Quantum behavior at a cusp, or how wave mechanics avoids a catastrophe

Abstract

The quantum mechanical system whose potential is the cusp catastrophe function is canonical for the uniform asymptotic analysis of the Stokes phenomenon for four transition points. It is shown that across the singularity which marks the bifurcation from a single to two modes, quantum eigenlevels vary smoothly. Also, their derivatives and nonadiabatic coupling matrix elements are continuous with respect to control parameters. These matrix elements, which are of interest for the three-body rearrangement problem, clearly exhibit the ridge effect, i.e. a peaking astride potential maxima.