Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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QUANTUM EXTENSIONS OF FOURIER-GAUSS AND FOURIER-MEHLER TRANSFORMS

  • Ji, Un-Cig (DEPARTMENT OF MATHEMATICS RESEARCH INSTITUTE OF MATHEMATICAL FINANCE CHUNGBUK NATIONAL UNIVERSITY)
  • Published : 2008.11.01
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Abstract

Noncommutative extensions of the Gross and Beltrami Laplacians, called the quantum Gross Laplacian and the quantum Beltrami Laplacian, resp., are introduced and their basic properties are studied. As noncommutative extensions of the Fourier-Gauss and Fourier-Mehler transforms, we introduce the quantum Fourier-Gauss and quantum Fourier- Mehler transforms. The infinitesimal generators of all differentiable one parameter groups induced by the quantum Fourier-Gauss transform are linear combinations of the quantum Gross Laplacian and quantum Beltrami Laplacian. A characterization of the quantum Fourier-Mehler transform is studied.

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References

  1. L. Accardi, A. Barhoumi, and U. C. Ji, Quantum Laplacians on generalized operators on Boson Fock space, Preprint, 2005
  2. L. Accardi, A. Barhoumi, and H. Ouerdiane, A quantum approach to Laplace operators, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006), no. 2, 215-248 https://doi.org/10.1142/S0219025706002329
  3. L. Accardi, H. Ouerdiane, and O. G. Smolyanov, Levy Laplacian acting on operators, Russian J. Math. Phys. 10 (2003), no. 4, 359-380
  4. L. Accardi and O. G. Smolyanov, Levy-Laplace operators in spaces of functions on rigged Hilbert spaces, Math. Notes 72 (2002), no. 1-2, 129-134 https://doi.org/10.1023/A:1019829424019
  5. D. M. Chung and U. C. Ji, Transformation groups on white noise functionals and their applications, Appl. Math. Optim. 37 (1998), no. 2, 205-223 https://doi.org/10.1007/s002459900074
  6. D. M. Chung, U. C. Ji, and N. Obata, Quantum stichastic analysis via white noise operators in weighted Fock space, Rev. Math. Phys. 14 (2002), no. 3, 241-272 https://doi.org/10.1142/S0129055X0200117X
  7. S. Dineen, Complex Analysis on Infinite Dimensional Spaces, Springer-Verlag, 1999
  8. L. Gross, Potential theory on Hilbert space, J. Funct. Anal. 1 (1967), 123-181 https://doi.org/10.1016/0022-1236(67)90030-4
  9. T. Hida, Analysis of Brownian Functionals, Carleton Math. Lect. Notes, no. 13, Carleton University, Ottawa, 1975
  10. T. Hida, Brownian Motion, Springer-Verlag, 1980
  11. T. Hida, H.-H. Kuo, and N. Obata, Transformations for white noise functionals, J. Funct. Anal. 111 (1993), no. 2, 259-277 https://doi.org/10.1006/jfan.1993.1012
  12. T. Hida, N. Obata, and K. Saito, Infinite dimensional rotations and Laplacians in terms of white noise calculus, Nagoya Math. J. 128 (1992), 65-93 https://doi.org/10.1017/S0027763000004220
  13. U. C. Ji and N. Obata, A unified characterization theorem in white noise theory, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 6 (2003), no. 2, 167-178 https://doi.org/10.1142/S0219025703001122
  14. U. C. Ji, Unitarity of Kuo's Fourier-Mehler transform, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 7 (2004), no. 1, 147-154 https://doi.org/10.1142/S0219025704001487
  15. U. C. Ji, Unitarity of generalized Fourier-Gauss transform, Stoch. Anal. Appl. 24 (2006), no. 4, 733-751 https://doi.org/10.1080/07362990600751837
  16. U. C. Ji, N. Obata, and H. Ouerdiane, Quantum Levy Laplacian and associated heat equation, J. Funct. Anal. 249 (2007), no. 1, 31-54 https://doi.org/10.1016/j.jfa.2007.04.006
  17. H.-H. Kuo, Fourier-Mehler transforms of generalized Brownian functionals, Proc. Japan Acad. Ser. A Math. Sci. 59A (1983), no. 7, 312-314
  18. H.-H. Kuo, On Laplacian operators of generalized Brownian functionals, Stochastic processes and their applications, 119-128, Lect. Notes in Math. Vol. 1203, (1986)
  19. H.-H. Kuo, Fourier transform in white noise calculus, J. Multivar. Anal. 31 (1989), no. 2, 311-327 https://doi.org/10.1016/0047-259X(89)90069-9
  20. H.-H. Kuo, Fourier-Mehler transform in white noise analysis, in: Gaussian Random Fields, the third Nagoya Levy Seminar (K.Ito and T. Hida, Eds.), pp. 257-271, World Scientific, 1991
  21. H.-H. Kuo, White Noise Distribution Theory, CRC Press, 1996
  22. H.-H. Kuo, N. Obata, and K. Saito, Levy Laplacian of generalized functions on a nuclear space, J. Funct. Anal. 94 (1990), no. 1, 74-92 https://doi.org/10.1016/0022-1236(90)90028-J
  23. Y. J. Lee, Integral transforms of analytic functions on abstract Wiener spaces, J. Funct. Anal. 47 (1982), no. 2, 153-164 https://doi.org/10.1016/0022-1236(82)90103-3
  24. P. Levy, Lecons D'Analyse Fonctionnelle, Gauthier-Villars, Paris, 1922
  25. N. Obata, White Noise Calculus and Fock Space, Lect. Notes in Math. Vol. 1577, Springer-Verlag, 1994
  26. N. Obata, An analytic characterization of symbols of operators on white noise functionals, J. Math. Soc. Japan 45 (1993), no. 3, 421-445 https://doi.org/10.2969/jmsj/04530421
  27. M. A. Piech, Parabolic equations associated with the number operator, Trans. Amer. Math. Soc. 194 (1974), 213-222 https://doi.org/10.2307/1996802
  28. J. Potthoff and L. Streit, A characterization of Hida distributions, J. Funct. Anal. 101 (1991), no. 1, 212-229 https://doi.org/10.1016/0022-1236(91)90156-Y
  29. K. Saito, A group generated by the Levy Laplacian and the Fourier-Mehler transform, Stochastic analysis on infinite-dimensional spaces, 274-288, Pitman Res. Notes Math. Ser., 310, Longman Sci. Tech., Harlow, 1994

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