Long-time dynamics of the Schrödinger-Poisson-Slater system

Journal of Statistical Physics

Volumn 114, Issue 1-2, 2004, Pages 179-204

  Sánchez, Óscar ^a   Soler, Juan ^a  

^a UNIVERSITY OF GRANADA   (Spain)

Author keywords

Asymptotic behaviour; Dispersion; Open quantum system; Schr dinger Poisson system; Stationary solutions; X approach

Indexed keywords

EID: 3542992111     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1023/b:joss.0000003109.97208.53     Document Type: Article

References (27)

1. R. Balian, From Microphysics to Macrophysics; Methods and Applications of Statistical Physics, Vols. I & II (Springer, 1991).
2. C. Bardos, The weak coupling limit of systems of N quantum particles, in Euroconference on Asymptotic Methods and Applications in Kinetic and Quantum-Kinetic Theory, Book of abstracts, L. L. Bonilla, J. Soler, and J. L. Vazquez, eds., Granada, Spain, September 2001.
3. O. Bokanowski, J. L. López, and J. Soler, On a exchange interaction model for quantum transport: The Schrödinger-Poisson-Slater system, to appear in Math. Mod. Meth. Appl. Sci.
4. I. Catto and P. L. Lions, Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. Part 1: A necessary and sufficient condition for the stability of general molecular system, Comm. Partial Differential Equations 17:1051-1110 (1992).
5. T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Métodos Matemáticos 26, 2nd ed. (Universidade Federal do Rio de Janeiro, 1993).
6. 2 solutions to the Schrödinger-Poisson system: Existence, uniqueness, time behaviour, and smoothing effects, Math. Mod. Meth. Appl. Sci. 7:1051-1083 (1997).
7. R. M. Dreizler and E. K. U. Gross, Density Functional Theory (Springer, Berlin, 1990).
8. R. S. Ellis, Entropy, Large Deviations, and Statistical Mechanics (Springer, New York, 1985).
9. I. Gasser, R. Illner, P. A. Markowich, and C. Schmeiser, Semiclassical, t → ∞ asymptotics and dispersive effects for Hartree-Fock systems, Math. Modelling and Numer. Anal. 32:699-713 (1998).
10. P. Gerard, Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var. 3:213-233 (1998).
11. M. K. Harbola and V. Sahni, Quantum-mechanical interpretation of the exchange-correlation potential of Kohn-Sham density-functional theory, Phys. Rev. Lett. 62:448-292 (1989).
12. R. Illner, 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. 17:349-376 (1994).
13. C. E. Kening, G. Ponce, and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106:617-633 (2000).
14. C. LeBris and E. Cancès, On the time-dependent Hartree-Fock equations coupled with a classical nuclear dynamics, Math. Mod. Meth. Appl. Sci. 9:963-990 (1999).
15. E. H. Lieb, Existence and uniqueness of the minimizing solution of choquard's nonlinear equation, Stud. Appl. Math. 57:93-105 (1977).
16. E. H. Lieb and S. Oxford, Improved lower bound on the indirect Coulomb energy, Int. J. Quant. Chem. 19:427-439 (1981)
17. E. H. Lieb, Thomas-Fermi and related theories of atoms and molecules, Rev. Modern Phys. 53:603-641 (1981).
18. E. H. Lieb, Thomas-Fermi Theory, Kluwer Encyclopedia of Mathematics, Supplement, Vol. II, pp. 311-313 (2000).
19. E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, Vol. 14 (American Mathematical Society, Providence, Rhode Island, 2001).
20. E. H. Lieb and B. Simon, The Thomas-Fermi theory of atoms, molecules, and solids, Adv. Math. 23:22-116 (1977).
21. P. L. Lions, The concentration-compactness principle in the Calculus of Variations. The locally compact case, Ann. Inst. H. Poincaré 1:109-145 & 223-283 (1984).
22. P. L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Commun. Math. Phys. 109:33-97 (1987).
23. P. Markowich, C. Ringhofer, and C. Schmeiser, Semiconductor Equations (Springer-Verlag, New York, 1990).
24. R. Penrose, On gravity's role in quantum state reduction, Gen. Relativity Gravitation 28:581 (1996).
25. A. Puente and L. Serra, Oscillation modes of two-dimensional nanostructures within the time-dependent local-spin-density approximation, Phys. Rev. Lett. 83:3266-3269 (1999).
26. E. Ruíz Arriola and J. Soler, A variational approach to the Schrödinger-Poisson system: Asymptotic behaviour, breathers, and stability, J. Stat. Phys. 103:1069-1105 (2001).
27. N. Zou, M. Willander, I. Linnerud, U. Hanke, K. A. Chao, and Y. M. Galperin, Suppresion of intrinsic bistability by the exchange-correlation effect in resonant-tunneling structures, Phys. Rev. B 49:2193-2196 (1994).