Higher-dimensional fractional time-independent Schrödinger equation via fractional derivative with generalised pseudoharmonic potential

Abstract

In this paper, we obtain approximate bound-state solutions of N-dimensional time-independent fractional Schrödinger equation for the generalised pseudoharmonic potential which has the form $V(r^{\alpha })=a_1r^{2\alpha } + ({a_2}/{r^{2\alpha }})+a_3$. Here $\alpha \;(0<\alpha <1)$ acts like a fractional parameter for the space variable r. The entire study consists of the Jumarie-type fractional derivative and the elegance of Laplace transform. As a result, we can successfully express the approximate bound-state solution in terms of Mittag–Leffler function and fractionally defined confluent hypergeometric function. Our study may be treated as a generalisation of all previous works carried out on this topic when $\alpha =1$ and N arbitrary. We provide numerical result of energy eigenvalues and eigenfunctions for a typical diatomic molecule for different $\alpha$ close to unity. Finally, we try to correlate our work with a Cornell potential model which corresponds to $\alpha = {1}$ $/$ ${2}$ with $a_3=0$ and predicts the approximate mass spectra of quarkonia.