SOLVING A FRACTIONAL NONLINEAR SCHR¨ ODINGER EQUATION WITH SINGULAR CONDITIONS

SOLVING A FRACTIONAL NONLINEAR SCHR¨ ODINGER EQUATION WITH SINGULAR CONDITIONS

A. Benmerrous, M. Elomari

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Abstract

In this paper the time-fractional Schr¨odinger equations with singular potentials are studied. Dirac function or even higher-order singularities are allowed. Using the Gronwall lemma and Laplace transforms, we give and prove the existence and uniqueness of the integral solution of the problem in Colombeau’s algebra.

Keywords

Schr¨odinger equations, Caputo derivative, Generalised solution, Semigroup, Singular potential.