Remarks on Time-dependent Schrödinger Equations, Bound States, and Coherent States

Abstract

It has been well known from the beginning of quantum theory that there exist deep connections between the time evolution of a classical Hamiltonian system and the bound states for the Schrodinger equation, in particular in the semiclassical regime. These connections are well understood for integrable systems (Bohr—Sommerfeld quantization rules). But for more intricate systems (like classically chaotic Hamiltonian) the mathematical analysis of the bound states is much more difficult and there are few rigorous mathematical results. In this paper our goal is to revisit some of these results and to show that they can be proven, and sometimes improved, by using essentially two technics: the Wigner—Weyl calculus and the propagation of observables on one side, the propagation of coherent states on the other side. We want to emphasize that in our approach we get rather explicit estimates in terms of classical dynamics