Clotilde Fermanian-Kammerer

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.

Nonlinear Coherent States and Ehrenfest Time for Schrödinger Equation

We consider the propagation of wave packets for the nonlinear Schrödinger equation, in the semi-classical limit. We establish the existence of a critical size for the initial data, in terms of the Planck constant: if the initial data are too small, the nonlinearity is negligible up to the Ehrenfest time. If the initial data have the critical size, then at leading order the wave function propagates like a coherent state whose envelope is given by a nonlinear equation, up to a time of the same order as the Ehrenfest time. We also prove a nonlinear superposition principle for these nonlinear wave packets.

Wigner Measure Propagation and Conical Singularity for General Initial Data

We study the evolution of Wigner measures of a family of solutions of a Schrödinger 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.

Single switch surface hopping for molecular quantum dynamics

The aim of this text is to present a surface hopping approximation for molecular quantum dynamics obeying a Schrödinger equation with crossing eigenvalue surfaces. After motivating Schrödinger equations with matrix valued potentials, we describe the single switch algorithm and present some numerical results. Then we discuss the algorithm’s mathematical justification and describe extensions to more general situations, where three eigenvalue surfaces intersect or the eigenvalues are of multiplicity two. We emphasize the generality of this surface hopping approximation for non-adiabatic transitions.

A Nonlinear Landau-Zener Formula

We consider a system of two coupled ordinary differential equations which appears as an envelope equation in Bose–Einstein Condensation. This system can be viewed as a nonlinear extension of the celebrated model introduced by Landau and Zener. We show how the nonlinear system may appear from different physical models. We focus our attention on the large time behavior of the solution. We show the existence of a nonlinear scattering operator, which is reminiscent of long range scattering for the nonlinear Schrödinger equation, and which can be compared with its linear counterpart.

Dispersive estimates for the Schrödinger 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 Schrodinger 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 Schrodinger operator on a space of the same dimension k as the radical of the canonical skew-symmetric form, which suggests a decay rate of 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.