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MEAN-FIELD APPROXIMATION OF QUANTUM SYSTEMS AND CLASSICAL LIMIT

S. GRAFFI

Dipartimento di Matematica, Università di Bologna, Italy

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A. MARTINEZ

Dipartimento di Matematica, Università di Bologna, Italy

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M. PULVIRENTI

Dipartimento di Matematica, Università di Roma "La Sapienza", Italy

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https://doi.org/10.1142/S0218202503002386Cited by:38 (Source: Crossref)

Abstract

We prove that, for a smooth two-body potential, the quantum mean-field approximation to the nonlinear Schrödinger equation of the Hartree type is stable at the classical limit h → 0, yielding the classical Vlasov equation.

Keywords:

AMSC: 60K35, 81T27, 82C21, 82C22

References

W. Braun and K. Hepp, Comm. Math. Phys. 56, 101 (1997), DOI: 10.1007/BF01611497. Crossref, Web of Science, Google Scholar
R. L. Dobrushin, Sov. J. Funct. Anal. 13, 115 (1979), DOI: 10.1007/BF01077243. Crossref, Web of Science, Google Scholar
H. Neunzert, Kinetic Theory and Boltzmann Equation, Springer Lecture Notes 207, ed. C. Cercignani (Springer, 1984) pp. 60–110. Crossref, Google Scholar
H. Spohn, Rev. Mod. Phys. 53, 569 (1980). Google Scholar
C. Bardos, F. Golse and N. Mauser, Methods Appl. Anal. 2, 275 (2000). Crossref, Web of Science, Google Scholar
C. Bardoset al., C. R. Acad. Sci. Paris 334, 515 (2002). Crossref, Web of Science, Google Scholar
H. Narnhofer and G. Sewell, Comm. Math. Phys. 79, 9 (1981), DOI: 10.1007/BF01208282. Crossref, Web of Science, Google Scholar
E. Grenier, Proc. Amer. Math. Soc. 126, 523 (1998), DOI: 10.1090/S0002-9939-98-04164-1. Crossref, Web of Science, Google Scholar