Mathematical and computational methods for semiclassical Schrödinger equations

Shi Jin; Peter Markowich; Christof Sparber

doi:10.1017/S0962492911000031

Mathematical and computational methods for semiclassical Schrödinger equations

Shi Jin^*, Peter Markowich, Christof Sparber

^*Corresponding author for this work

Computer, Electrical and Mathematical Sciences and Engineering

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

118 Scopus citations

Abstract

We consider time-dependent (linear and nonlinear) Schrödinger equations in a semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive models whose solutions exhibit high-frequency oscillations. The design of efficient numerical methods which produce an accurate approximation of the solutions, or at least of the associated physical observables, is a formidable mathematical challenge. In this article we shall review the basic analytical methods for dealing with such equations, including WKB asymptotics, Wigner measure techniques and Gaussian beams. Moreover, we shall give an overview of the current state of the art of numerical methods (most of which are based on the described analytical techniques) for the Schrödinger equation in the semiclassical regime.

Original language	English (US)
Pages (from-to)	121-209
Number of pages	89
Journal	Acta Numerica
Volume	20
DOIs	https://doi.org/10.1017/S0962492911000031
State	Published - Apr 2011

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: Partially supported by NSF grant no. DMS-0608720, NSF FRG grant DMS-0757285,a Van Vleck Distinguished Research Prize and a Vilas Associate Award from the Universityof Wisconsin–Madison.Supported by a Royal Society Wolfson Research Merit Award and by KAUST througha Investigator Award KUK-I1-007-43.Partially supported by the Royal Society through a University Research Fellowship.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

Numerical Analysis
General Mathematics

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10.1017/S0962492911000031

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title = "Mathematical and computational methods for semiclassical Schr{\"o}dinger equations",

abstract = "We consider time-dependent (linear and nonlinear) Schr{\"o}dinger equations in a semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive models whose solutions exhibit high-frequency oscillations. The design of efficient numerical methods which produce an accurate approximation of the solutions, or at least of the associated physical observables, is a formidable mathematical challenge. In this article we shall review the basic analytical methods for dealing with such equations, including WKB asymptotics, Wigner measure techniques and Gaussian beams. Moreover, we shall give an overview of the current state of the art of numerical methods (most of which are based on the described analytical techniques) for the Schr{\"o}dinger equation in the semiclassical regime.",

author = "Shi Jin and Peter Markowich and Christof Sparber",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUK-I1-007-43 Acknowledgements: Partially supported by NSF grant no. DMS-0608720, NSF FRG grant DMS-0757285,a Van Vleck Distinguished Research Prize and a Vilas Associate Award from the Universityof Wisconsin–Madison.Supported by a Royal Society Wolfson Research Merit Award and by KAUST througha Investigator Award KUK-I1-007-43.Partially supported by the Royal Society through a University Research Fellowship. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

year = "2011",

month = apr,

doi = "10.1017/S0962492911000031",

language = "English (US)",

volume = "20",

pages = "121--209",

journal = "Acta Numerica",

issn = "0962-4929",

publisher = "Cambridge University Press",

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AU - Jin, Shi

AU - Markowich, Peter

AU - Sparber, Christof

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUK-I1-007-43 Acknowledgements: Partially supported by NSF grant no. DMS-0608720, NSF FRG grant DMS-0757285,a Van Vleck Distinguished Research Prize and a Vilas Associate Award from the Universityof Wisconsin–Madison.Supported by a Royal Society Wolfson Research Merit Award and by KAUST througha Investigator Award KUK-I1-007-43.Partially supported by the Royal Society through a University Research Fellowship. This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2011/4

Y1 - 2011/4

N2 - We consider time-dependent (linear and nonlinear) Schrödinger equations in a semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive models whose solutions exhibit high-frequency oscillations. The design of efficient numerical methods which produce an accurate approximation of the solutions, or at least of the associated physical observables, is a formidable mathematical challenge. In this article we shall review the basic analytical methods for dealing with such equations, including WKB asymptotics, Wigner measure techniques and Gaussian beams. Moreover, we shall give an overview of the current state of the art of numerical methods (most of which are based on the described analytical techniques) for the Schrödinger equation in the semiclassical regime.

AB - We consider time-dependent (linear and nonlinear) Schrödinger equations in a semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive models whose solutions exhibit high-frequency oscillations. The design of efficient numerical methods which produce an accurate approximation of the solutions, or at least of the associated physical observables, is a formidable mathematical challenge. In this article we shall review the basic analytical methods for dealing with such equations, including WKB asymptotics, Wigner measure techniques and Gaussian beams. Moreover, we shall give an overview of the current state of the art of numerical methods (most of which are based on the described analytical techniques) for the Schrödinger equation in the semiclassical regime.

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U2 - 10.1017/S0962492911000031

DO - 10.1017/S0962492911000031

M3 - Article

AN - SCOPUS:79956066352

SN - 0962-4929

VL - 20

SP - 121

EP - 209

JO - Acta Numerica

JF - Acta Numerica

ER -

Mathematical and computational methods for semiclassical Schrödinger equations

Abstract

Bibliographical note

ASJC Scopus subject areas

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