Multi-scale semiclassical approximations for Schrödinger propagators on manifolds
Exponential representations of the Schrödinger propagator are constructed for an interacting quantum system on a semi-Riemannian manifold \(M\). For a local fundamental solution of the time dependent Schrödinger equation, the higher-order WKB approximation is re-expanded in the scaling parameter \(\varepsilon\) responsible for the gauge invariant derivative expansion. The geometrical methods which establish the consistency between the WKB and \(\varepsilon\) expansions involve a perturbative analysis of the classical dynamics on \(M\) which employs the Green function of the inhomogeneous geodesic deviation equation. The analysis applies when both propagator arguments lie in a region which is convex with respect to the classical motion. A multi-scale analysis in the parameters \(\hbar\), \(\varepsilon\), charge \(\lambda\) and time displacement \(\Delta t\) is performaned, leading to simple basic coefficient functions and their recurrence relations. These propagator representations display explicitly the interplay between the local geometry of the manifold and the effects of the background electromagnetic fields.
scientific article; zbMATH DE number 62641