Mathematical and computational methods for semiclassical Schrödinger equations

Semiclassical Singularity Propagation Property for Schrödinger Equations

Abstract We consider Schrödinger equations with variable coefficients, and it is supposed to be a long-range type perturbation of the flat Laplacian on Rn. We characterize the wave front set of solutions to Schrödinger equations in terms of the initial state. Then it is shown that the singularity propagates following the classical flow, and it is formulated in a semiclassical setting. Methods a...

Multiple Semiclassical Standing Waves for a Class of Nonlinear Schrödinger Equations

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.

Stability of Spectral Eigenspaces in Nonlinear Schrödinger Equations

We consider the time-dependent non linear Schrödinger equations with a double well potential in dimensions d = 1 and d = 2. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest two eigenvalues of the linear operator is almost invariant for any time.

The Vlasov-Poisson Equations as the Semiclassical Limit of the Schrödinger-Poisson Equations: A Numerical Study

In this paper, we numerically study the semiclassical limit of the SchrödingerPoisson equations as a selection principle for the weak solution of the VlasovPoisson in one space dimension. Our numerical results show that this limit gives the weak solution that agrees with the zero diffusion limit of the Fokker-Planck equation. We also numerically justify the multivalued solution given by a momen...