Solutions of a particle with fractional $\delta$-potential in a fractional dimensional space

TL;DR

This paper develops a Fourier transform method in fractional dimensional space to solve the Schrödinger equation with Riesz fractional derivatives, providing explicit solutions for a fractional delta potential and extending quantum mechanics to fractional spaces.

Contribution

It introduces a novel Fourier transform approach in fractional dimensions to solve fractional Schrödinger equations with delta potentials, deriving explicit wave functions and energy spectra.

Findings

Eigen solutions exist if 0<λ<α

Solutions reduce to standard quantum mechanics for λ=1 and α=2

Explicit wave functions expressed via Fox H-functions

Abstract

A Fourier transformation in a fractional dimensional space of order $\la$ ($0<\la\leq 1$) is defined to solve the Schr\"odinger equation with Riesz fractional derivatives of order $\a$. This new method is applied for a particle in a fractional $\delta$-potential well defined by $V(x) =- \gamma\delta^{\la}(x)$, where $\gamma>0$ and $\delta^{\la}(x)$ is the fractional Dirac delta function. A complete solutions for the energy values and the wave functions are obtained in terms of the Fox H-functions. It is demonstrated that the eigen solutions are exist if $0< \la<\a$. The results for $\la= 1$ and $\a=2$ are in exact agreement with those presented in the standard quantum mechanics.