On The Space-Time Fractional Schr\"{o}dinger Equation with time independent potentials

TL;DR

This paper investigates the fractional Schrödinger equation with time-independent potentials using space and time fractional derivatives, extending known results and solving for specific potentials like Dirac-delta and linear potentials.

Contribution

It introduces a separation of variables approach for the fractional Schrödinger equation with specific potentials, generalizing existing solutions to include fractional derivatives.

Findings

Derived solutions for Dirac-delta potential cases.

Extended fractional Schrödinger equation results to linear potentials.

Unified framework encompassing known and new fractional quantum models.

Abstract

This paper is about the fractional Schr\"{o}dinger equation (FSE) expressed in terms of the quantum Riesz-Feller space fractional and the Caputo time fractional derivatives. The main focus is on the case of time independent potential fields as a Dirac-delta potential and a linear potential. For such type of potential fields the separation of variables method allows to split the FSE into space fractional equation and time fractional one. The results obtained in this paper contain as particular cases already known results for FSE in terms of the quantum Riesz space fractional derivative and standard Laplace operator.