Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice. III: From ab initio models to WKB for Schrödinger-Poisson

DOI10.1016/j.jcp.2005.05.020zbMath1081.81041OpenAlexW2020289390MaRDI QIDQ2572802

Norbert J. Mauser, Laurent Gosse

Publication date: 4 November 2005

Published in: Journal of Computational Physics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.jcp.2005.05.020

zbMATH Keywords

homogenizationperiodic potentialsemiclassical limitmoment methodVlasov-Poisson equationnonstrictly hyperbolic systemsweakly nonlinear equationHartree ansatz

Mathematics Subject Classification ID

Statistical mechanics of crystals (82D25) Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics (81Q05) Hyperbolic conservation laws (35L65) Finite difference methods for initial value and initial-boundary value problems involving PDEs (65M06) Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory (81Q20)

Related Items (12)

A Bloch decomposition-based stochastic Galerkin method for quantum dynamics with a random external potential ⋮ A field-space-based level set method for computing multi-valued solutions to 1D Euler-Poisson equations ⋮ Semi-classical limit of Schrödinger--Poisson equations in space dimension \(n\geqslant 3\) ⋮ Multi-scale dynamics of the interaction between waves and mean flows: From nonlinear WKB theory to gravity-wave parameterizations in weather and climate models ⋮ Methods for solving singular perturbation problems arising in science and engineering ⋮ Impurity bands and quasi-Bloch waves for a one-dimensional model of modulated crystal ⋮ Computational high frequency waves through curved interfaces via the Liouville equation and geometric theory of diffraction ⋮ Bloch decomposition-based Gaussian beam method for the Schrödinger equation with periodic potentials ⋮ An accelerated algorithm for 2D simulations of the quantum ballistic transport in nanoscale MOSFETs ⋮ A Bloch band based level set method for computing the semiclassical limit of Schrödinger equations ⋮ Mathematical and computational methods for semiclassical Schrödinger equations ⋮ THE VLASOV–POISSON EQUATIONS AS THE SEMICLASSICAL LIMIT OF THE SCHRÖDINGER–POISSON EQUATIONS: A NUMERICAL STUDY

Cites Work

• Unnamed Item
• Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice. I: Homogeneous problems
• Using \(K\)-branch entropy solutions for multivalued geometric optics computations
• Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice. II: Impurities, confinement and Bloch oscillations
• Transport equation and Cauchy problem for BV vector fields
• Semi-classical asymptotics in solid state physics
• On Wigner measures
• A kinetic formulation for multi-branch entropy solutions of scalar conservation laws
• Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs Wigner
• Semi-classical limit of a Schrödinger equation for a stratified material
• Derivation of the Euler equations from a caricature of Coulomb interaction
• Derivation of the Schrödinger-Poisson equation from the quantum \(N\)-body problem
• The Schrödinger-Poisson-\(X\alpha\) equation
• On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime
• Semiclassical limit of the nonlinear Schrödinger-Poisson equation with subcritical initial data.
• Finite moment problems and applications to multiphase computations in geometric optics
• Semiclassical asymptotics for weakly nonlinear Bloch waves
• Effective one particle quantum dynamics of electrons: a numerical study of the Schrödinger-Poisson-X\(_\alpha\) model
• A level set method for the computation of multivalued solutions to quasi-linear hyperbolic PDEs and Hamilton-Jacobi equations
• Computational high-frequency wave propagation using the level set method with applications to the semi-classical limit of Schrödinger equations
• Some new results in multiphase geometrical optics
• Generalization of Band Theory to Include Self-Energy Corrections
• Averaged Multivalued Solutions for Scalar Conservation Laws
• Semiclassical Mechanics
• Recovering a function from a finite number of moments
• Sticky Particles and Scalar Conservation Laws
• A Wigner-function approach to (semi)classical limits: Electrons in a periodic potential
• Homogenization limits and Wigner transforms
• Computational high frequency wave propagation
• Semiclassical asymptotics in magnetic Bloch bands
• Quantum mechanics of electrons in crystal lattices
• Convergence of a Semi-Lagrangian Scheme for the One-Dimensional Vlasov--Poisson System
• Semiclassical limit for the Schrödinger‐Poisson equation in a crystal
• The limit from the Schrödinger‐Poisson to the Vlasov‐Poisson equations with general data in one dimension
• The evolution operator of the Hartree-type equation with a quadratic potential
• TWO MOMENT SYSTEMS FOR COMPUTING MULTIPHASE SEMICLASSICAL LIMITS OF THE SCHRÖDINGER EQUATION
• Coupling one-dimensional time-dependent classical and quantum transport models
• Identification of Asymptotic Decay to Self-Similarity for One-Dimensional Filtration Equations
• HANS BETHE'S CONTRIBUTIONS TO SOLID-STATE PHYSICS
• (Semi)Classical Limit of the Hartree Equation with Harmonic Potential
• Semiclassical limit for the Schrödinger equation with a short scale periodic potential.
• Renormalized solutions to the Vlasov equation with coefficients of bounded variation

This page was built for publication: Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice. III: From ab initio models to WKB for Schrödinger-Poisson