On the New Form of Representing Cauchy Problem for Schrodinger Equation on the Real Time

The study examines Cauchy problem for the one-dimensional Schrodinger equation ψ'_t(f,x) = iHψ(f,x) with the Hamiltonian -H of the form -Hf = 1/2f" + Vf, where potential V is a real-valued differentiable function bounded with its derivative. This equation has been studied from the creation of quantum mechanics, and it still appears to be a good model case for various methods of solving partial differential equations. In this paper we study the problem of representability of the solution of Cauchy problem in the form of the quasi-Feynman formula, and provide a positive answer to this problem. The quasi-Feynman formula constructed in the paper is a new type of expression, similar to the Feynman formula. It includes multiple integrals of an infinitely increasing multiplicity. The quasi-Feynman formulas are easier to prove (compared to the Feynman formulas) but they provide lengthier expression for the solution. The paper may be of interest to the ones who work in the fields of functional analysis and mathematical physics.

References

[1] Berezin F.A., Shubin M.A. Uravnenie Shredingera [Shrodinger equation]. Moscow, Moscow State University Publ., 1983. 392 p.

[2] Smolyanov O.G., Tokarev A.G., Truman A. Hamiltonian Feynman path integrals via the Chernoff formula. J. Math. Phys., 2002, vol. 43, iss. 10, pp. 5161-5171.

[3] Feynman R.P. Space-time approach to nonrelativistic quantum mechanics. Reviews of Modern Physics, 1948, vol. 20, iss. 2, pp. 367-387.

[4] Feynman R.P. An operation calculus having applications in quantum electrodynamics. Phys. Rev., 1951, vol. 84, pp. 108-128.

[5] Smolyanov O.G. Feynman formulae for evolutionary equations. Trends in stochastic analysis. London Mathematical Society Lecture Notes Series, 2009, vol. 353.

[6] Smolyanov O.G. Schrodinger type semigroups via Feynman formulae and all that. Proceedings of the Quantum Bio-Informatics V. Tokyo University of Science, Japan, 7-12 March 2011. World Sc. 2013.

[7] Butko Ya.A. Feynman formulae for evolution semigroups. Nauka i obrazovanie. MGTU im. N.E. Baumana [Science & Education of the Bauman MSTU. Electronic Journal], 2014, no. 3, pp. 95-132. DOI: 10.7463/0314.0701581 Available at: http://technomag.neicon.ru/en/doc/701581.html

[8] Plyashecnik A.S. Feynman formula for Schrodinger-Type equations with time- and space-dependent coefficients. Russian Journal of Mathematical Physics, 2012, vol. 19, no. 3, pp. 340-359.

[9] Plyashecnik A.S. Feynman formulas for second-order parabolic equations with variable coefficients. Russian Journal of Mathematical Physics, 2013, vol. 20, no. 3, pp. 377-379.

[10] Remizov I.D. Solution of a Cauchy problem for a diffusion equation in a Hilbert space by a Feynman formula. Russian Journal of Mathematical Physics, 2012, vol. 19, no. 3, pp. 360-372.

[11] Remizov I.D. Solution to a parabolic differential equation in Hilbert space via Feynman formula-I. Model. i analiz inform. sistem [Modelling and Analusis of Information Systems], 2015, vol. 22. no 3. pp. 337-355 (in Russ.).

[12] Remizov I.D. Quasi-Feynman formulas - a method of obtaining the evolution operator for the Schrodinger equation. Journal of Functional Analysis, 2016, vol. 270, pp. 4540-4557.

[13] Pazy A. Semigroups of linear operators and applications to partial differential equations. N.Y., Springer-Verlag, 1983. 277 p.

[14] Engel K.-J., Nagel R. One-parameter semigroups for linear evolution equations. N.Y., Springer, 2000. 586 p.

[15] Engel K.-J., Nagel R. A short course on operator semigroups. N.Y., Springer Science + Business Media, 2006. 243 p.

[16] Chernoff Paul R. Note on product formulas for operator semigroups. Journal of Functional Analysis, 1968, vol. 2, iss. 2, pp. 238-242.

[17] Bogachev V.I., Smolyanov O.G. Deystvitelniy i funkcionalniy analiz: universitetskiy kurs [Real and functional analysis: University course]. Izhevsk, NIC "Regulyarnaya i haotichnaya dinamika" [National Research Center "Regular and Chaotic Dynamics"], 2009. 724 p.

[18] Orlov Yu.N., Sakbaev V.Zh., Smolyanov O.G. Feynman formulas as a method of averaging random Hamiltonians. Proceedings of the Steklov Institute of Mathematics, August 2014, vol. 285, iss. 1, pp. 222-232.

[19] Smolyanov O.G., Weizs H. Vacker, Wittich O. Chernoffs theorem and discrete time approximations of Brownian motion on manifolds. Potential Analysis, February 2007, vol. 26, iss. 1, pp. 1-29.

[20] Sobolev S.L. Nekotorye primeneniya funktsional’nogo analiza v matematicheskoy fizike [Some applications of functional analysis in mathematical physics]. Moscow, Nauka Publ., 1998. 336 p.

[21] Pavlova M.F., Timerbaev M.R. Prostranstva Soboleva (teoremy vlozheniya) [Sobolev spaces (Embedding theorems)]. Kazan, Kazan Federal University Publ., 2010. 123 p.