EFFECTIVE ONE PARTICLE QUANTUM DYNAMICS OF ELECTRONS: A NUMERICAL STUDY OF THE SCHRODINGER-POISSON-Xα MODEL*

Communications in Mathematical Sciences

Volumn 1, Issue 4, 2003, Pages 809-828

  Bao, Weizhu ^a   Mauser, N J ^b   Stimming, H P ^b  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^b UNIVERSITY OF VIENNA   (Austria)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 11244292274     PISSN: 15396746     EISSN: 19450796     Source Type: Journal    
DOI: 10.4310/CMS.2003.v1.n4.a8     Document Type: Article

References (37)

1. [Ba1] V. Bach, Accuracy of mean field approximations for atoms and molecules, Commun. Math. Phys., 155:295-310, 1993.
2. [B] J. Bourgain, The nonlinear Schrodinger equation, AMS, 1999.
3. [BaG] C. Bardos, F. Golse, and N.J. Mauser, Weak coupling limit of the N-particle Schrodinger equation, Mathematical Analysis and Applications, 7(2):275-293, 2000.
4. [BM2] C. Bardos, F. Golse, A. Gottlieb, and N.J. Mauser, Mean field dynamics offermions and the time-dependent Hartree-Fock equation, J. d. Mathematiques Pures et Appl., 82(6):665-683, 2003.
5. [BM3] C. Bardos, A. Gottlieb, F. Golse, and N.J. Mauser, Derivation of the time-dependent Hartree-Fock equation: the Coulomb interaction case, manuscript.
6. [BMY] C. Bardos, L. Erdos, F. Golse, N.J. Mauser, and H.T. Yau, Derivation of the Schrodinger-Poisson equation from the quantum N -particle Coulomb problem, C. R. Acad. Sci., t. 334(6):515-520, Serie I Math., 2002.
7. [BM1] O. Bokanowski and N.J. Mauser, Local approximation for the Hartree-Fock exchange potential: a deformation approach, Math.Meth., and Mod.in the Appl.Sci., 9(6):941-961, 1999.
8. [BGM] O. Bokanowski, B. Gr6bert, and N.J. Mauser, Local density approximation for the energy of a periodic coulomb model, Math.Meth., and Mod.in the Appl.Sci., 13(8):1185-1217, 2003.
9. [BJM1] W. Bao, S. Jin, and P.A. Markowich, Time-splitting spectral approximations for the Schroodinger equation in the semiclassical regime, J. Comp. Phys., 175(2):487-524, 2002.
10. [BJM2] W. Bao, S. Jin, P.A. Markowich, Numerical studies of time-splitting spectral discretisations of nonlinear Schrodinger equations in the semiclassical regime, SIAM J. Sci. Comp., 25(1):27-64, 2003,
11. [BJ] W. Bao and D. Jaksch, An explicit unconditionally stable numerical method for solving damped nonlinear Schrodinger equations with a focusing nonlinearity, SIAM J. Numer. Anal., 41(4):1406-1426, 2003.
12. [BJaM] W. Bao, D. Jaksch, and P.A. Markowich, Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comput. Phys., 187(1):318-342, 2003.
13. [Caz] T. Cazenave, Introduction to nonlinear Schrodinger equations, Textos de Metodos Matematicos 26, Rio de Janeiro, Instituto de Matematica-UFRJ, 1996.
14. [Car] R. Carles, Remarques sur les mesures de Wigner, Serie I, 332:981-984, C.R.A.S. Paris, 2001.
15. [Co] J.W.D. Conolly, The Xα method in Semi-empirical methods of electronic structure calculations, ed., by G.A. Segal, Plenum Press, 1977.
16. [D1] P.A.M. Dirac, Note on exchange phenomena in the Thomas-Fermi atom, Proc. Cambridge Philos. Soc., 26:376-385, 1931.
17. [G1] R. Gaspar, Uber eine approximation des Hartree-Fockschen potentials durch eine universelle Potentialfunktion, Acta. Phys., Hungarica, 3:263-285, 1954.
18. [GMM] P. Gerard, P.A. Markowich, N.J. Mauser, and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure and Appl. Math., 50:321-377, 1997.
19. [GV] J. Ginibre and G. Velo, An introduction to nonlinear Schrodinger equations, Hokkaido Univ., Technical Report Series in Math., 43:80-128, 1996.
20. [KPV] C. Kenig, G. Ponce, and L. Vega, On the IVP for the nonlinear Schrodinger equations, AMS Contemp. Math., 189:353-367, 1995.
21. [KS] W. Kohn and L.J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev., A(140):1133, 1965.
22. [Lio1] P.L. Lions, Solution of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109:33-97, 1987.
23. [LP] P.L. Lions and T. Paul, Sur les Mesures de Wigner, Revista Mat. Iberoamericana, 9:553618, 1993.
24. [M1] N.J. Mauser, Quantum steady states: The Bloch-Poisson model, Proc. "Numsim91", Berlin, Ed., P. Deuflhard, Techn. Rep., 91-8, ZIB, 62-67, 1991.
25. [M2] N.J. Mauser, The Schrodinger-Poisson-Xα model, Appl. Math. Lett., 14:759-763, 2001.
26. [M3] N.J. Mauser, (Semi)classical limits for weakly nonlinear Schrodinger equation, Sem. Ecole Polytechnique, 1-12, 2002.
27. [MM1] P.A. Markowich and N.J. Mauser, The Classical limit of a self-consistent quantum-vlasov equation in 3-D, Math. Mod. and Meth. in Appl. Sciences, 9:109-124, 1993.
28. [MPPS] P.A. Markowich, P. Pietra, C. Pohl, and H.P. Stimming, A Wigner-measure analysis of the Dufort-Frankel scheme for the Schrodinger equation, SIAM J. Numer. Anal., 40(4):1281-1310, 2002.
29. [MPP] P.A. Markowich, P. Pietra, and C. Pohl, A Wigner measure approach to the analysis of difference methods for the Schrodinger equation, ENUMATH 97 (Heidelberg), 453-460, World Sci. Publishing, River Edge, NJ, 1998.
30. [MK] P. Miller and S. Kamvissis, On the semiclassical limit of the focusing nonlinear Schrodinger equation, Phys. Lett., A247(1-2):75-86, 1998.
31. [PY] R.G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules, Oxford university press, 1989.
32. [Sla] J.C. Slater, A simplification of the Hartree-Fock method, Phys. Rev., 81(3):385-390, 1951.
33. [SSJM] E. Skovsen, H. Stapelfeldt, S. Juhl, and K. Molmer, Quantum state tomography of dissociating molecules, Phys. Rev. Lett., 91(9):406-410, 2003.
34. [Ste1] H. Steinrück, The one-dimensional Wigner-Poisson problem and its relation to the Schroodinger-Poisson problem SIAM J. Math. Anal., 22(4):957-972, 1991.
35. [Stim1] H.P. Stimming, The IVP for the Schrodinger-Poisson-Xα equation in one dimension, manuscript, 2003.
36. [WH] J.A.C. Weideman and B.M. Herbst, Split-step methods for the solution of the nonlinear Schroodinger equation, SIAM J. Numer. Anal., 23(3):485-507, 1986.
37. [ZZM] P. Zhang, Y. Zheng, and N.J. Mauser, The limit from the Schrodinger-Poisson to Vlasov-Poisson equation with general data in one dimension, Comm. Pure and Appl. Math., 55(5):582-632, 2002.