Path integral solution of the Dirichlet problem

A scheme for functional integration developed by Cartier/DeWitt-Morette is first reviewed and then employed to construct the path integral representation for the solution of the Dirichlet problem in terms of first exit time. The path integral solution is then applied to calculate the fixed-energy point-to-point transition amplitude both in configuration and phase space. The path integral solution can also be derived using physical principles based on Feynman's original reasoning. We check that the Fourier transform in energy of the fixed-energy point-to-point transition amplitude gives the well known time-dependent transition amplitude, and calculate the WKB approximation. (C) 1997 Academic Press.