Feynman Formulas Representation of Semigroups Generated by Parabolic Difference-Differential Equations


We establish that the Laplas operator with perturbation by symmetrised linear hall of displacement argument operators is the generator of unitary group in the Hilbert space of square integrable functions. The representation of semigroup of Cauchy problem solutions for considered functional differential equation is given by the Feynman formulas.

Share and Cite:

V. Sakbaev and A. Yaakbarieh, "Feynman Formulas Representation of Semigroups Generated by Parabolic Difference-Differential Equations," American Journal of Computational Mathematics, Vol. 2 No. 4, 2012, pp. 295-301. doi: 10.4236/ajcm.2012.24040.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] A. D. Myshkis, “Mixed Functional Differential Equations,” Journal of Mathematical Science, Vol. 129, No. 5, 2005, pp. 4111-4226. doi:10.1007/s10958-005-0345-2
[2] A. L. Skubachevski, “On Some Properties of Elliptic and Parabolic Functional-Differential Equations,” Russian Mathematical Surveys, Vol. 51, No. 1, 1996, pp. 169-170. doi:10.1070/RM1996v051n01ABEH002765
[3] M. A. Vorontsov, Yu. D. Dumarevskii, D. V. Pruidze, V. I. Shmal’gauzen, “Izvestiya: The Academy of Sciences of the USSR,” Atmospheric and Oceanic Physics, Vol. 52, No. 2, 1988.
[4] Y. A. Butko, R. L. Shilling and O. G. Smolyanov, “Feynman Formulae for Feller Semigroups,” Doklady Mathematics, Vol. 82, No. 2, 2010, pp. 679-683. doi:10.1134/S1064562410050017
[5] A. L. Skubachevskii and R. V. Shamin “First Mixed Problem for a Parabolic Difference-Differential Equation,” Mathematical Notes, Vol. 66, No. 1, 1999, pp. 113- 119. doi:10.1007/BF02674077
[6] O. G. Smolyanov and H. von Weizsacker, “Feynman Formulas Generated by Self-Adjoint Extensions of the Laplacian,” Doklady Mathematics, Vol. 79, No. 3, 2009, pp. 335-338. doi:10.1134/S1064562409030090
[7] V. Zh. Sakbaev and O. G. Smolyanov, “Dynamics of a Quantum Particle with Discontinuous Position-Dependent Mass,” Doklady Mathematics, Vol. 82, No. 1, 2010, pp. 630-633. doi:10.1134/S1064562410040332
[8] R. P. Chernoff, “Note on Product Formulas for Operator Semigroups,” Journal of Functional Analysis, Vol. 2, No. 2, 1968, pp. 238-242. doi:10.1016/0022-1236(68)90020-7
[9] E. V. Dynkin, “Markovskie Processy,” Fizmatgiz, Moscow, 1963.
[10] O. G. Smolyanov and E. T. Shavgulidze “Kontinual’nye Integraly,” MGU, Moscow, 1990.
[11] Yu. L. Daleckij and S. V. Fomin, “Mery i Differencial’nye Uravneniya v Beskonechnomernih Prostranstvah,” Nauka, Moscow, 1983.
[12] V. Zh. Sakbaev and O. G. Smolyanov, “Diffusion and Quantum Dynamics of Particles with Position-Dependent Mass,” Doklady Mathematics, Vol. 86, No. 1, 2012, pp. 460-463.

Copyright © 2022 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.