Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice I. Homogeneous problems

Laurent Gosse; Peter A. Markowich

doi:10.1016/j.jcp.2003.12.004

Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice I. Homogeneous problems

Laurent Gosse^*, Peter A. Markowich

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

25 Scopus citations

Abstract

We present a computational approach for the WKB approximation of the wave function of an electron moving in a periodic one-dimensional crystal lattice. We derive a nonstrictly hyperbolic system for the phase and the intensity where the flux functions originate from the Bloch spectrum of the Schrödinger operator. Relying on the framework of the multibranch entropy solutions introduced by Brenier and Corrias, we compute efficiently multiphase solutions using well adapted and simple numerical schemes. In this first part we present computational results for vanishing exterior potentials which demonstrate the effectiveness of the proposed method.

Original language	English (US)
Pages (from-to)	387-417
Number of pages	31
Journal	Journal of Computational Physics
Volume	197
Issue number	2
DOIs	https://doi.org/10.1016/j.jcp.2003.12.004
State	Published - Jul 1 2004
Externally published	Yes

Keywords

Homogenization
Moment method
Non-strictly hyperbolic systems
Periodic potential
Semiclassical limit
Vlasov equation

ASJC Scopus subject areas

Numerical Analysis
Modeling and Simulation
Physics and Astronomy (miscellaneous)
General Physics and Astronomy
Computer Science Applications
Computational Mathematics
Applied Mathematics

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10.1016/j.jcp.2003.12.004

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title = "Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice I. Homogeneous problems",

abstract = "We present a computational approach for the WKB approximation of the wave function of an electron moving in a periodic one-dimensional crystal lattice. We derive a nonstrictly hyperbolic system for the phase and the intensity where the flux functions originate from the Bloch spectrum of the Schr{\"o}dinger operator. Relying on the framework of the multibranch entropy solutions introduced by Brenier and Corrias, we compute efficiently multiphase solutions using well adapted and simple numerical schemes. In this first part we present computational results for vanishing exterior potentials which demonstrate the effectiveness of the proposed method.",

keywords = "Homogenization, Moment method, Non-strictly hyperbolic systems, Periodic potential, Semiclassical limit, Vlasov equation",

author = "Laurent Gosse and Markowich, {Peter A.}",

note = "Funding Information: Both authors were partially supported by the EEC network #HPRN-CT-2002-00282, the “Modelling and Computations in Wave Propagation” contract #HPMD-CT-2001-00121 and the Wittgenstein 2000 Award of P.A. Markowich funded by the Austrian Research Fund FWF. Also support from the Austrian FWF Project P14876 No. 4 is acknowledged. ",

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AU - Gosse, Laurent

AU - Markowich, Peter A.

N1 - Funding Information: Both authors were partially supported by the EEC network #HPRN-CT-2002-00282, the “Modelling and Computations in Wave Propagation” contract #HPMD-CT-2001-00121 and the Wittgenstein 2000 Award of P.A. Markowich funded by the Austrian Research Fund FWF. Also support from the Austrian FWF Project P14876 No. 4 is acknowledged.

PY - 2004/7/1

Y1 - 2004/7/1

N2 - We present a computational approach for the WKB approximation of the wave function of an electron moving in a periodic one-dimensional crystal lattice. We derive a nonstrictly hyperbolic system for the phase and the intensity where the flux functions originate from the Bloch spectrum of the Schrödinger operator. Relying on the framework of the multibranch entropy solutions introduced by Brenier and Corrias, we compute efficiently multiphase solutions using well adapted and simple numerical schemes. In this first part we present computational results for vanishing exterior potentials which demonstrate the effectiveness of the proposed method.

AB - We present a computational approach for the WKB approximation of the wave function of an electron moving in a periodic one-dimensional crystal lattice. We derive a nonstrictly hyperbolic system for the phase and the intensity where the flux functions originate from the Bloch spectrum of the Schrödinger operator. Relying on the framework of the multibranch entropy solutions introduced by Brenier and Corrias, we compute efficiently multiphase solutions using well adapted and simple numerical schemes. In this first part we present computational results for vanishing exterior potentials which demonstrate the effectiveness of the proposed method.

KW - Homogenization

KW - Moment method

KW - Non-strictly hyperbolic systems

KW - Periodic potential

KW - Semiclassical limit

KW - Vlasov equation

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U2 - 10.1016/j.jcp.2003.12.004

DO - 10.1016/j.jcp.2003.12.004

M3 - Article

AN - SCOPUS:3242737651

SN - 0021-9991

VL - 197

SP - 387

EP - 417

JO - Journal of Computational Physics

JF - Journal of Computational Physics

IS - 2

ER -

Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice I. Homogeneous problems

Abstract

Keywords

ASJC Scopus subject areas

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