Abstract
This paper reports on the characterization of the quantum white noise (QWN) Gross Laplacian based on nuclear algebra of white noise operators acting on spaces of entire functions with \(\theta \)-exponential growth of minimal type. First, we use extended techniques of rotation invariance operators, the commutation relations with respect to the QWN-derivatives and the QWN-conservation operator. Second, we employ the new concept of QWN-convolution operators. As application, we study and characterize the powers of the QWN-Gross Laplacian. As for their associated Cauchy problem it is solved using a QWN-convolution and Wick calculus.
Similar content being viewed by others
References
Accardi, L., Ouerdiane, H., Smolyanov, O.G.: Lévy Laplacian acting on operators. Russ. J. Math. Phys. 10(4), 359–380 (2003)
Accardi, L., Smolyanov, O.G.: Transformations of Gaussian measures generated by the Lévy Laplacian and generalized traces. Dokl. Akad. Nauk SSSR 350, 5–8 (1996)
Barhoumi, A., Ben Ammou, B.K., Rguigui, H.: Operator theory: quantum white noise approach. Quantum Stud. Math. Found. 2(2), 221–241 (2015)
Barhoumi, A., Lanconelli, A., Rguigui, H.: QWN-convolution operators with application to differential equations. Random Oper. Stoch. Equ. 22(4), 195–211 (2014)
Barhoumi, A., Ouerdiane, H., Rguigui, H.: QWN-Euler operator and associated cauchy problem. Infinite Dimens. Anal. Quantum Probab. Relat. Top. 15(1), 1250004 (20 pages) (2012)
Barhoumi, A., Ouerdiane, H., Rguigui, H.: Stochastic heat equation on algebra of generalized functions. Infinite Dimens. Anal. Quantum Probab. Relat. Top. 15(4), 1250026 (18 pages) (2012)
Ettaieb, A., Ouerdiane, H., Rguigui, H.: Powers of quantum white noise derivatives. Infinite Dimens. Anal. Quantum Probab. Relat. Top. 17, 1450018 [16 pages] (2014)
Ettaieb, A., Khalifa, N.T., Ouerdiane, H., Rguigui, H.: Higher powers of analytic operators and associated *-Lie algebra. Infinite Dimens. Anal. Quantum Probab. Relat. Top. 19(2), 1650013 (20 pages) (2016)
Gannoun, R., Hachaichi, R., Ouerdiane, H., Rezgi, A.: Un théorème de dualité entre espace de fonction holomorphes à croissance exponentielle. J. Funct. Anal. 171, 1–14 (2000)
Gross, L.: Abstract Wiener spaces. In: Proceedings of the Fifth Berkeley Symposium Mathematical Statistics and Probability, vol. 2, pp. 31–42 (1967)
Hida, T.: A role of the Lévy Laplacian in the causal calculus of generalized white noise functionals. In: Cambanis, S., et al. (eds.) Stochastic Processes: A Festschrift in Honour of Gopinath Kallianpur. Springer, Berlin (1992)
Hida, T., Obata, N., Saitô, K.: Infinite dimensional rotation and laplacians in terms of white noise calculus. Nagoya Math. J. 128, 65–93 (1992)
Ji, U.C., Obata, N.: Annihilation-derivative, creation-derivative and representation of quantum martingales. Commun. Math. Phys. 286, 751–775 (2009)
Ji, U.C., Obata, N., Ouerdiane, H.: Analytic characterization of generalized Fock space operators as two-variable entire function with growth condition. Infinite Dimens. Anal. Quantum Probab. Relat. Top. 5(3), 395–407 (2002)
Kuo, H.H.: On Laplacian Operator of Generalized Brownian Functionals. Lecture Notes in Mathematics 1203, pp. 119–128 (1986)
Kuo, H.H.: White Noise Distribution Theory. CRC Press, Boca Raton (1996)
Obata, N.: White Noise Calculus and Fock Spaces. Lecture Notes in Mathematics 1577. Spriger, bERLIN (1994)
Obata, N.: Quantum white noise calculus based on nuclear algebras of entire function. Trends Infin. Dimens. Anal. Quantum Probab. (Kyoto 2001) RIMS 1278, 130–157 (2001)
Ouerdiane, H., Rguigui, H.: QWN-conservation operator and associated wick differential equation. Commun. Stoch. Anal. 6(3), 437–450 (2012)
Piech, M.A.: Parabolic equations associated with the number operator. Trans. Am. Math. Soc. 194, 213–222 (1974)
Rebei, H.: On the one-mode quadratic Weyl operators. J. Math. Anal. Appl. 439(1), 135–153 (2016)
Rguigui, H.: Quantum Ornstein–Uhlenbeck semigroups. Quantum Stud. Math. Found. 2(2), 159–175 (2015)
Rguigui, H.: Quantum \(\lambda \)-potentials associated to quantum Ornstein–Uhlenbeck semigroups. Chaos Solitons Fractals 73, 80–89 (2015)
Rguigui, H.: Characterization of the QWN-conservation operator. Chaos Solitons Fractals 84, 41–48 (2016)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by ILWOO CHO.
Rights and permissions
About this article
Cite this article
Rguigui, H. Characterization Theorems for the Quantum White Noise Gross Laplacian and Applications. Complex Anal. Oper. Theory 12, 1637–1656 (2018). https://doi.org/10.1007/s11785-018-0773-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11785-018-0773-x
Keywords
- QWN-Gross Laplacian
- QWN-convolution operators
- Rotation invariance operators
- Wick product
- Cauchy problem