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The modulation mapping for magnetic symbols and operators

Authors: Marius Mantoiu and Radu Purice
Journal: Proc. Amer. Math. Soc. 138 (2010), 2839-2852
MSC (2010): Primary 35S05, 47L15; Secondary 47L65, 47L90
Published electronically: April 2, 2010
MathSciNet review: 2644897
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Abstract | References | Similar Articles | Additional Information

Abstract: We extend the Bargmann transform to the magnetic pseudodifferential calculus, using gauge-covariant families of coherent states. We also introduce modulation mappings, a first step towards adapting modulation spaces to the magnetic case.

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  • 1. S.T. Ali, J-P. Antoine, J-P. Gazeau and U. A. Müller, Coherent States and Their Generalizations: A Mathematical Overview, Rev. Math. Phys. 7 (1990), 1013-1104. MR 1359988 (97b:81040)
  • 2. D. Beltită and I. Beltită, Magnetic Pseudodifferential Weyl Calculus on Nilpotent Lie Groups, Ann. Global Anal. Geom. 36 (3) (2009), 293-322. MR 2544305
  • 3. D. Beltită and I. Beltită, Uncertainty Principles for Magnetic Structures on Certain Coadjoint Orbits, J. Geom. Phys. 60 (1) (2010), 81-95.
  • 4. A. Boulkhemair, Remarks on a Wiener Type Pseudodifferential Algebra and Fourier Integral Operators, Math. Res. Lett. 4 (1) (1997), 53-67. MR 1432810 (98f:47057)
  • 5. H. G. Feichtinger, On a New Segal Algebra, Monatsh. Mat. 92 (4) (1981), 269-289. MR 643206 (83a:43002)
  • 6. H. G. Feichtinger, Modulation Spaces on Locally Compact Abelian Groups, in Proceedings of ``International Conference on Wavelets and Applications 2002'', 99-140, Chenai, India. Updated version of a technical report, University of Vienna, 1983.
  • 7. G. Fendler, K. Gröchenig and M. Leinert, Convolution-Dominated Operators on Discrete Groups, Int. Eq. Op. Th. 61 (2008), 493-509. MR 2434338 (2009m:47085)
  • 8. G. B. Folland, Harmonic Analysis in Phase Space, Princeton Univ. Press, Princeton, NJ, 1989. MR 983366 (92k:22017)
  • 9. K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser Boston Inc., Boston, MA, 2001. MR 1843717 (2002h:42001)
  • 10. K. Gröchenig, Time-Frequency Analysis of Sjöstrand Class, Revista Mat. Iberoam. 22 (2) (2006), 703-724. MR 2294795 (2008b:35308)
  • 11. K. Gröchenig, Composition and Spectral Invariance of Pseudodifferential Operators on Modulation Spaces, J. Anal. Math. 98 (2006), 65-82. MR 2254480 (2007f:47044)
  • 12. K. Gröchenig, A Pedestrian Approach to Pseudodifferential Operators, in C. Heil, editor, Harmonic Analysis and Applications, Birkhäuser, Boston, 2006. In honour of John J. Benedetto. MR 2249309 (2007i:35257)
  • 13. K. Gröchenig and C. Heil, Modulation Spaces and Pseudodifferential Operators, Int. Eq. Op. Th. 34 (1999), 439-457. MR 1702232 (2001a:47051)
  • 14. K. Gröchenig and J. Toft, Localization Operator Representation of Modulation Spaces, preprint.
  • 15. B. C. Hall, Quantum Mechanics in Phase Space, Contemporary Mathematics, 214, Amer. Math. Soc., Providence, RI, 1997, 47-62. MR 1601217 (99e:22015)
  • 16. C. Heil, J. Ramanathan, and P. Topiwala, Singular Values of Compact Pseudodifferential Operators, J. Funct. Anal. 150 (2) (1997), 426-452. MR 1479546 (98k:47102)
  • 17. V. Iftimie, M. Măntoiu and R. Purice, Magnetic Pseudodifferential Operators, Publ. RIMS 43 (2007), 585-623. MR 2361789 (2009h:35469)
  • 18. V. Iftimie, M. Măntoiu and R. Purice, A Beals-Type Criterion for Magnetic Pseudodifferential Operators, Commun. PDE, to appear.
  • 19. M. V. Karasev and T. A. Osborn, Symplectic Areas, Quantization and Dynamics in Electromagnetic Fields, J. Math. Phys. 43 (2002), 756-788. MR 1878969 (2002m:81120)
  • 20. N. P. Landsman, Classical Behaviour in Quantum Mechanics: A Transition Probability Approach, Int. J. of Mod. Phys. B (1996), 1545-1554. MR 1405182 (97g:81003)
  • 21. N. P. Landsman, Poisson Spaces with a Transition Probability, Rev. Math. Phys. 9 (1997), 29-57. MR 1426545 (98b:81014)
  • 22. N. P. Landsman, Mathematical Topics between Classical and Quantum Mechanics, Springer-Verlag, New York, 1998. MR 1662141 (2000g:81081)
  • 23. M. Lein, M. Măntoiu and S. Richard, Magnetic Pseudodifferential Operators with Coefficients in $ C^*$-algebras, preprint arXiv:0901.3704, to appear in Proc. of the RIMS.
  • 24. M. Măntoiu and R. Purice, The Magnetic Weyl Calculus, J. Math. Phys. 45 (4) (2004), 1394-1417. MR 2043834 (2005j:81085)
  • 25. M. Măntoiu and R. Purice, Strict Deformation Quantization for a Particle in a Magnetic Field, J. Math. Phys. 46 (5) (2005). MR 2142981 (2006b:81139)
  • 26. M. Măntoiu and R. Purice, The Mathematical Formalism of a Particle in a Magnetic Field, Proceedings of the Conference QMath. 9, Giens, France, Lecture Notes in Math., vol. 690, Springer-Berlin, 2006. MR 2235706 (2007c:81098)
  • 27. M. Măntoiu, R. Purice and S. Richard, Twisted Crossed Products and Magnetic Pseudodifferential Operators, Proceedings of the Conference of Sinaia (Romania), Theta Foundation, 2005. MR 2238287 (2008d:46096)
  • 28. M. Măntoiu, R. Purice and S. Richard, Spectral and Propagation Results for Magnetic Schrödinger Operators; a $ C^*$-Algebraic Approach, J. Funct. Anal. 250 (2007), 42-67. MR 2345905 (2009a:46128)
  • 29. M. A. Rieffel, Deformation Quantization for Actions of $ \mathbb{R}^d$, Memoirs Amer. Math. Soc. 106 (1993), no. 506. MR 1184061 (94d:46072)
  • 30. J. Sjöstrand, An Algebra of Pseudodifferential Operators, Math. Res. Lett. 1 (2) (1994), 185-192. MR 1266757 (95b:47065)
  • 31. J. Sjöstrand, Wiener Type Algebras of Pseudodifferential Operators, in Séminaire sur les Équations aux Dérivées Partielles, 1994-1995, Exp. No. IV, 21, École Polytech., Palaiseau, 1995. MR 1362552 (96j:47049)
  • 32. J. Toft, Subalgebras to a Wiener Type Algebra of Pseudo-differential Operators, Ann. Inst. Fourier (Grenoble) 51 (5) (2001), 1347-1383. MR 1860668 (2002h:47071)
  • 33. D. Williams, Crossed Products of $ C^*$-Algebras, Math. Surveys and Monographs, 134, American Mathematical Society, Providence, RI, 2007. MR 2288954 (2007m:46003)

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Additional Information

Marius Mantoiu
Affiliation: Departamento de Matematicas, Universidad de Chile, Las Palmeras 3425, Casilla 653, Santiago, Chile

Radu Purice
Affiliation: Institute of Mathematics Simion Stoilow of the Romanian Academy, P.O. Box 1-764, Bucharest, RO-70700, Romania

Keywords: Magnetic field, pseudodifferential operator, phase space, modulation mapping, crossed product algebra, coherent states, Bargmann transform
Received by editor(s): July 30, 2009
Received by editor(s) in revised form: December 2, 2009
Published electronically: April 2, 2010
Additional Notes: The first author is partially supported by Núcleo Cientifico ICM P07-027-F “Mathematical Theory of Quantum and Classical Magnetic Systems” and by the Chilean Science Foundation Fondecyt under grant no. 1085162. His interest in modulation spaces was raised by a very enjoyable visit to the University of Vienna in February 2009.
The second author acknowledges partial support from contract no. 2-CEx 06-11-18/2006.
Communicated by: Marius Junge
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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