Multi-scale semiclassical approximations for Schrödinger propagators on manifolds
type: Item
instance of: scholarly article
title: Multi-scale semiclassical approximations for Schrödinger propagators on manifolds
zbMATH Open document ID: 0747.58060
DOI: 10.1016/0003-4916(92)90063-R
author: S. Singh
published in: Annals of Physics
publication date: 1992-09-27T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 58Z05
Mathematics Subject Classification ID: 81Q20
zbMATH DE Number: 62641
zbMATH Keywords: semiclassical approximations
zbMATH Keywords: Schrödinger propagator
zbMATH Keywords: semi-Riemannian manifold
zbMATH Keywords: WKB approximation
MaRDI profile type: Publication
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full work available at URL: 0003-4916(92)90063-r
OpenAlex ID: W2024091818
description: scientific article; zbMATH DE number 62641