The magnetic Weyl calculus

Journal of Mathematical Physics

Volumn 45, Issue 4, 2004, Pages 1394-1417

  M̌ntoiu, Marius ^a   Purice, Radu ^a  

^a INSTITUTE OF MATHEMATICS   (Romania)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 2342421680     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1668334     Document Type: Article

References (32)

1. Avron, J., Herbst, I., and Simon, B., "Schrödinger operators with magnetic fields. I. General theory," Duke Math. J. 45, 847-883 (1978).
2. Anderson, R. F., "The Weyl functional calculus," J. Funct. Anal. 4, 240-267 (1969).
3. Anderson, R. F., "The multiplicative Weyl functional calculus," J. Funct. Anal. 9, 423-440 (1972).
4. Antonets, M. A., "The classical limit for the Weyl quantization," Lett. Math. Phys. 2, 241-245 (1978).
5. Bellissard, J., "C*-Algebras in Solid State Physics: 2D-Electrons in a Uniform Magnetic Field," in Operator Algebras and Applications, edited by D. E. Evans and M. Takesaki (Cambridge University Press, Cambridge, 1988), Vol. 2.
6. Bellissard, J., "Gap Labbeling Theorems for Schrödinger Operators," in Number Theory and Physics, edited by J.-M. Luck, P. Moussa, and M. Waldschmidt, Springer Preceedings in Physics Vol. 47 (Springer, Berlin, 1993).
7. Boutet de Monvel, L., Grigis, A., and Helffer, B., Parametrixes d'opérateurs pseudo-différentiels àcaractéristiques multiples, Journées: Équations aux Dérivées Partielles de Rennes, 93-121. Asterisque, No. 34-35, Soc. Math. France, Paris, 1976.
8. Cycon, H. L., Froese, R. H., Kirsch, W., and Simon, B., Schrödinger operators with application to quantum mechanics and global geometry, Texts and Monographs in Physics. Springer Study Edition (Springer-Verlag, Berlin, 1987).
9. Estrada, R., Gracia-Bondía, J. M., and Várilly, J. C., "On asymptotic expansions of twisted products," J. Math. Phys. 30, 2789-2796 (1989).
10. Folland, G. B., Harmonic Analysis in Phase Space (Princeton University Press, Princeton, NJ, 1989).
11. Georgescu, V., and Iftimovici, A., "C* Algebras of Energy Observables: II. Graded Symplectic Algebras and Magnetic Hamiltonians," preprint 01-99 at http://www.ma.utexas.edu/mp-arc/
12. Gérard, C., Martinez, A., and Sjöstrand, J., "A mathematical approach to the effective Hamiltonian in perturbed periodic problems," Commun. Math. Phys. 142, 217-244 (1991).
13. Gracia-Bondía, J. M., and Várilly, J. C., "Algebra of distributions suitable for phase-space quantum mechanics. I," J. Math. Phys. 29, 869-879 (1988).
14. Gracia-Bondía, J. M., and Várilly, J. C., "Algebra of distributions suitable for phase space quantum mechanics. II. Topologies on the Moyal algebra," J. Math. Phys. 29, 880-887 (1988).
15. Grossman, A., Lupias, G., and Stein, E. M., "An algebra of pseudo-differential operators and quantum mechanics in phase space," Ann. Inst. Fourier 18, 343-368 (1969).
16. Helffer, B., and Sjöstrand, J., "Equation de Schrödinger avec champ magnetique et équation de Harper," Lect. Notes Phys. 345, 118-197(1989).
17. Helffer, B. and Sjöstrand, J., "On diamagnetism and de Haas-van Alphen effect," Ann. I.H.P. Phys. Theor. 52, 303-375 (1990).
18. Hörmander, L., "The Weyl calculus of pseudo-differential operators," Commun. Pure Appl. Math. 32, 359-443 (1979).
19. Hörmander, L., The Analysis of Partial Differential Operators (Springer-Verlag, New York, 1985), Vol. III.
20. Leinfelder, H., "Gauge invariance of Schrödinger operators and related spectral properties," J. Oper. Theory 9, 163-179 (1983).
21. Leinfelder, H., and Simader, C. G., "Schrödinger operators with singular magnetic vector potentials," Math. Z. 176, 1-19 (1981).
22. Luttinger, J. M., "The effect of the magnetic field on electrons in a periodic potential," Phys. Rev. 84, 814-817 (1951).
23. Mǎntoiu, M., and Purice, R., The Algebra of Observables in a Magnetic Field, Mathematical Results in Quantum Mechanics (Taxco, 2001), Contemporary Mathematics (AMS, Providence, RI, 2002), Vol. 307, pp. 239-245.
24. Mohamed, A., and Raǐkov, G. D., "On the Spectral Theory of the Schrödinger Operator with Electromagnetic Potential," in Pseudo-differential Calculus and Mathematical Physics, Math. Top., 5 (Akademie Verlag, Berlin, 1994), pp. 298-390.
25. Moyal, J. E., "Quantum mechanics as a statistical theory," Proc. Cambridge Philos. Soc. 45, 99-124 (1949).
26. Nenciu, G., "Dynamics of band electrons in electric and magnetic fields: Rigorous justification of the effective Hamiltonians," Rev. Mod. Phys. 63, 91-128 (1991).
27. Reed, M., and Simon, B., Methods of Modern Mathematical Physics, I: Functional Analysis (Academic, New York, 1972).
28. Shubin, M. A., Pseudodifferential Operators and Spectral Theory, Springer Series in Soviet Mathematics (Springer-Verlag, Berlin, 1987).
29. Panati, G., Spohn, H., and Teufel, S., Effective Dynamics for Bloch Electrons: Peierls Substitution and Beyond, preprint 02-516 at http://www.ma.utexas.edu/mp-arc/
30. Trèves, F., Topological Vector Spaces, Distributions and Kernels (Academic, New York, 1967).
31. Zak, J., "Magnetic translation groups," Phys. Rev. 134, A1602-A1607 (1965); 134, A1607-A1611 (1965).
32. Zak, J., "Magnetic translation groups," Phys. Rev. 134, A1602-A1607 (1965); 134, A1607-A1611 (1965).