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doi: 10.3934/dcdss.2020246

Feynman path formula for the time fractional Schrödinger equation

Hassan Emamirad and  Arnaud Rougirel

Laboratoire de Mathématiques, Université de Poitiers. teleport 2, BP 179, 86960 Chassneuil du Poitou, Cedex, France

Received  February 2019 Revised  April 2019 Published  January 2020

In this paper, we define $ E_ \alpha(t^ \alpha A) $, where $ A $ is the generator of an uniformly bounded ($ C_0 $) semigroup and $ E_ \alpha(z) $ the Mittag-Leffler function. Since the mapping $ t\mapsto E_ \alpha(t^ \alpha A) $ has not the semigroup property, we cannot use the Trotter formula for representing the Feynman operator calculus. Thus for the Hamiltonian $ H_ \alpha = -\frac{{\hbar_ \alpha2}}{{2m}}\Delta +V(x) $, we express $ E_ \alpha(t^ \alpha H_ \alpha ) $ by subordination principle of the Feynman path integral and we retrieve the corresponding Green function.

Keywords: Feynman path formula, time fractional in the sense of Caputo, Mittag-Leffler function, Wright function.
Mathematics Subject Classification: 35Q41, 81Q30, 26A33.
Citation: Hassan Emamirad, Arnaud Rougirel. Feynman path formula for the time fractional Schrödinger equation. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020246
References:
[1]

B. N. Achar, B. T. Yale and J. Hanneken, Time fractional Schrodinger equation revisited, Adv. Math. Phys., 2013 (2013), Art. ID 290216, 11 pp.  Google Scholar

[2]

E. Bajlekova, Fractional Evolution Equations in Banach Spaces, , Einhoven University of Technology, Ph.D. Dissertation, 2001.  Google Scholar

[3]

P. Chernoff, Semigroup product formulas and addition of unbounded operators,, Bull. Amer. Math. Soc., 76 (1970), 395-398.  doi: 10.1090/S0002-9904-1970-12489-2.  Google Scholar

[4]

H. Emamirad and A. Rougirel, A functional calculus approach for rational approximation with nonuniform partitions,, Discrete Contin. Dyn. Syst., 22 (2008), 955-972.  doi: 10.3934/dcds.2008.22.955.  Google Scholar

[5]

H. Emamirad and A. Rougirel, Solution operators of three time variables for fractional linear problems, Math. Meth. Appl. Sci., 40 (2017), 1553-1572.  doi: 10.1002/mma.4079.  Google Scholar

[6]

H. Emamirad and A. Rougirel, Time fractional linear problem in $L^2(\mathbb R^d)$,, Bull. Sci. Math., 144 (2018), 1-38.  doi: 10.1016/j.bulsci.2018.01.002.  Google Scholar

[7]

H. Emamirad and A. Rougirel, Time fractional Schrödinger equation,, J. Evol. Equ., 19 (2019), 1-15.   Google Scholar

[8]

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, Emended edition. Emended and with a preface by Daniel F. Styer. Dover Publications, Inc., Mineola, NY, 2010.  Google Scholar

[9]

D. Fujiwara, Rigorous Time Slicing Approach to Feynman Path Integrals, Springer Heidelberg, New York, 2017. doi: 10.1007/978-4-431-56553-6.  Google Scholar

[10]

T. L. Gill and W. W. Zachary, Functional Analysis and the Feynman Operator Calculus, Springer Heidelberg, New York, 2016. doi: 10.1007/978-3-319-27595-6.  Google Scholar

[11]

J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, New York, 1985.  Google Scholar

[12] G. W. Johnson and M. L. Lapidus, The Feynman Integral and Feynman's Operational Calculus,, The Clarendon Press, Oxford University Press, New York, 2000.   Google Scholar
[13]

F. Mainardi, Y. Luchko and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation,, Frac. Calc. Appl. Anal., 4 (2001), 153-192.   Google Scholar

[14]

M. Naber, Time fractional Schrödinger equation,, J. Math. Phys., 45 (2004), 3339-3352.  doi: 10.1063/1.1769611.  Google Scholar

[15]

J. Peng and K. Li, A note on property of the Mittag-Leffler function, J. Math. Anal. Appl., 370 (2010), 635-638.  doi: 10.1016/j.jmaa.2010.04.031.  Google Scholar

[16]

J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, Boston, Berlin, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar

show all references

References:
[1]

B. N. Achar, B. T. Yale and J. Hanneken, Time fractional Schrodinger equation revisited, Adv. Math. Phys., 2013 (2013), Art. ID 290216, 11 pp.  Google Scholar

[2]

E. Bajlekova, Fractional Evolution Equations in Banach Spaces, , Einhoven University of Technology, Ph.D. Dissertation, 2001.  Google Scholar

[3]

P. Chernoff, Semigroup product formulas and addition of unbounded operators,, Bull. Amer. Math. Soc., 76 (1970), 395-398.  doi: 10.1090/S0002-9904-1970-12489-2.  Google Scholar

[4]

H. Emamirad and A. Rougirel, A functional calculus approach for rational approximation with nonuniform partitions,, Discrete Contin. Dyn. Syst., 22 (2008), 955-972.  doi: 10.3934/dcds.2008.22.955.  Google Scholar

[5]

H. Emamirad and A. Rougirel, Solution operators of three time variables for fractional linear problems, Math. Meth. Appl. Sci., 40 (2017), 1553-1572.  doi: 10.1002/mma.4079.  Google Scholar

[6]

H. Emamirad and A. Rougirel, Time fractional linear problem in $L^2(\mathbb R^d)$,, Bull. Sci. Math., 144 (2018), 1-38.  doi: 10.1016/j.bulsci.2018.01.002.  Google Scholar

[7]

H. Emamirad and A. Rougirel, Time fractional Schrödinger equation,, J. Evol. Equ., 19 (2019), 1-15.   Google Scholar

[8]

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, Emended edition. Emended and with a preface by Daniel F. Styer. Dover Publications, Inc., Mineola, NY, 2010.  Google Scholar

[9]

D. Fujiwara, Rigorous Time Slicing Approach to Feynman Path Integrals, Springer Heidelberg, New York, 2017. doi: 10.1007/978-4-431-56553-6.  Google Scholar

[10]

T. L. Gill and W. W. Zachary, Functional Analysis and the Feynman Operator Calculus, Springer Heidelberg, New York, 2016. doi: 10.1007/978-3-319-27595-6.  Google Scholar

[11]

J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, New York, 1985.  Google Scholar

[12] G. W. Johnson and M. L. Lapidus, The Feynman Integral and Feynman's Operational Calculus,, The Clarendon Press, Oxford University Press, New York, 2000.   Google Scholar
[13]

F. Mainardi, Y. Luchko and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation,, Frac. Calc. Appl. Anal., 4 (2001), 153-192.   Google Scholar

[14]

M. Naber, Time fractional Schrödinger equation,, J. Math. Phys., 45 (2004), 3339-3352.  doi: 10.1063/1.1769611.  Google Scholar

[15]

J. Peng and K. Li, A note on property of the Mittag-Leffler function, J. Math. Anal. Appl., 370 (2010), 635-638.  doi: 10.1016/j.jmaa.2010.04.031.  Google Scholar

[16]

J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, Boston, Berlin, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar

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