The Wigner function for two dimensional tori: Uniform approximation and projections

Abstract

The Wigner representation of a quantum state, corresponding to a classically integrable Hamiltonian, has been shown to be intimately tied to a classical phase space torus of the same energy. The fact that the semiclassical approximation of the Wigner function there derived turns out to be singular on the torus, as well as on the “Wigner caustic” which contains it, is due to well known limitations of the stationary phase method. The uniform approximation, here derived, does indeed ascribe to the Wigner function a high amplitude along the Wigner caustic, but this is modulated by rapid oscillations except at the torus itself. Asymptotic expansion away from the torus leads back to the semiclassical approximation. Close to the torus the Wigner function is described by a simple transitional approximation which can be resolved into a product of Wigner functions corresponding to one dimensional tori. These results permit one to explicitly project the Wigner function onto any (Lagrangian) coordinate plane so as to obtain the corresponding wave intensity.