Semiclassical Singularity Propagation Property for Schrödinger Equations

Propagation of the Homogeneous Wave Front Set for Schrödinger Equations∗

In this paper we study the propagation of singularity for Schrödingertype equations with variable coefficients. We introduce a new notion of wave propagation set, the homogeneous wave front set, and it propagates along straight lines with finite speed away from x 6= 0. Then we show that it is related to the wave front set in a natural way. These results may be considered as a refinement of the ...

Standing waves for nonlinear Schrödinger equations with singular potentials

We study semiclassical states of nonlinear Schrödinger equations with anisotropic type potentials which may exhibit a combination of vanishing and singularity while allowing decays and unboundedness at infinity. We give existence of spike type standing waves concentrating at the singularities of the potentials. © 2008 Elsevier Masson SAS. All rights reserved. Résumé Nous étudions les états semi...

Multiple Semiclassical Standing Waves for a Class of Nonlinear Schrödinger Equations

Semiclassical Models for the Schrödinger Equation with Periodic Potentials and Band Crossings∗

The Bloch decomposition plays a fundamental role in the study of quantum mechanics and wave propagation in periodic media. Most of the homogenization theory developed for the study of high frequency or semiclassical limit for these problems assumes no crossing of the Bloch bands, resulting in classical Liouville equations in the limit along each Bloch band. In this article, we derive semiclassi...

Soliton Dynamics for Fractional Schrödinger Equations

We investigate the soliton dynamics for the fractional nonlinear Schrödinger equation by a suitable modulational inequality. In the semiclassical limit, the solution concentrates along a trajectory determined by a Newtonian equation depending of the fractional diffusion parameter.