Path integral solution of the Dirichlet problem
The paper elaborates a further development of a functional integration scheme suggested by Cartier and De Witt-Morette. In particular, results of the paper together with Cartier/De Witt-Morette's results provide the functional integral solution of elliptic and parabolic second order PDE's. The path integral solution of the Dirichlet problem in terms of first exit time is constructed and then applied to calculate the fixed-energy point-to-point transition amplitude both in configuration and phase space. It is shown that the Fourier transform in energy of the fixed-energy point-to-point transition amplitude gives the well-known transition amplitude obtained by using physical principles based on Feynman's original reasoning. The WKB approximation is also calculated.
scientific article; zbMATH DE number 1217308
scientific article