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A note on the non-homogeneous initial boundary problem for an Ostrovsky-Hunter type equation
Feynman path formula for the time fractional Schrödinger equation
Laboratoire de Mathématiques, Université de Poitiers. teleport 2, BP 179, 86960 Chassneuil du Poitou, Cedex, France |
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.
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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