Strong phase-space semiclassical asymptotics

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

Wigner and Husimi transforms have long been used for the phase-space reformulation of Schr̈odinger-type equations and the study of the corresponding semiclassical limits. Most of the existing results provide approximations in appropriate weak topologies. In this work we are concerned with semiclassical limits in the strong topology, i.e., approximation of Wigner functions by solutions of the Liouville equation in L2 and Sobolev norms. The results obtained improve the state of the art and highlight the role of potential regularity, especially through the regularity of theWigner equation. It must be mentioned that the strong convergence can be shown up to O(log 1/ε ) time-scales, which is well known to be, in general, the limit of validity of semiclassical asymptotics. © 2011 Society for Industrial and Applied Mathematics.
Original languageEnglish
Pages (from-to)2116-2149
Number of pages34
JournalSIAM Journal on Mathematical Analysis
Volume43
Issue number5
DOIs
Publication statusPublished - 2011

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Semiclassical Limit
Phase Space
Regularity
Liouville equation
Topology
Liouville Equation
Wigner Function
Weak Topology
Approximation
Applied mathematics
Reformulation
Strong Convergence
Time Scales
Transform
Norm

Cite this

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abstract = "Wigner and Husimi transforms have long been used for the phase-space reformulation of Schr̈odinger-type equations and the study of the corresponding semiclassical limits. Most of the existing results provide approximations in appropriate weak topologies. In this work we are concerned with semiclassical limits in the strong topology, i.e., approximation of Wigner functions by solutions of the Liouville equation in L2 and Sobolev norms. The results obtained improve the state of the art and highlight the role of potential regularity, especially through the regularity of theWigner equation. It must be mentioned that the strong convergence can be shown up to O(log 1/ε ) time-scales, which is well known to be, in general, the limit of validity of semiclassical asymptotics. {\circledC} 2011 Society for Industrial and Applied Mathematics.",
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Strong phase-space semiclassical asymptotics. / Athanassoulis, Agissilaos; Paul, Thierry.

In: SIAM Journal on Mathematical Analysis, Vol. 43, No. 5, 2011, p. 2116-2149.

Research output: Contribution to journalArticle

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AU - Paul, Thierry

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AB - Wigner and Husimi transforms have long been used for the phase-space reformulation of Schr̈odinger-type equations and the study of the corresponding semiclassical limits. Most of the existing results provide approximations in appropriate weak topologies. In this work we are concerned with semiclassical limits in the strong topology, i.e., approximation of Wigner functions by solutions of the Liouville equation in L2 and Sobolev norms. The results obtained improve the state of the art and highlight the role of potential regularity, especially through the regularity of theWigner equation. It must be mentioned that the strong convergence can be shown up to O(log 1/ε ) time-scales, which is well known to be, in general, the limit of validity of semiclassical asymptotics. © 2011 Society for Industrial and Applied Mathematics.

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