For Dynkin labels of weights in 5 and 10 of SU(5), 10 and 16 of SO(10) and 27 ofE, we find it instructive to (a) explōit the Cartan decom position into a nonsemi simple symmetric subalgebra and coset space, (b) introduce a set of annihilation and creation operators, that may act on an invariant 10>, to represent suitably the shift-action and weight-vectors.

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 $V(r^{\alpha}) = a_{1}r^{2\alpha} + (a_{2}/r^{2\alpha}) + a_{3}$. 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 workscarried 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 α 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.

$D$-dimensional fractional Klein–Gordon equation with fractional vector and scalar potential has been studied. Both fractional potentials are taken as attractive Coulomb-type with different multiplicative parameters, namely $v$ and $s$. Jumarie-type definitions for fractional calculus have been used. We have succeeded in achieving Whittaker-type classical differential equation in fractional mode for the required eigenfunction. Fractional Whittaker equation has been manipulated using the behaviour of the eigenfunction at asymptotic distance and origin. This manipulation delivers fractional-type confluent hypergeometric equation to solve. Power series method has been employed to do the task. All the obtained results agree with the existing results in literature when fractional parameter $\alpha$ is unity. Finally, we furnish numerical results with a few eigenfunction graphs for different spatial dimensions and fractional parameters.