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) ẋ ẋ +Qij(t)x ẋ + 1 2 Rij(t)x x + Si(t) x . 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. PACS codes: 02.30.Wd; 03.65.-w; 03.65.Sq