München 1997 – scientific programme

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MP: Theoretische und Mathematische Grundlagen der Physik

MP 8: Endliche Quantensysteme

MP 8.1: Fachvortrag

Thursday, March 20, 1997, 14:00–14:20, HS 110

Semi-classical scattering in a gauge field — •Raymond Brummelhuis, P. Levy-Bruhl, and J. Nourrigat — Univ. de Reims, dept. de math., BP 1039, 51687 REIMS, FRANCE

We study the semi-classical limit of stationary scattering theory for a Schrödinger operator in a non-commutative gauge field, associated to some compact Lie-group and it’s representations. As is well-known by the work of several authors during the 80’s, taking the semi-classical limit here not only means letting ‘Planck’s constant’ ℏ tend to 0, but also letting at the same time grow the representation which occurs in the definition of such a Schrödinger operator, by taking it’s maximal weight equal to 1/ℏ times some fixed weight (1/ℏ being limited to the positive integers). An example is given by the Pauli Hamiltonian of a spin L particle with mass and charge proportional to L , with 1/L as semi-classical parameter.

Our main result is a complete asymptotic expansion of the scatte- ring amplitude (or, more precisely, of it’s matrix elements with respect to the coherent states associated to the group), together with an explicit formula for the main term coefficient in terms of the Wong flow, which is the classical Hamilton flow associated to the quantum system under consideration.