Semiclassics around a phase space caustic: an illustration using the Nelson Hamiltonian

TL;DR

This paper discusses the semiclassical approximation of the coherent-state propagator near phase space caustics, demonstrating the effectiveness of a uniform approximation through application to the Nelson Hamiltonian.

Contribution

It provides a practical application of a previously derived uniform approximation to handle divergences at phase space caustics in semiclassical propagator calculations.

Findings

The uniform approximation accurately resolves divergences at phase space caustics.

Application to the Nelson Hamiltonian illustrates the method's effectiveness.

The approach improves semiclassical calculations near caustics.

Abstract

The semiclassical formula for the coherent-state propagator is written in terms of complex classical trajectories of an equivalent classical system. Depending on the parameters involved, more than one trajectory may contribute to the calculation. Eventually, however, two contributing trajectories coalesce, characterizing what is called phase space caustic. In this case, the usual semiclassical formula for the propagator diverges, so that a uniform approximation is required to avoid this singularity. In this paper, we present a non-trivial application illustrating this scenario, and showing the accuracy of the uniform formula that we have previously derived.