Semiclassical limit of quantum dynamics with rough potentials and well‐posedness of transport equations with measure initial data

Abstract

In this paper we study the semiclassical limit of the Schrödinger equation. Under mild regularity assumptions on the potential U , which include Born-Oppenheimer potential energy surfaces in molecular dynamics, we establish asymptotic validity of classical dynamics globally in space and time for "almost all" initial data, with respect to an appropriate reference measure on the space of initial data. In order to achieve this goal we prove existence, uniqueness, and stability results for the flow in the space of measures induced by the continuity equation.

Finite volume schemes for the approximation via characteristics of linear convection equations with irregular data

“…Works that are related to the framework of measures are [7,8,6,14,5]. In [9], the approximation of (6) has been done by an explicit nite volume scheme, in the case where the functions X are C 1 solutions to (5); this scheme is identical, in the simplied case of a pure transport (1), to that presented in Section 2. The convergence proof of this scheme is provided in [9], under the condition that the space step tends to zero faster than the time step (this convergence result is again provided here under more general hypotheses made on X, in Section 2.2).…”

“…Note that many recent studies have been performed on the resolution of equation (6), exploring in particular the existence and uniqueness of a solution, depending on the regularity of a [3,4]. Works that are related to the framework of measures are [7,8,6,14,5]. In [9], the approximation of (6) has been done by an explicit nite volume scheme, in the case where the functions X are C 1 solutions to (5); this scheme is identical, in the simplied case of a pure transport (1), to that presented in Section 2.…”

Strong and Weak Semiclassical Limit for Some Rough Hamiltonians

“…Coulomb/piecewise smooth potential etc) as opposed to a general smoothness class [17,13], and/or need some additional, non-trivial condition (non-concentration, non-interference etc) [17,8]. Another type of results is for whole random populations of initial data [2] (which in particular can be even weaker than a result applying to "almost all" initial data; i.e. there might be no way to just choose "almost any" initial datum at t = 0 and keep track of it; the conclusions apply the population as a whole).…”

“…More specifically, semiclassical limits with rough potentials were considered recently in [2] where the hypotheses on the potential are, roughly speaking, that it has a generic part with BV loc gradient, plus possibly a repulsive Coulomb part. The setting involves a random population of initial data, and it is shown that the corresponding population of solutions at a later time tends weakly to the push-forward by the Liouville equation of a population of measures.…”

“…to a delta function in phase-space, a fact consistent with the weak limit strategy and the fact that the flow is not defined everywhere. The present work therefore concerns less general potentials than those treated in [2,8], (still giving rise to ill-posed classical dynamics of course), but can deal with initial data concentrating to point-supported measures.…”