PATH-INTEGRAL SOLUTION OF THE DIRICHLET PROBLEM

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