Quantum caustics for systems with quadratic Lagrangians in multi-dimensions

We study quantum caustics (i.e., the quantum analogue of the classical singularity in the Dirichlet boundary problem) in d-dimensional systems with quadratic Lagrangians of the form L=((1)/(2))Pij(t) xixj+Qij(t) xixj+((1)/(2))Rij(t) xixj+Si(t) xi. Based on Schulman's procedure in the path-integral we derive the transition amplitude on caustics in a closed form for generic multiplicity f, and thereby complete the previous analysis carried out for the maximal multiplicity case (f=d). The unitarity relation, together with the initial condition, fulfilled by the amplitude is found to be a key ingredient for determining the amplitude, which reduces to the well-known expression with Van Vleck determinant for the non-caustics case (f=0). Multiplicity dependence of the caustics phenomena is illustrated by examples of a particle interacting with external electromagnetic fields