Wigner Measure Propagation and Conical Singularity for General Initial Data

We study the evolution of Wigner measures of a family of solutions of a Schr\"odinger equation with a scalar potential displaying a conical singularity. Under a genericity assumption, classical trajectories exist and are unique, thus the question of the propagation of Wigner measures along these trajectories becomes relevant. We prove the propagation for general initial data.Comment: 24 pages, 1 figur

Defect measures on graded lie groups

In this article, we define a generalisation of microlocal defect measures (also known as H-measures) to the setting of graded nilpotent Lie groups. This requires to develop the notions of homogeneous symbols and classical pseudo-differential calculus adapted to this setting and defined via the representations of the groups. Our method relies on the study of the C *-algebra of 0-homogeneous symbols. Then, we compute microlocal defect measures for concentrating and oscillating sequences, which also requires to investigate the notion of oscillating sequences in graded Lie groups. Finally, we discuss compacity compactness approaches in the context of graded nilpotent Lie groups

Analysis of the Energy Decay of a Degenerated Thermoelasticity System

In this paper, we study a system of thermoelasticity with a degenerated second order operator in the Heat equation. We analyze the evolution of the energy density of a family of solutions. We consider two cases: when the set of points where the ellipticity of the Heat operator fails is included in a hypersurface and when it is an open set. In the first case and under special assumptions, we prove that the evolution of the energy density is the one of a damped wave equation: propagation along the rays of geometric optic and damping according to a microlocal process. In the second case, we show that the energy density propagates along rays which are distortions of the rays of geometric optic.Comment: 28 page

A kinetic model for the transport of electrons in a graphene layer

In this article, we propose a new numerical model for computation of the transport of electrons in a graphene device. The underlying quantum model for graphene is a massless Dirac equation, whose eigenvalues display a conical singularity responsible for non adiabatic transitions between the two modes. We first derive a kinetic model which takes the form of two Boltzmann equations coupled by a collision operator modeling the non-adiabatic transitions. This collision term includes a Landau-Zener transfer term and a jump operator whose presence is essential in order to ensure a good energy conservation during the transitions. We propose an algorithmic realization of the semi-group solving the kinetic model, by a particle method. We give analytic justification of the model and propose a series of numerical experiments studying the influences of the various sources of errors between the quantum and the kinetic models

Long-time dynamics of completely integrable Schr\"odinger flows on the torus

In this article, we are concerned with long-time behaviour of solutions to a semi-classical Schr\"odinger-type equation on the torus. We consider time scales which go to infinity when the semi-classical parameter goes to zero and we associate with each time-scale the set of semi-classical measures associated with all possible choices of initial data. We emphasize the existence of a threshold: for time-scales below this threshold, the set of semi-classical measures contains measures which are singular with respect to Lebesgue measure in the "position" variable, while at (and beyond) the threshold, all the semi-classical measures are absolutely continuous in the "position" variable.Comment: 41 page

Semiclassical Completely Integrable Systems : Long-Time Dynamics And Observability Via Two-Microlocal Wigner Measures

We look at the long-time behaviour of solutions to a semi-classical Schr\"odinger equation on the torus. We consider time scales which go to infinity when the semi-classical parameter goes to zero and we associate with each time-scale the set of semi-classical measures associated with all possible choices of initial data. On each classical invariant torus, the structure of semi-classical measures is described in terms of two-microlocal measures, obeying explicit propagation laws. We apply this construction in two directions. We first analyse the regularity of semi-classical measures, and we emphasize the existence of a threshold : for time-scales below this threshold, the set of semi-classical measures contains measures which are singular with respect to Lebesgue measure in the "position" variable, while at (and beyond) the threshold, all the semi-classical measures are absolutely continuous in the "position" variable, reflecting the dispersive properties of the equation. Second, the techniques of two- microlocal analysis introduced in the paper are used to prove semiclassical observability estimates. The results apply as well to general quantum completely integrable systems.Comment: This article contains and develops the results of hal-00765928. arXiv admin note: substantial text overlap with arXiv:1211.151

Dispersive estimates for the Schr\"odinger operator on step 2 stratified Lie groups

The present paper is dedicated to the proof of dispersive estimates on stratified Lie groups of step 2, for the linear Schr\"odinger equation involving a sublaplacian. It turns out that the propagator behaves like a wave operator on a space of the same dimension p as the center of the group, and like a Schr\"odinger operator on a space of the same dimension k as the radical of the canonical skew-symmetric form, which suggests a decay with exponant -(k+p-1)/2. In this article, we identify a property of the canonical skew-symmetric form under which we establish optimal dispersive estimates with this rate. The relevance of this property is discussed through several examples

On the time evolution of Wigner measures for Schrodinger equations

In this survey, our aim is to emphasize the main known limitations to the use of Wigner measures for Schrodinger equations. After a short review of successful applications of Wigner measures to study the semi-classical limit of solutions to Schrodinger equations, we list some examples where Wigner measures cannot be a good tool to describe high frequency limits. Typically, the Wigner measures may not capture effects which are not negligible at the pointwise level, or the propagation of Wigner measures may be an ill-posed problem. In the latter situation, two families of functions may have the same Wigner measures at some initial time, but different Wigner measures for a larger time. In the case of systems, this difficulty can partially be avoided by considering more refined Wigner measures such as two-scale Wigner measures; however, we give examples of situations where this quadratic approach fails.Comment: Survey, 26 page