by allowing the paths striking the boundary to jump to other locations on the boundary. Analytic continuation in time results in the Green’s functions of the Schrödinger equation satisfying the boundary condition characterizing the self-adjoint Hamiltonian. 1 I.