Fractional Green's function for the time-dependent scattering problem in the Space-time-fractional quantum mechanics

TL;DR

This paper develops a fractional Green's function framework for the space-time-fractional Schrödinger equation, enabling analysis of quantum scattering with new analytical tools and approximate solutions.

Contribution

It introduces a novel fractional Green's function expressed via Fox's H-function and series, advancing the mathematical tools for fractional quantum scattering analysis.

Findings

Derived the integral form of the fractional Green's function.

Expressed the Green's function in terms of Fox's H-function and series.

Obtained asymptotic formulas and approximate scattering wave functions.

Abstract

Integral form of the space-time-fractional Schr\"odinger equation for the scattering problem in the fractional quantum mechanics is studied in this paper. We define the fractional Green's function for the space-time fractional Schrodinger equation and express it in terms of Fox's H-function and in a computable series form. The asymptotic formula of the Green's function for large argument is also obtained, and applied to study the fractional quantum scattering problem. We get the approximate scattering wave function with correction of every order.