Wigner Measure Propagation and Conical Singularity for General Initial Data

@article{FermanianKammerer2013WignerMP,
  title={Wigner Measure Propagation and Conical Singularity for General Initial Data},
  author={Clotilde Fermanian-Kammerer and Patrick G{\'e}rard and Caroline Lasser},
  journal={Archive for Rational Mechanics and Analysis},
  year={2013},
  volume={209},
  pages={209-236}
}
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. 
Semiclassical analysis of the Schrödinger equation with conical singularities
In this article we study the propagation of Wigner measures linked to solutions of the Schr{\"o}dinger equation with potentials presenting conical singularities and show that they are transported by
Semiclassical diffraction by conormal potential singularities
We establish propagation of singularities for the semiclassical Schr\"odinger equation, where the potential is conormal to a hypersurface. We show that semiclassical wavefront set propagates along
Effective mass theorems with Bloch modes crossings.
We study a Schr{\"o}dinger equation modeling the dynamics of an electron in a crystal in the asymptotic regime of small wavelength comparable to the characteristic scale of the crystal. Using Floquet
Wavepackets in inhomogeneous periodic media: Effective particle-field dynamics and Berry curvature
We consider a model of an electron in a crystal moving under the influence of an external electric field: Schrodinger’s equation with a potential which is the sum of a periodic function and a general
Semiclassical analysis of the Schrödinger equation with singular potentials
In the first part of this thesis we study the propagation of Wigner measures linked to solutions of the Schrodinger equation with potentials presenting conical singularities and show that they are
Wavepackets in Inhomogeneous Periodic Media: Propagation Through a One-Dimensional Band Crossing
AbstractWe consider a model of an electron in a crystal moving under the influence of an external electric field: Schrödinger’s equation in one spatial dimension with a potential which is the sum of
Computing Semiclassical Quantum Expectations by Husimi Functions
In this thesis we study the quantum propagation of Heisenberg observables for the time-dependent semiclassical Schrödinger equation. We develop a numerical approximation for the evolution of
Wave dynamics in locally periodic structures by multiscale analysis
Wave dynamics in locally periodic structures by multiscale analysis Alexander B. Watson We study the propagation of waves in spatially non-homogeneous media, focusing on Schrödinger’s equation of
Regularized semiclassical limits: linear flows with infinite Lyapunov exponents
Semiclassical asymptotics for linear Schr\"odinger equations with non-smooth potentials give rise to ill-posed formal semiclassical limits. These problems have attracted a lot of attention in the
...
...

References

SHOWING 1-10 OF 27 REFERENCES
Propagation through conical crossings: An asymptotic semigroup
We consider the standard model problem for a conical intersection of electronic surfaces in molecular dynamics. Our main result is the construction of a semi‐group in order to approximate the Wigner
Semiclassical limit of quantum dynamics with rough potentials and well‐posedness of transport equations with measure initial data
In this paper we study the semiclassical limit of the Schrodinger equation. Under mild regularity assumptions on the potential U, which include Born‐Oppenheimer potential energy surfaces in molecular
STRONG AND WEAK SEMICLASSICAL LIMIT FOR SOME ROUGH HAMILTONIANS
We present several results concerning the semiclassical limit of the time-dependent Schrödinger equation with potentials whose regularity does not guarantee the uniqueness of the underlying classical
Sur les mesures de Wigner
We study the properties of the Wigner transform for arbitrary functions in L2 or for hermitian kernels like the so-called density matrices. And we introduce some limits of these transforms for
Propagation through Generic Level Crossings: A Surface Hopping Semigroup
TLDR
A surface hopping semigroup, which asymptotically describes nuclear propagation through crossings of electron energy levels, is constructed and convergence to the true solution is proved with an error of the order of $\varepsilon^{1/8}$, where $\varpsilon$ is the semiclassical parameter.
Homogenization limits and Wigner transforms
We present a theory for carrying out homogenization limits for quadratic functions (called “energy densities”) of solutions of initial value problems (IVPs) with anti-self-adjoint (spatial)
Semiclassical limit for mixed states with singular and rough potentials
We consider the semiclassical limit for the Heisenberg-von Neumann equation with a potential which consists of the sum of a repulsive Coulomb potential, plus a Lipschitz potential whose gradient
Semiclassical analysis of generic codimension 3 crossings
We study a family of solutions to a system of two equations which display a codimension 3 generic crossing: the crossing set is a symplectic submanifold of the phase space. We consider initial data
A semi-classical picture of quantum scattering
— This article is devoted to some singularly perturbed semi-classical asymptotics. It corresponds to a critical case where standard semi-classical techniques do not apply any more. We show how the
...
...