Self-consistent relaxation-time models in quantum mechanics

Communications in Partial Differential Equations

Volumn 21, Issue 3-4, 1996, Pages 473-506

  Arnold, Anton ^a,b  

^a PURDUE UNIVERSITY   (United States)

^b TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0012307177     PISSN: 03605302     EISSN: None     Source Type: Journal    
DOI: 10.1080/03605309608821193     Document Type: Article

References (31)

1. A. Arnold, "The relaxation-time Wigner equation", to appear in the proceedings of the workshop on 'Mathematical Problems in Semiconductor Physics', Rome 1993, Pitman Research Notes in Mathematics, 1995. Technical Report #254, Center for Applied Math, Purdue University, October 1994.
2. A. Arnold, "The relaxation-time von Neumann-Poisson equation", to appear in the proceedings of 'ICIAM 95', Hamburg 1995, ZAMM, 1996.
3. A. Arnold, P. A. Markowich, and N. Mauser, "The one-dimensional periodic Bloch-Poisson equation", Math. Mod. Meth. Appl. Sci., 1 (1991) pp. 83-112.
4. 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) pp. 595-613.
5. m)", Technical Report, Department of Mathematics, Linköping University.
6. F. Brezzi and P. A. Markowich, "The three-dimensional Wigner-Poisson problem: Existence, uniqueness and approximation", Math. Meth. Appl. Sci., 14 (1991) pp. 35-61.
7. F. A. Buot and K. L. Jensen, "Lattice Weyl-Wigner formulation of exact many-body quantum-transport theory and applications to novel solid-state quantum-based devices", Phys. Rev. B, 42 (1990) pp. 9429-9457.
8. R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, 1, Springer-Verlag, Berlin, New York, 1990.
9. R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, 5, Springer-Verlag, Berlin, New York, 1992.
10. D. Ferry and H. Grubin, "Quantum transport in semiconductor devices", to appear in Solid State Physics, Academic Press, NY, 1995.
11. W. R. Frensley, "Wigner function model of a resonant-tunneling semiconductor device", Phys. Rev., B36 (1987) pp. 1570-1580.
12. A. Friedman, Partial Differential Equations, Holt-Rhinehart-Winston, New York, 1969.
13. J. Ginibre and G. Velo, "On a class of non linear Schrödinger equations with non local interaction", Math. Z., 170 (1980) pp. 109-136.
14. R. Illner, H. Lange, and P. F. Zweifel, "Global existence, uniqueness and asymptotic behavior of solutions of the Wigner-Poisson and Schrödinger-Poisson systems", Math. Meth. Appl. Sc., 17 (1994) pp. 349-376.
15. N. Kluksdahl, A. M. Kriman, D. K. Ferry, and C. Ringhofer, "Self-consistent study of the resonant tunneling diode", Phys. Rev. B, 39 (1989) pp. 7720-7735.
16. E. H. Lieb, "Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation", Stud. Appl. Math., 57 (1977) pp. 93-105.
17. E. H. Lieb and W. E. Thirring, "Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities", in Studies in Mathematical Physics, (E. H. Lieb, B. Simon, and A. S. Wightman, eds.), essays in honor of Valentine Bargmann, Princeton Univ. Press, 1976.
18. P. L. Lions and T. Paul, "Sur les mesures de Wigner", Rev. Math. Iberoam., 9 (1993) pp. 553-61.
19. P. A. Markowich, "On the equivalence of the Schrödinger and the quantum Liouville equations", Math. Meth. Appl. Sci., 11 (1989) pp. 459-469.
20. P. A. Markowich and N. J. Mauser, "The classical limit of a self-consistent quantum-Vlasov equation", Math. Meth. Mod. Appl. Sc., 3 (1993) pp. 109-124.
21. P. A. Markowich, C. Ringhofer, and C. Schmeiser, Semiconductor Equations, Springer Verlag, Wien, New York (1990).
22. F. Nier, "A stationary Schrödinger-Poisson system arising from the modeling of electronic devices", Forum Math., 2 (1990) pp. 489-510.
23. A. Pazy, "Semigroups of linear operators and applications to partial differential equations", in Applied Mathematical Sciences, (F. John, J. E. Marsden, and L. Sirovich, eds.), 44, Springer-Verlag, New York, 1983.
24. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, Academic Press, New York and London, 1972.
25. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II: Fourier analysis, self-adjointness, Academic Press, New York, San Francisco, and London, 1975.
26. B. Simon, "Quantum mechanics for Hamiltonians defined as quadratic forms", in Princeton Series in Physics, Princeton University Press, 1971.
27. B. Simon, Trace Ideals and Applications, Cambridge University Press, 1978.
28. H. Steinrück, "The one-dimensional Wigner-Poisson problem and its relation to the Schrödinger-Poisson problem", SIAM J. Math. Anal., 22 (1991) pp. 957-972.
29. N. D. Suh, M. R. Feix, and P. Bertrand, "Numerical simulation of the quantum Liouville-Poisson system", J. Comput. Phys., 94 (1991) pp. 403-441.
30. R. Temam, "Infinite-dimensional dynamical systems in mechanics and physics", in Applied Mathematical Sciences, (F. John, J. E. Marsden, and L. Sirovich, eds.), 68 Springer-Verlag, New York, 1988.
31. E. P. Wigner, "On the quantum correction for thermodynamic equilibrium", Phys. Rev., 40 (1932) pp. 749-759.