A Wigner-function approach to (semi)classical limits: Electrons in a periodic potential

Abstract

A rigorous derivation of the semiclassical Liouville equation for electrons which move in a crystal lattice (without the influence of an external field) is presented herein. The approach is based on carrying out the semiclassical limit in the band-structure Wigner equation. The semiclassical macroscopic densities are also obtained as limits of the corresponding quantum quantities.

On Time-Splitting Spectral Approximations for the Schrödinger Equation in the Semiclassical Regime

“…This limit process was considered rigorously in [9,13]. We remark that the definition and analysis of Wigner transforms can easily be adapted to x-periodic wave functions (by replacing Fourier transforms by Fourier series); for the sake of simplicity we shall, however, consider only the whole space case (1.1), (1.2) here.…”

“…The analysis of the so-called semiclassical limit is a mathematically rather complex issue. Much progress has been made recently in this area, particularly by the introduction of tools from microlocal analysis, such as defect measures [8], H-measures [19], and Wigner measures [7,9,13]. These techniques have provided powerful technical tools for exploiting properties of the Schrödinger equation in the semiclassical limit regime, allowing the passage to the limit ε → 0 in the macroscopic densities by revealing an underlying kinetic structure.…”

Numerical Methods for the Nonlinear Schrödinger Equation with Nonzero Far-field Conditions

“…Much progress has been made recently in analytical understanding semiclassical limits of the linear Schödinger equation (i.e. f (ρ) ≡ 0 in (1.1)), particularly by the introduction of tools from microlocal analysis, such as defect measures [13], H-measures [30], and Wigner measures [12,14,24]. These techniques have not been successfully extended to the semiclassical limit of the NLS, except that the 1D defocusing (cubically) NLS (1.1) was solved by using techniques of inverse scattering [16,17].…”

Multiphase semiclassical approximation of the one-dimensional harmonic crystal: I. The periodic case

“…We shall try to show that the smaller ε, the closer the solutions' observables become. However, only weak convergence of observables can generally be hoped for [17], hence we shall look at the L 1 (R) norm of the antiderivative of the difference between position densities:…”

“…[17]) and ensure that for any fixed t and κ ∈B, there exists a complete set of eigenfunctions Ψ n,κ (t, .) ∈ L 2 (0, 2π/k) with countably many eigenvalues…”