The IVP for the Schrödinger-Poisson-Xα equation in one dimension

Mathematical Models and Methods in Applied Sciences

Volumn 15, Issue 8, 2005, Pages 1169-1180

  Stimming, H P ^a  

^a UNIVERSITY OF VIENNA   (Austria)

Author keywords

Nonlinear Schr dinger equation; Quantum semiconductor modeling; Semiclassical limit; Time dependent density functional theory

Indexed keywords

EID: 31144462554     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202505000698     Document Type: Article

References (22)

1. A. Arnold and F. Nier, The two-dimensional Wigner-Poisson problem for an electron gas in the charge neutral case, Math. Meth. Appl. Sci. 14 (1991) 595-613.
2. J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations (Amer. Math. Soc., 1999).
3. C. Bardos, F. Golse and N. J. Mauser, Weak coupling limit of the N-particle Schrödinger equation, Math. Anal. Appl. 7 (2000) 275-293.
4. C. Bardos, F. Golse, A. Gottlieb and N. J. Mauser, Mean field dynamics of fermions and the time-dependent Hartree-Fock equation, J. Math. Pures Appl. 82 (2003) 665-683.
5. C. Bardos, L. Erdös, F. Golse, N. J. Mauser and H.-T. Yau, Derivation of the Schrödinger-Poisson equation from the quantum N-particle Coulomb problem, C. R. Acad. Sci. 334 (2002) 515-520.
6. O. Bokanowski and N. J. Mauser, Local approximation for the Hartree-Fock exchange potential: A deformation approach, Math. Meth. Mod. Appl. Sci. 9 (1999) 941-961.
7. W. Bao, N. J. Mauser and H. P. Stimming, Effective one particle quantum dynamics of electrons: A numerical study of the Schrödinger-Poisson- Xα model, Comm. Math. Sci. 1 (2003) 809-831.
8. R. Carles, Geometric optics with caustic crossing for some nonlinear Schrödinger equations, Indiana Univ. Math. J. 49 (2000) 475-551.
9. R. Carles, Remarques sur les mesures de Wigner, C. R. Acad. Sci. 332 (2001) 981-984.
10. T. Cazenave, Introduction to Nonlinear Schrödinger Equations, Textos de Métodos Matemáticos, Vol. 26 (Instituto de Matemática - UFRJ, 1996).
11. D. Fujiwara, A construction of the fundamental solution for the Schrödinger equation, J. Anal. Math. 35 (1979) 41-96.
12. d) for the Schrödinger equation with a time-dependent potential, Math. Ann. 299 (1994) 117-125.
13. S. Jin, C. D. Levermore and D. McLaughlin, The semiclassical limit of the defocusing NLS hierarchy, Comm. Pure Appl. Math. 52 (1999) 613-654.
14. S. Kamvissis, K. T.-R. McLaughlin and P. D. Miller, Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrdinger Equation, Annals of Mathematics Studies, Vol. 154 (Princeton Univ. Press, 2003).
15. P. L. Lions and T. Paul, Sur les Mesures de Wigner, Rev. Mat. Iberoamer. 9 (1993) 553-618.
16. N. J. Mauser, The Schrödinger-Poisson-Xα model, Appl. Math. Lett. 14 (2001) 759-763.
17. P. A. Markowich and N. J. Mauser, The classical limit of a self-consistent quantum-Vlasov equation in 3D, Math. Mod. Meth. Appl. Sci. 9 (1993) 109-124.
18. I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math. 155 (2004) 451-513.
19. J. C. Slater, A simplification of the Hartree-Fock method, Phys. Rev. 81 (1951) 385-390.
20. C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (Springer, 1999).
21. H. Steinrück, The one-dimensional Wigner-Poisson problem and its relation to the Schrödinger-Poisson problem, SIAM J. Math. Anal. 22 (1991) 957-972.
22. P. Zhang, Y. Zheng, N. J. Mauser, The limit from the Schrödinger- Poisson to the Vlasov-Poisson equations with general data in one dimension, Comm. Pure Appl. Math. 55 (2002) 582-632.