SEPARATION OF VARIABLES IN PATH-INTEGRALS AND PATH INTEGRAL SOLUTION OF 2 POTENTIALS ON THE POINCARE UPPER HALF-PLANE

The author discusses how to separate variables in path integrals. It is assumed that a one-dimensional problem with potential V(x) has an exact solution with energy levels Elambda and wavefunctions Psi lambda. In order to perform the separation of variables, a time transformation is performed back and forth in the path integral which allows one to insert the path integral solution corresponding to the potential V(x). The author illustrates the method by discussing some specific potential problems on the Poincare upper half-plane.