On the Dynamics of WKB Wave Functions Whose Phase are Weak KAM Solutions of H–J Equation

Abstract

In the framework of toroidal Pseudodifferential operators on the flat torus T n := (R/2π Z) n we begin by proving the closure under composition for the class ofwe exhibit the toroidal version of the equation for the Wigner transform of the solution of the Schrödinger equation. Moreover, we prove the convergence (in a weak sense) of the Wigner transform of the solution of the Schrödinger equation to the solution of the Liouville equation on T n × R n written in the measure sense. These results are applied to the study of some WKB type wave functions in the Sobolev space H 1 (T n ; C) with phase functions in the class of Lipschitz continuous weak KAM solutions (positive and negative type) of the Hamilton-Jacobi equation 12 |P + ∇ x v(P, x)| 2 + V (x) =H (P) for P ∈ Z n with > 0, and to the study of the backward and forward time propagation of the related Wigner measures supported on the graph of P + ∇ x v.

Schrödinger dynamics and optimal transport of measures on the torus

“…as firstly shown in [13] within the euclidean setting (and many others under various assumptions, see [3] and the references therein) and recently in [16] within the toroidal setting. The second main ingredient is that all the semiclassical measures of ϕ as in (2) take the form…”

A weak KAM approach to the periodic stationary Hartree equation

“…To conclude, we recall that the application of KAM theory or weak KAM theory into the semiclassical Analysis of Schrödinger operators can be given for various problems: like the study of WKB quasimodes, Wigner measures or the asymptotics of the spectrum (see [3,4,6,8,21,23,[34][35][36]43] and references therein). For the use of Fourier Integral Operators and solutions of Hamilton-Jacobi equation to represent the unitary operator solving the quantum dynamics we address to [19,22] and references therein.…”

Difference equations and pseudo-differential operators onZn

“…The toroidal quantization has since led to many further developments and applications, see e.g. [LNJP16,PZ14b,PZ14a,Car14], to mention a few. So, the described link leads to a way of transferring results from the toroidal setting to the lattice.…”