Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile

TL;DR

This paper investigates how measures supported on Lipschitz graphs evolve under Hamiltonian flows, analyzing the structure, singularities, and atoms of the transported measures, with applications to the classical limit of quantum mechanics.

Contribution

It provides new estimates on the number of folds and singularities in the transported measures, extending understanding of measure evolution with rough momentum profiles.

Findings

Bounded the number of folds in the support of transported measures.

Characterized conditions for the emergence of atoms and singularities.

Provided examples demonstrating the sharpness of the results.

Abstract

Consider in the phase space of classical mechanics a Radon measure that is a probability density carried by the graph of a Lipschitz continuous (or even less regular) vector field. We study the structure of the push-forward of such a measure by a Hamiltonian flow. In particular, we provide an estimate on the number of folds in the support of the transported measure that is the image of the initial graph by the flow. We also study in detail the type of singularities in the projection of the transported measure in configuration space (averaging out the momentum variable). We study the conditions under which this projected measure can have atoms, and give an example in which the projected measure is singular with respect to the Lebesgue measure and diffuse. We discuss applications of our results to the classical limit of the Schr\"{o}dinger equation. Finally we present various examples and counterexamples showing that our results are sharp.