Quantum extensions of Fourier-Gauss and Fourier-Mehler transforms
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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.
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.
참고문헌 (29)
- L. Accardi, "Quantum Laplacians on generalized operators on Boson Fock space, Preprint", ~ (2005)
- L. Accardi, "A quantum approach to Laplace operators", Infinite Dimensional Analysis Quantum Probability and Related Topics, 9(2), pp.215~248 (2006)
- L. Accardi, "L´evy Laplacian acting on operators", Russian Journal of Mathematical Physics, 10(4), pp.359~380 (2003)
- L. Accardi, "L´evy-Laplace operators in spaces of functions on rigged Hilbert spaces", Mathematical Notes, 72(1), pp.129~134 (2002)
- D. M. Chung, "Transformation groups on white noise functionals and their applications", Applied Mathematics & Optimization, 37(2), pp.205~223 (1998)
- D. M. Chung, "Quantum stichastic analysis via white noise operators in weighted Fock space", Reviews in Mathematical Physics, 14(3), pp.241~272 (2002)
- S. Dineen, "Complex Analysis on Infinite Dimensional Spaces" (1999), Springer-Verlag
- L. Gross, "Potential theory on Hilbert space", Journal of Functional Analysis, 1, pp.123~181 (1967)
- Analysis of Brownian Functionals, Carleton Math. Lect. Notes, "T. Hida", Carleton University, ~ (1975)
- T. Hida, "Brownian Motion" (1980), Springer–Verlag
- T. Hida, "Transformations for white noise functionals", Journal of Functional Analysis, 111(2), pp.259~277 (1993)
- T. Hida, "Infinite dimensional rotations and Laplacians in terms of white noise calculus", Nagoya Mathematics Journal, 128, pp.65~93 (1992)
- U. C. Ji, "A unified characterization theorem in white noise theory", INFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS, 6(2), pp.167~178 (2003)
- U. C. Ji, "Unitarity of Kuo’s Fourier-Mehler transform", INFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS, 7(1), pp.147~154 (2004)
- U. C. Ji, "Unitarity of generalized Fourier-Gauss transform", Stochastic Analysis and Applications, 24(4), pp.733~751 (2006)
- U. C. Ji, "Quantum L´evy Laplacian and associated heat equation", Journal of Functional Analysis, 249(1), pp.31~54 (2007)
- H.-H. Kuo, "Fourier-Mehler transforms of generalized Brownian functionals", Proc. Japan Acad. Ser. A Math. Sci., 통권 : 59A, pp.312~314 (1983)
- H.-H. Kuo, "On Laplacian operators of generalized Brownian functionals, Stochastic processes and their applications", Lecture Notes in Mathematics, 1203, pp.119~128 (1986)
- H.-H. Kuo, "Fourier transform in white noise calculus", Journal of Multivariate Analysis, 31(2), pp.311~327 (1989)
- H.-H. Kuo, "Fourier-Mehler transform in white noise analysis, in: Gaussian Random Fields, the third Nagoya L´evy Seminar", pp.257~271 (1991), World Scientific
- H.-H. Kuo, "White Noise Distribution Theory" (1996), CRC Press
- H.-H. Kuo, "L´evy Laplacian of generalized functions on a nuclear space", Journal of Functional Analysis, 94(1), pp.74~92 (1990)
- Y. J. Lee, "Integral transforms of analytic functions on abstract Wiener spaces", Journal of Functional Analysis, 47(2), pp.153~164 (1982)
- P. L´evy, "Lecons D’Analyse Fonctionnelle" (1922), Gauthier-Villars
- N. Obata, "White Noise Calculus and Fock Space, Lect. Notes in Math. Vol. 1577" (1994), Springer-Verlag
- N. Obata, "An analytic characterization of symbols of operators on white noise functionals", Journal of the Mathematical Society of Japan, 45(3), pp.421~445 (1993)
- M. A. Piech, "Parabolic equations associated with the number operator", Transactions of the American Mathematical Society, 194, pp.213~222 (1974)
- J. Potthoff, "A characterization of Hida distributions", Journal of Functional Analysis, 101(1), pp.212~229 (1991)
- K. Saitˆo, "A group generated by the L´evy Laplacian and the Fourier-Mehler transform, Stochastic analysis on infinite-dimensional spaces", Pitman Research Notes in Mathematics Series, 310, pp.274~288 (1994), Longman Sci. Tech.