Fractional Schrödinger equation with singular potentials of higher order. (English) Zbl 1527.35363
Summary: In this paper the space-fractional Schrödinger equations with singular potentials are studied. Delta like or even higher-order singularities are allowed. By using the regularising techniques, we introduce a family of ‘weakened’ solutions, calling them very weak solutions. The existence, uniqueness and consistency results are proved in an appropriate sense. Numerical simulations are done, and a particles accumulating effect is observed in the singular cases. From the mathematical point of view a “splitting of the strong singularity” phenomena is also observed.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35R11 | Fractional partial differential equations |
Keywords:
Schrödinger equation; fractional Laplacian; generalised solution; singular potential; regularisation; numerical analysis; distributional coefficient; very weak solutionReferences:
| [1] | Altybay, A.; Ruzhansky, M.; Tokmagambetov, N., Wave equation with distributional propagation speed and mass term: Numerical simulations, Appl. Math. E-Notes, 19, 552-562 (2019) · Zbl 1432.35145 |
| [2] | Altybay, A.; Ruzhansky, M.; Sebih, M.; Tokmagambetov, N., Fractional Klein-Gordon equation with singular mass, Chaos Solitons Fractals, 143, 110579 (2021) · Zbl 1498.35364 |
| [3] | Al-Raeei, M.; Sayem El-Daher, M., A numerical method for fractional Schrödinger equation of Lennard-Jones potential, Phys. Lett. A, 383, 26, 125831 (2019) · Zbl 1476.81032 |
| [4] | Cho, Y.; Hwang, G.; Hajaiej, H.; Ozawa, T., On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity, Funkcial. Ekvac., 56, 2, 193-224 (2013) · Zbl 1341.35138 |
| [5] | Díaz, J. I.; Gomez-Castro, D.; Vazquez, J. L., The fractional Schrödinger equation with general nonnegative potentials. The weighted space approach, Nonlinear Anal., 177, 325-360 (2018) · Zbl 1402.35093 |
| [6] | El Aidi, M., On the decay at infinity of solutions of fractional Schrödinger equations, Complex Var. Elliptic Equ., 65, 2, 141-151 (2019) · Zbl 1507.35320 |
| [7] | Ebert, M. R.; Reissig, M., Methods for Partial Differential Equations (2018), Birkhäuser · Zbl 1503.35003 |
| [8] | Friedlander, F. G.; Joshi, M., Introduction to the Theory of Distributions (1998), Cambridge University Press · Zbl 0971.46024 |
| [9] | Garetto, C.; Ruzhansky, M., Hyperbolic second order equations with non-regular time dependent coefficients, Arch. Rational Mech. Anal., 217, 1, 113-154 (2015) · Zbl 1320.35181 |
| [10] | Garetto, C., On the wave equation with multiplicities and space-dependent irregular coefficients (2020), Preprint arXiv:2004.09657 |
| [11] | Gomez-Castro, D.; Vazquez, J. L., The fractional Schrödinger equation with singular potential and measure data, Discrete Contin. Dyn. Syst., 39, 12, 7113-7139 (2019) · Zbl 1425.35212 |
| [12] | Guo, X.; Xu, M., Some physical applications of fractional Schrödinger equation, J. Math. Phys., 47, 082104 (2006) · Zbl 1112.81028 |
| [13] | Laskin, N., Fractional quantum mechanics and Levy path integrals, Phys. Lett. A, 268, 298-305 (2000) · Zbl 0948.81595 |
| [14] | Laskin, N., Fractional Schrödinger equation, Phys. Rev. E, 66, 056108 (2002) |
| [15] | Laskin, N., Fractional Quantum Mechanics (2018), World Scientific Publishing Co. Pte. Ltd. · Zbl 1425.81007 |
| [16] | Liemert, A.; Kienle, A., Fractional Schrödinger equation in the presence of the linear potential, Matematics, 4, 2, 1-14 (2016) · Zbl 1398.35273 |
| [17] | Longhi, S., Fractional Schrödinger equation in optics, Opt. Lett., 6, 1117-1120 (2015) |
| [18] | Lenzi, E. K.; Ribeiro, H. V.; dos Santos, M. A.F.; Rossato, R.; Mendes, R. S., Time dependent solutions for a fractional Schrödinger equation with delta potentials, J. Math. Phys., 54, 082107 (2013) · Zbl 1284.81118 |
| [19] | Munoz, J. C.; Ruzhansky, M.; Tokmagambetov, N., Wave propagation with irregular dissipation and applications to acoustic problems and shallow water, J. Math. Pure Appl., 123, 127-147 (2019) · Zbl 1418.35277 |
| [20] | de Oliveira, E. C.; Costa, F. S.; Vaz, J., The fractional Schrödinger equation for delta potentials, J. Math. Phys., 51, 123517 (2010) · Zbl 1314.81082 |
| [21] | de Oliveira, E. C.; Vaz, J., Tunneling in fractional quantum mechanics, J. Phys. A, 44, 185303 (2011) · Zbl 1215.81115 |
| [22] | Rozmej, P.; Bandrowski, B., On fractional Schrödinger equation, Computational Methods in Science and Technology, 16, 2, 191-194 (2010) |
| [23] | Ruzhansky, M.; Tokmagambetov, N., Wave equation for operators with discrete spectrum and irregular propagation speed, Arch. Rational Mech. Anal., 226, 3, 1161-1207 (2017) · Zbl 1386.35263 |
| [24] | Ruzhansky, M.; Tokmagambetov, N., On a very weak solution of the wave equation for a Hamiltonian in a singular electromagnetic field, Math. Notes, 103, 5-6, 856-858 (2018) · Zbl 1400.35174 |
| [25] | Schwartz, L., Sur l’impossibilité de la multiplication des distributions, C. R. Acad. Sci. Paris, 239, 847-848 (1954) · Zbl 0056.10602 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
