Propagators and Scattering Theory

Christian Gérard, Izabella Łaba

Fundamentals Geometrical methods I The Mourre theory Basic propagation estimates Geometrical methods II Wave operators and scattering theory Open problems Appendix Bibliography Index.

J S Briggs, James M. Feagin

Conventional scattering theory is incomplete in that it does not adequately describe the behaviour of the wave function at macroscopic distances from the scattering reaction volume. In scattering experiments particles are incident from sources at macroscopic distance and measured at macroscopic distance from the microscopic reaction volume. Standard experimental procedure is to describe asymptotic particle motion by classical mechanics. However, this transition to classical mechanics is not accounted for in standard quantum scattering theory. Hence, we revisit conventional scattering theory with the introduction of semiclassical propagators to describe the wave function in both incident and outgoing asymptotic scattering channels. This description leads to the manifestation of classical trajectories in the quantum wave function both before and after the collision. It includes the case of extraction with external fields as in modern multi-particle detectors. We derive the asymptotic Kemble imaging theorem (IT) limit to demonstrate how formal quantum scattering theory contains implicit classical mechanics threaded throughout the asymptotic wave functions.

Günther Ludwig

Marcos Mariño

It presents the unitary time evolution operator and its integral kernel in the space representation, also known as the quantum-mechanical propagator. It introduces the resolvent operator and its analytical properties, and it discusses in some detail scattering theory in one dimension.

Harald Friedrich