Asymptotic behavior to the 3-D Schrödinger/Hartree-Poisson and Wigner-Poisson systems

Mathematical Models and Methods in Applied Sciences

Volumn 10, Issue 6, 2000, Pages 923-943

  Lopez J L ^a   Soler, J ^a  

^a UNIVERSITY OF GRANADA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034392770     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202500000471     Document Type: Article

References (26)

1. A. Arnold, Self-consistent relaxation-time models in quantum mechanics, Commun. Part Differential Equation 21 (1996) 473-506.
2. A. Arnold and F. Nier, The two-dimensional Wigner-Poisson problem for an electron gas in the charge neutral case, Math. Methods Appl. Sci. 14 (1991) 595-613.
3. n), Annali della Scuola Norm. Sup. di Pisa 4 (1975) 523-555.
4. H. Brezis, Analyse Fonctionnelle (Masson, 1983).
5. F. Brezzi and P. A. Markowich, The three-dimensional Wigner-Poisson problem: existence, uniqueness and approximation, Math. Methods Appl. Sci. 14 (1991) 35-61.
6. J. A. Carrillo, J. Soler and J. L. Vázquez, Asymptotic behaviour and self-similarity for the three-dimensional Vlasov-Poisson-Fokker-Planck system, J. Funct. Anal. 141 (1996) 99-132.
7. 2-solutions to the Schödinger-Poisson system: Existence, uniqueness, time behavior and smoothing effects, Math. Models Methods Appl. Sci. 7 (1997) 1051-1083.
8. J. P. Dias and M. Figueira, Conservation laws and time decay for the solutions of some nonlinear Schrödinger-Hartree equations and systems, J. Math. Anal. Appl. 84 (1981) 486-508.
9. L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, Conference Board of the Mathematical Sciences Vol. 74 (AMS, Providence, 1990).
10. P. Gerard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Commun. Pure Appl. Math. 50 (1997) 323-379.
11. R. T. Glassey, Asymptotic behavior of solutions to certain nonlinear Schrödinger-Hartree equations, Commun. Math. Phys. 53 (1977) 9-18.
12. N. Hayashi, P. I. Naumkin and T. Ozawa, Scattering theory for the Hartree equation, SIAM J. Math. Anal. 29 (1998) 1256-1267.
13. R. Illner, P. F. Zweifel and H. Lange, Global existence, uniqueness and asymptotic behaviour of solutions of the Wigner-Poisson and Schrödinger-Poisson systems, Math. Methods Appl. Sci. 17 (1994) 349-376.
14. S. Kamin and J. L. Vázquez, Fundamental solutions and asymptotic behaviour for the p-Laplacian equation, Rev. Mat. Iberoamericana 4 (1988) 339-354.
15. O. Korolev and S. Kruzhkov, On the imbedding theory of anisotropic function spaces, Sov. Math. Dokl. 32 (1985) 829-832.
16. P. L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat.Iberoamericana 9 (1993) 553-618.
17. J. L. López and J. Soler, Scaling limits in the 3-D Schrödinger-Poisson system, Appl. Math. Lett. 10 (1997) 61-65.
18. J. L. López and J. Soler, Space-time Wigner transforms and high frequency limits to the 3-D Schrödinger-Poisson system, to appear in J. Differential Int. Equations.
19. P. A. Markowich, On the equivalence of the Schrödinger and the quantum Liouville equations, Math. Methods Appl. Sci. 11 (1989) 459-469.
20. P. A. Markowich and N. J. Mauser, The classical limit of a self-consistent quantum-Vlasov equation in 3-D, Math. Models Methods Appl. Sci. 3 (1993) 109-124.
21. P. A. Markowich, N. J. Mauser and F. Poupaud, A Wignerfunction approach to (semi)classical limits: electrons in a periodic potential, J. Math. Phys. 35 (1994) 1066-1094.
22. P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations (Springer-Verlag, Vienna, 1990).
23. H. Spohn, Long time asymptotics for quantum particles in a periodic potential, Phys. Rev. Lett. 77 (1996) 1198-1201.
24. H. Steinrück, The one-dimensional Wigner-Poisson problem and its relation to the Schrödinger-Poisson problem, SIAM J. Math. Anal. 22 (1991) 957-972.
25. E. P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932) 749-759.
26. P. F. Zweifel, The Wigner transform and the Wigner-Poisson system, Transp. Theory Stat. Phys. 22 (1993) 459-484.