# Time independent fractional Schrödinger equation for generalized Mie-type
potential in higher dimension framed with Jumarie type fractional derivative

Tapas Das<EMAIL_ADDRESS>Kodalia Prasanna Banga High School (H.S), South
24 Parganas 700146, India  Uttam Ghosh<EMAIL_ADDRESS>Susmita Sarkar
<EMAIL_ADDRESS>Department of Applied Mathematics, University of
Calcutta, Kolkata, India  Shantanu Das<EMAIL_ADDRESS>Reactor Control
System Design Section (E & I Group) BARC Mumbai India

###### Abstract

In this paper we obtain approximate bound state solutions of $N$-dimensional
fractional time independent Schrödinger equation for generalised Mie-type
potential, namely
$V(r^{\alpha})=\frac{A}{r^{2\alpha}}+\frac{B}{r^{\alpha}}+C$. Here
$\alpha(0<\alpha<1)$ acts like a fractional parameter for the space variable
$r$. When $\alpha=1$ the potential converts into the original form of Mie-type
of potential that is generally studied in molecular and chemical physics. The
entire study is composed with Jumarie type fractional derivative approach. The
solution is expressed via Mittag-Leffler function and fractionally defined
confluent hypergeometric function. To ensure the validity of the present work,
obtained results are verified with the previous works for different potential
parameter configurations, specially for $\alpha=1$. At the end, few numerical
calculations for energy eigenvalue and bound states eigenfunctions are
furnished for a typical diatomic molecule.
KEYWORDS: Fractional Schrödinger equation, fractional Laplace transform,
generalized Mie-type potential, bound state solutions, Mittag-Leffler
function.
MSC(2010): 26A33, 44A10, 35Q40

##  1 Introduction

Although the classical calculus provides quite promising tool to model and
explain different dynamical phenomenon, still there are uncountable complex
systems in nature which demand the generalization of classical calculus. For
example, the non-linear oscillation of earthquake [1], the fluid dynamics
traffic model [2], the transport of chemical contaminant through porous rock
[3], the dynamics of visco-elastic materials [4], and more in biology,
continuum and statistical mechanics, anomalous transport, magneto-
thermoelastic diffusion [5-8] are only possible to model elegantly, at least
from mathematical point of view, if concern differential equations are
developed with arbitrary order derivatives (commonly known as fractional order
derivatives).
Addition to the above list, quantum mechanics is also one of the most
celebrated area of study where fractional derivatives are employed for
remodelling the Schrödinger equation [9]. Basically being a non-relativistic
equation, Schrödinger equation is an essential tool for the study of atoms,
nuclei and molecules and their spectral behaviours through proper potential
function which describes the nature of bonding of the quantum particles. One
may ask ‘why we will remodel Schrödinger equation via fractional derivatives?’
‘Does this change will describe any new phenomena?’ Obviously there is a
physical reason behind this. There are lots of similarities between standard
diffusion equation and basic Schrödinger equation. Now when non-Brownian path
are taken in the path integral approach, then resulting diffusion equation
emerges with fractional derivatives which properly describes many complex
diffusion phenomenon [10]. As Schrödinger equation is an outcome of Feynman
path integral formalism one may expect same type of outcome for Schrödinger
equation over the non-Brownian path. Based on this theoretical view fractional
Schrödinger equation has been developed by Laskin [11]. After then Guo and Xu
[12], Dong and Xu [13] studied the space fractional Schrödinger equation with
few specific potential models. More recent works on the one dimensional
fractional Schrödinger equation can be found in reference [14-17].
Motivated by these studies and others in this paper we find the approximate
bound state solution of time independent fractional Schrödinger equation for
generalized Mie-type potentials in $N$-dimensions. The idea of higher
dimension is not fancy in physics. String theory, the only self-consistent
quantum theory of gravity, needs more extra spatial dimensions over the usual
three dimensions to support the theoretical structure. We do not feel the
addition dimensions neither they have been discovered from any high energy
particle physics experiments but we can not say that they do not exit. Perhaps
we could be restrained to 3D world, which is in fact of part of a more
complicated mutidimensional universe possibly up to 10 or more spacial
dimensions [18-19]. The advantage of higher dimensional study is, it provides
a general treatment of the model in such a manner that one can obtain required
results in lower dimensions just dialling appropriate $N$. The Mie-type
potentials [20] originally defined as

$\displaystyle
V(r)=D_{0}\left[\frac{a_{0}}{b_{0}-a_{0}}\left(\frac{r_{0}}{r}\right)^{b_{0}}-\frac{b_{0}}{b_{0}-a_{0}}\left(\frac{r_{0}}{r}\right)^{a_{0}}\right]\,,$

where $D_{0}$ is the interaction energy between two atoms in a molecular
system at equilibrium distance $r=r_{0}$. Here $a_{0}$ and $b_{0}$ are
restricted to take integer values. In this paper we will overrule the
restriction for $a_{0}$ and $b_{0}$ and proposed a more general form of the
potential by adding a constant term $C$. If $b_{0}=2\alpha$, $a_{0}=\alpha$
$(0<\alpha<1)$ then the potential gets a nicer form as

$\displaystyle V(r^{\alpha})=\frac{A}{r^{2\alpha}}+\frac{B}{r^{\alpha}}+C\,,$
(1.1)

where $A=D_{0}r_{0}^{2\alpha}$ and $B=-2D_{0}r_{0}^{\alpha}$. This form is
customary and flexible in terms mathematical point of view. When $\alpha=1$
with $A=-D_{0}r_{0}^{2},B=2D_{0}r_{0}$ and $C=-D_{0}$, we have the modified
Kartzer potential and similarly for $A=D_{0}r_{0}^{2},B=-2D_{0}r_{0}$ and
$C=0$ we have Kartzer-Fues potential [21].
To make this present work self contained it is organized as follows: In the
next section we briefly outline Jumarie type fractional derivative and Laplace
transform. In section 3 we construct the $N$-dimensional Schrödinger equation
via fractional Laplacian operator in hypersherical coordinate. Bound state
spectrum for the Mie type potential is obtained in section 4. Section 5 is
devoted for the discussion where theoretical as well as numerical results are
discussed with few eigenfunctions plotting. Finally the conclusion of the work
appears in the section 6.

##  2 Outline of fractional derivative and Laplace transform

###  2.1 Fractional order derivative of Jumarie type

Jumarie [22] defined the fractional order derivative by modifying the left-
Riemann-Liouville (RL) fractional derivative in the following form for a
continuous function $f(x)$ (but not necessarily differentiable) in the
interval $a$ to $x$, with $f(x)=0$ for $x<a$

$\,{}_{a}^{J}D_{x}^{\alpha}[f(x)]=f^{(\alpha)}(x)=\begin{cases}\frac{1}{\Gamma(-\alpha)}\int_{a}^{x}(x-\xi)^{-\alpha-1}f(\xi)d\xi,&\alpha<0,\\\
\frac{1}{\Gamma(-\alpha)}\frac{d}{dx}\int_{a}^{x}(x-\xi)^{-\alpha}(f(\xi)-f(a))d\xi,&0<\alpha<1,\\\
(f^{(\alpha-n)}(x))^{(n)},&n\leq\alpha<n+1.\end{cases}$ (2.1)

It is customary to take the start point of the interval as $a=0$ and use the
symbol $\,{}_{0}^{J}D_{x}^{\alpha}[f(x)]$ for $f^{(\alpha)}(x)$. Here from in
the rest of the paper we will always denote the fractional derivative
$f^{(\alpha)}(x)\equiv\frac{d^{\alpha}f(x)}{dx^{\alpha}}$ as
$\,{}_{0}^{J}D_{x}^{\alpha}[f(x)]$ with Jumarie sense. In the above
definition, the first expression is just the Riemann-Liouville fractional
integration, the second expression is known as modified Riemann-Liouville
derivative of order $0<\alpha<1$ because of the involvement of $f(a=0)$. The
third line definition is for the range $n\leq\alpha<n+1$. Apart from the
integral type of definition we can also express fractional derivative via
fractional difference. Let $f:\Re\rightarrow\Re$, denotes a continuous (but
not necessarily differentiable) function such that $x\rightarrow f(x)$ for all
$\in\Re$. If $h>0$ denotes a constant discretization span with forward
operator $FW(h)f(x)=f(x+h)$; then the right hand fractional difference of
$f(x)$ of order $\alpha$ ($0<\alpha<1$) is defined by the expression [23]

$\displaystyle\bigtriangleup^{\alpha}f(x)=(FW(h)-1)^{\alpha}f(x)=\sum_{i=0}^{\infty}(-1)^{i}{{\alpha}\choose{i}}f[x+(\alpha-i)h]\,,$
(2.2)

where generalized binomial coefficients
$\frac{\Gamma(\alpha-i)}{\Gamma(-\alpha)\Gamma(i+1)}={{i-\alpha-1}\choose{i}}=(-1)^{i}{{\alpha}\choose{i}}$.
These equalities being readily established from the definition of a binomial
coefficient and generalization of factorials with Gamma function
$\,{}^{n}C_{r}={{n}\choose{r}}=\frac{n!}{r!(n-r)!}$. Then the Jumarie
fractional derivative is defined as

$\displaystyle f^{(\alpha)}(x)=\lim_{h\downarrow
0}\frac{\bigtriangleup^{\alpha}[f(x)-f(0)]}{h^{\alpha}}=\frac{d^{\alpha}f(x)}{dx^{\alpha}}\,.$
(2.3)

This definition is close to the standard definition of derivatives for
beginner’s study. Following this definition it is clear that the $\alpha$-th
derivative of a constant for $0<\alpha<1$ is zero. Few results for Jumarie
type derivative are listed below depending on the characteristics of given
function $(f[u(x)])$ [24]

$\,{}_{0}^{J}D_{x}^{\alpha}(f[u(x)])$
$\displaystyle=f_{u}^{(\alpha)}(u)(u_{x}^{{}^{\prime}})^{\alpha}\,,$ (2.4a)
$\,{}_{0}^{J}D_{x}^{\alpha}(f[u(x)])$
$\displaystyle=(f/u)^{1-\alpha}(f^{{}^{\prime}}_{u}(u))^{\alpha}u^{\alpha}(x)\,,$
(2.4b) $\,{}_{0}^{J}D_{x}^{\alpha}(f[u(x)])$
$\displaystyle=(1-\alpha)!u^{\alpha-1}f_{u}^{(\alpha)}(u)u^{\alpha}(x)\,,$
(2.4c) $\,{}_{0}^{J}D_{x}^{\alpha}(x^{\beta})$
$\displaystyle=\frac{\Gamma(1+\beta)}{\Gamma(1+\beta-\alpha)}x^{\beta-\alpha}\,.$
(2.4d)

In fractional calculus solution of any linear fractional differential
equation, composed with Jumarie derivative, can be easily obtained in terms of
Mittag-Leffler function of one parameter [25] which is defined as

$\displaystyle
E_{\alpha}(z)=\sum_{\kappa=0}^{\infty}\frac{z^{\kappa}}{\Gamma(\alpha\kappa+1)}\,,\,\,(\alpha>0)\,.$
(2.5)

or more general form [26]
$E_{\alpha,\beta}(z)=\sum_{\kappa=0}^{\infty}\frac{z^{\kappa}}{\Gamma(\alpha\kappa+\beta)}$.
Clearly $E_{\alpha,1}(z)=E_{\alpha}(z)$ and $E_{1,1}(z)=E_{1}(z)=e^{z}$. We
provide few derivative rules [27-28] associated with the Mittag-Leffler
function and its trigonometric counterparts.

$\,{}_{0}^{J}D_{x}^{\alpha}[E_{\alpha}(ax^{\alpha})]$
$\displaystyle=aE_{\alpha}(ax^{\alpha})\,,$ (2.6a)
$\,{}_{0}^{J}D_{x}^{\beta}[E_{\alpha}(ax^{\alpha})]$
$\displaystyle=x^{\alpha-\beta}E_{\alpha,\alpha-\beta+1}(x^{\alpha})\,,$
(2.6b) $\,{}_{0}^{J}D_{x}^{\alpha}[cos_{\alpha}(ax^{\alpha})]$
$\displaystyle=-asin_{\alpha}(ax^{\alpha})\,,$ (2.6c)
$\,{}_{0}^{J}D_{x}^{\alpha}[sin_{\alpha}(ax^{\alpha})]$
$\displaystyle=acos_{\alpha}(ax^{\alpha})\,,$ (2.6d)

where one parameter fractional sine and cosine function are defined as follows
[29]
$cos_{\alpha}(x^{\alpha})=\sum_{\kappa=0}^{\infty}(-1)^{\kappa}\frac{x^{2\kappa\alpha}}{\Gamma(1+2\alpha\kappa)}$
and
$sin_{\alpha}(x^{\alpha})=\sum_{\kappa=0}^{\infty}(-1)^{\kappa}\frac{x^{(2\kappa+1)\alpha}}{\Gamma(1+(2\kappa+1)\alpha)}$
with
$E_{\alpha}(ix^{\alpha})=cos_{\alpha}(x^{\alpha})+isin_{\alpha}(x^{\alpha})$.

###  2.2 Laplace transformation of fractional differ-integrals

In general Laplace transform $F(s)$ or $\mathcal{L}$ of a function $f(x)$ is
defined as [30]

$\displaystyle
F(s)=\mathcal{L}\left\\{f(x)\right\\}=\int_{0}^{\infty}e^{-sx}f(x)dx\,.$ (2.7)

If there is some constant $\sigma\in\Re$ such that $|e^{-\sigma
x}f(x)|\leq\mathcal{M}$ for sufficiently large $x$, the above definition will
exist for Re $[s]>\sigma$. The following are the well known derivative
properties of Laplace transform when $n$ is an integer.

$\displaystyle\mathcal{L}\left\\{f^{(n)}(x)\right\\}$
$\displaystyle=s^{n}F(s)-\sum_{k=0}^{n-1}s^{n-1-k}{f^{(k)}(0)}\,,$ (2.8a)
$\displaystyle\mathcal{L}\left\\{x^{n}f(x)\right\\}$
$\displaystyle=(-1)^{n}F^{(n)}(s)\,,$ (2.8b)

where the superscript $(n)$ denotes the $n$-th derivative with respect to $x$
for $f^{(n)}{(x)}$, and with respect to $s$ for $F^{(n)}(s)$. Now if $n$
becomes non integer, say $\alpha$, then above two rules are generalized as
[31]

$\displaystyle\mathcal{L}\left\\{\frac{d^{\alpha}f(x)}{dx^{\alpha}}\right\\}$
$\displaystyle=s^{\alpha}F(s)-\sum_{k=0}^{n-1}s^{k}\frac{d^{\alpha-1-k}f(x)}{dx^{\alpha-1-k}}\Bigg{|}_{x=0}\,,$
(2.9a) $\displaystyle\mathcal{L}\left\\{x^{\alpha}f(x)\right\\}$
$\displaystyle=-\tau\frac{d^{\alpha}F(s)}{ds^{\alpha}}\,,$ (2.9b)

where, $n$ is the largest integer such that $(n-1)<\alpha\leq n$ and
$\tau=-\frac{cosec((\alpha-\delta)\pi)}{cosec(-\delta\pi)}$ with $-1<\delta<0$
(see Appendix-2). The sum in the Eq.(2.9a) is zero when $\alpha\leq 0$. The
Eq.(2.9b) shows that
$\mathcal{L}\left\\{x^{\alpha}f(x)\right\\}\neq\pm\frac{d^{\alpha}F(s)}{ds^{\alpha}}$
but when $\alpha=1$, (that makes $\tau=1$) this is consistent with Eq.(2.8b)
if we take $n=1$. Choosing the initial condition $f(0)=0$ (frequently appears
in quantum mechanical problems) it is easy to have
$\mathcal{L}\left\\{\frac{d^{\alpha}f(x)}{dx^{\alpha}}\right\\}=s^{\alpha}F(s)$.
Under this circumstance one can generate
$\mathcal{L}\left\\{x^{\alpha}\frac{d^{\beta}f(x)}{dx^{\beta}}\right\\}$ as
follows

$\displaystyle\mathcal{L}\left\\{x^{\alpha}\frac{d^{\beta}f(x)}{dx^{\beta}}\right\\}$
$\displaystyle=-\tau\frac{d^{\alpha}}{ds^{\alpha}}\mathcal{L}\left\\{\frac{d^{\beta}f(x)}{dx^{\beta}}\right\\}\,,$
$\displaystyle=-\tau\frac{d^{\alpha}}{ds^{\alpha}}[s^{\beta}F(s)]$
$\displaystyle=-\tau\Big{[}F(s)\frac{\Gamma(1+\beta)}{\Gamma(1+\beta-\alpha)}s^{\beta-\alpha}+s^{\beta}\frac{d^{\alpha}F(s)}{ds^{\alpha}}\Big{]}\,,$
(2.10)

where we have used the rule $(uv)^{\alpha}=u^{(\alpha)}v+v^{(\alpha)}u$ in
Jumarie sense. Beside the above, the following operational formulae are well
established [25,31]

$\displaystyle\mathcal{L}\left\\{x^{\alpha}\right\\}$
$\displaystyle=\frac{\Gamma(1+\alpha)}{s^{\alpha}}\,,$ (2.11a)
$\displaystyle\mathcal{L}\left\\{E_{\alpha}(ax^{\alpha})\right\\}$
$\displaystyle=\frac{s^{\alpha-1}}{s^{\alpha}-a}\,,$ (2.11b)
$\displaystyle\mathcal{L}\left\\{E_{\alpha}^{(k)}(-x^{\alpha})\right\\}$
$\displaystyle=\frac{s^{\alpha+k-1}}{s^{\alpha}+1}\,,$ (2.11c)
$\displaystyle\mathcal{L}\left\\{x^{\alpha
k+\beta-1}E_{\alpha,\beta}^{(k)}(\pm ax^{\alpha})\right\\}$
$\displaystyle=\frac{k!s^{\alpha-\beta}}{(s^{\alpha}\mp a)^{k+1}}\,,$ (2.11d)

##  3 Construction of fractional Schrödinger equation in $N$-dimension

In this subsection we will define the Cartesian coordinates $x_{i}$ in
$N$-dimensional space as

$\displaystyle x_{1}$
$\displaystyle=r^{\alpha}cos_{\alpha}\theta_{1}^{\alpha}sin_{\alpha}\theta_{2}^{\alpha}\cdots
sin_{\alpha}\phi^{\alpha}\,,$ $\displaystyle x_{2}$
$\displaystyle=r^{\alpha}sin_{\alpha}\theta_{1}^{\alpha}sin_{\alpha}\theta_{2}^{\alpha}\cdots
sin_{\alpha}\phi^{\alpha}\,,$ $\displaystyle x_{b}$
$\displaystyle=r^{\alpha}cos_{\alpha}\theta_{b-1}^{\alpha}sin_{\alpha}\theta_{b}^{\alpha}\cdots
sin_{\alpha}\phi^{\alpha}\,,$ (3.1) $\displaystyle x_{N}$
$\displaystyle=r^{\alpha}cos_{\alpha}\phi^{\alpha}\,,$

where $b\in[3,N-1]$ and $\phi^{\alpha}=\theta_{N-1}^{\alpha}$. Here
$r^{\alpha}$ and $\theta_{b}^{\alpha}$ are the hyperspherical coordinates in
$N$-dimensions. Clearly when $N=3$ and $\alpha=1$ the above equations convert
into usual three dimensional coordinate system $(r,\theta,\varphi)$. Denoting
$\theta_{1}=\varphi$ , $\theta_{2}(=\phi)=\theta$ we can have $x_{1}\equiv
x=rsin\theta cos\varphi,x_{2}\equiv y=rsin\theta sin\varphi,x_{3}\equiv
z=rcos\theta$. Proceeding further to the present hyperspherical coordinates
the sum of the squares of Eqs.(3.1) provides

$\displaystyle r^{2\alpha}=\sum_{i=1}^{\infty}x_{i}^{2}\,,$ (3.2)

where we have used $E_{\alpha}(\pm ix^{\alpha})=cos_{\alpha}(x^{\alpha})\pm
isin_{\alpha}(x^{\alpha})$ and
$E_{\alpha}(i(x+y)^{\alpha})=E_{\alpha}(ix^{\alpha})\times
E_{\alpha}(iy^{\alpha})$ with Jumarie [29] sense. Thus $r^{\alpha}$ is the
fractional radius of a $N$-dimensional sphere. Now following the usual
expression of Laplacian operator in curvilinear coordinate [32], the
fractional Laplacian operator in terms of hyperspherical coordinates can be
written as

$\displaystyle\nabla_{N}^{2\alpha}=\frac{1}{h_{\alpha}}\sum_{k=0}^{N-1}\,{}_{0}^{J}D_{\theta_{k}}^{\alpha}\Big{(}\frac{h_{\alpha}}{h_{k}^{2\alpha}}\,_{0}^{J}D_{\theta_{k}}^{\alpha}\Big{)}\,,$
(3.3)

where the symbol
$\,{}_{0}^{J}D_{\theta_{k}}^{\alpha}=\frac{\partial^{\alpha}}{\partial\theta_{k}^{\alpha}}\Big{(}\text{or}\,\frac{d^{\alpha}}{d\theta_{k}^{\alpha}}\Big{)}$
denotes Jumarie fractional derivative operator and

$\displaystyle
r^{\alpha}=\theta_{0}^{\alpha}\,,\,\;h_{\alpha}=\prod_{k=0}^{N-1}h_{k}^{\alpha}\,,\,\;h_{k}^{2\alpha}=\sum_{i=1}^{N}[\,_{0}^{J}D_{\theta_{k}}^{\alpha}(x_{i})]^{2}\,.$
(3.4)

Now using the formula Eq.(2.4d), Eq.(2.6c) and Eq.(2.6d) it is not hard to
find

$\displaystyle h_{0}^{\alpha}$ $\displaystyle=\alpha!\,,$ $\displaystyle
h_{1}^{\alpha}$
$\displaystyle=r^{\alpha}sin_{\alpha}\theta_{2}^{\alpha}sin_{\alpha}\theta_{3}^{\alpha}\cdots
sin_{\alpha}\phi^{\alpha}\,,$ $\displaystyle h_{2}^{\alpha}$
$\displaystyle=r^{\alpha}sin_{\alpha}\theta_{3}^{\alpha}sin_{\alpha}\theta_{4}^{\alpha}\cdots
sin_{\alpha}\phi^{\alpha}\,,$ (3.5) $\displaystyle h_{p}^{\alpha}$
$\displaystyle=r^{\alpha}sin_{\alpha}\theta_{p+1}^{\alpha}sin_{\alpha}\theta_{p+2}^{\alpha}\cdots
sin_{\alpha}\phi^{\alpha}\,,$ $\displaystyle h_{N-1}^{\alpha}$
$\displaystyle=r^{\alpha}\,.$

So we have

$\displaystyle
h_{\alpha}=\alpha!(r^{\alpha})^{N-1}sin_{\alpha}\theta_{2}^{\alpha}sin^{2}_{\alpha}\theta_{3}^{\alpha}sin^{3}_{\alpha}\theta_{4}^{\alpha}\cdots
sin^{N-3}_{\alpha}\theta_{N-2}^{\alpha}sin^{N-2}_{\alpha}\phi^{\alpha}\,.$
(3.6)

This helps to write the Eq.(3.3) as

$\nabla_{N}^{2\alpha}=\frac{1}{(\alpha!)^{2}}\frac{1}{(r^{\alpha})^{N-1}}\frac{\partial^{\alpha}}{\partial
r^{\alpha}}\Big{[}(r^{\alpha})^{N-1}\frac{\partial^{\alpha}}{\partial
r^{\alpha}}\Big{]}-\frac{\Lambda^{2\alpha}_{N-1}}{r^{2\alpha}}\,,$ (3.7)

where $\Lambda^{2\alpha}_{N-1}$ is fractional hyperangular momentum operator.
The explicit form is

$\displaystyle\Lambda^{2\alpha}_{N-1}=-\Bigg{[}\sum_{k=1}^{N-2}\frac{1}{sin^{2}_{\alpha}\theta^{\alpha}_{k+1}sin^{2}_{\alpha}\theta^{\alpha}_{k+2}\cdots
sin^{2}_{\alpha}\phi^{\alpha}}\Big{(}\frac{1}{sin^{k-1}_{\alpha}\theta^{\alpha}_{k}}\frac{\partial^{\alpha}}{\partial\theta_{k}^{\alpha}}sin^{k-1}_{\alpha}\theta^{\alpha}_{k}\frac{\partial^{\alpha}}{\partial\theta_{k}^{\alpha}}\Big{)}+$
$\displaystyle\frac{1}{sin^{N-2}_{\alpha}\phi^{\alpha}}\frac{\partial^{\alpha}}{\partial\phi^{\alpha}}\Big{(}sin^{N-2}_{\alpha}\phi^{\alpha}\frac{\partial^{\alpha}}{\partial\phi^{\alpha}}\Big{)}\Bigg{]}\,.$
(3.8)

Before going further here we can verify the form of usual three dimensional
Laplacian operator when $\alpha=1$. Earlier we have denoted
$\theta_{1}=\varphi$ , $\theta_{2}(=\phi)=\theta$. Then Eq.(3.7) gives
$\nabla_{3}^{2}\equiv\nabla^{2}=\frac{1}{r^{2}}\frac{\partial}{\partial
r}\Big{(}r^{2}\frac{\partial}{\partial
r}\Big{)}-\frac{\Lambda_{2}^{2}}{r^{2}}$, where $\Lambda_{2}^{2}$ can be
obtained form Eq.(3.8) as
$\Lambda_{2}^{2}=-\Big{[}\frac{1}{sin\theta}\frac{\partial}{\partial\theta}(sin\theta\frac{\partial}{\partial\theta})+\frac{1}{sin^{2}\theta}\frac{\partial^{2}}{\partial\varphi^{2}}\Big{]}$.
So this concludes the verification that

$\displaystyle\nabla^{2}=\frac{1}{r^{2}}\frac{\partial}{\partial
r}\Big{(}r^{2}\frac{\partial}{\partial
r}\Big{)}+\frac{1}{sin\theta}\frac{\partial}{\partial\theta}(sin\theta\frac{\partial}{\partial\theta})+\frac{1}{sin^{2}\theta}\frac{\partial^{2}}{\partial\varphi^{2}}\,.$

Now we are in a position to write the $N$-dimensional fractional time
independent Schrödinger equation for a diatomic molecule of reduced mass
(centre of mass coordinate system) $M=\frac{m_{1}m_{2}}{m_{1}+m_{2}}$ where
$m_{1}$ and $m_{2}$ are the masses of constituent particles forming the
molecule. If we choose the natural unit $\hbar=c=1$ then the form of the
equation in large-$N$ expansion [33] will be

$\displaystyle\Big{[}\nabla_{N}^{2\alpha}+2M(\mathcal{E}_{\alpha}-V(r^{\alpha}))\Big{]}\psi(r^{\alpha},\Omega^{\alpha}_{N})=0\,,$
(3.9)

where $\mathcal{E}_{\alpha}$ and $V(r^{\alpha})$ are the fractional energy and
potential respectively. The term $\Omega^{\alpha}_{N}$ within the argument of
$\psi$ denotes angular variables
$\theta_{1}^{\alpha},\theta_{2}^{\alpha},\theta_{3}^{\alpha}\cdots\theta_{N-2}^{\alpha},\phi^{\alpha}$.
Taking the solution by means of separation variable technique
$\psi(r^{\alpha},\Omega^{\alpha}_{N})=R(r^{\alpha})Y(\Omega^{\alpha}_{N})$ and
adopting the eigenvalue equation for $Y(\Omega^{\alpha}_{N})$ as (see
Appendix-1 for more)

$\displaystyle\Lambda^{2\alpha}_{N-1}Y(\Omega^{\alpha}_{N})=\ell(\ell+N-2)|_{N>1}Y(\Omega^{\alpha}_{N})\,,$
(3.10)

where $\ell$ is orbital angular momentum quantum number, we have the
fractional order hyperradial or in short ‘radial’ equation

$\displaystyle\Bigg{[}\frac{d^{2\alpha}}{dr^{2\alpha}}+\frac{\Gamma(1+\alpha(N-1))}{\Gamma(1+\alpha(N-2))}\frac{1}{r^{\alpha}}\frac{d^{\alpha}}{dr^{\alpha}}-\frac{\ell(\ell+N-2)(\alpha!)^{2}}{r^{2\alpha}}$
$\displaystyle+2M(\alpha!)^{2}(\mathcal{E}_{\alpha}-V(r^{\alpha}))\Bigg{]}R(r^{\alpha})=0\,.$
(3.11)

Here in deriving Eq.(3.11) from Eq.(3.9), we have expanded Eq.(3.7) by means
of Jumarie derivative rules. It is worth to mention that, $\ell$ can take
quantized values $0,1,2,3\cdots$ only.

##  4 Bound state spectrum of fractional Mie-type potential

Inserting the potential (1.1) into Eq.(3.11) and using the following
abbreviations:

$\displaystyle\nu_{\alpha}(\nu_{\alpha}+1)$
$\displaystyle=\ell(\ell+N-2)(\alpha!)^{2}+2M(\alpha!)^{2}A\,,$ (4.1a)
$\displaystyle-\epsilon_{\alpha}^{2}$
$\displaystyle=2M(\alpha!)^{2}(\mathcal{E}_{\alpha}-C)\,,$ (4.1b)
$\displaystyle-\beta_{\alpha}$ $\displaystyle=2M(\alpha!)^{2}B\,,$ (4.1c)

we have

$\,{}_{0}^{J}D_{r}^{2\alpha}[R(r^{\alpha})]+\frac{\Gamma(1+\alpha(N-1))}{\Gamma(1+\alpha(N-2))}\frac{1}{r^{\alpha}}\,_{0}^{J}D_{r}^{\alpha}[R(r^{\alpha})]+\Big{[}\frac{\beta_{\alpha}}{r^{\alpha}}-\frac{\nu_{\alpha}(\nu_{\alpha}+1)}{r^{2\alpha}}-\epsilon_{\alpha}^{2}\Big{]}R(r^{\alpha})=0\,.$
(4.2)

We choose the bound state eigenfunctions
$\psi(r^{\alpha},\Omega^{\alpha}_{N})$ that are vanishing for $r\rightarrow 0$
and $r\rightarrow\infty$. Since this type of initial conditions are associated
with $R(r^{\alpha})$, we take the predetermined solutions of Eq.(4.2) as

$\displaystyle R(r^{\alpha})=(r^{\alpha})^{-k}f(r^{\alpha})|_{k>0}\,.$ (4.3)

Here the term $(r^{\alpha})^{-k}$ ensures the fact that
$R(r\rightarrow\infty)=0$. The unknown function $f(r^{\alpha})$ is expected to
behave like $f(r\rightarrow 0)=0$. After deriving
$\,{}_{0}^{J}D_{r}^{2\alpha}[f(r^{\alpha})]$,
$\,{}_{0}^{J}D_{r}^{\alpha}[f(r^{\alpha})]$ and performing little calculation
on Eq.(4.2) we have

$\,{}_{0}^{J}D_{r}^{2\alpha}f(r^{\alpha})+\frac{Q_{1}(\alpha,k,N)}{r^{\alpha}}\,_{0}^{J}D_{r}^{\alpha}f(r^{\alpha})+\Big{[}\frac{Q_{2}(\alpha,k,N,\nu_{\alpha})}{r^{2\alpha}}+\frac{\beta_{\alpha}}{r^{\alpha}}-\epsilon_{\alpha}^{2}\Big{]}f(r^{\alpha})=0\,,$
(4.4)

where

$\displaystyle Q_{1}(\alpha,k,N)$ $\displaystyle=\frac{2\Gamma(1-\alpha
k)}{\Gamma(1-\alpha
k-\alpha)}+\frac{\Gamma(1+\alpha(N-1))}{\Gamma(1+\alpha(N-2))}\,,$ (4.5a)
$\displaystyle Q_{2}(\alpha,k,N,\nu_{\alpha})$
$\displaystyle=\frac{\Gamma(1-\alpha k)}{\Gamma(1-\alpha
k-\alpha)}\Bigg{[}\frac{\Gamma(1-\alpha k-\alpha)}{\Gamma(1-\alpha
k-2\alpha)}+\frac{\Gamma(1+\alpha(N-1))}{\Gamma(1+\alpha(N-2))}\Bigg{]}-\nu_{\alpha}(\nu_{\alpha}+1)\,.$
(4.5b)

Finding the solution of Eq.(4.4) is a difficult task due to the strong
singular term $\frac{Q_{2}(\alpha,k,N,\nu_{\alpha})}{r^{2\alpha}}$. To ease
out the situation we will study the Eq.(4.4) in transformed space (Laplace)
with a parametric restriction

$\displaystyle Q_{2}(\alpha,k,N,\nu_{\alpha})=0\,.$ (4.6)

It is important to mention here that, above condition is not a mandatory or
essential to apply the Laplace transform on Eq.(4.4). The imposed condition
only helps to reduce the tenacious mathematical steps. Denoting the solution
of Eq.(4.6) for $k(>0)$ as $k_{\alpha}^{*}$, we can rewrite Eq.(4.4) as

$\displaystyle
r^{\alpha}\,_{0}{}^{J}D_{r}^{2\alpha}g(r)+Q_{1}(\alpha,k_{\alpha}^{*},N)\,_{0}^{J}D_{r}^{\alpha}g(r)+(\beta_{\alpha}-\epsilon_{\alpha}^{2}r^{\alpha})g(r)=0\,,$
(4.7)

where $f(r^{\alpha})$ is replaced with $g(r)$. Now defining
$\mathcal{L}\left\\{g(r)\right\\}=\zeta(s)$ and using the rules of Laplace
transform, mentioned in subsection (2.2) with $g(0)=0$, it is easy to obtain
the following fractional differential equation (see Appendix-2)

$\,{}_{0}^{J}D_{s}^{\alpha}\zeta(s)+\eta(s^{\alpha})\zeta(s)=0\,,$ (4.8)

where

$\displaystyle\eta{(s^{\alpha})}$
$\displaystyle=\frac{\lambda_{1}}{s^{\alpha}+\epsilon_{\alpha}}+\frac{\lambda_{2}}{s^{\alpha}-\epsilon_{\alpha}}\,,$
(4.9a) $\displaystyle\lambda_{1}$
$\displaystyle=\frac{1}{2}\Big{[}\gamma_{\alpha}+\frac{\beta_{\alpha}}{\tau\epsilon_{\alpha}}\Big{]}\,,$
(4.9b) $\displaystyle\lambda_{2}$
$\displaystyle=\frac{1}{2}\Big{[}\gamma_{\alpha}-\frac{\beta_{\alpha}}{\tau\epsilon_{\alpha}}\Big{]}\,,$
(4.9c) $\displaystyle\gamma_{\alpha}$
$\displaystyle=\frac{\Gamma(1+2\alpha)}{\Gamma(1+\alpha)}-\frac{Q_{1}(\alpha,k_{\alpha}^{*},N)}{\tau}\,.$
(4.9d)

Exact solution of Eq.(4.8) is very complicated and tedious in fractional
domain. The good news is we can approximate the solution very near to the
$\alpha\approx 1.00$ (see Appendix-2). Adopting Jumarie sense integration with
the definition of ‘$\alpha$-logarithmic’ function [34]i.e,
$\int\frac{d^{\alpha}t}{t}=Ln_{\alpha}(\frac{t}{C}),t=E_{\alpha}(Ln_{\alpha}t)$
where $C$ denotes a constant such that $(\frac{t}{C})>0$, we have the
approximate solution in transformed space as

$\displaystyle\zeta(s)=C_{1}(s^{\alpha}+\epsilon_{\alpha})^{-\gamma_{\alpha}}\Big{(}\frac{s^{\alpha}-\epsilon_{\alpha}}{s^{\alpha}+\epsilon_{\alpha}}\Big{)}^{-\lambda_{2}}\,,$
(4.10)

where $C_{1}$ is the integration constant. The second factor of Eq.(4.10) is a
multivalued function when the power $-\lambda_{2}$ is a non integer. The
quantum mechanical eigenfunction must be single valued in nature. So we must
take

$\displaystyle-\lambda_{2}=\frac{1}{2}\Big{[}\frac{\beta_{\alpha}}{\tau\epsilon_{\alpha}}-\gamma_{\alpha}\Big{]}=n\,,n=0,1,2,3,\ldots$
(4.11)

The inverse transform of Eq.(4.10) will provide the solution of the problem in
actual space. To that aim, we expand Eq.(4.10) with help of Eq.(4.11) as

$\displaystyle\zeta(s)$
$\displaystyle=C_{1}(s^{\alpha}+\epsilon_{\alpha})^{-\gamma_{\alpha}}\Big{[}1-\frac{2\epsilon_{\alpha}}{s^{\alpha}+\epsilon_{\alpha}}\Big{]}^{n}\,,$
$\displaystyle=C_{1}\sum_{j=0}^{n}\frac{n!}{j!(n-j)!}(-1)^{j}(2\epsilon_{\alpha})^{j}(s^{\alpha}+\epsilon_{\alpha})^{-(\gamma_{\alpha}+j)}\,,$
$\displaystyle=C_{1}\sum_{j=0}^{n}\frac{n!}{j!(n-j)!}(-1)^{j}(2\epsilon_{\alpha})^{j}\frac{1}{(s^{\alpha}+\epsilon_{\alpha})^{m_{j}+1}}\,,\quad\text{where
$m_{j}=(\gamma_{\alpha}+j-1)$}\,.$ (4.12)

Using the formula given by Eq.(2.11d) for $\alpha=\beta$ we can find the
inverse of Eq.(4.12) quite easily.

$\displaystyle g(r)$
$\displaystyle=C_{1}\sum_{j=0}^{n}\frac{n!}{j!(n-j)!}(-1)^{j}(2\epsilon_{\alpha})^{j}\frac{1}{\Gamma(\gamma_{\alpha}+j)}r^{\alpha(\gamma_{\alpha}+j)-1}E_{\alpha}^{(m_{j})}(-\epsilon_{\alpha}r^{\alpha})\,,$
$\displaystyle=\frac{C_{1}}{\Gamma(\gamma_{\alpha})}r^{\alpha\gamma_{\alpha}-1}\sum_{j=0}^{n}\frac{n!}{j!(n-j)!}(-1)^{j}\frac{\Gamma(\gamma_{\alpha})}{\Gamma(\gamma_{\alpha}+j)}(2\epsilon_{\alpha}r^{\alpha})^{j}E_{\alpha}^{(m_{j})}(-\epsilon_{\alpha}r^{\alpha})\,,$
$\displaystyle=\mathcal{N}_{c}r^{\alpha\gamma_{\alpha}-1}E_{\alpha}^{(m_{j})}(-\epsilon_{\alpha}r^{\alpha})\,_{1}F_{1}(-n,\gamma_{\alpha},2\epsilon_{\alpha}r^{\alpha})\,,$
(4.13)

where
$E_{\alpha}^{(m_{j})}(-\epsilon_{\alpha}r^{\alpha})=\frac{d^{m_{j}}}{dr^{m_{j}}}E_{\alpha}(-\epsilon_{\alpha}r^{\alpha})=\sum_{p=0}^{\infty}\frac{(p+m_{j})!}{p!}\frac{(-\epsilon_{\alpha}r^{\alpha})^{p}}{\Gamma(\alpha
p+\alpha m_{j}+\alpha)}$[19] and $\,{}_{1}F_{1}$ is fractionally defined
confluent hypergeometric function i.e

$\,{}_{1}F_{1}(-n,\gamma_{\alpha},2\epsilon_{\alpha}r^{\alpha})=\sum_{j=0}^{n}\frac{n!}{j!(n-j)!}(-1)^{j}\frac{\Gamma(\gamma_{\alpha})}{\Gamma(\gamma_{\alpha}+j)}(2\epsilon_{\alpha}r^{\alpha})^{j}\,.$

Hence the complete radial eigenfunctions are

$\displaystyle R_{n\alpha N\ell}(r^{\alpha})=r^{-\alpha
k_{\alpha}^{*}}g(r)=\mathcal{N}_{c}r^{\alpha(\gamma_{\alpha}-k_{\alpha}^{*})-1}E_{\alpha}^{(m_{j})}(-\epsilon_{\alpha}r^{\alpha})\,_{1}F_{1}(-n,\gamma_{\alpha},2\epsilon_{\alpha}r^{\alpha})\,,$
(4.14)

where $\mathcal{N}_{c}=\frac{C_{1}}{\Gamma(\gamma_{\alpha})}$ acts like a
normalization constant. The energy eigenvalue equation of the potential model
comes out from Eq.(4.11) as

$\displaystyle\mathcal{E}_{n\alpha
N\ell}=C-\frac{M(\alpha!)^{2}}{2\tau^{2}}\Bigg{[}\frac{B}{n+\frac{1}{2}\frac{\Gamma(1+2\alpha)}{\Gamma(1+\alpha)}-\frac{Q_{1}(\alpha,k^{*}_{\alpha},N)}{2\tau}}\Bigg{]}^{2}\,.$
(4.15)

##  5 Discussion

Originally, Schrödinger equation is a second order differential equation with
non-constant coefficients under the potential model $V(r)$. In this study we
have generalized the potential model as $V(r^{\alpha})$ and eventually studied
fractional Schrödinger equation in multidimensional space. As an example
generalized Mie-type potential has been taken as $V(r^{\alpha})$ and solved
approximately for $\alpha\approx 1.00$. The solution is vary closed to the
exact one, which means we can express the grand solution as

$\displaystyle\Psi(r^{\alpha},\Omega^{\alpha}_{N})=\sum_{n}\mathcal{C}_{n}\psi_{n}(r^{\alpha},\Omega^{\alpha}_{N})\,,$

where $\sum_{n}$ helps to express the overall solution in terms of all
possible solutions i.e a linear combinational form via the constants
$C_{n}(n=1,2,3...)$. It has been noticed that for a particular value of
$\alpha(0<\alpha<1)$ the solution depends on the dimensionality as well as on
the potential variables $A,B,C$. The potential parameter $B$ is not a free
parameter as $A$ and $C$. For example, under any circumstances the parameter
$B$ can not be taken as zero otherwise it will force $\beta_{\alpha}$ to have
a zero value and spoil the quantization condition given by Eq.(4.11). Here we
have few special cases

###  5.1 Fractional Coulomb potential

For this case $A=C=0$ and $B\neq 0$ make the potential as
$V(r^{\alpha})=\frac{B}{r^{\alpha}}$. Immediately we have the energy
eigenvalue equation

$\displaystyle\mathcal{E}_{n\alpha
N\ell}=-\frac{M(\alpha!)^{2}}{2\tau^{2}}\Bigg{[}\frac{B}{n+\frac{1}{2}\frac{\Gamma(1+2\alpha)}{\Gamma(1+\alpha)}-\frac{Q_{1}(\alpha,k^{*}_{\alpha},N)}{2\tau}}\Bigg{]}^{2}\,,$
(5.1)

with radial eigenfunction

$\displaystyle R_{n\alpha
N\ell}(r^{\alpha})=\mathcal{N}_{c}r^{\alpha(\gamma_{\alpha}-k_{\alpha}^{*})-1}E_{\alpha}^{(m_{j})}(-\epsilon_{\alpha}r^{\alpha})\,_{1}F_{1}(-n,\gamma_{\alpha},2\epsilon_{\alpha}r^{\alpha})\,.$
(5.2)

In this case the values of $k_{\alpha}^{*}$ emerge from the Eq.(4.6) where
$\nu_{\alpha}(\nu_{\alpha}+1)=\ell(\ell+N-2)(\alpha!)^{2}$. Now when
$\alpha=1$ (as a result $\tau=1$), the Eq.(4.6) provides $k_{1}^{*}=\ell+N-2$
and hence $Q_{1}(1,k_{1}^{*},N)=-2\ell-N+3$. The energy eigenvalue becomes

$\displaystyle\mathcal{E}_{n1N\ell}=-\frac{M}{2}\Bigg{[}\frac{B}{n+\ell+\frac{N-1}{2}}\Bigg{]}^{2}\,.$
(5.3)

The radial eigenfunctions come out as

$\displaystyle R_{n1N\ell}(r)=\mathcal{N^{{}^{\prime}}}r^{\ell}e^{-\epsilon
r}\,_{1}F_{1}(-n,2\ell+N-1,2\epsilon r)\,,$ (5.4)

where $\mathcal{N^{{}^{\prime}}}$ is new normalization constant viz
$\mathcal{N^{{}^{\prime}}}=\mathcal{N}_{c}(-\epsilon)^{m_{j}}$ The results are
similar with the previous works [19,35].

###  5.2 Mie-type potential in $N$ dimension when $\alpha=1$

Under this condition it is easy to rewrite Eq.(4.6) as
$k(k+1)-k(N-1)-\nu(\nu+1)=0$. For simplicity we write the positive solution of
$k$ as $k_{1}^{*}=k_{\ell N}$. This enable us to find
$Q_{1}(1,k_{1}^{*},N)=-2k_{\ell N}+N-1$. Hence the energy eigenvalues for this
situation emerge as

$\displaystyle\mathcal{E}_{n1N\ell}=C-\frac{M}{2}\Bigg{[}\frac{B}{n+k_{\ell
N}+\frac{3-N}{2}}\Bigg{]}^{2}\,,$ (5.5)

with the radial eigenfunctions

$\displaystyle R_{n1N\ell}(r)=\mathcal{N^{{}^{\prime}}}r^{k_{\ell
N}+2-N}e^{-\epsilon r}\,_{1}F_{1}(-n,2k_{\ell N}-N+3,2\epsilon r)\,.$ (5.6)

$\mathcal{N^{{}^{\prime}}}$ acts the normalization constant as previous. These
all outcomes in this subsection are well matched with the results [36].
Apart from the theoretical aspects, we have examined the present model
numerically also. For a typical diatomic molecule ($M=0.31GeV,D_{0}=2\times
10^{-9}GeV,r_{0}=10^{5}GeV^{-1}$) energy eigenvalues have been derived in
table-1. This clearly shows bound state energy ($E<0$) is possible for
$\alpha(0<\alpha<1)$ to some extent. In case of dimension $N=3$, physically
significant energy eigenvalues come out in the range $0.85<\alpha<1$ for
$n=1,2$ states with $\ell=1$. Corresponding eigenfunctions have been shown in
FIG.1 and FIG.2. The eigenfunctions are continuous and well behaved, but for
lower $\alpha$, such as near $\alpha=0.85$, the graph looses its periodicity
significantly and tries to blow up in the selected range. As the dimension $N$
increases, the quantum mechanical well behaved eigenfunctions are only
possible from $\alpha=0.95$ to $\alpha=1$. These are displayed in
FIG.3,FIG.4,FIG.5,FIG.6 respectively. In these cases, going to the further
lower $\alpha$ near $0.90$ the eigenfunction losses its well behaved property
for $N=4,5$. This is due to the value of the $k_{\alpha}^{*}$ originated from
the condition $Q_{2}(\alpha,k,N,\nu_{\alpha})=0$. It is important to mention
here that, the function $Q_{2}$ is very crucial to stabilize the entire model
of the present study. The solution of $Q_{2}(\alpha,k,N,\nu_{\alpha})=0$, i.e
$k_{\alpha}^{*}$, has several possibilities but only bound states are possible
for those $k_{\alpha}^{*}$ for which the difference
$(\gamma_{\alpha}-k_{\alpha}^{*})$ is not so large to make the factor
$r^{\alpha(\gamma_{\alpha}-k_{\alpha}^{*})-1}$, in Eq.(4.14), a sharp
increasing one.

Fractional Kartzer-Fues potential
Different potential parameters for a typical diatomic molecule and $\tau$
against $\alpha$ $\alpha$ $A=D_{0}r_{0}^{2\alpha}$ $B=-2D_{0}r_{0}^{\alpha}$
$\tau$ $0.70$ $0.0200$ $-1.2649\times 10^{-5}$ $1.7013$ $0.75$ $0.0632$
$-2.2494\times 10^{-5}$ $1.4142$ $0.80$ $0.2000$ $-4.0\times 10^{-5}$ $1.2361$
$0.85$ $0.6325$ $-7.1131\times 10^{-5}$ $1.2223$ $0.90$ $2.000$ $-1.2649\times
10^{-4}$ $1.0515$ $0.95$ $6.3246$ $-2.2494\times 10^{-4}$ $1.0125$ $1.0$
$20.0$ $-4.0\times 10^{-4}$ $1.0$

Table 1: $D_{0}=2\times 10^{-9}GeV$, $M=0.31GeV$, $r_{0}=10^{5}GeV^{-1}$,
$C=0$, $\delta=-0.5$, $\ell=1$

Energy spectrum of the Molecule (eV unit)

$N$ $\alpha$ $k_{\alpha}^{*}$ $Q_{1}$ $\gamma_{\alpha}$ $\mathcal{E}(n=1)$
$\mathcal{E}(n=2)$ $0.70$ $1.401540$ $-24.0199$ $15.4856$ $-8.9562\times
10^{-5}$ $-7.212\times 10^{-5}$ $0.75$ $1.277020$ $-9.2051$ $7.9555$ $-0.0013$
$-8.9706\times 10^{-4}$ $0.80$ $2.280294$ $-3.9385$ $4.7212$ $-0.0121$
$-0.0072$ $3$ $0.85$ $2.118725$ $-3.2152$ $4.2640$ $-0.0463$ $-0.0266$ $0.90$
$2.030419$ $-3.1805$ $4.7679$ $-0.1753$ $-0.1045$ $0.95$ $5.452018$ $-5.1659$
$6.9670$ $-0.3536$ $-0.2364$ $1.0$ $4.327531$ $-6.6551$ $8.6551$ $-0.8456$
$-0.5994$ $0.70$ $8.293139$ $-9.9419$ $7.2108$ $-3.2277\times 10^{-4}$
$-2.1788\times 10^{-4}$ $0.75$ $1.324563$ $-61.9221$ $45.2324$ $-5.7474\times
10^{-5}$ $-5.29\times 10^{-5}$ $0.80$ $2.382292$ $-6.7296$ $6.9792$ $-0.0068$
$-0.0045$ $4$ $0.85$ $6.463783$ $-5.1534$ $5.8497$ $-0.0295$ $-0.0187$ $0.90$
$2.128283$ $-4.8426$ $6.3486$ $-0.1152$ $-0.0750$ $0.95$ $5.529740$ $-5.3053$
$7.1047$ $-0.3430$ $-0.2306$ $1.0$ $5.0496913$ $-7.0994$ $9.0994$ $-0.7792$
$-0.5595$ $0.70$ $8.352522$ $-12.3138$ $8.6049$ $-2.4348\times 10^{-4}$
$-1.7235\times 10^{-4}$ $0.75$ $7.687194$ $-8.5297$ $7.4779$ $-0.0014$
$-9.7326\times 10^{-4}$ $0.80$ $7.095835$ $-6.6828$ $6.9413$ $-0.0068$
$-0.0046$ $5$ $0.85$ $4.425374$ $-5.4828$ $6.1192$ $-0.0276$ $-0.0177$ $0.90$
$7.1076947$ $-5.7343$ $7.1966$ $-0.0950$ $-0.0641$ $0.95$ $5.6810119$
$-5.6611$ $7.4561$ $-0.3180$ $-0.2167$ $1.0$ $5.818564$ $-7.6371$ $9.6371$
$-0.7089$ $-0.5162$

Figure 1: n=1 state eigenfunctions in N=3 dimension for fractional Kartzer-
Fues potential @ $\alpha=1.0,0.95,0.90,0.85$ Figure 2: n=2 state
eigenfunctions in N=3 dimension for fractional Kartzer-Fues potential @
$\alpha=1.0,0.95,0.90,0.85$ Figure 3: n=1 state eigenfunctions in N=4
dimension for fractional Kartzer-Fues potential @ $\alpha=1.0,0.95$ Figure 4:
n=2 state eigenfunctions in N=4 dimension for fractional Kartzer-Fues
potential @ $\alpha=1.0,0.95$ Figure 5: n=1 state eigenfunctions in N=5
dimension for fractional Kartzer-Fues potential @ $\alpha=1.0,0.95$ Figure 6:
n=2 state eigenfunctions in N=5 dimension for fractional Kartzer-Fues
potential @ $\alpha=1.0,0.95$ Figure 7: Kartzer-Fues potential variation with
$r$

##  6 Conclusion

In this paper, we have studied approximate bound state solutions of
$N$-dimensional fractional Schrödinger equation for generalised Mie-type
potential, namely
$V(r^{\alpha})=\frac{A}{r^{2\alpha}}+\frac{B}{r^{\alpha}}+C$, where
$\alpha(0<\alpha<1)$ acts like a fractional parameter. We have framed the
entire study by Jumarie type derivative rules. Applying the separation
variable method, $N$-dimensional fractional Schrödinger equation has been
separated into hyperradial(or radial) and hyperangular equations which contain
the fractional order derivatives. The radial fractional order differential
equation, coupled with the eigenvalues of hyperangular part, is found
complicated due to the strong singular term $\frac{1}{r^{2\alpha}}$. To solve
this, after introducing a parametric restriction to remove the strong singular
term, we have constructed the replica of radial equation in transformed space
(Laplace space) which is comparatively easy to tackle, as it contains the
lower order fractional derivative of transformed variable. Inverse transform
of the transformed equation provides us the radial eigenfunction in actual
space with one parameter Mittag-Leffler function and fractional confluent
hypergeometric function. Energy eigenvalue has been obtained via a
quantization condition which also prevent the eigenfunction to turned into a
multivalued one. The results are verified for (i) Coulomb potential in $N$ and
$3$ dimensions (ii) Mie-type potential in $3$ dimensions. We have also
furnished numerical results as well as few eigenfunctions for typical diatomic
molecular system and studied their variation with $\alpha$. This present study
encourages to use fractional Schrödinger equation for different potential
models and disclose the hidden physics behind it for $0<\alpha<1$. Before
ending the conclusion we have drawn FIG.7, the Kartzer-Fues potentials for
different $\alpha$. There is an interesting physics lying beneath of the FIG.7
and associated eigenfunctions. As $\alpha$ is lowered the strength of the
potential becomes smaller i.e tending to zero. In other word we can interpret
that the under examined particle is somehow becoming a free particle type.
Under such circumstance, the eigenfunctions specially for $(N>3)$ and lower
$\alpha$ tend to approach to the eigenfunction of a infinite spherical well
type problem or the quantum mechanical box problem of a free particle. Though
the periodicity losses, the spatial spreading of the eigenfunctions become
narrower eventually that signifies that the probability of finding the
particle in the specified range becomes higher. Now since the uncertainty in
position of the particle decreases, according to uncertainty principle the
uncertainty in momentum will increase considerably. This will also make the
energy more uncertain. This is what we have achieved in TABLE 1. It is clear
in table that as we go to the higher dimension with lower $\alpha$ the energy
eigenvalues become less prominent. More work in the aspect of uncertainty
principle via fractional Schrödinger equation need to be carried out.

## References

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## Appendix-1

A1-1: N dimensional space in ordinary sense
The relations between the Cartesian coordinates $x_{i}$ and the hyperspherical
coordinates $r$ and $\theta_{b}$ in $N$ dimensional space are defined

$\displaystyle x_{1}$ $\displaystyle=rcos\theta_{1}sin\theta_{2}\cdots
sin\theta_{N-1}\,,$ $\displaystyle x_{2}$
$\displaystyle=rsin\theta_{1}sin\theta_{2}\cdots sin\theta_{N-1}\,,$
$\displaystyle x_{b}$ $\displaystyle=rcos\theta_{b-1}sin\theta_{b}\cdots
sin\theta_{N-1}\,,$ $\displaystyle x_{N}$ $\displaystyle=rcos\theta_{N-1}\,,$

where $b\in[3,N-1]$. These generate $r^{2}=\sum_{i=1}^{N}x_{i}^{2}$. The
volume element of the configuration space is calculated as
$\prod_{a=1}^{N}dx_{a}=r^{N-1}drd\Omega$ where
$\Omega=\prod_{a=1}^{N-1}(sin\theta_{a})^{a-1}d\theta_{a}$ with
$r\in(0,\infty),\theta_{1}\in(-\pi.\pi)$ and
$\theta_{c}\in(0,\pi),c\in(2,N-1)$.
A1-2: The hyperspherical harmonics $Y(\Omega^{\alpha}_{N})$
Ordinary spherical harmonic $Y(\theta,\varphi)$, in case of Schrödinger
Hydrogen atom problem, is eigenfunction of angular momentum operator
$\Lambda^{2}$. The eigenvalue is characterized by the angular momentum quantum
number $\ell$ and dimension $N$. We know in two dimension the eigenvalue of
angular momentum operator is $\ell^{2}\equiv\ell(\ell+2-2)$, for three
dimension it is $\ell(\ell+1)\equiv\ell(\ell+3-2)\cdots$ So in $N$ dimension
the eigenvalue is taken as $\ell(\ell+N-2)$. As $\ell$ is a quantum number it
is restricted to take integer values like $0,1,2,3,\cdots$. In fractional
Schrödinger equation, for radial symmetry fractional potential problem in $N$
dimension, the spherical harmonic $Y(\Omega^{\alpha}_{N})$ is named as
hyperspherical harmonic via fractional generalised coordinates. In this study
we proposed that the eigenvalue equation for generalised angular momentum
operator is of the form

$\displaystyle\Lambda^{2\alpha}Y(\Omega^{\alpha}_{N})=\ell(\ell+N-2)Y(\Omega^{\alpha}_{N})\,,$
(A1-1)

where eigenvalue $\ell(\ell+N-2)$ is independent of fractional parameter
$\alpha$.

## Appendix-2

  1. 1.

Proof of Eq.(2.9b)
let us take $\phi(s)=\mathcal{L}\left\\{f(x)\right\\}=s^{\delta}$ with
$-1<\delta<0$. Then

$\displaystyle\frac{d^{\alpha}}{ds^{\alpha}}\mathcal{L}\left\\{f(x)\right\\}$
$\displaystyle=\frac{d^{\alpha}[s^{\delta}]}{ds^{\alpha}}$
$\displaystyle=\frac{\Gamma(\delta+1)}{\Gamma(\delta-\alpha+1)}s^{\delta-\alpha}$
$\displaystyle=\frac{\Gamma(\delta+1)}{\Gamma(\delta-\alpha+1)}\mathcal{L}\left\\{\frac{x^{\alpha-\delta-1}}{\Gamma(\alpha-\delta)}\right\\}$
$\displaystyle=\frac{\Gamma(\delta+1)\Gamma(-\delta)}{\Gamma(\delta-\alpha+1)\Gamma(\alpha-\delta)}\mathcal{L}\left\\{x^{\alpha}f(x)\right\\}$
$\displaystyle=\frac{cosec(-\delta\pi)}{cosec((\alpha-\delta)\pi)}\mathcal{L}\left\\{x^{\alpha}f(x)\right\\}\,,$

where we have used $\Gamma(-y)\Gamma(y+1)=-\pi cosec(\pi y)$ and
$\mathcal{L}\left\\{x^{m}\right\\}=\frac{(m+1)}{s^{m+1}}$ in the above steps.
So we can have

$\displaystyle\mathcal{L}\left\\{x^{\alpha}f(x)\right\\}=-\tau\frac{d^{\alpha}}{ds^{\alpha}}\mathcal{L}\left\\{f(x)\right\\}=-\tau\frac{d^{\alpha}}{ds^{\alpha}}\phi(s)\,,$
(A2-1)

where $\tau=-\frac{cosec((\alpha-\delta)\pi)}{cosec(-\delta\pi)}$ and when
$\alpha=1$ it makes $\tau=1$.
The above relation is valid for a function $f(x)$ such that
$f(x)=\frac{x^{-\delta-1}}{\Gamma(-\delta)};\,\,\ \delta\neq 0$ for which the
Laplace transform is $\phi(s)=s^{\delta}$. The above example suggest that it
is futile to seek generalization of classical formula as in classical calculus
& Laplace transform i.e
$\mathcal{L}\left\\{x^{n}f(x)\right\\}=(-1)^{n}\phi^{(n)}(s)$. However, for
$n$ as non integer we use the expression
$\mathcal{L}\left\\{x^{\alpha}f(x)\right\\}=-\tau\frac{d^{\alpha}}{ds^{\alpha}}\phi(s)$.
A simple criterion for a function $f(x)\sim x^{p}$ is fractionally
differintegrable when $p>-1$, comes form a fact that
$D^{\alpha}x^{p}=\frac{\Gamma(p+1)}{\Gamma(p+1-\alpha)}x^{p-\alpha}$ to be
finite one needs $\frac{\Gamma(p+1)}{\Gamma(p+1-\alpha)}$ should exist; that
is when $p+1>0$ and $p+1-\alpha>0$, i.e we are excluding $p+1=0$ and
$p+1-\alpha=0$, where the Gamma function is blowing out, With these two we say
$p>-1$ as condition for a function $f(x)\sim x^{p}$ as fractionally
differintegrable. For our case $\mathcal{L}\left\\{f(x)\right\\}=s^{\delta}$
is fractionally differintegrable if $\delta>-1$ and our
$f(x)=\frac{x^{-\delta-1}}{\Gamma(-\delta)}=\mathcal{L}^{-1}\left\\{s^{\delta}\right\\}$,
where $\delta\neq 0$, thus we take $-1<\delta<0$ to have values of $\tau$.

  2. 2.

Derivation of Eq.(4.10)
Let us write the Eq.(4.7) in a general way and find the solution with given
initial condition. So the problem is find the solution of

$\displaystyle
x^{\alpha}\frac{d^{2\alpha}y(x)}{dx^{2\alpha}}+c_{1}\frac{d^{\alpha}y(x)}{dx^{\alpha}}+(c_{2}-c_{3}^{2}x^{\alpha})y(x)=0\,,\,\,c_{i}(i=1,2,3)\,\mbox{are
constants}$ (A2-2)

with the initial condition $y(0)=0$.
Let the Laplace transform of $y(x)$ is taken as $z(t)$ i.e
$z(t)=\mathcal{L}\left\\{y(x)\right\\}$. Following section-2 we have these

$\displaystyle\mathcal{L}\left\\{\frac{d^{\alpha}y(x)}{dx^{\alpha}}\right\\}$
$\displaystyle=t^{\alpha}z(t)\,,$ (A2-3a)
$\displaystyle\mathcal{L}\left\\{x^{\alpha}y(x)\right\\}$
$\displaystyle=-\tau\frac{d^{\alpha}z(t)}{dt^{\alpha}}\,,$ (A2-3b)
$\displaystyle\mathcal{L}\left\\{x^{\alpha}\frac{d^{2\alpha}y(x)}{dx^{2\alpha}}\right\\}$
$\displaystyle=-\tau
z(t)\frac{\Gamma(1+2\alpha)}{\Gamma(1+\alpha)}t^{\alpha}-\tau
t^{2\alpha}\frac{d^{\alpha}z(t)}{dt^{\alpha}}\,.$ (A2-3c)

Now taking the Laplace transform on Eq.(A2-2) and using above three results we
have

$\displaystyle-\tau
z(t)\frac{\Gamma(1+2\alpha)}{\Gamma(1+\alpha)}t^{\alpha}-\tau
t^{2\alpha}\frac{d^{\alpha}z(t)}{dt^{\alpha}}+c_{1}t^{\alpha}z(t)+c_{2}z(t)+c_{3}^{2}\tau\frac{d^{\alpha}z(t)}{dt^{\alpha}}=0\,.$
(A2-4)

Rearrangement of the transformed Equation (A2-4) yields

$\displaystyle(t^{2\alpha}-c_{3}^{2})\frac{d^{\alpha}z(t)}{dt^{\alpha}}+(bt^{\alpha}-c)z(t)$
$\displaystyle=0\,,$
$\displaystyle\frac{d^{\alpha}z(t)}{dt^{\alpha}}+\eta(t^{\alpha})z(t)$
$\displaystyle=0\,.$ (A2-5)

where $b=\frac{\Gamma(1+2\alpha)}{\Gamma(1+\alpha)}-\frac{c_{1}}{\tau}$,
$c=\frac{c_{2}}{\tau}$ and
$\eta(t^{\alpha})=\frac{bt^{\alpha}-c}{t^{2\alpha}-c_{3}^{2}}$. Now writing
$\eta(t^{\alpha})=\frac{\lambda_{1}}{t^{\alpha}+c_{3}}+\frac{\lambda_{2}}{t^{\alpha}-c_{3}}$
with $\lambda_{1}=\frac{b}{2}+\frac{c}{2c_{3}}$ and
$\lambda_{2}=\frac{b}{2}-\frac{c}{2c_{3}}$ Eq.(A2-5) can be expressed as

$\displaystyle\frac{d^{\alpha}z(t)}{z(t)}=-\Big{(}\frac{\lambda_{1}}{t^{\alpha}+c_{3}}+\frac{\lambda_{2}}{t^{\alpha}-c_{3}}\Big{)}dt^{\alpha}\,.$

Proceeding to the integration of the last step we can write

$\displaystyle\int\frac{d^{\alpha}z(t)}{z(t)}=-\Big{(}\int_{0}^{t}\frac{\lambda_{1}d\xi^{\alpha}}{\xi^{\alpha}+c_{3}}+\int_{0}^{t}\frac{\lambda_{2}d\xi^{\alpha}}{\xi^{\alpha}-c_{3}}\Big{)}\,.$
(A2-6)

Using the definition of fractional logarithmic function in Jumarie sense
$\int\frac{d^{\alpha}t}{t}=Ln_{\alpha}(\frac{t}{C}),t=E_{\alpha}(Ln_{\alpha}t)$
where $C$ denotes a constant such that $(\frac{t}{C})>0$ and $Ln_{\alpha}t$
denotes the inverse function of the Mittag-Leffler function. The integral form
of $Ln_{\alpha}t$ is given by
$Ln_{\alpha}t=\frac{1}{(1-\alpha)!}\int_{0}^{t}(\frac{d\xi}{\xi})^{\alpha}$.
So we have

$\displaystyle\int\frac{d^{\alpha}z(t)}{z(t)}=Ln_{\alpha}\Big{(}\frac{z(t)}{C}\Big{)}\,.$
$\displaystyle\int\frac{dt^{\alpha}}{t^{\alpha}+c_{3}}$
$\displaystyle=\int_{0}^{t}\frac{d\xi^{\alpha}}{\xi^{\alpha}+c_{3}}$
$\displaystyle=\int_{\sqrt[\alpha]{c_{3}}}^{{\sqrt[\alpha]{t^{\alpha}+c_{3}}}}\frac{d(u^{\alpha}-c_{3})}{u^{\alpha}}\,\,\
[\mbox{where}\,\,\xi^{\alpha}+c_{3}=u^{\alpha}]$
$\displaystyle=\int_{\sqrt[\alpha]{c_{3}}}^{{\sqrt[\alpha]{t^{\alpha}+c_{3}}}}\frac{du^{\alpha}}{u^{\alpha}}$
$\displaystyle=\int_{0}^{{\sqrt[\alpha]{t^{\alpha}+c_{3}}}}\frac{du^{\alpha}}{u^{\alpha}}-\int_{0}^{{\sqrt[\alpha]{c_{3}}}}\frac{du^{\alpha}}{u^{\alpha}}$
$\displaystyle=(1-\alpha)!\Big{[}Ln_{\alpha}({\sqrt[\alpha]{t^{\alpha}+c_{3}}})-Ln_{\alpha}(\sqrt[\alpha]{c_{3}})\Big{]}$

Using $Ln_{\alpha}x^{y}=y^{\alpha}Ln_{\alpha}x$ we write

$\displaystyle\int_{0}^{t}\frac{d\xi^{\alpha}}{\xi^{\alpha}+c_{3}}=(1-\alpha)!\Big{[}Ln_{\alpha}({\sqrt[\alpha]{t^{\alpha}+c_{3}}})-Ln_{\alpha}(\sqrt[\alpha]{c_{3}})\Big{]}$
$\displaystyle=\frac{(1-\alpha)!}{\alpha^{\alpha}}[Ln_{\alpha}(t^{\alpha}+c_{3})-Ln_{\alpha}c_{3}]\,.$
(A2-7)

Similarly

$\displaystyle\int_{0}^{t}\frac{d\xi^{\alpha}}{\xi^{\alpha}-c_{3}}=\frac{(1-\alpha)!}{\alpha^{\alpha}}[Ln_{\alpha}(t^{\alpha}-c_{3})-Ln_{\alpha}c_{3}]\,.$
(A2-8)

Hence From Eq.(A2-6) we can get

$\displaystyle Ln_{\alpha}\Big{(}\frac{z(t)}{C}\Big{)}$
$\displaystyle=-\lambda_{1}\frac{(1-\alpha)!}{\alpha^{\alpha}}Ln_{\alpha}(t^{\alpha}+c_{3})-\lambda_{2}\frac{(1-\alpha)!}{\alpha^{\alpha}}Ln_{\alpha}(t^{\alpha}-c_{3})+c_{4}$
$\displaystyle Ln_{\alpha}\Big{(}\frac{z(t)}{C}\Big{)}$
$\displaystyle=-\Big{(}\frac{(1-\alpha)!}{\alpha^{\alpha}}\Big{)}[\lambda_{1}Ln_{\alpha}(t^{\alpha}+c_{3})+\lambda_{2}Ln_{\alpha}(t^{\alpha}-c_{3})]+c_{4}\,,$
(A2-9)

where
$c_{4}=-\frac{(1-\alpha)!}{\alpha^{\alpha}}(\lambda_{1}+\lambda_{2})Ln_{\alpha}(c_{3})$.
We have these identities in Jumarie sense
$Ln_{\alpha}(x^{y})=y^{\alpha}Ln_{\alpha}x$ and
$(Ln_{\alpha}(uv))^{\frac{1}{\alpha}}=(Ln_{\alpha}u)^{\frac{1}{\alpha}}+(Ln_{\alpha}v)^{\frac{1}{\alpha}}$
Here we can make an approximation for $\alpha\approx 1.00$ as
$Ln_{\alpha}(x^{y})\approx yLn_{\alpha}x$ and
$(Ln_{\alpha}(uv))\approx(Ln_{\alpha}u)+(Ln_{\alpha}v)$. Using these
approximations we write the Eq.(A2-9)

$\displaystyle Ln_{\alpha}\Big{(}\frac{z(t)}{C}\Big{)}$
$\displaystyle=-\Big{(}\frac{(1-\alpha)!}{\alpha^{\alpha}}\Big{)}[\lambda_{1}Ln_{\alpha}(t^{\alpha}+c_{3})+\lambda_{2}Ln_{\alpha}(t^{\alpha}-c_{3})]+c_{4}$
$\displaystyle\approx-[Ln_{\alpha}((t^{\alpha}+c_{3})^{\lambda_{1}})+Ln_{\alpha}((t^{\alpha}-c_{3})^{\lambda_{2}})+ln_{\alpha}c_{5}]\,\,;c_{5}=Ln_{\alpha}c_{4}$
$\displaystyle\approx-
Ln_{\alpha}[c_{5}((t^{\alpha}+c_{3})^{\lambda_{1}})((t^{\alpha}-c_{3})^{\lambda_{2}})]$

Hence from above we write the approximate solution for $\alpha\approx 1.00$ as
following

$\displaystyle
z(t)=C^{{}^{\prime}}(t^{\alpha}+c_{3})^{\lambda_{1}}(t^{\alpha}-c_{3})^{\lambda_{2}}\,,$
(A2-10)

where $C^{{}^{\prime}}=-Cc_{5}$ is any arbitrary suitable constant. In using
the Laplace technique we assume $z(t)\sim t^{\delta}$ i.e ‘the power law
function’ so that is our first approximation and we applied the Laplace
relation to get the equation A2-4. The solution yields in terms of Mittag-
Leffler function is also sum of power law function. Thus our first
approximation is valid to get the final solution in terms of fractional
Logarithmic function. Finally too after further simplification and using
$\alpha\approx 1.00$ we do get a solution A2-10 i nterms of power law
functions. Now for calculation of $\tau$ we chose arbitrarily a value of
$\delta=-0.5$ i.e a mid value of $-1<\delta<0$ for different values of
$\alpha$. The any other number would have given different values but the
analysis would have been similar.