Quantum caustics for systems with quadratic Lagrangians in multi-dimensions

We study quantum caustics (i.e., the quantum analogue of the classical sing ularity in the Dirichlet boundary problem) in d-dimensional systems with qu adratic Layrangians of the from L = 1/2 P-ij (t) x(i)x(j) + Q(ij) (t) x(i)x (j) + 1/2 R-ij (t) x(i)x(j +) S-i(t) x(t). Based on Schulman's procedure in the path-integral we derive the transition amplitude on caustics in a clos ed form for generic multiplicity f, and thereby complete the previous analy sis carried out for the maximal multiplicity case (f = d). The unitarity re lation, together with the initial condition, fulfilled by the amplitude is found to be a hey ingredient for determining the amplitude. which reduces t o the well-known expression with Van Vleck determinant for the non-caustics case (f = 0). Multiplicity dependence of the caustics phenomena is illustr ated by examples of ii particle interacting with external electromagnetic f ields. (C) 2000 Academic Press.