Fractional Schrödinger Equation with Singular Potentials of Higher Order

Altybay A. Ruzhansky M. Sebih M.E. Tokmagambetov N.
February 2021 Elsevier Ltd

Reports on Mathematical Physics
2021 #87 Issue 1 129 - 144 pp.

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.

distributional coefficient , fractional Laplacian , generalised solution , numerical analysis , regularisation , Schrödinger equation , singular potential , very weak solution

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Institute of Mathematics and Mathematical Modeling, 125 Pushkin str., 050010
Al-Farabi Kazakh National University, 71 Al-Farabi ave., Almaty, 050040, Kazakhstan
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, 281 Krijgslaan, Ghent, S8 9000, Belgium
School of Mathematical Sciences, Queen Mary University of London, Mile End Rd, Stepney, London, E1 4NS, United Kingdom
Laboratory of Analysis and Control of Partial Differential Equations, Djillali Liabes University, PO. Box 89, Sidi Bel Abbes, 22000, Algeria
Laboratory of Geomatics, Ecology and Environment, University Mustapha Stambouli of Mascara, Mascara, 29000, Algeria

Institute of Mathematics and Mathematical Modeling
Al-Farabi Kazakh National University
Department of Mathematics: Analysis
School of Mathematical Sciences
Laboratory of Analysis and Control of Partial Differential Equations
Laboratory of Geomatics

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