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

Wigner measures and codimension two crossings

This article gives a semiclassical description of nucleonic propagation through codimension two crossings of electronic energy levels. Codimension two crossings are the simplest energy level crossings, which affect the Born–Oppenheimer approximation in the zeroth order term. The model we study is a two-level Schrödinger equation with a Laplacian as kinetic operator and a matrix-valued linear potential, whose eigenvalues cross, if the two nucleonic coordinates equal zero. We discuss the case of well-localized initial data and obtain a description of the wavefunction’s two-scaled Wigner measure and of the weak limit of its position density, which is valid globally in time

Propagation through Generic Level Crossings: A Surface Hopping Semigroup

We construct a surface hopping semigroup, which asymptotically describes nuclear propagation through crossings of electron energy levels. The underlying time-dependent Schrödinger equation has a matrix-valued potential, whose eigenvalue surfaces have a generic intersection of codimension two, three, or five in Hagedorn's classification. Using microlocal normal forms reminiscent of the Landau–Zener problem, we prove convergence to the true solution with an error of the order $\varepsilon^{1/8}$, where $\varepsilon$ is the semiclassical parameter. We present numerical experiments for an algorithmic realization of the semigroup illustrating the convergence of the algorithm

Single switch surface hopping for molecular dynamics with transitions

A trajectory surface hopping algorithm is proposed, which stems from a mathematically rigorous analysis of propagation through conical intersections of potential energy surfaces. Since nonadiabatic transitions are only performed when a classical trajectory attains one of its local minimal surface gaps, the algorithm is called single switch surface hopping. Numerical experiments for a two mode Jahn–Teller system are presented, which illustrate the asymptotic justification of the method as well as its good performance in the physically relevant parameter range

Semiclassical approximations for Hamiltonians with operator-valued symbols

We consider the semiclassical limit of quantum systems with a Hamiltonian given by the Weyl quantization of an operator valued symbol. Systems composed of slow and fast degrees of freedom are of this form. Typically a small dimensionless parameter $\varepsilon\ll 1$ controls the separation of time scales and the limit $\varepsilon\to 0$ corresponds to an adiabatic limit, in which the slow and fast degrees of freedom decouple. At the same time $\varepsilon\to 0$ is the semiclassical limit for the slow degrees of freedom. In this paper we show that the $\varepsilon$-dependent classical flow for the slow degrees of freedom first discovered by Littlejohn and Flynn, coming from an \epsi-dependent classical Hamilton function and an $\varepsilon$-dependent symplectic form, has a concrete mathematical and physical meaning: Based on this flow we prove a formula for equilibrium expectations, an Egorov theorem and transport of Wigner functions, thereby approximating properties of the quantum system up to errors of order $\varepsilon^2$. In the context of Bloch electrons formal use of this classical system has triggered considerable progress in solid state physics. Hence we discuss in some detail the application of the general results to the Hofstadter model, which describes a two-dimensional gas of non-interacting electrons in a constant magnetic field in the tight-binding approximation.Comment: Final version to appear in Commun. Math. Phys. Results have been strengthened with only minor changes to the proofs. A section on the Hofstadter model as an application of the general theory was added and the previous section on other applications was remove

Nonlinear coherent states and Ehrenfest time for Schrodinger equation

We consider the propagation of wave packets for the nonlinear Schrodinger 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.Comment: 27 page

Localized instabilities of the Wigner equation as a model for the emergence of Rogue Waves

In this paper, we model Rogue Waves as localized instabilities emerging from homogeneous and stationary background wavefields, under NLS dynamics. This is achieved in two steps: given any background Fourier spectrum P(k), we use the Wigner transform and Penrose’s method to recover spatially periodic unstable modes, which we call unstable Penrose modes. These can be seen as generalized Benjamin–Feir modes, and their parameters are obtained by resolving the Penrose condition, a system of nonlinear equations involving P(k). Moreover, we show how the superposition of unstable Penrose modes can result in the appearance of localized unstable modes. By interpreting the appearance of an unstable mode localized in an area not larger than a reference wavelength λ0 as the emergence of a Rogue Wave, a criterion for the emergence of Rogue Waves is formulated. Our methodology is applied to δ spectra, where the standard Benjamin–Feir instability is recovered, and to more general spectra. In that context, we present a scheme for the numerical resolution of the Penrose condition and estimate the sharpest possible localization of unstable modes. Keywords: Rogue Waves; Wigner equation; Nonlinear Schrodinger equation; Penrose modes; Penrose conditio

Energy transport by acoustic modes of harmonic lattices

We study the large scale evolution of a scalar lattice excitation u which satisfies a discrete wave equation in three dimensions, u(t)(gamma) = -Sigma gamma' alpha(gamma - gamma') u(t)(gamma'), where gamma, gamma' epsilon Z(3) are lattice sites. We assume that the dispersion relation. associated to the elastic coupling constants alpha(gamma - gamma') is acoustic; i. e., it has a singularity of the type |k| near the vanishing wave vector, k = 0. To derive equations describing the macroscopic energy transport, we employ a related multiscale Wigner transform and a scale parameter epsilon > 0. The spatial and temporal scales of the Wigner transform are related to the corresponding lattice parameters via a scaling by e. In the continuum limit, which is achieved by sending the parameter e to 0, the Wigner transform disintegrates into three different limit objects: the Wigner transform of a rescaled weak-L-2 limit, an H-measure, and a Wigner measure. The first two provide the finer resolution of the energy concentrating at k = 0 so that a set of closed evolution equations may arise. We demonstrate that these three limit objects satisfy a set of decoupled transport equations: a wave equation for the weak limit, a geometric optics transport equation for the H-measure limit, and a dispersive transport equation for the standard limiting Wigner measure. This yields a complete characterization of macroscopic energy transport in harmonic lattices with regular acoustic dispersion relations