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The limit from the Schrodinger-Poisson to the Vlasov-Poisson equations with general data in one dimension
Zhang, P; Zheng, YX; Mauser, NJ
2002-05-01
Source PublicationCOMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
ISSN0010-3640
Volume55Issue:5Pages:582-632
AbstractWe deal with the classical limit of the Schrodinger-Poisson system to the Vlasov-Poisson equations as the Planck constant epsilon goes to zero. This limit is also frequently called the "semiclassical limit." The coupled Schrodinger-Poisson system for the wave functions {psi(j)(epsilon)(t, x) is transformed to the Wigner-Poisson equations for a "phase space function" f(epsilon)(t, x, epsilon). For the case of the so-called completely mixed state, i.e., j = 1, 2,..., infinity, under additional assumptions on the potential, this classical limit was solved in 1993 by P.-L, Lions and T. Paul in [23] and, independently, by P. A. Markowich and N. J. Mauser in [26] with strong assumptions on the initial data. The so-called pure state case, where only one or a finite number of wave functions {psi(j)(epsilon)(t, x)} are considered, has been open up to now. We prove here for general initial data (the pure-state as well as the mixed-state case) of the wave functions in one space dimension that the Wigner measure f(t, x, epsilon), which is a weak limit of f(epsilon)(t, x, epsilon) as c tends to 0, satisfies the classical one-dimensional Vlasov-Poisson equations. As a by-product, we have improved the decay assumption on the initial data of one-dimensional Vlasov-Poisson equations in [38] for the existence of global weak solutions with measures as initial data. The equations we regard are widely used in quantum/classical transport and semiconductor theory as a nonlinear one-particle ("mean field") approximation of the linear N-electron Schrodinger/Hamilton equation.
Language英语
WOS Research AreaMathematics
WOS SubjectMathematics, Applied ; Mathematics
WOS IDWOS:000174507300002
PublisherJOHN WILEY & SONS INC
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Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/17035
Collection中国科学院数学与系统科学研究院
Corresponding AuthorZhang, P
Affiliation1.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100080, Peoples R China
2.Penn State Univ, Dept Math, University Pk, PA 16802 USA
3.Univ Vienna, Inst Math, A-1090 Vienna, Austria
Recommended Citation
GB/T 7714
Zhang, P,Zheng, YX,Mauser, NJ. The limit from the Schrodinger-Poisson to the Vlasov-Poisson equations with general data in one dimension[J]. COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS,2002,55(5):582-632.
APA Zhang, P,Zheng, YX,&Mauser, NJ.(2002).The limit from the Schrodinger-Poisson to the Vlasov-Poisson equations with general data in one dimension.COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS,55(5),582-632.
MLA Zhang, P,et al."The limit from the Schrodinger-Poisson to the Vlasov-Poisson equations with general data in one dimension".COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS 55.5(2002):582-632.
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