-
Previous Article
On the effects of the exterior matrix hostility and a U-shaped density dependent dispersal on a diffusive logistic growth model
- DCDS-S Home
- This Issue
-
Next Article
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 $L^2(\mathbb R^d)$,, 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. 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. |
[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 $L^2(\mathbb R^d)$,, 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. 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. |
[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. |
[1] |
Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020443 |
[2] |
Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (2) : 583-621. doi: 10.3934/cpaa.2020282 |
[3] |
Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017 |
[4] |
Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296 |
[5] |
Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117 |
[6] |
Raimund Bürger, Christophe Chalons, Rafael Ordoñez, Luis Miguel Villada. A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function. Networks & Heterogeneous Media, 2021 doi: 10.3934/nhm.2021004 |
[7] |
Biao Zeng. Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021001 |
[8] |
Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070 |
[9] |
Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021011 |
[10] |
Madhurima Mukhopadhyay, Palash Sarkar, Shashank Singh, Emmanuel Thomé. New discrete logarithm computation for the medium prime case using the function field sieve. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020119 |
[11] |
Kateřina Škardová, Tomáš Oberhuber, Jaroslav Tintěra, Radomír Chabiniok. Signed-distance function based non-rigid registration of image series with varying image intensity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1145-1160. doi: 10.3934/dcdss.2020386 |
[12] |
Olivier Ley, Erwin Topp, Miguel Yangari. Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021007 |
[13] |
Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 |
[14] |
Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu, Nguyen H. Tuan. On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020109 |
[15] |
Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106 |
[16] |
Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312 |
[17] |
Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1749-1762. doi: 10.3934/dcdsb.2020318 |
[18] |
Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021002 |
[19] |
Sören Bartels, Jakob Keck. Adaptive time stepping in elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 71-88. doi: 10.3934/dcdss.2020323 |
[20] |
Simone Göttlich, Elisa Iacomini, Thomas Jung. Properties of the LWR model with time delay. Networks & Heterogeneous Media, 2020 doi: 10.3934/nhm.2020032 |
2019 Impact Factor: 1.233
Tools
Metrics
Other articles
by authors
[Back to Top]