Semiclassical limit for mixed states with singular and rough potentials

Abstract

We consider the semiclassical limit for the Heisenberg-von Neumann equation with a potential which consists of the sum of a repulsive Coulomb potential, plus a Lipschitz potential whose gradient belongs to BV ; this assumption on the potential guarantees the well posedness of the Liouville equation in the space of bounded integrable solutions. We find sufficient conditions on the initial data to ensure that the quantum dynamics converges to the classical one. More precisely, we consider the Husimi functions of the solution of the Heisenberg-von Neumann equation, and under suitable assumptions on the initial data we prove that they converge, as ε → 0, to the unique bounded solution of the Liouville equation (locally uniformly in time).

WKB Analysis of Bohmian Dynamics

“…The Wigner measure (also called semiclassical defect measure) is a well-established tool in semiclassical analysis, which allows us to efficiently describe the classical limit of quantum mechanical observables; cf. [2,24,28,42,50]. For completeness, the definition of the Wigner measure and its main properties will be recalled in Section 4.2.…”

Strong and Weak Semiclassical Limit for Some Rough Hamiltonians

“…In most cases the conclusions make use of a particular form of singularity (e.g. 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.…”

“…The classical propagation problem for the random family of measures is shown to be well-posed by virtue of the Ambrosio-Di Perna-Lions theory [1,7]. In [8] similar core ideas were used with deterministic mixed states. In that context, an averaging condition forbids initial data concentrating e.g.…”

“…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.…”

Almost everywhere well-posedness of continuity equations with measure initial data

“…In [4], relying also on some a priori estimates of [3] (see also [9]), the authors consider a potential U which can be written as the sum of a repulsive Coulomb potential U s plus a bounded Lipschitz iteration term U b with ∇U b ∈ B V loc . We observe that in this case Eq.…”