Generalized Lagrangian States and Their Propagation in Bargmann Space.

Abstract

The purpose of this work is to introduce and investigate the concept of a generalized Lagrangian state. The author begins with a detailed discussion of relevant background material that focuses on the elementary theory of the Heisenberg group, Bargmann space, and quantization schemes. With the general background in place, the notion of a generalized Lagrangian state, which is an integral in the sense of manifolds that is oscillatory in the limit ћ→0, is presented. These objects are then studied asymptotically with the method of stationary phase, and some basic semiclassical properties are shown. The action of the ring of pseudodifferential operators with polynomial symbol on generalized Lagrangian states is meticulously studied first in one dimension, and then in the general case. One of the main results of this work is to show that when certain types of generalized Lagrangian states are propagated quantum mechanically, with respect to a polynomial Hamiltonian, they evolve into another such generalized Lagrangian state modulo an error term that is semiclassically negligible. Some applications of generalized Lagrangian states to problems in physics and chemistry are presented. The problem of semiclassically approximating the operator kernel of the quantum propagator (with respect to a polynomial Hamiltonian) on Bargmann space by a generalized Lagrangian state is investigated both formally and then rigorously. A class of elements of Bargmann space called semiclassically localized states is introduced, and a rigorous operator estimate is proven with such elements taken as initial conditions. An interesting connection between generalized Lagrangian states and the Hermann-Kluk propagator is discussed. Finally, it is shown that the problem of approximating the kernel of a product of propagators in a ‘forward-backward’ propagation fits into the general framework established to investigate a single propagator.