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December  2020, 13(12): 3391-3400. 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  December 2020 Early access  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 and Continuous Dynamical Systems - S, 2020, 13 (12) : 3391-3400. 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.

[2]

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

[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.

[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.

[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.

[6]

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

[7]

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

[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.

[9]

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

[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.

[11]

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

[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. 
[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. 

[14]

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

[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.

[16]

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

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.

[2]

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

[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.

[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.

[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.

[6]

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

[7]

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

[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.

[9]

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

[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.

[11]

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

[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. 
[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. 

[14]

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

[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.

[16]

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

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