Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile

Abstract

Consider a monokinetic probability measure on the phase space $${{\bf R}^N_{x} \times {\bf R}^N_{\xi}}$$ , i.e. $${\mu^{\rm {in}} = \rho^{\rm {in}}(x)\delta(\xi - U^{\rm {in}}(x))}$$ where U in is a vector field on R N and ρ in a probability density on R N . Let Φ t be a Hamiltonian flow on R N × R N . In this paper, we study the structure of the transported measure $${\mu(t) := \Phi_t\#\mu^{\rm {in}}}$$ and of its integral in the ξ variable denoted ρ(t). In particular, we give estimates on the number of folds in $${\Phi_t({\rm graph of} U^{\rm {in}})}$$ , on which μ ( t) is concentrated. We explain how our results can be applied to investigate the classical limit of the Schrödinger equation by using the formalism of Wigner measures. Our formalism includes initial momentum profiles U in with much lower regularity than required by the WKB method. Finally, we discuss a few examples showing that our results are sharp.