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\begin{document}

\title{Study on energy and information-entropic measures of Hulth\selectlanguage{ngerman}én potential
in D dimension by an intuitive approximation to centrifugal term\selectlanguage{english}}



\author[1]{D Nath}%
\author[2]{Amlan Roy}%
\affil[1]{Vivekananda College Thakurpukur}%
\affil[2]{Indian Institute of Science Education and Research Kolkata}%


\vspace{-1em}



  
  \date{June 10, 2024}


\begingroup
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\selectlanguage{english}
\begin{abstract}
%%1234567890 %%1234567890 %%1234567890 %%1234567890 %%1234567890 %%1234567890 %%1234567890 %%1234567890 %%1234567890 %%1234567890
Energy spectrum as well as various information theoretic measures are considered for Hulth\'en potential in $D$ dimension. 
For a given $\ell \neq 0$ state, analytic expressions are derived, following a simple intuitive approximation for accurate representation of 
centrifugal term, within the conventional Nikiforov-Uvarov method. This is derived from a linear combination of two widely 
used Greene-Aldrich and Pekeris-type approximations. Energy, wave function, normalization constant, expectation value in $r$ and $p$ 
space, Heisenberg uncertainty relation, entropic moment of order $\bar{\alpha}$, Shannon entropy, R\'enyi entropy, disequilibrium, majorization as 
well as four selected complexity measures like LMC (L\'opez-Ruiz, Mancini, Calbert), shape R\'enyi complexity, Generalized R\'enyi 
complexity and R\'enyi complexity ratio are offered for different screening parameters ($\delta$). The effective potential is 
described quite satisfactorily throughout the whole domain. Obtained results are compared with theoretical energies available in literature, 
which shows excellent agreement. Performance of six different approximations to centrifugal term is critically discussed. An approximate 
analytical expression for \emph{critical screening} for a specific state in arbitrary dimension is offered. Additionally, some inter-dimensional 
degeneracy occurring in two states, at different dimension for a particular $\delta$ is also uncovered. 
 
\vspace{10mm}
{\bf PACS:} 02.60.-x, 03.65.Ca, 03.65.Ge, 03.65.-w             \\
{\bf Keywords:} Hulth\'en potential, R\'enyi complexity ratio, Statistical complexity, Majorization, Pekeris approximation, Greene-Aldrich 
approximation.%
\end{abstract}%



\sloppy


\altaffiliation{Corresponding author. Email: debrajn@gmail.com}

\altaffiliation{Corresponding author. Email: akroy@iiserkol.ac.in, akroy6k@gmail.com.}

\section*{introduction}
An important model potential is due to Hulth\'en \hyperref[csl:1]{(L.\ Hulth\'en, 1942)},

\begin{equation}
V(r)=-\frac{Z e^2 \delta e^{-\delta r}}{(1-e^{-\delta r })}, 

\end{equation}
where $Z$ is a constant (in the context of atoms, it can be identified with atomic number) and $\delta$ ($> 0$) is a dimensionless, 
screening parameter, that determines the range of potential. It represents an important short-range potential with many 
applications in nuclear and particle physics, atomic physics, solid-state physics, chemical physics \cite{hulthen1957, lam1978, 
durand1981, mishra1989, vandijk1989}, etc. It behaves like Coulomb potential for small (near origin) $r \to 0$, and for large $r$ ($\gg  1$, 
the asymptotic limit), it decays monotonically exponentially to zero. Another distinctive feature is its capacity to offer 
a \emph{limited} number of bound states, lower than that of Coulomb potential, due to the existence of screening parameter. 
For small $\d$, this gives rise to Coulomb potential. Moreover, it constitutes a special case of Eckart potential 
\hyperref[csl:2]{(C.~Eckart, 1303)}.

The bound and scattering states were investigated by a number of research workers, in both non-relativistic and realistic 
domain. The Schr\"odinger equation (SE) can be solved analytically \emph{only} for states with zero angular momentum \hyperref[csl:3]{(1974)}. 
For $\ell \neq 0$ states, therefore, approximations are necessary. 
Various approximate elegant methods were developed for accurate calculation of eigenvalues, eigenfunctions, critical 
screening constant and other properties of this potential, for arbitrary-$\ell$ states. Here we can mention some 
of the promising ones, like variational method utilizing a multitude of basis functions \hyperref[csl:4]{(O.\ A.\ Gomes \& J.\ R.\ Mohallem, 1994}; \hyperref[csl:5]{Y.\ P.\ Varshni, 4682}; \hyperref[csl:6]{C.~Stubbins, 1993)}, 
perturbation theory \hyperref[csl:7]{(P.\ Matthys \& H.\ De Meyer, 1168)}, Pad\'e approximation \hyperref[csl:8]{(C.\ S.\ Lai, 1981}; \hyperref[csl:9]{E.\ R.\ Vrscay, 1433)}, shifted 1/N expansion having different 
flavour \hyperref[csl:10]{(B.\ Roy \& R.\ Roychoudhury, 3051}; \hyperref[csl:11]{A.\ Z.\ Tang \& F.\ T.\ Chan, 1987}; \hyperref[csl:12]{missing citation)}, dynamical group analysis \hyperref[csl:7]{(P.\ Matthys \& H.\ De Meyer, 1168)}. Some other methods include: 
an asymptotic iteration \hyperref[csl:13]{(O.~Bayrak \& I.~Boztosun, 1152)} with a traditional Greene-Aldrich-type centrifugal approximation \hyperref[csl:14]{(R.~L.~Greene \& C.~Aldrich, 2363)}, 
more suitable for smaller screening constant, NU method \hyperref[csl:15]{(S.~M.~Ikhdair, 2009)}, SUSY quantum mechanics and shape 
invariance \hyperref[csl:16]{(H.~I.~Ahmadov \& M.~V.~Qocayeva, 1550)}, combined SUSY and first-order perturbation theory \hyperref[csl:17]{(B.~G\"on\"ul \& M.~Ko\c{c}ak, 2000)}. Energies, radial wave functions were 
deduced by means of a host of approximations to centrifugal term \hyperref[csl:18]{(C.-S.~Jia \& P.-Q.~Wang, 4779}; \hyperref[csl:19]{C.-S.~Jia \& T.~Chen, 4519}; \hyperref[csl:20]{J.~Stanek, 2011)}. Simple quasi-analytic wave 
functions were proposed \hyperref[csl:21]{(S.~H.~Patil, 3153)}, employing the asymptotic and threshold properties, through virial relation. 
Eigenvalues, eigenfunctions as well as ladder operators were constructed \hyperref[csl:22]{(M.~R.~Setare \& E.~Karimi, 1381)}. Accurate numerical schemes for
energies and density moments, such as Ritz \hyperref[csl:23]{(M.~A.~N\'u\~{n}ez, 3620)} and generalized pseudospectral (GPS) \hyperref[csl:24]{(A.~K.~Roy, 2005)} were developed for 
low- and high-lying states. Solution in $D$ dimension, as well as inter-dimensional degeneracies were  
discussed by an exponential approximation to centrifugal term within NU method \hyperref[csl:25]{(D.~Agboola, 653AD)}, a quantization rule 
\hyperref[csl:26]{(X.-Y.~Gu \& J.-Q.~Sun, 221AD)}, etc.

The Heisenberg uncertainty relation of two non-commutative observables is a fundamental property in quantum mechanics. It propounds a certain 
fundamental limit to the accuracy with which the values corresponding to a pair of complementary variables can be predicted. For spherically 
symmetric central potentials, it is often termed as \emph{radial uncertainty} relation. For large variation in distribution function, however, 
these relations can not be properly used. In such occasions, the information-theoretic measure of uncertainty relations appears more appropriate. 
Usually, Shannon and R\'enyi entropic uncertainty relations are employed, where R\'enyi entropy ($R$) \hyperref[csl:27]{(A.~R\'enyi, 1970)} generalizes Shannon entropy 
($S$) \hyperref[csl:28]{(C.~E.~Shannon, 1948)}. In $R$, the entropic moments of a distribution function is used and it is called one-parameter generalized entropy. The entropic moment of a density function $f({\bf r})$ of order $\bar{\a}$ is defined by: $\mathcal{W}^{(\bar{\a})}_{f}=\left\langle f^{\bar{\a}-1}
({\bf r})\right\rangle=\ds\int_{R^D}\left[f({\bf r})\right]^{\bar{\a}}d^Dr$. It is an important quantity for a given entropic functional. Different 
orders lead to various applications in sciences; a particular case of interest is recognized as disequilibrium \hyperref[csl:29]{(M.~J.~W.~Hall, 2602)} of a density 
function $f({\bf r})$, given as: $\mathcal{D}_f=\mathcal{W}^{(2)}_f=\ds\int_{R^D}\left[f({\bf r})\right]^2\,d^Dr$, that occurs for $\bar{\a}=2$. 
Moreover, it is used in linear entropy, $1-\mathcal{D}_f$, as a measure of \emph{entanglement} in a quantum system. Sometimes it is termed as 
\emph{self-similarity} , and signifies the average position of density function. The moments are closely connected to Thomas-Fermi kinetic 
$\big( T_{TF}=\f{3}{10}(3\pi^2)^{\f{2}{3}}\,\mathcal{W}_f^{\f{5}{3}}\big)$ and Dirac exchange 
$\big(E_{D}=\f{3^{\f{4}{3}}}{4\pi^{\f{1}{3}}}\, \mathcal{W}_f^{\f{4}{3}}\big)$ energy functionals, with wide-spread ramifications 
in the realm of density functional theory for atoms and molecules \cite{R.G.Parr.BOOK,
effective.domain,dn.ijqc.rcr}. The $R$ of order $\bar{\a}$ is 
represented as: $\mathcal{R}^{(\bar{\a})}_{f} =\f{1}{1-\bar{\a}}\ln\left[\mathcal{W}^{(\bar{\a})}_{f}\right]$. Tsallis entropy ($T$) \hyperref[csl:30]{(C.~Tsallis, 1988)} 
is yet another related measure, which is connected to $R$. Both $R$ and $T$ have further importance in various composite measure of \emph{complexity} 
(to be discussed later). The $T$ of order $\bar{\a}$ is expressed as: $\mathcal{T}^{(\bar{\a})}_{f}= \f{1}{\bar{\a}-1}\left[1-\mathcal{W}^{(\bar{\a})}_{f}
\right]$. This has attracted much attention in the past few decades in biological, physicochemical and engineering works \hyperref[csl:31]{(E.~Morin, 2008}; \hyperref[csl:32]{K.~Kaneko \& I.~Tsuda, 2001)}. 
In the limiting case of $\bar{\a}\rightarrow 1$, these two entropies lead to $\mathcal{S}_{f}=\ds-\int_{R^D}f({\bf r})\ln\left[f({\bf r})\right]d^Dr$, which is the most important quantity to quantify measure of uncertainty and degree of entanglement. In the other limit $(\bar{\a}\rightarrow\infty)$, $R$ offers the maximum value of a given distribution 
over a domain, i.e., \(\lim\limits_{\bar{\a}\rightarrow\infty}R_f^{(\bar{\a})} = -\ln\sup_{{\bf r}}f ({\bf r})\). On the other hand, 
$\mathcal{R}_{f}^{(1)}-\mathcal{R}_{f}^{(2)}$ is known as the structural entropy \hyperref[csl:33]{(I.~Varga \& J.~Pipek, 262AD)} of a distribution. We note that, estimation of $S$ 
offers important applications in quantifying neural activity, generic diversity, network anomaly detection etc. Usually, $R$ is 
employed as a bound on $S$, and in many occasions, it replaces $S$ as a measure of randomness. Diverse applications of $R$ are recorded in 
literature, including unsupervised learning, password guessability, source adaptation, image registration, etc. They offer a host of 
other interesting properties, which could be found in the references \hyperref[csl:34]{(P.~Ghosh \& D.~Nath, 2646}; \hyperref[csl:35]{D.~Nath, 2681}; \hyperref[csl:36]{I.~Bialynicki-Birula, 521AD}; \hyperref[csl:37]{C.~Beck \& F.~Schl\"ogl, 1995}; \hyperref[csl:38]{K.~\.Zyczkowski \& Information Dyn. {\bf 10}, 2003}; \hyperref[csl:39]{1951}; \hyperref[csl:40]{J.~C.~Baez, 1102}; \hyperref[csl:41]{P.~Jizba \& T.~Arimitsu, 2004}; \hyperref[csl:42]{2015}; \hyperref[csl:43]{1197}; \hyperref[csl:44]{2016}; \hyperref[csl:45]{2009)}.

Thus it is clear that the analysis and interpretation of characteristic features of probability distribution has given rise to a 
large variety of information-theoretic functionals, each one designed to quantify different aspects of the distribution. Recently, 
a rather new concept of \emph{complexity} \hyperref[csl:46]{(K.~D.~Sen (Ed.) {\it Statistical Complexity: Applications in Electronic Structure}, 2011)} has emerged, which is defined in a number of ways and its quantification is not so 
straightforward. Broadly speaking, however, it simply estimates the ability of a system to organize and thus is a good indicator of structure 
or correlation. Several generalizations have been put forth along with an enormous amount of applications. To quantify it in natural 
systems in a quantitative way, a novel idea was proposed; it is maximum in a system which lies in half way of 
equilibrium and disequilibrium. In this work, we are concerned with a few such measures having a prominent role. For two distribution 
functions $f, g$ of order $(\bar{\a},\bar{\b)}$, 
R\'enyi complexity ratio (RCR) \hyperref[csl:35]{(D.~Nath, 2681)} is defined by, $\mathcal{C}^{(\bar{\a},\bar{\b})}_{(f,g)} =e^{\mathcal{R}^{(\bar{\a})}_f-\mathcal{R}^{(\bar{\b})}_g}$. The generalized R\'enyi complexity (GRC) of $f$ of order $(\bar{\a},\b)$ is 
expressed as, \hyperref[csl:34]{(P.~Ghosh \& D.~Nath, 2646}; \hyperref[csl:35]{D.~Nath, 2681}; \hyperref[csl:47]{1235}; \hyperref[csl:48]{231AD}; \hyperref[csl:49]{2014}; \hyperref[csl:50]{2016}; \hyperref[csl:51]{1950)} 
$\mathcal{C}^{(\bar{\a},\bar{\b})}_{f}=e^{\mathcal{R}^{(\bar{\a})}_f-\mathcal{R}^{(\bar{\b})}_f}$.
From this two-parameter GRC, one recovers a one-parameter shape R\'enyi complexity (SRC) of $f$
of order $\bar{\a}$ is given as \hyperref[csl:34]{(P.~Ghosh \& D.~Nath, 2646}; \hyperref[csl:35]{D.~Nath, 2681}; \hyperref[csl:47]{1235}; \hyperref[csl:51]{1950)} 
$\mathcal{C}^{(\bar{\a})}_{f}=\mathcal{D}_fe^{\mathcal{R}_f^{(\bar{\a})}}$. Further, this concept has given rise to the familiar 
LMC (L\'opez-Ruiz, Mancini, Calbert) complexity \hyperref[csl:52]{(1995)}, which is obtained from SRC, with $R$ replacing $S$. This has been 
formulated as a product of two factors, namely one accounting for order/disequilibrium and the other as disorder/uncertainty 
\hyperref[csl:34]{(P.~Ghosh \& D.~Nath, 2646}; \hyperref[csl:35]{D.~Nath, 2681}; \hyperref[csl:52]{1995}; \hyperref[csl:53]{2081}; \hyperref[csl:54]{2596}; \hyperref[csl:55]{C.~Anteneodo \& A.~R.~Plastino, 1996}; \hyperref[csl:56]{R.~G.~Catal\'an \& R.~L\'opez-Ruiz, 111AD)}, and, as such, 
given as, $\mathcal{C}^{(LMC)}_{f}=\mathcal{D}_fe^{\mathcal{S}_f}$. The LMC of continuous variable is defined in 
\hyperref[csl:55]{(C.~Anteneodo \& A.~R.~Plastino, 1996}; \hyperref[csl:56]{R.~G.~Catal\'an \& R.~L\'opez-Ruiz, 111AD)}. Of late, it has been argued that, RCR is the most general form of complexity for a 
given distribution function. Moreover, one can define GRC and all other complexities from RCR.

The above discussion suggests that, a considerable number of good-quality methods are available for low-lying (especially, $\ell =0$) 
states of this potential. While such attempts are rather limited for high-lying states belonging to $\ell \neq 0$. Also, a number of works 
resort to the approximation of centrifugal term from physical/mathematical consideration. Moreover, a majority of the articles are devoted
to the eigenvalues and eigenfunctions, mainly in 3D. Here we have several objectives. At first, we would like to make a detailed analysis 
of energy and test performance of a recently developed, simple approximation for centrifugal term in the context of this potential. Towards 
this goal, a total of six different approximations have been considered to solve the hyper-radial SE by NU method, for an arbitrary state, 
characterized by radial ($n_r$) and angular ($\ell$) quantum numbers. So far, this has been applied to shifted Deng-Fan, Manning-Rosen, P\"oschl-Teller and Eckart 
molecular potentials \hyperref[csl:57]{(D.~Nath \& A.~K.~Roy, 2661}; \hyperref[csl:58]{D.~Nath \& A.~K.~Roy, 2021}; \hyperref[csl:59]{D.~Nath \& A.~K.~Roy, 1389)}. This will extend the scope of applicability to a wider range of physical systems. 
Another objective is to report the results in D dimension, for which references are rather scanty \cite{Hulthen.greene15,
Hulthen.greene16,D.Hydrogen}. After that, we make a first time report on Heisenberg uncertainty product, entropic moments, $R$, $S$,
disequilibrium, majorization as well as a few composite measures of complexity like LMC, SRC, GRC and RCR. Sec.~\ref{sec2} summarizes the details 
of approximation employed, along with the essentials of NU method. Sec.~\ref{sec3} provides the analytic eigenvalues and eigenfunctions in $D$ 
dimension with a special consideration on degeneracy. Representative results are presented to compare and contrast the proposed approximations. 
Finally, information theoretic measures as mentioned above are derived in Sec.~\ref{sec4}. Their calculated results are offered in Sec.~\ref{sec4.3}. 
Reference theoretical results are quoted wherever possible; these exist only for energy spectrum, while information analysis practically remains
unreported so far. We finally offer some concluding remarks in Sec.~\ref{sec5}.

\section*{Methodology}\label{sec2}

\subsection*{The SE of a central potential in $D$ dimension}
At first, let us consider the single-particle, non-relativistic SE, as follows,  
\beq\label{Schro.Eq} 
-\f{\hbar^2}{2\bar{\mu}}{\bf\nabla}_D^2\psi+V({\bf r})\psi({\bf r})=E\psi({\bf r}),
\eeq
where $\overrightarrow{\bf\nabla}_D$ represents the $D$-dimensional gradient operator, ${\bf r}=(x_1,x_2,...,x_D)=(r,\theta_1,\theta_2,...,\theta_{D-1})=(r,\Om_{D-1}), r=|{\bf r}|=\sqrt{\sum\limits_{i=1}^Dx_i^2}$, $x_i=r\cos\theta_i\prod\limits_{k=1}^{i-1}\sin\theta_k$, 
$0\le\theta_i<\pi,~i=1,2,...,D-2$, and $0\le \theta_{D-1}<2\pi$, whereas, 
\beq
{\bf\nabla}_D=\left(\f{\partial}{\partial r},\f{1}{r}\f{\partial}{\partial\theta_1},\f{1}{r\sin\theta_1}\f{\partial}{\partial\theta_2},...,\f{1}{r\prod_{j=1}^{k-1}\sin\theta_{j}}\f{\partial}{\partial\theta_k},...,\f{1}{r\prod_{j=1}^{D-2}\sin\theta_{j}}\f{\partial}{\partial\theta_{D-1}}\right).
\eeq
One can define the Laplacian operator ${\bf\nabla}_D^2$ as, 
\beq
{\bf \nabla}_D^2=\f{1}{r^{D-1}}\f{\partial}{\partial r}\left(r^{D-1}\f{\partial}{\partial r}\right)-\f{\Lambda^2}{r^2},
\eeq
where $\Lambda$ refers to the $D$-dimensional generalized angular momentum operator, as below,  
\beq
\ds{\Lambda}^{2}=-\sum\limits_{i=1}^{D-1}\f{(\sin\theta_i)^{i+1-D}}{(\prod_{j=1}^{i-1}\sin\theta_{j})^2}\f{\partial}{\partial\theta_i}\left((\sin\theta_i)^{D-i-1}\f{\partial}{\partial\theta_i}\right),
\eeq
and $\bar{\mu}, r$, for a diatomic molecule, signify reduced mass and internuclear distance respectively. By supposing,
\beq
\psi_{n_r,\ell,\{\mu\}}({\bf r})=r^{-\f{D-1}{2}}R_{{n_r},\ell}(r)Y_{\ell,\{\mu\}}(\Om),
\eeq
to be the solution of Eq.~(\ref{Schro.Eq}), one gets the following $D$-dimensional hyper-radial SE, 
\beq\label{Eq.R} 
\f{d^2R_{n_r,\ell}}{dr^2}+\left[\f{2\bar{\mu}}{\hbar^2}E_{n_r, \ell}-\f{2\bar{\mu} }{\hbar^2}V(r)-
\f{L^{(\ell,D)}(L^{(\ell,D)}+1)}{r^2}\right]R_{n_r,\ell}=0,
\eeq
along with 
\beq
\ds{\Lambda}^{2} Y_{\ell,\{\mu\}}=\ell(\ell+D-2) Y_{\ell,\{\mu\}},
\eeq
where $R_{n_r,\ell}$ corresponds to the hyper-radial component of wave function, $\ell(\ell+D-2)$ implies the separation constant, 
while $L^{(\ell,D)}=\ell+\f{D-3}{2}$, and $\ell$ refers to orbital angular momentum quantum number. Then it appears that, 
\beq\label{Ylm} 
Y_{\ell,\{\mu\}}=N_{\ell,\{\mu\}}e^{im\theta_{D-1}}\prod\limits_{j=1}^{D-2}C_{\mu_j-\mu_{j+1}}^{\a_j+\mu_j+1}(\cos\theta_j)(\sin\theta_j)^{\mu_j+1},
\eeq
where $C_{i}^{j}(t)$ is the Gegenbauer polynomial in $t$ of degree $i$ with parameter $j$, whereas,   
\beq
N_{\ell,\{\mu\}}=\f{1}{2\pi}\prod\limits_{j=1}^{D-2}\f{(\a_j+\mu_j)(\mu_j-\mu_{j+1})!\G(\a_j+\mu_{j+1})^2}{\pi 2^{1-2\a_j-2\mu_{j+1}}\G(2\a_j+\mu_j+\mu_{j+1})},
\eeq
provides the normalization constant. Here, $(\ell,\{\mu\})=(\mu_1,\mu_2,...,\mu_{D-1})$, $\ell=\mu_1\ge\mu_2\ge...\ge\mu_{D-2}\ge|\mu_{D-1}|=|m|$, $\ell=0,1,2,...$, $m=0,\pm1,\pm2,...$, $\a_j=(D-j-1)/2$.
In order to obtain the general solution of resulting hyper-radial SE, Eq.~(\ref{Eq.R}), we invoke an exponential transformation of the form, 
$s=e^{-\d r}$. Further, we propose a set of functions $\left\{1,\f{s}{1-s},\f{s^2}{(1-s)^2}\right\}$ for which the centrifugal term is approximated in 
different forms, such as Greene-Aldrich \hyperref[csl:14]{(R.~L.~Greene \& C.~Aldrich, 2363)}, Pekeris-type \hyperref[csl:60]{(C.~L.~Pekeris, 1934)} and a simple novel combination of these two approximations, 
following the works of \hyperref[csl:57]{(D.~Nath \& A.~K.~Roy, 2661}; \hyperref[csl:58]{D.~Nath \& A.~K.~Roy, 2021)}.

\subsection*{Approximation to the centrifugal term}
Let us begin with the following generalized equation for approximating the centrifugal term,  
\beq\label{greene.app1} 
\f{1}{r^2}\approx f_i(r)=\d^2\left(x_{1i}+\f{x_{2i}s}{1-s}+\f{x_{3i}s^2}{(1-s)^2}\right),~i=1,2,3,4.
\eeq
The approximation $f_1(r)$ is identified with those from \hyperref[csl:14]{(R.~L.~Greene \& C.~Aldrich, 2363}; \hyperref[csl:61]{E.~D.~Filho \& R.~M.~Ricotta, 1613}; \hyperref[csl:17]{B.~G\"on\"ul \& M.~Ko\c{c}ak, 2000}; \hyperref[csl:62]{M.~Simsek \& H.~Egrifes, 4379}; \hyperref[csl:13]{O.~Bayrak \& I.~Boztosun, 1152}; \hyperref[csl:25]{D.~Agboola, 653AD}; \hyperref[csl:26]{X.-Y.~Gu \& J.-Q.~Sun, 221AD)}, for values of $x_{11}= 0,x_{21}=x_{31}=1$; $f_2(r)$ is related to those of \hyperref[csl:18]{(C.-S.~Jia \& P.-Q.~Wang, 4779}; \hyperref[csl:19]{C.-S.~Jia \& T.~Chen, 4519)}, with $x_{12}=0,x_{22}=\xi,x_{32}=1$ and $\xi$, an adjustable dimensionless parameter. On the other hand, $f_3(r), f_4(r)$ are considered in \hyperref[csl:63]{(W.-C.~Qiang \& W.-L.~Chen, 2053)} and \hyperref[csl:15]{(S.~M.~Ikhdair, 2009)}, for following values, namely, $x_{13}=\f{1}{12},x_{23}=x_{33}=1; x_{14}=0.0823058167837972,x_{24}=x_{34}=1$. In addition to that, 
$\f{1}{r^2}$ is approximated near $r=r_0$, in \hyperref[csl:60]{(C.~L.~Pekeris, 1934)} by the following, 
\beq\label{pekeris.app1} 
\f{1}{r^2}\approx f_5(r)=\d^2\left(x_{15}+\f{x_{25}s}{1-s}+\f{x_{35}s^2}{(1-s)^2}\right),
\eeq
where
\beq
\ba{l}
x_{15}=\f{1}{u^4}\left[(3+u)s_0^2+(2u-6)s_0+(3-3u+u^2)\right],\\
x_{25}=\f{2}{u^4}(1-s_0)^2\left[3+u+\f{2u-3}{s_0}\right],\\
x_{35}=-\f{1}{u^4}(1-s_0)^3\left[\f{3+u}{s_0}+\f{u-3}{s_0^2}\right],\\
s_0=e^{-u},~u=\d r_0,
\ea
\eeq
with $r_0$ being a positive real number having the dimension of length. The limiting case of $\d \to 0$ yields, 
\beq
\lim\limits_{x\rightarrow 0}x_{15}=\f{1}{12},\lim\limits_{x\rightarrow0}x_{25}=1,\lim\limits_{x\rightarrow0}x_{35}=1.
\eeq
Here, in this work, a new approximation to the centrifugal term, is proposed, much in the spirit of \hyperref[csl:57]{(D.~Nath \& A.~K.~Roy, 2661}; \hyperref[csl:58]{D.~Nath \& A.~K.~Roy, 2021}; \hyperref[csl:59]{D.~Nath \& A.~K.~Roy, 1389)}, 
\beq\label{app.convex} 
\f{1}{r^2}\approx  f_6(r)=\sum\limits_{j=1}^5\lam_j\,f_{j}(r)=\d^2\left(y_1+\f{y_2s}{1-s}+\f{y_3s^2}{(1-s)^2}\right),
\eeq
where
\beq
\ba{l}
y_i=\sum\limits_{j=1}^5\lam_jx_{ij},~i=1,2,3,
\ea
\eeq
with
\beq\label{sum.lam} 
\sum\limits_{j=1}^{5}\lam_j=1.
\eeq
The limiting case ($\d \to 0$) in this occasion produces,  
\beq
\ba{l}
\lim\limits_{\d\rightarrow0}y_1=\sum\limits_{j=1}^4\lam_jx_{1j}+\f{\lam_5}{12}, \\
\lim\limits_{\d\rightarrow0}y_2=\sum\limits_{j=1}^4\lam_jx_{2j}+\lam_5, \\
\lim\limits_{\d\rightarrow0}y_3=\sum\limits_{j=1}^4\lam_jx_{3j}+\lam_5.
\ea
\eeq
Clearly, all the approximations lead to exact centrifugal term, in the following limit,  
\beq
\lim\limits_{\d\rightarrow 0}f_i(r)=\f{1}{r^2},1=1,2,3,4,5,6,
\eeq
and the Hulth\'en potential then reduces to Coulomb potential, 
\beq
\lim\limits_{\d\rightarrow0}V(r)=\lim\limits_{\d\rightarrow0}-\frac{Z e^2 \delta e^{-\delta r}}{(1-e^{-\delta r })}=-\frac{Z e^2}{r}.
\eeq
`\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/f1/f1}
\caption{{\label{fig1.approx} Differences, $\ell(\ell+1)[\f{1}{r^2}-f_i]$, $i=1-6$, in six approximations, for 
$\ell=2$, $\delta=0.05$, $r_0=0.5$, $\xi=1.03$. For (F) and (L), $(\lam_1,\lam_2,\lam_3,\lam_4,\lam_5)$ correspond to $(0,0,0.99,0,0.01)$ 
 $(0,0,0.01,0,0.99)$. Top and bottom panels feature $r \to 0$ and $r \to r_0$ behavior. See text for details.%
}}
\end{center}
\end{figure}

In Fig.~\ref{fig1.approx}, we have plotted the differences between \emph{exact} centrifugal term $\f{\ell(\ell+1)}{r^2}$ and
approximations in the form, $\ell(\ell+1)f_i$, $i=1,2,3,4,5,6$, for $\ell=2$, $\delta=0.05$, $r_0=0.5$, $\xi=1.03$. 
Here the sixth approximations, $f_6^{(a)}$, $f_6^{(b)}$ correspond to $(\lam_1,\lam_2,\lam_3,\lam_4,\lam_5)=(0,0,0.99,0,0.01)$ and 
$(\lam_1,\lam_2,\lam_3,\lam_4,\lam_5)=(0,0,0.01,0,0.99)$ respectively, 
The first six top panels (A)-(F) illustrate the performance near $r \to 0$ region, while last six bottom panels (G)-(L) refer to those 
pertaining 
to $r \to r_0$ area. It is easily seen that, panels (B), (H) corresponding to $f_2$ approximation show qualitatively different behavior
from the rest, and also they record the maximum deviation from \emph{exact} centrifugal term.  The panels (C), (F), show that, $f_3$ and $f_6^{(a)}$ approximations are favorable in the former case, while in 
latter scenario, (I), (K), (L) suggest $f_3, f_5$ and $f_6^{(b)}$ to be more acceptable. The approximations $f_i,i=1,2,3,4$ are seen to be
suitable near origin and $f_5$ around $r=r_0$. On the contrary, however, our proposed approximation $f_6$ performs well near the origin as 
well as in the neighborhood of $r=r_0$. Thus, it remains quite accurate over the \emph{entire effective} domain of $r$ 
\hyperref[csl:57]{(D.~Nath \& A.~K.~Roy, 2661}; \hyperref[csl:58]{D.~Nath \& A.~K.~Roy, 2021}; \hyperref[csl:34]{P.~Ghosh \& D.~Nath, 2646}; \hyperref[csl:59]{D.~Nath \& A.~K.~Roy, 1389}; \hyperref[csl:35]{D.~Nath, 2681)}.

\section*{Eigenvalues and eigenfunctions in $D$ dimension}\label{sec3}

\subsection*{Solution of the hyper-radial SE by NU method} 
In this section, we will review the solutions of a second order differential equation of the form,  \beq\label{Eq.NU.gen} 
\f{d^2R_{n_r,\ell}}{ds^2}+\f{\widetilde{\tau}(s)}{\sigma(s)}\f{dR_{n_r,\ell}}{ds}+\f{\widetilde{\sigma}(s)}{\sigma^2(s)}R_{n_r,\ell}=0,
\eeq
where $\widetilde{\tau}(s)$, $\sigma(s)$, $\widetilde{\sigma}(s)$ are polynomials of degree one, two and two, in $s$, respectively. 
If we let, 
\beq
R_{n_r,\ell}(s)=\phi_1(s)\phi_2(s),
\eeq
be the solution of Eq.~(\ref{Eq.NU.gen}), then it is obtained \hyperref[csl:64]{(A.~F.~Nikiforov \& V.~B.~Uvarov, 1988}; \hyperref[csl:57]{D.~Nath \& A.~K.~Roy, 2661}; \hyperref[csl:58]{D.~Nath \& A.~K.~Roy, 2021)} that, 
\beq\label{Eq.y} 
\sigma(s)\phi_2^{''}(s)+\tau(s)\phi_2'(s)+\nu\, \phi_2(s)=0,
\eeq
and
\beq
\ba{l}
\phi_1(s)=\ds e^{\int\f{\pi(s)}{\sigma(s)}ds},
\ea
\eeq
where
\beq
\ba{ll}\label{pi.sigma} 
\pi(s)&=\f{\sigma'(s)-\widetilde{\tau}(s)}{2}\pm\sqrt{\left(\f{\sigma'(s)-\widetilde{\tau}(s)}{2}\right)^2-\widetilde{\sigma}(s)+k\,\sigma(s)},\\
\tau(s)&=\widetilde{\tau}(s)+2\pi(s),~
\bar{\nu}=k+\pi'(s),
\ea
\eeq
while $\bar{\nu}$ and $k$ are real constants. Since $\pi(s)$ is a polynomial in $s$, one needs to find $k$ in such a way that, $\left(\f{\sigma'(s)-\widetilde{\tau}(s)}{2}\right)^2-\widetilde{\sigma}(s)+k\,\sigma(s)$ is a square of a polynomial in $s$. Then the solutions of 
Eq.~(\ref{Eq.y}) are provided as,  
\beq
\phi_{2,n_r}(s)=\f{1}{\rho(s)}\f{d^{n_r}}{ds^{n_r}}\left[\sigma^{n_r}(s)\rho(s)\right],
\eeq
and eigenvalues are obtained from,  
\beq\label{eigen.nu} 
\nu_{n_r}=-{n_r}\,\tau'(s)-\f{{n_r}({n_r}-1)}{2}\sigma^{''}(s),~{n_r}=0,1,2,\dots, 
\eeq
where
\beq
\ba{l}
\rho(s)=\ds \left[\sigma(s)\right]^{-1}\,e^{\int\f{\tau(s)}{\sigma(s)}ds}.
\ea
\eeq

\begingroup
\squeezetable\selectlanguage{english}
\begin{table}
\caption{{\label{Table1} Energies, $E_{n_r,l}$, of Hulth\'en potential, for $\d=0.025,D=3,r_0=0.5$. The 8th and 9th columns correspond 
to $f_6^{(c)}$ and $f_6^{(d)}$ respectively. For details, see text.}}
\centering
\begin{ruledtabular}
\scalebox{0.9}{\begin{tabular}{c|ccccccccc}
$n_r,\ell$ & $-E_{n_r,\ell}(f_1)^{\dagger}$   &$-E_{n_r,\ell}(f_2)^{\ddagger}$    & $-E_{n_r,\ell}(f_3)$ & 
$-E_{n_r,\ell}(f_4)^{\S}$ & $-E_{n_r,\ell}(f_5)$ & $-E_{n_r,\ell}[f_6^{(c)}]$ & $-E_{n_r,\ell}[f_6^{(d)}]$& Ref.~\hyperref[csl:5]{(Y.\ P.\ Varshni, 4682)} & GPS$^{\P}$ \\ \hline
0,   1   &  0.1128125   &  0.1126344  &  0.1127604  &  0.1127611  & 0.1127604  &  0.1127604  &  0.1127604  &  0.1127605 & 0.1127605 \\
\hline
1,   1   &  0.0437587   &  0.0436848  &  0.0437066  &  0.0437072  & 0.0437066  &  0.0437066  &  0.0437066  &  0.0437069 & 0.0437069 \\
0,   2   &  0.0437587   &  0.0435371  &  0.0436024  &  0.0436044  & 0.0436024  &  0.0436024  &  0.0436024  &  0.0436030 & 0.0436030 \\
\hline
2,   1   &  0.0200000   &  0.0199625  &  0.0199479  &  0.0199486  & 0.0199479  &  0.0199479  &  0.0199479  &  0.0199489 & 0.0199489 \\
1,   2   &  0.0200000   &  0.0198877  &  0.0198438  &  0.0198457  & 0.0198438  &  0.0198438  &  0.0198438  &  0.0198462 & 0.0198462 \\
0,   3   &  0.0200000   &  0.0197756  &  0.0196875  &  0.0196913  & 0.0196875  &  0.0196875  &  0.0196875  &  0.0196911 & 0.0196911 \\
\hline
3,   1   &  0.0094531   &  0.0094325  &  0.0094010  &  0.0094017  & 0.0094010  &  0.0094010  &  0.0094010  &  0.0094036 & 0.0094036 \\
2,   2   &  0.0094531   &  0.0093913  &  0.0092969  &  0.0092988  & 0.0092969  &  0.0092969  &  0.0092969  &  0.0093037 & 0.0093037 \\
1,   3   &  0.0094531   &  0.0093298  &  0.0091406  &  0.0091445  & 0.0091406  &  0.0091406  &  0.0091406  &  0.0091521 & 0.0091521 \\
0,   4   &  0.0094531   &  0.009248x  &  0.0089323  &  0.0089387  & 0.0089323  &  0.0089323  &  0.0089323  &  0.0089465 & 0.0089465 \\
\hline
4,   1   &  0.0042014   &  0.0041899  &  0.0041493  &  0.0041499  & 0.0041493  &  0.0041493  &  0.0041493  &  0.0041548 & 0.0041548 \\
3,   2   &  0.0042014   &  0.0041671  &  0.0040451  &  0.0040471  & 0.0040451  &  0.0040451  &  0.0040451  &  0.0040606 & 0.0040606 \\
2,   3   &  0.0042014   &  0.0041329  &  0.0038889  &  0.0038927  & 0.0038889  &  0.0038889  &  0.0038889  &  0.0039168 & 0.0039168 \\
1,   4   &  0.0042014   &  0.0040876  &  0.0036806  &  0.0036870  & 0.0036806  &  0.0036806  &  0.0036806  &  0.0037201 & 0.0037201 \\
0,   5   &  0.0042014   & 0.0040313   & 0.0034201   & 0.0034298   & 0.0034202  & 0.0034201   & 0.0034202   & $\cdots$   & 0.0034654 \\
\end{tabular}}
\end{ruledtabular}
\begin{tabbing}
$^{\dagger}${Compares with \hyperref[csl:13]{(O.~Bayrak \& I.~Boztosun, 1152)}.} \hspace{0.2in}  \=
$^{\ddagger}${Compares with \hyperref[csl:18]{(C.-S.~Jia \& P.-Q.~Wang, 4779)}, when $\xi=1.03$.} \hspace{0.2in}  \=
$^{\S}${Compares with \hyperref[csl:15]{(S.~M.~Ikhdair, 2009)}.}  \hspace{0.2in}  \=
                $^{\P}$These are done here afresh. \\
\end{tabbing} 

\end{table}

Now, under the transformation, $s=e^{-\d r}$, Eq.(\ref{Eq.R}) reduces to, 
\beq
\ba{l}\label{Eq.R.NU.MR} 
\ds\f{d^2R_{n_r,\ell}}{ds^2}+\f{1-s}{s(1-s)}\f{dR_{n_r,\ell}}{ds}+\f{-As^2+Bs-C}{s^2(1-s)^2}R_{n_r,\ell}=0.
\ea
\eeq
In this case, some algebraic manipulation leads to,  
\beq
\ba{ll}
A &=\epsilon_2-\epsilon_3+C=\f{2\bar{\mu}Ze^2}{\d\hbar^2}-L^{(\ell,D)}(L^{(\ell,D)}+1)(y_2-y_3)+C,\\
B &=\epsilon_2+2C=\f{2\bar{\mu}Ze^2}{\d\hbar^2}-L^{(\ell,D)}(L^{(\ell,D)}+1) y_2+2C,\\
C &=\epsilon_0^2+L^{(\ell,D)}(L^{(\ell,D)}+1)y_1,\\
\epsilon_0&=\sqrt{-\f{2\bar{\mu} E}{\d^2\hbar^2}}.\\
\ea
\eeq
Therefore, the eigenvalues, $E_{n_r,l}$, are expressed as, 
\beq\label{Energy.Hul} 
\ba{ll}
E_{n_r,\ell}&=\f{\d^2\hbar^2L^{(\ell,D)}(L^{(\ell,D)}+1) y_1}{2\bar{\mu}}-\f{\hbar^2}{2\bar{\mu}}\left(\f{\f{\bar{\mu}e^2Z}{\hbar^2}-\d L^{(\ell,D)}(L^{(\ell,D)}+1)(y_2-y_3)/2}{n_r+L_1^{(\ell,D)}}-\f{\d(n_r+L_1^{(\ell,D)})}{2}\right)^2,
\ea
\eeq
where 
\beq
L_1^{(\ell,D)}=\f{1}{2}+\sqrt{\f{1}{4}+L^{(\ell,D)}(L^{(\ell,D)}+1)y_3}.
\eeq
So, the radial wave function of Hulth\'en potential are finally derived as, 
\beq
R_{n_r,\ell}(r)=N_{{n_r},\ell}\,s^{\sqrt{C_{n_r,\ell}}}(1-s)^{L_1^{(\ell,D)}}\,{}_2F_1\left(-{n_r},{n_r}+2\sqrt{C_{n_r,\ell}}+2L_1^{(\ell,D)};2\sqrt{C_{n_r,\ell}}+1;s\right),
\eeq
whereas the normalization constant is found to be (with the help of \hyperref[csl:65]{(missing citation)}),  %$P_n^{(\a,\b)}$ is the Jacobi polynomial of degree $n$ and 
\beq
N_{{n_r},l}=\left[\f{2\d\sqrt{C_{n_r,\ell}}(n_r+\sqrt{C_{n_r,\ell}}+L_1^{(\ell,D)})\G(n_r+2\sqrt{C_{n_r,\ell}}+1)\G(n_r+2\sqrt{C_{n_r,\ell}}+2L_1^{(\ell,D)})}{({n_r})!(n_r+\sqrt{C_{n_r,\ell}})\G(n_r+2L_1^{(\ell,D)})\left[\G(2\sqrt{C_{n_r,\ell}}+1)\right]^2}\right]^{\f{1}{2}},
\eeq
and $C_{n_r,\ell}=-\f{2\bar{\mu}E_{n_r,\ell}}{\d^2\hbar^2}+(\ell+\f{D-3}{2})(\ell+\f{D-1}{2})y_1$. At the end, the 
explicit form of eigenfunctions can be written as, 
\beq

\begin{aligned}
\psi_{n_r,\ell,\{\mu\}}({\bf r}) &= N_{{n_r},\ell}\,r^{\f{1-D}{2}}\,s^{\sqrt{C_{n_r,\ell}}}(1-s)^{L_1^{(\ell,D)}}\times  \\
          &  {}_2F_1\left(-{n_r},{n_r}+2\sqrt{C_{n_r,\ell}}+2L_1^{(\ell,D)};2\sqrt{C_{n_r,\ell}}+1;s\right)  Y_{\ell,\{\mu\}}(\Om),\\
&s=e^{-\d r}.

\end{aligned}
\eeq\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/f2/f2}
\caption{{\label{fig2.approx} Bound-state energies of Hulthe\'en potential, under $f_6^{(b)}$ approximation: (A) $\ell=2$, $D=3$ (B) $\d=0.02$, $D=3$ (C) $\ell=1$, $\d=0.02$ and $r_0=0.5$, $\xi=1.03$, $(\lam_1,\lam_2,\lam_3,\lam_4,\lam_5)=(0,0,0.01,0,0.99)$. For details, see text.%
}}
\end{center}
\end{figure}

It is to be noted that the approximation, Eq.~(\ref{app.convex}) is well defined for convex combination $(0\le \lam_j\le 1,~j=1(1)5,~\sum\limits_{j=1}^5\lam_j=1)$, and one may consider a linear combination (may be negative values) for different values of $\lam_j,~j=1(1)5$, if following relations are 
satisfied,  
\beq
\ba{l}\label{relation1} 
\sum\limits_{j=1}^5\lam_jx_{1j}\ge 0,\\
\f{1}{4}+(\ell+\f{D-3}{e})(\ell+\f{D-1}{2})\sum\limits_{j=1}^5\lam_jx_{3j}\ge 0.
\ea
\eeq

\begingroup
\squeezetable\selectlanguage{english}
\begin{table}
\caption{{\label{Table2} Energies, $E_{n_r,l}$ for $\d=0.025,r_0=0.5$ in higher dimension within $f_6^{(d)}$ approximation.}}
%The 8th and 9th columns are defined for $(\lam_1,\lam_2,\lam_3,\lam_4,\lam_5)=(0, 0, 0.1, 0,0.9)$ and $(0, 0, 0.1, 0,0.9)$ respectively.}
\centering
\begin{ruledtabular}
\scalebox{0.9}{\begin{tabular}{ccccc}
$n_r$& $\ell$ & $-E_{n_r,\ell}$ ($D=4$)   &$-E_{n_r,\ell}$ ($D=5$)& $-E_{n_r,\ell}$ ($D=6$)  \\ \hline
0 & 1  & 0.0678906  & 0.0436024  & 0.0290455  \\

1 & 1  & 0.0291757  & 0.0198437  & 0.0135455\\

0 & 2  & 0.0290455  & 0.0196875  & 0.0133632 \\

2 & 1  & 0.0136757  & 0.0092969  & 0.0061644 \\

1 & 2  & 0.0135455  & 0.0091406  & 0.0059821 \\

0 & 3  & 0.0133632  & 0.0089323  & 0.0057477  \\

3 & 1  & 0.0062945  & 0.0040451  & 0.0024072 \\

2 & 2  & 0.0061644  & 0.0038889  & 0.0022250 \\

1 & 3  & 0.0059821  & 0.0036806  & 0.0019906 \\

0 & 4  & 0.0057477  & 0.0034202  & 0.0017041 \\

4 & 1  & 0.0025374  & 0.0013760  & 0.0005556 \\

3 & 2  & 0.0024072  & 0.0012197  & 0.0003733 \\

2 & 3  & 0.0022250  & 0.0010114  & 0.0001389 \\

1 & 4  & 0.0019906  & 0.0007510  & $\cdots$     \\

0 & 5  & 0.0017041  & 0.0004385  & $\cdots$  \\
\end{tabular}}
\end{ruledtabular}

\end{table}

\subsection*{Energy spectrum and degeneracy}
At first, let us consider the eigenvalues calculated from various approximations, $f_1, f_2, f_3, f_4, f_5$ as well as $f_6^{(c)}$ and 
$f_6^{(d)}$ with $\lambda_1$--$\lambda_5$ values $(0,0,0.995,0,0.005)$ and $(0,0,0.1,0,0.9)$. It should be noted that, for a given 
calculation with $f_6$ approximation, we choose sets, $\lam_1$--$\lam_5$, in accordance with Eq.~(\ref{relation1}). As mentioned earlier, 
$f_6$ is a well-behaved approximation for the \emph{entire} effective domain of $r$; hence we consider the set of $\lambda_1$--$\lambda_5$ as 
$(0,0,0.995,0,0.005)$, for near origin, and $\lambda_1$--$\lambda_5$ as $(0,0,0.1,0,0.9)$, for near $r=r_0=0.5$. Throughout the whole work, 
the following parameters are set to unity: $Z=e=\hbar=\bar{\mu}=1$. Sample energies are given in table \ref{Table1} for representative states 
for a particular value of $\d=0.025$ in case of $D=3$. Accurate eigenvalues are available from various methods in the literature. Here we 
quote the variational results of \hyperref[csl:5]{(Y.\ P.\ Varshni, 4682)}, as well as those from accurate GPS 
\hyperref[csl:24]{(A.~K.~Roy, 2005}; \hyperref[csl:66]{A.~K.~Roy \& E.~I.~Proynov, 2008}; \hyperref[csl:67]{A.~K.~Roy, 1405}; \hyperref[csl:68]{A.~K.~Roy, 1450)} method, in last two columns of table 
\ref{Table1}. The latter has been quite successful for a number of physically and 
chemically important systems, and is done here freshly for the current purpose. As expected, $f_1, f_2, f_4$-approximated energies 
match quite well with the respective references supplied in the footnote. Furthermore, $f_6^{(c)}$ and $f_6^{(d)}$ produce practically 
identical results.

In Fig.~\ref{fig2.approx} (A) we have depicted energies with respect to $n_r$, for six selected values of $\d$, in 3D, 
for a fixed $\ell=2$. A general feature of energy spectrum is that, for low $n_r$, the ordering resembles that of 
Coulomb potential; however at higher $n_r$, significant deviations take place between the two, leading to complex level crossings, especially in 
the neighborhood of zero energy \hyperref[csl:5]{(Y.\ P.\ Varshni, 4682}; \hyperref[csl:24]{A.~K.~Roy, 2005)}. These are constructed by employing $f_6^{(b)}$. Thus, for a fixed $\d$, states 
belonging to a
particular $\ell$ grows with $n_r$. Also for a given $n_r$, in a particular $D$, energy rises with $\d$. Similarly, panel (B) displays the 
same $n_r$ variations in energy for $D=3$, having a specific $\d=0.02$ under the same centrifugal approximation, for $\ell$ ranging from 1--6. 
For a fixed $\d$, energy progresses with $\ell$ for a series of states having same $n_r$. Panel (C) reveals that, for a given $\d$, at a fixed $n_r$, 
energy enhances with $D$. For a fixed $\ell$, in a given $D$, it increases with $n_r$.

As already mentioned in Sec.~I, in the limiting case of ($\d\rightarrow0$), the effective potential closely resembles 
Coulomb potential. In that situation, one arrives at the following expression for energy, which is coincident on $D$-dimensional 
Coulombic potential. This holds true for all six approximations, $f_1$--$f_6$, in accordance with the findings of \hyperref[csl:69]{(S.~M.~Al-Jaber, 1289)}, 
\beq\label{D.coulomb} 
\lim\limits_{\d\rightarrow 0}E_{n_r,\ell}=-\f{2\bar{\mu}e^4Z^2}{\hbar^2(2n_r+2\ell+D-1)^2}.
\eeq

A very important quantity in connection to this (and other screening) potential, is the existence of so-called \emph{critical screening} , 
denoted $\d_c$. This signifies a certain particular value of $\d$, at which the binding energy of a level in question reduces to zero. Much attention
has been paid to estimate this parameter, both analytically and numerically (see, e.g., references \cite{patil1984,Hulthen.greene12,demiralp2005, 
roy2016}). Quite accurate values have been reported and sometimes finding them can pose great numerical instability, depending on the concerned 
method of calculation. 
For $\ell=0$ states, these are trivially obtained \hyperref[csl:5]{(Y.\ P.\ Varshni, 4682}; \hyperref[csl:25]{D.~Agboola, 653AD)} as $\d_c = 8/(2n+D-3)^2$ for $D \geq 3$, where the principal 
quantum number $n$ is defined in terms of $n_r, \ell$ as $n=n_r+\ell+1$.  But for $\ell \neq 0$, this is non-trivial. In present 
analysis, these are derived within $f_1$-$f_4$ and $f_6$ approximations, having $\lam_1$--$\lam_5$ as $(\lam_1, \lam_2, \lam_3, \lam_4,0)$ and 
satisfying Eqs.~(\ref{sum.lam}) and (\ref{relation1}), as, 
\beq\label{zero.energy2} 
\d_c=\f{2\bar{\mu}e^2Z}{\hbar^2\left[(n_r+L_1^{(\ell,D)})^2+L^{(\ell,D)}(L^{(\ell,D)}+1)(y_2-y_3)\pm2(n_r+L_1^{(\ell,D)})\sqrt{L^{(\ell,D)}(L^{(\ell,D)}+1)y_1}\right]}.
\eeq
It is true for all $f_i, i=1-6$, except $f_5$, where the analytical form is slightly difficult. However, $\d_c$ in latter as well as 
in other approximations, can be obtained numerically. Further, in the special case of $y_1=0$, it can be derived as,
\beq\label{zero.energy1} 
\d_c=\f{2\bar{\mu}e^2Z}{\hbar^2\left[(n_r+L_1^{(\ell,D)})^2+L^{(\ell,D)}(L^{(\ell,D)}+1)(y_2-y_3)\right]}.
\eeq
It is easy to verify that, for $f_1$ approximation this further reduces to the following expression,    
\beq\label{zero.energy} 
\d_c^{(1)}=\f{8\bar{\mu}e^2Z}{\hbar^2(2n_r+2\ell+D-1)^2},
\eeq
in accordance with the formula offered in \hyperref[csl:25]{(D.~Agboola, 653AD)}.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/f3deltac/f3deltac}
\caption{{\label{fig3.deltac} Estimate of $\d_c$ under various approximations, for selected states: (A)-(B): $f_1$, (C)-(D): 
$f_6^{(e)}$ with $\lam_1$-$\lam_5$ as $(0.49,0.49,-1.60203,1.62203,0)$, (E)-(F): $f_6^{(f)}$ with $\lam_1$-$\lam_5$ as $(0.4,0.4,0.1,0.1,0)$. 
In left panels: $D=3, \ell=1-6$, while for right panels $\ell=1, D=3-8$. In last panel, $\xi=1.03$. For details, see text.%
}}
\end{center}
\end{figure}

At this stage, a few words may be mentioned about the degeneracy present in this system. Our analysis reveals that, in all 
approximations involving $\lam_5=0$ and $y_1=0$, the states $(n_{r_{1}},\ell_1)$ in $D_i$ dimension and $(n_{r_{2}},\ell_2)$ in $D_j$ 
dimension, may be degenerate at a $\d=\d_{D_i,D_j}$, given by, 
\beq
\d_{D_i,D_j}=\f{2\bar{\mu}e^2Z\left[n_{r_{1}}+n_{r_{2}}+L_1^{(\ell_1,D_i)}+L_2^{(\ell_2,D_j)}\right]}{\hbar^2(A+B)},
\eeq
where
\beq
\ba{l}
A=\Big[n_{r_{1}}+n_{r_{2}}+L_1^{(\ell_1,D_i)}+L_1^{(\ell_2,D_j)}\Big] \Big[n_{r_{1}}+L_1^{(\ell_1,D_i)}\Big] \Big[n_{r_{2}}+L_1^{(\ell_2,D_j)} \Big],\\
B=(y_2-y_3)\bigg\{ L^{(\ell_1,D_i)} \Big(L^{(\ell_1,D_i)}+1\Big) \Big(n_{r_{2}}+L_1^{(\ell_2,D_j)}\Big) +L^{(\ell_2,D_j)} \Big(L^{(\ell_2,D_j)}+1\Big) 
\Big(n_{r_{1}}+L_1^{(\ell_1,D_i)} \Big) \bigg\}.\\
\ea
\eeq
Since $L^{(\ell,D)}=L^{(\ell-1,D+2)}$ and $L_1^{(\ell,D)}=L_1^{(\ell-1,D+2)}$, we find $E_{n_r,\ell}(D)=E_{n_r,\ell-1}(D+2)$ for all 
admissible $n_r,\ell,D$.
Under the special case of $f_1$ approximation, this further simplifies to (same as found in \hyperref[csl:25]{(D.~Agboola, 653AD)}), 
\beq\label{degenerate.energy} 
\ba{l}
\d_{{D_i,D_j}}^{(1)}=\f{8\bar{\mu}e^2Z}{\hbar^2\left(2n_{r_{1}}+2\ell_1+D_i-1\right)\left(2n_{r_{2}}+2\ell_2+D_j-1\right)}. 
\ea
\eeq
From Eqs.~(\ref{zero.energy}) and (\ref{degenerate.energy}), it is observed that $\d_{D_i,D_j}^{(1)}=\d_c^{(1)}$, in the special case of 
$D_i=D_j=D$, $n_{r_{1}}=n_{r_{2}}=n_r$ and $\ell_1=\ell_2=\ell$. Now, some illustrative energies in higher dimension ($D=4-6$) are reported for 
some representative $\ell \neq 0$ states in Table~\ref{Table2}, under $f_6^{(d)}$ approximation for $\d=0.025$. No reference results could be found for direct
comparison. The last two states do not exist in $D=6$, for this particular $\d$.

Next, Fig.~\ref{fig3.deltac} plots calculated $\d_c$'s with respect to $n_r$ at fixed $D=3$ for $\ell=1-6$ in left panels. 
Likewise, right panels show respective variations at a fixed $\ell=1$ for $D=3-8$. Segments (A), (B) 
are produced via $f_1$ approximation following the special case of Eq.~(\ref{zero.energy}), while (C), (D) from $f_6^{(e)}$ satisfying  
the particular case of Eq.~(\ref{zero.energy1}), where $(\lam_1,\lam_2,\lam_3,\lam_4,\lam_5)=(0.49,0.49,-1.60203,1.62203,0)$, and (E), (F) 
from $f_6^{(f)}$, obeying Eq.~(\ref{zero.energy2}), with $(\lam_1,\lam_2,\lam_3,\lam_4,\lam_5)=(0.4,0.4,0.1,0.1,0)$. As 
expected, (E), (F) results are slightly improved over the respective top four panels. Moreover these are in decent 
agreement with \hyperref[csl:70]{(S.~H.~Patil, 1984}; \hyperref[csl:17]{B.~G\"on\"ul \& M.~Ko\c{c}ak, 2000}; \hyperref[csl:71]{M.~Demiralp, 1380}; \hyperref[csl:72]{A.~K.~Roy, 2016)}, in $D=3$. To the best of our knowledge, however, no such 
results have been reported in higher dimension.

\section*{Information theoretic measures}\label{sec4}
This section is now devoted to the presentation of a number of information theoretical quantities in Hulth\'en potential. In all future proceedings, 
we resort to a simplified notation of \(L_1^{(\ell)}\) in place of \(L_1^{(\ell,D)}\), for brevity.

\subsection*{Expectation values and Heisenberg uncertainty relation}
The expectation value of a function $f({\bf r})$ with respect to density function $\rho({\bf r})$ is defined \hyperref[csl:73]{(1035)}, 
\beq
\langle f({\bf r})\rangle=\int_{R^D}f({\bf r})\rho({\bf r})d^Dr,
\eeq
where $d^Dr$ is given as $d^Dr=r^{D-1}dr\,d\Om_{D-1}$, with $d\Om_{D-1}=\left[\prod\limits_{j=1}^{D-2}\left(\sin\theta_j\right)^{2\a_j}\,d\theta_j\right]d\theta_{D-1}$.
From this, one can write, 
\beq
\langle f({\bf r})\rangle=\int_{R^D}f({\bf r})\om_{n,\ell}(r)\left|Y_{\ell,\{\mu\}}(\Om_{D-1})\right|^2dr\,d\Om_{D-1},
\eeq
If $f({\bf r})=f(r)$, i.e., independent of $\theta_1,\theta_2,...,\theta_{D-1}$, then the corresponding improper integral,  
\beq
\left\langle f(r)\right\rangle=\ds\int_0^{\infty}f(r)\om(r)dr 
\eeq
in $r$ space becomes a convergent improper integral in $s$ space, as given below,  
\beq
\left\langle f(r)\right\rangle=\ds\f{1}{\a}\int_0^{1}\f{\widetilde{f}(s)\widetilde{\om}(s)}{s}ds
\eeq
Where the following quantities have been defined in $s$ space,  
\beq
\widetilde{f}(s)=f(r),~\widetilde{\om}(s)=\om(r),s=e^{-\a r},
\eeq
in $s$ space, having singularity at point $s=0$, the nature of which depends on the real-valued function $f(r)$. In this scenario, one can 
derive following analytical expressions for certain expectation values, as provided below, 
\beq
\ba{ll}
\left\langle r\right\rangle&=-\ds\f{N_{n_r,\ell}^2}{\d}\sum\limits_{k=0}^{2n_r}\f{2B_{k+2,2}\left(c^{(\ell)}_0,2!c^{(\ell)}_1,\dots,(k+1)!c^{(\ell)}_k\right)}{(k+2)!}I_1,\\
\left\langle r^2\right\rangle&=\ds\f{N_{n_r,\ell}^2}{\d^2}\sum\limits_{k=0}^{2n_r}\f{2B_{k+2,2}\left(c^{(\ell)}_0,2!c^{(\ell)}_1,\dots,(k+1)!c^{(\ell)}_k\right)}{(k+2)!}I_2,\\
\left\langle\f{1}{r^2}\right\rangle&=\ds\d^2 N_{n_r,\ell}^2\sum\limits_{k=0}^{2n_r}\f{2B_{k+2,2}\left(c^{(\ell)}_0,2!c^{(\ell)}_1,\dots,(k+1)!c^{(\ell)}_k\right)}{(k+2)!}I_3,\\
\left\langle\f{1}{r}\right\rangle&=\ds\d  N_{n_r,\ell}^2\sum\limits_{k=0}^{2n_r}\f{2B_{k+2,2}\left(c^{(\ell)}_0,2!c^{(\ell)}_1,\dots,(k+1)!c^{(\ell)}_k\right)}{(k+2)!}I_4.
\ea
\eeq
In the above equation, the quantities, $I_1$-$I_4$ and $c_k^{(\ell)}$ are defined through following expressions \hyperref[csl:65]{(missing citation)}, 
\beq
\ba{ll}
I_1&=\ds\int_0^1 s^{2\sqrt{C}+k-1}(1-s)^{2L_1}\ln s \, ds=B(2\sqrt{C_{n_r,\ell}}+k,2L_1^{(\ell)}+1)\left[\varPsi(2\sqrt{C_{n_r,\ell}}+k)-\varPsi(2\sqrt{C_{n_r,\ell}}+k+2L_1^{(\ell)}+1)\right]\\
I_2&=\ds\int_0^1 s^{2\sqrt{C}+k-1}(1-s)^{2L_1}\left[\ln s\right]^2\, ds,\\
&=B(2\sqrt{C_{n_r,\ell}}+k,2L_1^{(\ell)}+1)\Big[\left(\varPsi(2\sqrt{C_{n_r,\ell}}+k)-\varPsi(2\sqrt{C_{n_r,\ell}}+k+2L_1^{(\ell)}+1)\right)^2+\varPsi'(2\sqrt{C_{n_r,\ell}}+k)\\
&-\varPsi'(2\sqrt{C_{n_r,\ell}}+k+2L_1^{(\ell)}+1)\Big],\\
I_3&=\ds\int_0^1 \f{s^{2\sqrt{C_{n_r,\ell}}+k-1}(1-s)^{2L_1^{(\ell)}}}{\left[\ln s\right]^2}\, ds=\f{1}{2}\sum\limits_{j=0}^{2L_1{(\ell)}}(-1)^j\binom{2L_1^{(\ell)}}{j}(j+2\sqrt{C_{n_r,\ell}}+k)^2\ln\left[j+2\sqrt{C_{n_r,\ell}}+k\right],\\
I_4&=\ds\int_0^1 \f{s^{2\sqrt{C_{n_r,\ell}}+k-1}(1-s)^{2L_1^{(\ell)}}}{\ln s}\, ds=\ds(-j)^{2L_1^{(\ell)}}\sum\limits_{j=0}^{2L_1{(\ell)}}(-1)^j\binom{2L_1^{(\ell)}}{j}\ln\left[2\sqrt{C_{n_r,\ell}}+k+2L_1^{(\ell)}-j\right],\\
c_k^{(\ell)}&=\left\{\ba{lll}\ds\f{(-n_r)_k(n_r+2\sqrt{C_{n_r,\ell}}+2L_1^{(\ell)})_k}{k!(2\sqrt{C_{n_r,\ell}}+1)_k},&k\le n_r,\\
0,&k>n_r,
\ea\right\},
\ea
\eeq
Furthermore, $\varPsi(x)=\f{\G'(x)}{\G(x)}$ indicates the digamma function, while  
\beq
\ds B_{m,\ell}(x_1,x_2,...,x_{m-\ell+1})=\ds\sum\limits_{\widehat{\pi}(m,\ell)}\f{\left(\f{x_{m-\ell+1}}{(m-\ell+1)!}\right)^{j_{m-\ell+1}}...\left(\f{x_2}{2!}\right)^{j_2}\left(\f{x_1}{1!}\right)^{j_1}}{(m!)^{-1}j_1!j_2!...j_{m-\ell+1}!}
\eeq
is the Bell polynomial and $\widehat{\pi}(m,\ell)$ denotes the set of partitions satisfying the condition below, 
\beq
\ds j_1+j_2+\dots+j_{m-\ell+1}=\ell,~j_1+2j_2+\dots+(m-\ell+1)j_{m-\ell+1}=m.
\eeq
Therefore, the root mean square (RMS) of $r$ is expressed as,   
\beq
\ba{ll}
\bigtriangleup r &  =\sqrt{\langle r^2\rangle-\langle r\rangle^2} \\
        = & \ds\f{N_{n_r,\ell}}{\d}\sqrt{\sum\limits_{k=0}^{2n_r}\f{2B_{k+2,2}\left(c^{(\ell)}_0,2!c^{(\ell)}_1,\dots,(k+1)!c^{(\ell)}_k\right)}{(k+2)!}I_2-\ds N_{n_r,\ell}^2\left(\sum\limits_{k=0}^{2n_r}\f{2B_{k+2,2}\left(c^{(\ell)}_0,2!c^{(\ell)}_1,\dots,(k+1)!c^{(\ell)}_k\right)}{(k+2)!}I_1\right)^2}.
\ea
\eeq
For the expectation values of $\langle p \rangle$ and $\langle p^2 \rangle$, analogous expressions, as in the following, can be derived, 
\beq
\ba{ll}
\left\langle p\right\rangle
&=-\f{i\hbar(D-1)N_{n_r,\ell}^2}{2}\sum\limits_{k=0}^{2n_r}\f{2B_{k+2,2}\left(c^{(\ell)}_0,2!c^{(\ell)}_1,\dots,(k+1)!c^{(\ell)}_k\right)}{(k+2)!}I_4,\\
\left\langle p^2\right\rangle
&=\ds I_5-\ds\f{\a\hbar^2(D-1)^2N_{n_r,\ell}^2}{4}\sum\limits_{k=0}^{2n_r}\f{2B_{k+2,2}\left(c^{(\ell)}_0,2!c^{(\ell)}_1,\dots,(k+1)!c^{(\ell)}_k\right)}{(k+2)!}I_3,
\ea
\eeq
where, following intermediate quantities $I_5$-$I_8$ have been introduced, 
\beq
\ba{ll}
\left.\ba{ll}
I_5&=\a\hbar^2N_{n_r,\ell}^2\left(I_6+I_7+I_8\right)\\
I_6&=\ds\sum\limits_{i,j=0}^{n_r}\f{(-n_r)_{(i,j)}(n_r+2\sqrt{C_{n_r,\ell}}+2L_1^{(\ell)})_{(i,j)}}{i!j!(2C_{n_r,\ell}+1)_{(i,j)}}\Big[C_{n_r,\ell}B\left(2\sqrt{C_{n_r,\ell}}+i+j,2L_1^{(\ell)}+1\right)\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ -2\sqrt{C_{n_r,\ell}}L_1^{(\ell)}B\left(2\sqrt{C_{n_r,\ell}}+i+j+1,2L_1^{(\ell)}\right) +L_1^{(\ell)2}B\left(2\sqrt{C_{n_r,\ell}}+i+j+2,2L_1^{(\ell)}-1\right)\Big],\\
I_7&=\ds\sum\limits_{i,j=0}^{n_r-1}\f{(-n_r+1)_{(i,j)}(n_r+2\sqrt{C_{n_r,\ell}}+2L_1+1)_{(i,j)}}{i!j!(2C_{n_r,\ell}+2)_{(i,j)}}B\left(2\sqrt{C_{n_r,\ell}}+i+j+2,2L_1^{(\ell)}+1\right),\\
I_8&=\ds\sum\limits_{i=0}^{n_r}\sum\limits_{j=0}^{n_r-1}\f{(-n_r)_{i}(-n_r+1)_j(n_r+2\sqrt{C_{n_r,\ell}}+2L_1^{(\ell)})_{i}(n_r+2\sqrt{C_{n_r,\ell}}+2L_1^{(\ell)}+1)_{j}}{i!j!(2C_{n_r,\ell}+1)_i(2C_{n_r,\ell}+2)_j}\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times\Big[2\sqrt{C_{n_r,\ell}}B\left(2\sqrt{C_{n_r,\ell}}+i+j+1,2L_1^{(\ell)}+1\right)2L_1^{(\ell)}B\left(2\sqrt{C_{n_r,\ell}}+i+j+2,2L_1^{(\ell)}\right)\Big],
\ea\right\},&n_r>0\\
I_5=\a\hbar^2N_{n_r,\ell}^2, I_6=1,I_7=0=I_8,&n_r=0.
\ea
\eeq\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/f4/f4}
\caption{{\label{Fig4} Radial density plot for various states in Hulth\'en potential for $\d=0.025$ in 3D $(\hbar=\bar{\mu}=Z=e=1)$: 
(A) $\ell=0$, (B) $\ell=1$, (C) $\ell=2$ and (D) $\ell=3$. Red, blue and black lines represent $n_r=0,1$, 2 respectively.%
}}
\end{center}
\end{figure}

Therefore, one can work out the RMS of $p$ as, 
\beq
\small{\bigtriangleup p = \sqrt{I_5+\hbar^2(D-1)^2N_{n_r,\ell}^2\left(N_{n_r,\ell}\left[\sum\limits_{k=0}^{2n_r}\f{B_{k+2,2}\left(c^{(\ell)}_0,2!c^{(\ell)}_1,\dots,(k+1)!c^{(\ell)}_k\right)}{(k+2)!}I_4\right]^2-\f{\a}{2}\sum\limits_{k=0}^{2n_r}\f{B_{k+2,2}\left(c^{(\ell)}_0,2!c^{(\ell)}_1,\dots,(k+1)!c^{(\ell)}_k\right)}{(k+2)!}I_3\right)}}.
\eeq
The ordinary moments $\left\langle r^k\right\rangle$ and $\left\langle p^k\right\rangle$ are used in radial uncertainty relations in 3D and 
multidimensional quantum systems. Note that, for central potentials, they are connected via the important relation \hyperref[csl:73]{(1035)}, 
$\langle p^2 \rangle \ge \ell(\ell+D-2)\langle r^{-2} \rangle $, assuming  (\(\hbar=\mu=1\)). 
For a general potential, one can write the following simplified relation \hyperref[csl:73]{(1035)}, 
\beq
\ba{ll}
\left\langle p^2\right\rangle&\ge \f{(D-2)^2}{4}\left\langle r^{-2}\right\rangle.\\
\ea
\eeq
Note that, the Heisenberg uncertainty product is $\bigtriangleup r\bigtriangleup p\ge\f{\hbar}{2}$. 
Lately, a modified Heisenberg-like uncertainty inequality \hyperref[csl:74]{(2020)} has been suggested as well: 
\beq
\left\langle r^2\right\rangle \left\langle p^2\right\rangle\ge \left(L^{(\ell,D)}+\f{3}{2}\right)^2.\eeq\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/f5/f5}
\caption{{\label{Fig5} Same as in Fig.~4, in higher dimension ($D=4$).%
}}
\end{center}
\end{figure}

\subsection*{Analytical results of linear and composite measures of radial density function}
In this section, we will present the linear and composite measures of radial density functions. The entropic moments of positive integral order 
$\bar{\a}\in\mathbb{N}$ of the density function $\om_{n_r,\ell}(r)$ can be expressed as, 
\beq
\mathcal{W}^{(\bar{\a})}_{\om_{n_r,\ell}}
=\ds\left[N_{n_r,\ell}\right]^{2\bar{\a}}\sum\limits_{k=0}^{2\bar{\a}n_r}\f{(2\bar{\a})!}{(k+2\bar{\a})!}B\left(2\bar{\a}\sqrt{C_{n_r,\ell}}+k,2\bar{\a}L_1^{(\ell)}+1\right) B_{k+2\bar{\a},2\bar{\a}}\left(c^{(\ell)}_0,2!c^{(\ell)}_1,\dots,(k+1)!c^{(\ell)}_k\right).
\eeq
One can conveniently define $R$ of $\om_{n_r,\ell}(r)$ of order $\bar{\a}\in\mathbb{N}$ as, 
\beq\label{renyi} 
\mathcal{R}^{(\bar{\a})}_{\om_{n_r,\ell}}=\ds\f{1}{1-\bar{\a}}\ln\left[\left[N_{n_r,\ell}\right]^{2\bar{\a}}\sum\limits_{k=0}^{2\bar{\a}n_r}\f{(2\bar{\a})!}{(k+2\bar{\a})!}B\left(2\bar{\a}\sqrt{C_{n_r,\ell}}+k,2\bar{\a}L_1^{(\ell)}+1\right) B_{k+2\bar{\a},2\bar{\a}}\left(c^{(\ell)}_0,2!c^{(\ell)}_1,\dots,(k+1)!c^{(\ell)}_k\right)\right],
\eeq
while disequilibrium of $\om_{n_r,\ell}(r)$ is derived as, 
\beq\label{disequi} 
\mathcal{D}_{n_r,\ell}=\ds N_{n_r,\ell}^4\sum\limits_{k=0}^{4n_r}\f{4!}{(k+4)!}B\left(4\sqrt{C_{n_r,\ell}}+k,4L_1^{(\ell)}+1\right) B_{k+4,4}\left(c^{(\ell)}_0,2!c^{(\ell)}_1,\dots,(k+1)!c^{(\ell)}_k\right).
\eeq
\begingroup          \squeezetable\selectlanguage{english}
\begin{table}
\caption{{\label{table3} Measures of Heisenberg uncertainties and their products in $3D$ and $4D$, in Hulth\'en potential.}}
\centering
\begin{ruledtabular}
\scalebox{0.9}{\begin{tabular}{cc|lll|lll}
&& \multicolumn{3}{c|}{$D=3$}&\multicolumn{3}{c}{$D=4$}\\ 
\hline
$n_r$& $\ell$ & $\bigtriangleup r_{n_r,\ell}$ & $\bigtriangleup p_{n_r,\ell}$ & $\bigtriangleup r_{n_r,\ell}$$\bigtriangleup p_{n_r,\ell}$ & $\bigtriangleup r_{n_r,\ell}$ & $\bigtriangleup p_{n_r,\ell}$ & 
$\bigtriangleup r_{n_r,\ell}$$\bigtriangleup p_{n_r,\ell}$\\

% &  & ($\AA$) & ($eV/c$)   & $\ge\f{\hbar}{2}=0.5$& ($\AA$) & ($eV/c$)  &$\ge\f{\hbar}{2}=0.5$\\ 
\hline 
%%%0&0&0.8661607398&0.0001041716235,&0.00009022937047&&1.500989628&1.805789972&2.710472018&\\
%%%1&0&2.45332756&1.034784741&2.538665923&&3.5823509&2.21458472&7.933419567& \\
%%%2&0&4.115912449&3.058018739&12.5865374&&5.343205199&8.029298934&42.90219181& \\\hline
0&1&2.24027  &  0.99995&2.24015       &3.07502&2.21808&6.82064 \\
1&1&4.90334&1.00386&4.92229           &6.40302&2.24352&14.36528\\
2&1&6.78815&3.87413&26.29821          &8.48735&9.00363&76.41690\\\hline
0&2&4.00252&1.00726&4.03159           &5.02603&2.24609&11.28890\\
1&2&8.08100&1.00094&8.08860           &9.95042&2.24798&22.36840 \\
2&2&10.48598&4.04240&42.38846         &12.84525&8.99754&115.5756 \\\hline
0&3&6.15572&1.00423&6.18179           &7.41084&2.24988&16.6735 \\
1&3&12.04117&1.00031&12.04495         &14.40695&2.24920&32.4042\\
2&3&15.65792&3.90745&61.18263         & 19.07948&8.44692&161.1629\\
\end{tabular}}
\end{ruledtabular}

\end{table}

Note that, $S$ of $\om_{n_r,\ell}$ can not be expressed analytically, and therefore, they are pursued numerically. For our current potential, 
RCR of $\om_{n_{{r}_1},\ell_1}$ and $\om_{n_{{r}_2},\ell_2}$ of orders $(\bar{\a},\bar{\b})$ are derived as, 
\beq\label{rcr} 
\mathcal{C}^{(\bar{\a},\bar{\b})}_{(n_{{r}_1},\ell_1),(n_{{r}_2},\ell_2)}=\f{\left[\left[N_{n_{{r}_1},\ell_1}\right]^{2\bar{\a}}\sum\limits_{k=0}^{2\bar{\a}n_r}\f{(2\bar{\a})!}{(k+2\bar{\a})!}B\left(2\bar{\a}\sqrt{C_{n_{{r}_1},\ell_1}}+k,2\bar{\a}L_1^{(\ell_1)}+1\right) B_{k+2\bar{\a},2\bar{\a}}\left(c^{(\ell_1)}_0,2!c^{(\ell_1)}_1,\dots,(k+1)!c^{(\ell_1)}_k\right)\right]^{\f{1}{1-\bar{\a}}}}{\left[\left[N_{n_{{r}_2},\ell_2}\right]^{2\bar{\b}}\sum\limits_{k=0}^{2\bar{\b}n_r}\f{(2\bar{\b})!}{(k+2\bar{\b})!}B\left(2\bar{\b}\sqrt{C_{n_{{r}_1},\ell_2}}+k,2\bar{\b}L_1^{(\ell_2)}+1\right) B_{k+2\bar{\b},2\bar{\b}}\left(c^{(\ell_2)}_0,2!c^{(\ell_2)}_1,\dots,(k+1)!c^{(\ell_2)}_k\right)\right]^{\f{1}{1-\bar{\b}}}}.
\eeq
Similarly, GRC of $\om_{n_r,\ell}$ of order $(\bar{\a},\bar{\b})$ can be expressed as below,  
\beq\label{grc} 
\mathcal{C}^{(\bar{\a},\bar{\b})}_{n_r,\ell}=\f{\left[\left[N_{n_r,\ell}\right]^{2\bar{\a}}\sum\limits_{k=0}^{2\bar{\a}n_r}\f{(2\bar{\a})!}{(k+2\bar{\a})!}B\left(2\bar{\a}\sqrt{C_{n_r,\ell}}+k,2\bar{\a}L_1^{(\ell)}+1\right) B_{k+2\bar{\a},2\bar{\a}}\left(c^{(\ell)}_0,2!c^{(\ell)}_1,\dots,(k+1)!c^{(\ell)}_k\right)\right]^{\f{1}{1-\bar{\a}}}}{\left[\left[N_{n_r,\ell}\right]^{2\bar{\b}}\sum\limits_{k=0}^{2\bar{\b}n_r}\f{(2\bar{\b})!}{(k+2\bar{\b})!}B\left(2\bar{\b}\sqrt{C_{n_r,\ell}}+k,2\bar{\b}L_1^{(\ell)}+1\right) B_{k+2\bar{\b},2\bar{\b}}\left(c^{(\ell)}_0,2!c^{(\ell)}_1,\dots,(k+1)!c^{(\ell)}_k\right)\right]^{\f{1}{1-\bar{\b}}}}.
\eeq
Proceeding further, one can find the following expression for SRC of $\om_{n_r,\ell}$ of order $\bar{\a}$, 
\beq\label{src} 
\ba{ll}
\mathcal{C}^{(\bar{\a})}_{n_r,\ell}&=\ds\left[N_{n_r,\ell}^4\sum\limits_{k=0}^{4n_r}\f{24B\left(4\sqrt{C_{n_r,\ell}}+k,4L_1^{(\ell)}+1\right)}{(k+4)!} B_{k+4,4}\left(c^{(\ell)}_0,2!c^{(\ell)}_1,\dots,(k+1)!c^{(\ell)}_k\right)\right]\\
&\ \ \ \ \times \left[\left[N_{n_r,\ell}\right]^{2\bar{\a}}\sum\limits_{k=0}^{2\bar{\a}n_r}\f{(2\bar{\a})!}{(k+2\bar{\a})!}B\left(2\bar{\a}\sqrt{C_{n_r,\ell}}+k,2\bar{\a}L_1^{(\ell)}+1\right) B_{k+2\bar{\a},2\bar{\a}}\left(c^{(\ell)}_0,2!c^{(\ell)}_1,\dots,(k+1)!c^{(\ell)}_k\right)\right]^{\f{1}{1-\bar{\a}}}.
\ea
\eeq
Likewise, for $\om_{n_r,\ell}$, LMC is found to be, 
\beq\label{lmc} 
\mathcal{C}^{(LMC)}_{n_r,\ell}=\ds\left[N_{n_r,\ell}^4\sum\limits_{k=0}^{4n_r}\f{24B\left(4\sqrt{C_{n_r,\ell}}+k,4L_1^{(\ell)}+1\right)}{(k+4)!} B_{k+4,4}\left(c^{(\ell)}_0,2!c^{(\ell)}_1,\dots,(k+1)!c^{(\ell)}_k\right)\right]e^{\mathcal{S}_{n_r,\ell}},
\eeq
where $\mathcal{S}_{n_r,\ell}$ needs to be calculated numerically.

\subsection*{Majorization effect on linear and composite measures}\label{sec4.3}
Let $\rho_1:M^D\rightarrow[0,1]$ and $\rho_2:M^D\rightarrow[0,1]$ be two continuous density functions. Then, following \hyperref[csl:35]{(D.~Nath, 2681}; \hyperref[csl:75]{H.~Joe, 1217}; \hyperref[csl:76]{D.~Nath, 2106)}, 
$\rho_1$ majorizes $\rho_2$ $\left(\rho_1\succ \rho_2\right)$, if $\ds\int\left[\rho_1({\bf r})-r_0\right]^+d{\bf r}\ge\ds\int\left[\rho_2({\bf r})-r_0\right]^+d{\bf r}$, holds for all $r_0\ge 0$, where $\left[y\right]^+=\max\left\{y,0\right\}$. Therefore, following 
\hyperref[csl:34]{(P.~Ghosh \& D.~Nath, 2646}; \hyperref[csl:35]{D.~Nath, 2681}; \hyperref[csl:76]{D.~Nath, 2106}; \hyperref[csl:77]{K.~Zyczkowski, 2003}; \hyperref[csl:78]{2010}; \hyperref[csl:79]{A.~Galindo, 2002)}, one can write, 
\beq
\rho_1\succ \rho_2\implies\left\{\ba{ll}\mathcal{W}^{(\bar{\a})}_{\rho_1}\le \mathcal{W}^{(\bar{\a})}_{\rho_2}&\mbox{if~}0<\bar{\a}<1\\ \mathcal{W}^{(\bar{\a})}_{\rho_1}\ge \mathcal{W}^{(\bar{\a})}_{\rho_2}&\mbox{if~}\bar{\a}>1\ea\right. ,
\eeq
and then it follows that, $\mathcal{R}_{\rho}^{(\bar{\a})}$ is a concave functional of $\rho$ for any $\bar{\a}>0$, which enables us to write,  
\beq
\rho_1\succ \rho_2\implies\left\{\ba{ll}\mathcal{R}_{\rho_1}^{(\bar{\a})}\le \mathcal{R}_{\rho_2}^{(\bar{\a})},&\bar{\a}>0\\ \mathcal{S}_{\rho_1}\le \mathcal{S}_{\rho_2},& \bar{\a}=1\ea\right. .
\eeq
It is well known that, $\mathcal{R}_f^{(\a)}$ is a non-increasing function of $\a$ and hence, $C_{(f,g)}^{(\a,\b)}$ satisfies the following 
inequalities \hyperref[csl:35]{(D.~Nath, 2681)}, 
\beq\label{majo.RCR} 
C^{(\a,\b)}_{(f,g)}\left\{\ba{ll}
\le 1, & \a\ge \b,~ f \succ g\\
\ge 1, & \a\le\b,~ f\prec g \\
\ea \right.,
\eeq
and one cannot say anything about the cases $\a>\b,~f\prec g$ and $\a<\b,~f\succ g$, but one can find lower and upper bounds of $C_{(f,g)}^{(\a,\b)}$ 
using majorization effect, Eq.~(\ref{majo.RCR}). The upper bound of $C^{(\a,\b)}_{(f,g)}=1$, exists, if $f\succ g$ and $\a>\b$, while the lower bound of 
$C^{(\a,\b)}_{(f,g)}=1$ exists, if $f\prec g$ and $\a<\b$. In general, RCR is a positive definite functional of density function; therefore, the lower 
bound of $C^{(\a,\b)}_{(f,g)}$ is zero and the upper bound may be defined by R\'enyi entropic bound \hyperref[csl:35]{(D.~Nath, 2681}; \hyperref[csl:80]{221AD)}. The LMC, SRC, GRC and 
RCR are all composite measures, being the product of two global information, \emph{viz.} , one is convex and another is a concave functional of the 
density function. Therefore, we can not say definitely about their nature. In what follows, these are investigated numerically for some representative 
states having $n_r=0-2,\ell=0-3$.

\begingroup           \squeezetable\selectlanguage{english}
\begin{table}
\caption{{\label{table4.sha.lmc.src} \(\mathcal{S}_{n_r, \ell}\), LMC complexity, $\mathcal{C}^{(LMC)}_{n_r,\ell}$, 
SRC $(\mathcal{C}^{(\bar{\a}_1)}_{n_r,\ell}$ \& $\mathcal{C}^{(\bar{\a}_2)}_{n_r,\ell})$, where $\bar{\a}_1=2.5>2$, $\bar{\a}_2=1.5<2, d_{n_r,\ell}=10000$.}}
\centering
\begin{ruledtabular}
\begin{tabular}{cc|cccc|cccc }
& & \multicolumn{4}{c|}{$D=3$}&\multicolumn{4}{c}{$D=4$}   \\ \hline 
%%& & \multicolumn{2}{c|}{$D=3$}&\multicolumn{4}{c}{$D=4$}\\ \hline
%%%& & & & & & & \\
$n_r$ & $\ell$  &  $ \mathcal{S}_{n_r, \ell}$  &   $\mathcal{C}^{(LMC)}_{n_r,\ell}>1$ & $\mathcal{C}^{(\bar{\a}_1)}_{n_r,\ell}<1$ & $\mathcal{C}^{(\bar{\a}_2)}_{n_r,\ell}>1$ &  $\mathcal{S}_{n_r, \ell}$ &    $\mathcal{C}^{(LMC)}_{n_r,\ell}>1$ & $\mathcal{C}^{(\bar{\a}_1)}_{n_r,\ell}<1$ & $\mathcal{C}^{(\bar{\a}_2)}_{n_r,\ell}>1$\\\hline
0 & 0 & 1.1546  &    1.1896      & 0.9558       & 1.0679  & 1.7362   &1.1819      & 0.9570       & 1.0655      \\
1 & 0 & 2.2359  &    1.2041      & 0.9507       & 1.0748  & 2.5960   &1.2029      & 0.9496       & 1.0754      \\
2 & 0 & 2.7796  &    1.2068      & 0.9534       & 1.0732  & 3.0336   &1.2173      & 0.9509       & 1.0773      \\\hline
0 & 1 & 2.1490  &    1.1780      & 0.9577       & 1.0644  & 2.3980   &1.1758      & 0.9581       & 1.0637     \\
1 & 1 & 2.8897  &    1.2000      & 0.9492       & 1.0750  & 3.1225   &1.1968      & 0.9494       & 1.0742      \\
2 & 1 & 3.2602  &    1.2260      & 0.9487       & 1.0807  & 3.4498   &1.2332      & 0.9467       & 1.0836      \\\hline
0 & 2 & 2.4130  &    1.1743      & 0.9583       & 1.0632  & 2.2592   &1.1734      & 0.9584       & 1.0630      \\
1 & 2 & 3.2357  &    1.1940      & 0.9496       & 1.0733  & 3.1827   &1.1917      & 0.9500       & 1.0726      \\
2 & 2 & 3.5335  &    1.2387      & 0.9449       & 1.0861  & 3.4503   &1.2426      & 0.9435       & 1.0880      \\\hline
0 & 3 & 2.0536  &    1.1729      & 0.9585       & 1.0628  & 1.8554   &1.1727      & 0.9586       & 1.0627      \\
1 & 3 & 3.0759  &    1.1900      & 0.9502       & 1.0720  & 3.0546   &1.1891      & 0.9504       & 1.0717      \\
2 & 3 & 3.3283  &    1.2450      & 0.9423       & 1.0893  & 3.4549   &1.2461      & 0.9415       & 1.0902      \\
\end{tabular}
\end{ruledtabular}

\end{table}

\subsection*{Numerical results}\label{sec4.results}
To verify the majorization effect on linear and composite measures, as delineated above, we will present sample numerical results on entropic moments with 
non-integral orders and $S_{n_r, \ell}$, which are then used for the calculation of RCR, GRC, SRC and LMC. For this purpose, we have adopted Lebesgue 
measure in $r$ space, as well as the concept of \emph{effective domain} \hyperref[csl:34]{(P.~Ghosh \& D.~Nath, 2646}; \hyperref[csl:35]{D.~Nath, 2681)}. In such a scenario, the improper integral 
$\left\langle f(r)\right\rangle_{n_r,\ell}$ is defined \hyperref[csl:34]{(P.~Ghosh \& D.~Nath, 2646}; \hyperref[csl:35]{D.~Nath, 2681)} in a finite closed bounded domain $[0,d_{n_r,\ell}]$, such that, 
\beq\label{improper.nl} 
\left\langle f(r)\right\rangle_{n_r,\ell}\approx\ds\int_0^{d_{n_r,\ell}} R_{n_r,\ell}^*(r)f(r)R_{n_r,\ell}(r)dr,
\eeq
and
\beq
\ba{ll}
1-\eps_{n_r,\ell}<\ds\int_0^{d_{n_r,\ell}} \om_{n_r,\ell}(r)dr\le 1,\\
0\le \ds\int_{d_{n_r,\ell}}^{\infty}\om_{n_r,\ell}(r)dr<\eps_{n_r,\ell},\\
\om_{n_r,\ell}(d_{n_r,\ell}),\left[R_{n_r,\ell}^*(r)f(r)R_{n_r,\ell}(r)\right]_{r=d_{n_r,\ell}}\approx 0.
\ea
\eeq
If $d_{n,\ell}$ increases, then $\eps_{n,\ell}$ decreases, and the integral, Eq.~(\ref{improper.nl}) converges to a fixed number. To find numerical 
values of different expectations and information theoretic measures, we consider a new set of normalized wave function, 
\beq
\widetilde{\om}_{n_r,\ell}(r,\eps_{n_r,\ell})=\f{\om_{n_r,\ell}(r)}{1-\eps_{n_r,\ell}},~r\in[0,d_{n_r,\ell}],~n_r=0,1,2.
\eeq
If $d_{n_r,\ell}\rightarrow\infty$, then $\eps_{n_r,\ell}=0$ and $\widetilde{\om}_{n_r,\ell}(r,\eps_{n_r,\ell})=\om_{n_r,\ell}(r)$. For numerical purposes
we use $\eps_{n_r,\ell}<10^{-7}$ and $\widetilde{\om}_{n_r,\ell}(r,\eps_{n_r,\ell})\approx\om_{n_r,\ell}(r,\eps_{n_r,\ell})$. For $n_r=0,1,2$ and
$\ell=0$, taking $d_{n,\ell}>55$, one gets $\eps_{n_r,\ell}<10^{-9}$ when $D=3$ and taking $d_{n_r,\ell}>75$, we find $\eps_{n_r,\ell}<10^{-10}$, when
$D=4$. An increase in $n_r$ or $\ell$ or $D$, leads to growth in $d_{n_r,\ell}$. To illustrate the localization effect of radial density functions on 
composite measures, we have chosen closed and bounded domain $[0,d_{n_r,\ell}]$ for various states having $(n_r=0,1,2,\ell=0,1,2,3$). Using the 
localization effect, one can define the majorization property of density functions \hyperref[csl:34]{(P.~Ghosh \& D.~Nath, 2646}; \hyperref[csl:35]{D.~Nath, 2681)}. To verify the effect of closed 
domain on density function, we have plotted density functions \(\om_{n_r,\ell}\) in Figs.~\ref{Fig4} and \ref{Fig5} in \(3D\) and \(4D\) respectively, for 
$n_r=0,1,2,~\ell=0,1,2,3$. Representative expectation values and Heisenberg uncertainty products are tabulated in Table~\ref{table3}. Note that, 
Tables~\ref{table3}, \ref{table4.sha.lmc.src}, \ref{table5.grc.rcr} and Figs.~\ref{Fig4}, \ref{Fig5} are generated using $\xi=1.03$, $\lam_1=0, 
\lam_2=0, \lam_3=0.995, \lam_4=0, \lam_5=0.005$, under a best set of parameter values corresponding to approximation \(f_6^{(c)}\).

\begingroup    \squeezetable\selectlanguage{english}
\begin{table}
\caption{{\label{table5.grc.rcr} Calculated composite measures of complexities: GRC $(\mathcal{C}^{(\bar{\a}_1,\bar{\b}_1)}_{n_r,\ell}$ \& $\mathcal{C}^{(\bar{\a}_2,\bar{\b}_2)}_{n_r,\ell} )$ and RCR $\left(\mathcal{C}^{(\bar{\a}_1,\bar{\b}_1)}_{(n_{{r}_1},\ell_1),(n_{{r}_2},\ell_2)} \right. $ $\&$ 
$\left. \mathcal{C}^{(\bar{\a}_2,\bar{\b}_2)}_{(n_1,\ell_1),(n_2,\ell_2)}\right)$, in 3D and 4D. $(\bar{\a}_1,\bar{\b}_1)=(2.5,1.5)$ $(\bar{\a}_2,\bar{\b}_2)=(1.75,2.75)$, where $\bar{\a}_1>\bar{\b}_1$ and $\bar{\a}_2<\bar{\b}_2$.}}
\centering
\begin{ruledtabular}
\scalebox{.95}{\begin{tabular}{cc|cc|cc | cc | cc |cc } 
& & \multicolumn{2}{c}{$D=3$} & \multicolumn{2}{c|}{$D=4$}  &    &  &  \multicolumn{2}{c}{$D=3$} & \multicolumn{2}{c}{$D=4$}  \\ \hline
$n_r$ & $\ell$  & $\mathcal{C}^{(\bar{\a}_1,\bar{\b}_1)}_{n_r,\ell}  $ & $\mathcal{C}^{(\bar{\a}_2,\bar{\b}_2)}_{n_r,\ell}  $ & $\mathcal{C}^{(\bar{\a}_1,\bar{\b}_1)}_{n_r,\ell}  $ & $\mathcal{C}^{(\bar{\a}_2,\bar{\b}_2)}_{n_r,\ell}  $  &  $(n_{{r}_1},\ell_1)$ & $(n_{{r}_2},\ell_2)$  & 
 $\mathcal{C}^{(\bar{\a}_1,\bar{\b}_1)}_{(n_{{r}_1},\ell_1),(n_{{r}_2},\ell_2)}$ & $\mathcal{C}^{(\bar{\a}_2,\bar{\b}_2)}_{(n_{{r}_1},\ell_1),(n_{{r}_2},\ell_2)}$
& $\mathcal{C}^{(\bar{\a}_1,\bar{\b}_1)}_{(n_{{r}_1},\ell_1),(n_{{r}_2},\ell_2)}$ & $\mathcal{C}^{(\bar{\a}_2,\bar{\b}_2)}_{(n_{{r}_1},\ell_1),(n_{{r}_2},\ell_2)}$ \\ 
  &  & $< 1$  & $>1$  & $< 1$  & $>1$  &    &    &    &    &    &    \\ \hline
0 & 0    & 0.89503      & 1.09700     &0.89818      & 1.09394     
& (0,0)  & (0,0)        & 0.89503     &  1.09696    &  0.89818   & 1.09394      \\
1 & 0 & 0.88454      & 1.10860     &0.88296      & 1.11122    
& (1,0)&(0,0)  & 2.59345        &   3.20546      &  2.06859      & 2.55077      \\
2 & 0 & 0.88832      & 1.10280     &0.88269      & 1.10868     
&  (2,0)&(0,0)  & 4.46938        &   5.50291      &  3.17098      & 3.90557      \\ \hline
0 & 1 & 0.89977      & 1.09242     &0.90071      & 1.09153   
&  (0,1)&(0,1)  & 0.89977        &   1.09242      &  0.90071      & 1.09153      \\
1 & 1 & 0.88304      & 1.11183     &0.88380      & 1.11150    
&  (1,1)&(0,1)  & 1.82649        &   2.24825      &  1.69136      & 2.07865      \\
2 & 1 & 0.87784      & 1.11394     &0.87363      & 1.11864     
&  (2,1)&(0,1)  & 2.58885        &   3.19430      &  2.27686      & 2.81550      \\\hline
0 & 2 & 0.90130      & 1.09098     &0.90167      & 1.09063    
&  (0,2)&(0,2)  & 0.90130        &   1.09098      &  0.90167 & 1.09063  \\
1 & 2 & 0.88476      & 1.11078     &0.88568      & 1.10999   
&  (1,2)&(0,2)  & 1.60814        &   1.97370      &  1.55459  & 1.90588  \\
2 & 2 & 0.87008      & 1.12276     &0.86721      & 1.12622    
& (2,2)&(0,2)  & 2.09747        &   2.59930      &  1.99608  & 2.47848  \\\hline
0 & 3 & 0.90188      & 1.09043     &0.90197      & 1.09035   
&  (0,3)&(0,3)  & 0.90188        &   1.09043      &  0.90197  & 1.09035  \\
1 & 3 & 0.88642      & 1.10931     &0.88688      & 1.10888    
& (1,3)&(0,3)  & 1.52039        &   1.86246      &  1.50063  & 1.83743  \\
2 & 3 & 0.86506      & 1.12896     &0.86359      & 1.13095    
& (2,3)&(0,3)  & 1.94872        &   2.42357      &  1.94649   & 2.42376  \\
\end{tabular}}
\end{ruledtabular}

\end{table}

Now, it is noticed that, 
\beq
\ds\int\left[\om_{0,\ell}(r)-r_0\right]^+dr\ge\ds\int\left[\om_{1,\ell}(r)-r_0\right]^+dr\ge\ds\int\left[\om_{2,\ell}(r)-r_0\right]^+dr,~0<r_0<1.
\eeq
Therefore, 
\beq
\ba{ll}
\om_{0,\ell}\succ\om_{1,\ell}\succ\om_{2,\ell}&\implies\left\{\ba{l} 
\mathcal{D}_{0,\ell}>\mathcal{D}_{1,\ell}>\mathcal{D}_{2,\ell},\\ \mathcal{S}_{0,\ell}<\mathcal{S}_{1,\ell}<\mathcal{S}_{2,\ell},\\ \mathcal{R}_{0,\ell}^{(\bar{\a})}<\mathcal{R}_{1,\ell}^{(\bar{\a})}<\mathcal{R}_{2,\ell}^{(\bar{\a})},\ea\right..
\ea
\eeq
It is easily observed that, $\mathcal{S}_{n_r,\ell}$ of the density function $\om_{n_r,\ell}$ over a domain $[0,\infty)$ is equal to 
$\mathcal{S}_{\widetilde{\om}_{n_r,\ell}}$ over the domain $[0,d_{n,\ell}]$. It is verified that, $\widetilde{\om}_{0,\ell}\succ\widetilde{\om}_{1,\ell}\succ\widetilde{\om}_{2,\ell}$, for $\ell=0-3$, and they are continuous, bounded functions. Therefore, entropy $\mathcal{S}_{\widetilde{\om}_{n_r,\ell}}$ can 
be expressed as a limit of a sum \hyperref[csl:33]{(I.~Varga \& J.~Pipek, 262AD}; \hyperref[csl:36]{I.~Bialynicki-Birula, 521AD)}, as below, 
\beq
\mathcal{S}_{n_r,\ell}\approx\mathcal{S}_{\widetilde{\om}_{n_r,\ell}}=-\ds\lim\limits_{\D d\rightarrow0}\sum\limits_{i=1}^k\widetilde{\om}_{n_r,\ell}(r_i,\eps_{n_r,\ell})\ln\left[\widetilde{\om}_{n_r,\ell}(r_i,\eps_{n_r,\ell})\right]\D d,
\eeq
where $k\D d=d_{n_r,\ell},r_i\in[(i-1)\D d,i\D d]$. The numerically estimated values of $S_{n_{r}, \ell}$ are reported in 
Table~\ref{table4.sha.lmc.src}. From this table and 
making use of Eq.~(\ref{disequi}), one can then obtain \(\mathcal{C}^{(LMC)}_{n,\ell}\), which are presented in Table~\ref{table4.sha.lmc.src}. Similarly, 
the calculated SRC \(\mathcal{C}^{(\bar{\a})}_{n_r,\ell}\), for orders \(\bar{\a}=2.5,1.5\), in \(D=3,4\) are displayed in 
Table~\ref{table4.sha.lmc.src}. On the other hand, GRC's \(\mathcal{C}^{(\bar{\a},\bar{\b})}_{n_r,\ell}\), for order 
\((\bar{\a},\bar{\b})=(2.5,1.5),(1.75,2.75)\) are produced in Table~\ref{table5.grc.rcr}. Finally, specimen RCR, 
\(\mathcal{C}^{(\bar{\a},\bar{\b})}_{(n_{{r}_1},\ell_1),(n_{{r}_2},\ell_2)}\), of order \((\bar{\a},\bar{\b})=(2.5,1.5),(1.75,2.75)\) are given in 
Table~\ref{table5.grc.rcr}. It is well-known that LMC is a product of two functions, of which one is increasing and the other one decreasing. With 
respect to the quantum number \(n_r\) it has an optimum value \hyperref[csl:54]{(2596)}. Here we have found the following relation to hold, namely, 
\beq
\ba{ll}
\mathcal{C}^{LMC}_{n_r,\ell}<\mathcal{C}^{LMC}_{n_r+1,\ell},\ell\ge 0,\\
\mathcal{C}^{LMC}_{n_r,\ell}>\mathcal{C}^{LMC}_{n_r,\ell+1},n_r=0,1,\\
\mathcal{C}^{LMC}_{n_r,\ell}<\mathcal{C}^{LMC}_{n_r,\ell+1},n_r=2.
\ea
\eeq
It is also found that,  
\beq
\ba{ll}
1>\mathcal{C}^{(\bar{\a})}_{n_r,\ell}\ge \mathcal{C}^{(\bar{\a})}_{n_r+1,\ell},\ell>0,\bar{\a}>2\\
1<\mathcal{C}^{(\bar{\a})}_{n_r,\ell}\le \mathcal{C}^{(\bar{\a})}_{n_r+1,\ell},\ell>0,\bar{\a}<2\\
\ea
\eeq
and 
\beq
\ba{ll}
1>\mathcal{C}^{(\bar{\a},\bar{\b})}_{n_r,\ell}\ge \mathcal{C}^{(\bar{\a},\bar{\b})}_{n_r+1,\ell},\ell>0,\bar{\a}>\bar{\b},\\
1<\mathcal{C}^{(\bar{\a},\bar{\b})}_{n_r,\ell}\le \mathcal{C}^{(\bar{\a},\bar{\b})}_{n_r+1,\ell},\ell>0,\bar{\a}<\bar{\b},\\ 
\ea
\eeq
Additionally, our numerical results satisfy the inequality, $\mathcal{C}^{(\bar{\a},\bar{\b})}_{(n_{r},\ell),(0,\ell)} <\mathcal{C}^{(\bar{\a},\bar{\b})}_{(n_{r}+1,\ell),(0,\ell)}$, for $n_r=0,1$ and $\ell=0-3$.

\section*{Conclusion}\label{sec5}
Accurate analytical expressions for eigenvalues and eigenfunctions are derived for Hulth\'en potential, in an arbitrary $\ell \neq 0$ 
state. For this, we have proposed a new and improved approximation of the centrifugal term. It is generated from a suitable linear 
combination of two frequently used approximations, namely Greene-Aldrich and Pekeris. This facilitates an accurate representation of the potential
in \emph{both short and long range} . The relevant SE is solved in $D$ dimension within the general rubric of NU method. 
Our calculated energies are compared with existing theoretical results, which reveals quite favorable agreement.  
The generalized wave function as well as normalization constant are also reported for a given state. A total of six different approximations are 
critically compared and contrasted for their performances in energy. 
An approximate analytic expression for \emph{critical screening} for a particular state is offered in a given dimension. Furthermore, we also 
derive the  screening parameter, where an \emph{inter-dimensional} degeneracy occurs, i.e., two states belonging to different dimensions produce 
same energy.

In the later part, we deal with certain linear and composite information theoretic measures under the best solution amongst the
newly proposed approximations. Towards this goal, analytical closed-form expressions for Heisenberg uncertainty measures \(\bigtriangleup r\) and 
\(\bigtriangleup p\) were derived. Apart from that, analytical forms of $R$, RCR, GRC, SRC were offered for positive integral order, whereas, 
$S$ and LMC complexity were defined numerically, over a closed bounded domain. For numerical presentation, we defined a new set of normalized 
radial density functions over a closed and bounded domain, which are practically same as that of the original density functions. Besides that, 
majorization effect on linear and composite measures are verified for \emph{admissible} bound states, which critically depends on \(\d\).
The success of the method will be further enhanced by its application to other related properties of interest, as well as other important screened 
potentials in quantum systems (such as plasma). Some of these may be taken up in future.

\section*{Acknowledgement}
AKR gratefully acknowledges financial support from MATRICS, DST-SERB, New Delhi (sanction order: MTR/2019/000012) and partial assistance from DST-SERB (sanction order: CRG/2019/000293) is appreciated.

\section*{Author Contribution}
\noindent
Debraj Nath: Conceptualization, Investigation, Methodology, Writing-original draft. \\
Amlan K. Roy: Conceptualization, Investigation, Methodology, Writing-original draft.

\section*{Data availability statement}
All data generated or analyzed during this study are included in this article.

\section*{ORCID}
\noindent Debraj Nath: https://orcid.org/0000-0001-9937-7032\\
\noindent Amlan K. Roy: https://orcid.org/0000-0001-5555-8915

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