# PDM Klein-Gordon oscillators in cosmic string spacetime in magnetic and
Aharonov-Bohm flux fields within the Kaluza-Klein theory

Omar Mustafa<EMAIL_ADDRESS>Department of Physics, Eastern
Mediterranean University, G. Magusa, North Cyprus, Mersin 10 - Turkey.

###### Abstract

Abstract: In the cosmic string spacetime and within Kaluza-Klein theory (KKT)
backgrounds (indulging magnetic and Aharonov-Bohm flux fields), we introduce
and study position-dependent mass (PDM) Klein-Gordon (KG) oscillators. The
effective PDM is introduced as a deformation/defect in the momentum operator.
We show that there are four different ways to obtain KG-oscillator. Two of
which are readily known and the other two are obtained as byproducts of PDM
settings. Next, we provide a thorough analysis on the corresponding spectra
under different parametric effects, including the curvature parameter’s
effect. Such analysis is used as a reference/lead model which is used in the
discussion of different PDM KG-oscillators models: a mixed power-law and
exponential type PDM model that yields a pseudo-confined PDM KG-oscillator in
cosmic string spacetime within KKT (i.e., the PDM KG-oscillators are confined
in their own PDM manifested Cornell-type confinement), and a PDM KG-oscillator
confined in a Cornell-type potential. Moreover, we extend our study and
discuss a non-Hermitian $\mathcal{PT}$-symmetric PDM-Coulombic-type KG-
particle model in cosmic string spacetime within KKT.

PACS numbers: 05.45.-a, 03.50.Kk, 03.65.-w

Keywords: Cosmic string spacetime backgrounds, Kaluza-Klein theory (KKT),
Klein-Gordon (KG) oscillator. position-dependent mass (PDM) KG-particles,
$\mathcal{PT}$-symmetric PDM

##  I Introduction

Topological defects as products of disclination, dislocation, and dispiration
in media or in cosmic string spacetime manifestly affect the physical
properties of the system at hand R1 ; R2 . They appear as monopole, strings,
and walls in gravitation R3 , as vortices in superconductors or superfuilds,
domain walls in magnetic materials, solitons in polymers, and disclinations or
dislocations in solids and liquid crystals R4 ; R5 ; Geusa 2001 . Studies of
quantum mechanical relativistic particles in spacetime with cosmic strings as
topological defects are ample in the literature. It has been shown, for
example, that the energy spectrum in a curved spacetime is different than that
in the flat Minkowski spacetime R7 ; R8 ; R9 ; R10 ; R11 ; Ahmed 1 ; Ahmed 2 ;
Santos 2018 ; R12 . The Dirac and Klein-Gordon particles in a Gödel-type
spacetime background at positive, negative , and zero curvatures are studied
in Figueiredo 1992 . Nevertheless, the Dirac oscillator of Moshinsky and
Szczepaniak Moshinsky 1989  has encouraged research activities on Klein-Gordon
(KG) oscillator Bruce 1993 ; Dvoeg 1994 . For example, the KG-oscillator in
the Gödel and Gödel-type spacetime backgrounds (e.g., Moshinsky 1989 ; Bruce
1993 ; Dvoeg 1994 ; Das 2008 ; Carvalho 2016 ; Garcia 2017 ; Vitoria 2016 ;
Vitoria1 2016 ; Ahmed1 2018 ; Lutfuoglu 2020 ), in cosmic string spacetime
within Kaluza-Klein theory (KKT) backgrounds (e.g., Ahmed 2 ; Ahmed1 2018 ;
Ahmed1 2020 ; Ahmed1 2021 ; Boumal 2014 ; R13 ), in Som-Raychaudhuri Wang 2015
, in the (2+1)-dimensional Gürses spacetime backgrounds (e.g., Gurses 1994 ;
Ahmed 2019 ; Ahmed1 2019 ; Ahmed2 2019 ), etc.

Position-dependent mass (PDM), on the other hand, have attracted researchers
attention over the years after the introduction of the Mathews-Lakshmanan
oscillator M-L 1974 , both in classical and quantum mechanics M-L 1974 ; von
Roos ; Carinena Ranada Sant 2004 ; Mustafa 2019 ; Mustafa1 2020 ; Mustafa
arXiv ; Mustafa Phys.Scr. 2020 ; Mustafa Habib 2007 ; Mustafa Algadhi 2019 ;
Khlevniuk 2018 ; Mustafa 2015 ; Dutra Almeida 2000 ; dos Santos 2021 ;
Nabulsi1 2020 ; Nabulsi2 2020 ; Nabulsi3 2021 ; Quesne 2015 ; Tiwari 2013 .
Such a PDM concept does not necessarily mean that the mass is literally
position dependent. PDM means that the mass becomes effectively position-
dependent. It has been shown that the PDM concept is, in fact, a metaphoric
manifestation of coordinate deformation/transformation Mustafa1 2020 ; Mustafa
arXiv ; Mustafa Phys.Scr. 2020 ; Khlevniuk 2018 . The coordinate
deformation/transformation, in effect, changes the form of the canonical
momentum in classical and the momentum operator in quantum mechanics (e.g.,
Mustafa1 2020 ; Mustafa arXiv ; Mustafa Algadhi 2019 ; dos Santos 2021  and
related references therein). In classical mechanics, for example, negative the
gradient of the potential force field is shown to be related to the time
derivative of the pseudo-momentum (also called Noether momentum)
$\pi\left(x\right)=\sqrt{m\left(x\right)}\dot{x}$ Mustafa arXiv , and is no
longer given by the time derivative of the canonical momentum
$p=m\left(x\right)\dot{x}$. In quantum mechanics, however, the PDM momentum
operator is constructed by Mustafa and Algadhi Mustafa Algadhi 2019  and is
used to find the PDM creation and annihilation operators for the PDM-
Schrödinger oscillator Mustafa1 2020 . It would be interesting, therefore, to
investigate the effects of PDM on the KG-oscillator (in particular, and KG-
particles in general) in cosmic string spacetime background within KKT.

Attempts were, in fact, made to include PDM settings in the Dirac and/or KG
relativistic equations Mustafa Habib 2008 ; Mustafa Habib1 2007 ; Vitoria
Bakke 2016  through the assumption that $m\longrightarrow
m+m\left(r\right)+S\left(r\right)=M\left(r\right)$, where $m$ denotes the rest
mass energy, $m\left(r\right)$ PDM, $S\left(r\right)$ the Lorentz scalar
potential interaction, and $M\left(r\right)$ denotes effective PDM in the
relativistic wave equation at hand. However, in the current methodical
proposal, we stay away from such PDM perception and adopt the fact that PDM is
a manifestation of coordinate deformation/transformation and, therefore, use
the PDM-quantum mechanical operator Mustafa Algadhi 2019 ; Mustafa1 2020 ; dos
Santos 2021  (see (6) below). Under such settings, we shall study PDM KG-
oscillator in cosmic string spacetime within KKT.

The organization of our methodical proposal is in order. In section 2, we
introduce the PDM KG-oscillator in cosmic string spacetime within KKT. Where
the effective PDM is introduced as a defect/deformation in the momentum
operator. Moreover, we show that there are four different ways to obtain KG-
oscillators. Two of which are readily known and the other two (new) are
byproducts of PDM settings. In section 3, we recycle/recollect the KG-
oscillators in cosmic string spacetime in magnetic and Aharonov-Bohm flux
fields within the KKT, and provide a thorough analysis on the corresponding
spectra under different parametric effects. In so doing, we provide a
reference/lead model that is to be used in the following sections on different
PDM KG-oscillators models in cosmic string spacetime within KKT. In section 4,
we discuss a mixed power-law and exponential type PDM model that yields a
pseudo-confined PDM KG-oscillator in cosmic string spacetime within KKT. This
confinement is introduced as a byproduct of the PDM settings of the KG-
oscillator (i.e., the PDM KG-oscillators are confined in their own PDM
manifested confinement), hence the notion ”pseudo-confined PDM KG-
oscillators”. In section 5, we discuss a confined PDM KG-oscillator-III (the
third of the four KG-oscillators reported in section 2) in cosmic string
spacetime within KKT. In section 6, we use a $\mathcal{PT}$-symmetric PDM-
Coulombic-type KG-particle model in cosmic string spacetime within KKT.
Hereby, we follow the Bender and Boettcher Bender 1998  relaxation of the
Hermiticity condition on quantum Hamiltonians that have resulted in the
introduction of the nowadays called $\mathcal{PT}$-symmetric quantum
mechanics, which yields real spectra (for more details on this issue the
reader may refer to Znojil Levai 2000 ; Mustafa Znojil 2002  and references
cited therein). Where, $\mathcal{P}$ denotes space reflection:
$x\rightarrow-x$, $\mathcal{T}$ mimics the time-reversal: $i\rightarrow-i$,
and a $\mathcal{PT}$-symmetric function would satisfy $f(x)=|f(-x)|^{*}$. We
give our concluding remarks in section 7.

##  II PDM KG-oscillators in cosmic string spacetime in magnetic and Aharonov-
Bohm flux fields within KKT

The Kaluza-Klein theory proposes an extra compact space dimension while
introducing pure gravity in the new $(4+1)$-dimensional spacetime. The
idealized assumption in the KKT is that the topological defect carries in its
core a magnetic field with a magnetic flux $\Phi$ (i.e., Aharonov-Bohm flux
field) that vanish outside the defect Geusa 2001 ; Furtado 2000 . Under such
assumption, the magnetic field is not introduced via minimal coupling but is
rather treated as a topological defect within the framework of the Kaluza-
Klein theory. The corresponding cosmic string spacetime metric (in
$G=\hbar=c=1$ units) is given by

$ds^{2}=-dt^{2}+dr^{2}+\alpha^{2}\,r^{2}\,d\varphi^{2}+dz^{2}+(du-\kappa\,A_{\mu}\,dx^{\mu})^{2},$
(1)

where the fifth dimension $u$ is a space-like and it varies as
$0<u<2\,\pi\,a$, where $a$ is the radius of the compact dimension of $u$.
Here, $\kappa$ is the Kaluza constant, and $\alpha=(1-4\,\tilde{\mu})$ with
the linear mass density of the string $\tilde{\mu}$ Santos 2018 . The
parameter $\alpha$ assumes values $0<\alpha<1$ for positive curvature (in
general relativity for cosmic string with positive mass density),
$1<\alpha<\infty$ for negative curvature (in the geometric theory of defects
in condensed matter), and $\alpha=1$ corresponds to Minkowski flat spacetime.
Moreover, the coordinates ranges are $\varphi\in[0,2\pi]$, $r\in[0,\infty]$,
and $(z,t)\in[-\infty,\infty]$. We now assume that the electromagnetic four-
vector potential, indulging the Aharonov-Bohm flux field $\Phi$, within KKT is
given by

$A_{\mu}=(0,0,A_{\varphi},0)\,;\,\,A_{\varphi}=\kappa^{-1}\,\left[\frac{1}{2\,\alpha}\,B_{\circ}\,r+\frac{\Phi}{2\,\pi\,\alpha\,r}\right],$
(2)

so that

$\kappa\,A_{\mu}\,dx^{\mu}=\kappa\,A_{\varphi}\,\alpha\,r\,d\varphi=\left(\frac{1}{2}\,B_{\circ}\,r^{2}+\frac{\Phi}{2\,\pi}\right)\,d\varphi,$
(3)

where a uniform magnetic field is manifested in the $z$-direction through
$\mathbf{B=\nabla\times A}=(\kappa^{-1}B_{\circ}/\alpha)\,\hat{z}$, and the
Aharonov-Bohm flux field $\Phi$ is indulged in the second term of
$A_{\varphi}$ Ahmed 1 ; Geusa 2001 ; Furtado 2000 ; Mustafa Z 2020 . Here, we
have used $\mathbf{A}=\mathbf{A}_{1}+\mathbf{A}_{2}$ so that
$\mathbf{A}_{1}=\left(0,\frac{\kappa^{-1}}{2\,\alpha}\,B_{\circ}\,r,0\right)$,
and $\mathbf{A}_{2}=\left(0,\frac{\kappa^{-1}\,\Phi}{2\,\pi\,\alpha
r},0\right)$, with $\mathbf{\nabla\times
A}_{1}=(\kappa^{-1}B_{\circ}/\alpha)\,\hat{z}$, and $\mathbf{\nabla\times
A}_{2}=0$. Hence, the cosmic string in $(4+1)$-dimensional spacetime metric
within KKT reads

$ds^{2}=-dt^{2}+dr^{2}+\alpha^{2}\,r^{2}\,d\varphi^{2}+dz^{2}+\left[du+\tilde{A}_{\varphi}\,d\varphi\right]^{2};\,\tilde{A}_{\varphi}=-\left(\frac{1}{2}\,B_{\circ}\,r^{2}+\frac{\Phi}{2\,\pi}\right).$
(4)

Then the corresponding covariant and contravariant metric tensors in this
case, respectively, read

$g_{\mu\nu}=\left(\begin{tabular}[]{ccccc}$-1\vskip 3.0pt plus 1.0pt minus
1.0pt$&$\,0\,$&$0$&$\,0$&$0$\\\ $0$&$1\vskip 3.0pt plus 1.0pt minus
1.0pt$&$0$&$0$&$0$\\\
$0$&$\,0$&$\,\left(\alpha^{2}\,r^{2}+\tilde{A}_{\varphi}^{2}\right)\,$&$0$&$\tilde{A}_{\varphi}$\\\
$0$&$0$&$0$&$1$&$0$\\\
$0$&$0$&$\tilde{A}_{\varphi}$&$0$&$1$\end{tabular}\right)\Leftrightarrow
g^{\mu\nu}=\left(\begin{tabular}[]{ccccc}$-1\vskip 3.0pt plus 1.0pt minus
1.0pt$&$\,0\,$&$0$&$0$&$0$\\\ $0$&$\,1\vskip 3.0pt plus 1.0pt minus
1.0pt$&$0$&$0$&$0$\\\
$0$&$\,0$&$\,\frac{1}{\alpha^{2}\,r^{2}}\,$&$-0$&$-\frac{\tilde{A}_{\varphi}}{\alpha^{2}\,r^{2}}$\\\
$0$&$0$&$-0\,$&$\,1$&$0$\\\
$0$&$0$&$-\frac{\tilde{A}_{\varphi}}{\alpha^{2}\,r^{2}}$&$0$&$\left(1+\frac{\tilde{A}_{\varphi}^{2}}{\alpha^{2}\,r^{2}}\right)$\end{tabular}\right)$
(5)

where $\det\left(g\right)=-\alpha^{2}\,r^{2}.$

Very recently, however, Mustafa and Algadhi Mustafa Algadhi 2019  have
introduced the PDM-momentum operator

$\mathbf{\hat{p}}\left(\mathbf{r}\right)=-i\,\left(\mathbf{\nabla-}\frac{\mathbf{\nabla}m\left(\mathbf{r}\right)}{4\,m\left(\mathbf{r}\right)}\right)\Longleftrightarrow
p_{j}=-i\,\left(\partial_{j}-\frac{\partial_{j}\,m\left(\mathbf{r}\right)}{4\,m\left(\mathbf{r}\right)}\right),$
(6)

which is used to study the torsion effects on the PDM Klein-Gordon oscillator
in a cosmic string spacetime background Mustafa arXiv1 . In the current study,
we shall use such PDM-momentum operator Mustafa Algadhi 2019  and consider a
PDM KG-oscillator with a uniform magnetic field and Aharonov-Bohm flux field
within the KKT. Moreover, we shall use the assumption that
$m(\textbf{r})=m(r)$ (i.e., the PDM function is only radially dependent).
Under such settings, the momentum operator would take the PDM form Mustafa
arXiv1  so that

$\tilde{p}_{\mu}\longrightarrow-i\,\partial_{\mu}+i\,\mathcal{F}_{\mu}\,{;}\quad\mathcal{F}_{\mu}=\left(0,\eta\,r+\frac{m^{\prime}\left(r\right)}{4\,m\left(r\right)},0,0\right)\Longrightarrow\mathcal{F}_{r}=\eta\,r+\frac{m^{\prime}\left(r\right)}{4\,m\left(r\right)},$
(7)

which is to be used to construct the PDM KG-oscillator in the presence of a
Lorentz scalar potential $S(r)$ as follows

$\frac{1}{\sqrt{-g}}\,\left(\partial_{\mu}+\mathcal{F}_{\mu}\right)\,\left[\sqrt{-g}\,g^{\mu\nu}\,\left(\partial_{\nu}-\mathcal{F}_{\nu}\right)\,\Psi\right]=(m+S(r))^{2}\,\Psi.$
(8)

This would result, with
$\Psi=e^{i\,(-E\,t+l\,\varphi+k_{z}\,z+Q\,u)}\,R\left(r\right)$, in

$R^{\prime\prime}\left(r\right)+\frac{1}{r}R^{\prime}\left(r\right)+\left[\lambda-\frac{\tilde{\gamma}^{2}}{r^{2}}-\omega^{2}r^{2}-\left(\mathcal{F}_{r}^{\prime}+\frac{\mathcal{F}_{r}}{r}+\mathcal{F}_{r}^{2}\right)-2mS\left(r\right)-S\left(r\right)^{2}\right]R\left(r\right)=0,$
(9)

and the substitution $R\left(r\right)=U\left(r\right)/\sqrt{r}$ would imply

$U^{\prime\prime}\left(r\right)+\left[\lambda-\frac{\left(\tilde{\gamma}^{2}-1/4\right)}{r^{2}}-\omega^{2}r^{2}-\left(\mathcal{F}_{r}^{\prime}+\frac{\mathcal{F}_{r}}{r}+\mathcal{F}_{r}^{2}\right)-2mS\left(r\right)-S\left(r\right)^{2}\right]U\left(r\right)=0,$
(10)

where

$\lambda=E^{2}-Q^{2}-k_{z}^{2}-m^{2}-2\omega\tilde{\gamma}\,;\,\,\tilde{\gamma}^{2}=\left(\frac{\ell}{\alpha}+\frac{\Phi
Q}{2\,\alpha\,\pi}\right)^{2}\,;\,\,\omega=\frac{QB_{\circ}}{2\alpha}.$ (11)

Eventually, with $\mathcal{F}_{r}$ of (7), we obtain

$U^{\prime\prime}\left(r\right)+\left[\tilde{\lambda}-\frac{\left(\tilde{\gamma}^{2}-1/4\right)}{r^{2}}-\tilde{\omega}^{2}\,r^{2}-2\,m\,S\left(r\right)-S\left(r\right)^{2}+M\left(r\right)\right]\,U\left(r\right)=0,$
(12)

where $\tilde{\lambda}=\lambda-2\,\eta$,
$\tilde{\omega}^{2}=\omega^{2}+\eta^{2}$, and

$M\left(r\right)=\frac{3}{16}\,\left(\frac{m^{\prime}\left(r\right)}{m\left(r\right)}\right)^{2}-\frac{1}{4}\,\frac{m^{\prime\prime}\left(r\right)}{m\left(r\right)}-\frac{m^{\prime}\left(r\right)}{4\,r\,m\left(r\right)}-\eta\,r\,\frac{m^{\prime}\left(r\right)}{2\,m\left(r\right)}.$
(13)

Under the current proposal settings, it could be interesting to report that
there are four feasible models that yield KG-oscillators:

  1. 1.

KG-oscillator-I: is obtained by setting $\mathcal{F}_{r}=\eta\,r$,
$B_{\circ}=0=\Phi$ (i.e., no magnetic fields), $S(r)=0$ (i.e., no confinement)
and $m(r)=1$ (i.e., constant mass settings as in Mirza et al’s recipe Mirza
2004 . This would result in the KG-oscillator with
$V_{eff}(r)=\eta^{2}\,r^{2}+2\,\eta$.

  2. 2.

KG-Oscillator-II: is obtained by setting $\eta=0$, $m(r)=1$, $S(r)=0$ and
$B_{\circ}\neq 0$, which implies a KG-oscillator with
$V_{eff}(r)=\omega^{2}\,r^{2}$,

  3. 3.

KG-oscillator-III: is obtained by setting $\eta=0=B_{\circ}$, $S(r)=0$ and an
effective PDM in the form of

$m(r)=A\,e^{2\,\Omega\,r^{2}},$ (14)

that yields a shifted by a constant PDM KG-oscillator with
$V_{eff}(r)=-M(r)=\Omega^{2}\,r^{2}+2\,\Omega$. However, one should notice the
resemblance between this KG-oscillator-III and the KG-oscillator-I above.
Consequently, the KG- oscillator-I of Mirza et al’s recipe Mirza 2004  or KG-
oscillator-III may very well be considered as byproducts of PDM settings

  4. 4.

KG-oscillator-IV: is obtained by setting $\eta=0=B_{\circ}$, $S(r)=0$ and an
effective PDM of complex settings in the form of

$m\left(r\right)=C\,{\frac{{{I}_{0}\left(\frac{\Omega\,{r}^{2}}{2}\right)}^{4}}{{\Omega}^{4}{r}^{8}\left[{{I}_{0}\left(\frac{\Omega\,{r}^{2}}{2}\right)}{{K}_{1}\left(\frac{\Omega\,{r}^{2}}{2}\right)}+{{I}_{1}\left(\frac{\Omega\,{r}^{2}}{2}\right)}{{K}_{0}\left(\frac{\Omega\,{r}^{2}}{2}\right)}\right]^{4}}},$
(15)

where $I_{i}(x)$ and $K_{i}(x)$ are the modified Bessel functions. would
result $M(r)=-\Omega^{2}\,r^{2}$, in (13), and consequently yielding a PDM KG-
oscillator with $V_{eff}(r)=\Omega^{2}\,r^{2}$.

Notably, one may figure out that the KG-oscillator-I by Mirza’s et al recipe
Mirza 2004  turns out to be nothings but a byproduct of the PDM setting in KG-
oscillator-III of (14). Effectively, the topological defect in the momentum
operator suggested by Mirza et al. may very well be explained as an effective
topological defect of the effective PDM settings. That is, if one sets
$\eta=0$ (i.e., dismissing Mirza’s term) then the effective frequency of the
emerging PDM KG-oscillator is now given by $\Omega$. Consequently, the current
methodical proposal may mark a new and more general PDM KG-particles (not only
KG-oscillators) in cosmic string spacetime or in the geometric theory of
topological defects in condensed matter.

In the forthcoming sections, we start with the most simplistic KG-oscillator
that would serve as a reference model to check on the exact solvability of the
more general and more complicated models to be discussed in the sequential
sections. Yet, the four KG-oscillator models above are exactly solvable within
the lines discussed in the following section.

##  III KG-oscillator in cosmic string spacetime within KKT: recycled and
analyzed

In this section, we assume the Lorentz scalar potential $S(r)=0$ and the PDM
function $m\left(r\right)=1$ (i.e., constant mass settings). Therefore,
equation (12) becomes

$U^{\prime\prime}(r)+\left[\tilde{\lambda}-\frac{\left(\tilde{\gamma}^{2}-1/4\right)}{r^{2}}-\tilde{\omega}^{2}\,r^{2}\right]\,U(r)=0,$
(16)

where $\tilde{\lambda}=\lambda-2\,\eta$,
$\tilde{\omega}^{2}=\omega^{2}+\eta^{2}$ (this KG-oscillator covers both KG-
oscillator-I and/or II, above). Moreover, this equation resembles the two-
dimensional radial Schrödinger harmonic oscillator with an effective
oscillation frequency $\tilde{\omega}$ and an irrational magnetic quantum
number $\tilde{\gamma}$. Consequently, it inherits its textbook eigenvalues

$\tilde{\lambda}=2\,\tilde{\omega}\,\left(2\,n_{r}+\left|\tilde{\gamma}\right|+1\right)\Longrightarrow\lambda-2\,\eta=2\,\tilde{\omega}\,\left(2\,n_{r}+\left|\tilde{\gamma}\right|+1\right),$
(17)

where $n_{r}=0,1,2,\cdots$ is the radial quantum number (denotes number of
nodes in the wave function) (c.f., e.g., Furtado 2000 ; Bezerra 2019 ; Bakke
2010 ; Mota 2014 ). Hence, the KG-oscillator of (16) admits exact energy
levels given by

$E^{2}_{n_{r},l}=2\,\tilde{\omega}\,\left(2\,n_{r}+\left|\frac{\ell}{\alpha}+\frac{\,\Phi\,Q}{2\,\alpha\,\pi}\right|+1\right)+\frac{Q\,B_{\circ}}{\alpha}\left(\frac{\ell}{\alpha}+\frac{\Phi\,Q}{2\,\alpha\,\pi}\right)+k_{z}^{2}+m^{2}+Q^{2}+2\,\eta,$
(18)

and the corresponding unnormalized radial eigenfunctions are given by

$U\left(r\right)\sim
r^{\left|\tilde{\gamma}\right|+1/2}\exp\left(-\frac{\tilde{\omega}r^{2}}{2}\right)L_{n_{r}}^{\left|\tilde{\gamma}\right|}\left(\tilde{\omega}r^{2}\right)\Longleftrightarrow
R\left(r\right)\sim
r^{\left|\tilde{\gamma}\right|}\exp\left(-\frac{\tilde{\omega}r^{2}}{2}\right)L_{n_{r}}^{\left|\tilde{\gamma}\right|}\left(\tilde{\omega}r^{2}\right),$
(19)

Figure 1:  For $m=k_{z}=Q=\Phi=B_{\circ}=\eta=1$ we plot the energy levels
$E_{n_{r},\ell}$ so that in (a) $E_{n_{r},0}$ at $0<\alpha<10$ for
$n_{r}=0,1,2,3,4$, and (b) $E_{n_{r},\ell}$ at $0<\alpha<3$ for $n_{r}=0,2$
and $\ell=0,\pm 1,\pm 2$. For $m=k_{z}=Q=B_{\circ}=\eta=1$, $\Phi=0$ we plot
in (c) $E_{n_{r},\ell}$ at $0<\alpha<3$ for $n_{r}=0,2$ and $\ell=0,\pm 1,\pm
2$.

where
$L_{n_{r}}^{\left|\tilde{\gamma}\right|}\left(\tilde{\omega}r^{2}\right)$ are
the associated Laguerre polynomials that are, up to a multiplicity constant,
related to the confluent hypergeometric polynomials
${}_{1}F_{1}(-n_{r},1+|\tilde{\gamma}|,\tilde{\omega}\,r^{2})$. Nevertheless,
one should notice that for $S(r)\neq 0$ and/or $M(r)\neq 0$, the solution of
such a more general problem should collapse back to (18) and (19) for the case
$S(r)=0$ and $M(r)=0$. Moreover, it is obvious that square the energies in
(18) (i.e., $E^{2}$) resembles Landau-type energies-squared with an irrational
magnetic quantum number $\tilde{\gamma}$ that indulges within the Aharonov-
Bohm flux field effect. It is interesting to know that this is manifestly
introduced by the coupling between the compact Kaluza-Klein fifth-dimension
$u$ and the $\varphi$ coordinate. As such, we may argue that the additional
compact dimension of the Kaluza-Klein theory offers quasi-Landau energy levels
(e.g., Geusa 2001 ; Bezerra 2019 ; Zeinab 2020 ). Moreover, one should be
aware that the four possible KG-oscillators’ settings discussed above, with
some straightforward parametric substitutions, admit exact solvability as in
(18) and (19).

In Fig. 1 we plot the energy levels $E_{n_{r},\ell}$ against the parameter
$\alpha$. In Fig.1(a) we observe that as $\alpha$ increases within the range
$0<\alpha<1$ (i.e., cosmic string with positive mass density), the energy gap
shrinks and the energy levels separate apart from clustering near
$\alpha\approx 0$. Whereas, as $\alpha$ increases within the range
$1<\alpha<\infty$, the energy gap stabilizes and the energy levels
$E_{n_{r},0}$ maintain constant (but not equal) spacing as $\alpha$ grows up
from 1. Fig. 1(b), shows that for a given $n_{r}$ the energy levels
$E_{n_{r},\ell}$ would cluster and batch up (without levels crossings) around
the corresponding $E_{n_{r},0}$ energy level (i.e., corresponding S-state) for
$\alpha>>1$ within the range $1<\alpha<\infty$. Whereas, a different trend of
batching up (clustering) is obvious as $\alpha\rightarrow 0$. That is, state
of the same magnetic quantum number $\ell$ (i.e., states with the same color)
batch up and cluster together. Two different trends of clustering, as
$\alpha\rightarrow 0$ and as $\alpha\rightarrow\infty$, would necessarily
imply that energy levels crossings are eminent and unavoidable in the process.
In Fig. 1(c), we set the magnetic flux field $\Phi=0$ and found that the
batching-up phenomenon has nothings to do with the Aharonov-Bohm flux field
$\Phi$, but it is rather a characterization of the geometric theory of
topological defects in condensed matter as well as in cosmic string spacetime,
as $\alpha\rightarrow 0$ and as $\alpha\rightarrow\infty$ from $\alpha=1$,
respectively. Notably, at $\alpha=1$ (i.e., Minkowski flat spacetime) the
energy levels maintain their regular quasi-Landau-type structure.
Mathematically speaking, two terms in the energy equation (18), the first and
the second terms, determine the fate of the energy levels. If we temporarily
set $\eta=0$ in (18) we get the first two terms as

Figure 2:  For $m=k_{z}=Q=B_{\circ}=\eta=1$ we plot the energy levels
$E_{n_{r},\ell}$ so that in (a) $E_{n_{r},0}$ with $\alpha=0.5<1$ (positive
curvature) for $n_{r}=0,1,2,3,4$ at different values of $\Phi$, (b)
$E_{0,\ell}$ with $\alpha=0.5$ for $\ell=0,\pm 1,\pm 2$ at different values of
$\Phi$, and (c) $E_{0,\ell}$ with $\alpha=1.5>1$ (negative curvature) for
$\ell=0,\pm 1,\pm 2$ at different values of $\Phi$.

${Q\,B_{\circ}}\,\left(\frac{2\,n_{r}}{\alpha}+\left|\frac{\ell}{\alpha^{2}}+\frac{\,\Phi\,Q}{2\,\alpha^{2}\,\pi}\right|+\frac{1}{\alpha}\right)+{Q\,B_{\circ}}\left(\frac{\ell}{\alpha^{2}}+\frac{\Phi\,Q}{2\,\alpha^{2}\,\pi}\right).$
(20)

We may see two asymptotic behaviours for (20): (i) when $\alpha>>1$ we get
$Q\,B_{\circ}(2\,n_{r}+1)/\alpha$ as the dominant term where the contribution
of $\ell$ becomes insignificant (and hence for a given $n_{r}$ the energy
levels $E_{n_{r},\ell}$ would cluster and batch up around the corresponding
$E_{n_{r},0}$ energy level), and (ii) as $\alpha\rightarrow 0$ we get

${Q\,B_{\circ}}\,\left(\left|\frac{\ell}{\alpha^{2}}+\frac{\,\Phi\,Q}{2\,\alpha^{2}\,\pi}\right|\right)+{Q\,B_{\circ}}\left(\frac{\ell}{\alpha^{2}}+\frac{\Phi\,Q}{2\,\alpha^{2}\,\pi}\right).$
(21)

as the dominating terms (hence states with the same $\ell=+j;\,j=0,1,2,\cdots$
value would cluster and batch up irrespective of the values of $n_{r}$, same
for states with $\ell=-j$).

To study the effect of Aharonov-Bohm flux field $\Phi$, we plot Figure 2. We
observe that for $\Phi>0$ the energy levels are boosted up for positive
energies, $E_{+}$, or boosted down for negative energies, $E_{-}$.
Consequently, the energy gap widens up more rapidly than that for $\Phi<0$.
This is mainly attributed to the competition between the first and second
terms of (18) for $\ell=0$ in the case of Fig. 2(a) and becomes more sever in
Fig. 2(b) as the magnetic quantum number $\ell$ gets more involved. This would
in effect explain the energy levels shifts (i.e., to the positive $\Phi$
region for negative $\ell$ and to the negative $\Phi$ region for positive
$\ell$). Hence, energy levels crossings are unavoidable in the process.
Similar trend of behaviour is observed in Fig. 2(c) but with less sever
shifting for $\alpha=3/2>1$ (negative curvature) than that for $\alpha=0.5<1$
(positive curvature) in 2(b).

##  IV A pseudo-confined PDM KG-oscillator in cosmic string spacetime within
KKT

A mixed exponentially growing and power-law type PDM in the form of

$m(r)=a\,\frac{e^{b\,r}}{r^{\sigma}}$ (22)

would yield

$M(r)=-\frac{\sigma^{2}}{16\,r^{2}}-\frac{1}{2}\eta\,b\,r-\frac{(\sigma-2)\,b}{8\,r}-\frac{b^{2}}{16}-\frac{1}{2}\sigma\,\eta.$
(23)

This in effect, would result that equation (12), with $S(r)=0$, now reads

$U^{\prime\prime}\left(r\right)+\left[\mathcal{E}-\frac{\left(\tilde{\beta}^{2}-1/4\right)}{r^{2}}-\tilde{\omega}^{2}\,r^{2}-\eta\,\tilde{a}\,r-\frac{\tilde{b}}{r}\right]\,U\left(r\right)=0,$
(24)

and

$\mathcal{E}=E^{2}-Q^{2}-{k_{z}}^{2}-m^{2}-2\,\omega\,\tilde{\gamma}-\frac{1}{2}(\sigma+4)\,\eta-\frac{b^{2}}{16},$
(25)

where

$\tilde{a}=b/2,\,\,\tilde{b}=b(\sigma-2)/8,\,\,\tilde{\omega}^{2}=\omega^{2}+\eta^{2},\,\,\tilde{\beta}^{2}=\tilde{\gamma}^{2}+\sigma^{2}/16.$
(26)

Obviously, such PDM has introduced its own Cornell-type (i.e.,
$\eta\,\tilde{a}\,r+{\tilde{b}}/{r}$) confinement as a PDM-byproduct. Hence,
the notion of pseudo-confined PDM KG-oscillator is introduced. Moreover, this
differential equation admits a solution in the form of biconfluent Heun
function given by

$U(r)=\mathcal{N}\,r^{|\tilde{\beta}|+1/2}\,exp\left(-\frac{1}{2}\tilde{\omega}\,r^{2}+\frac{\tilde{a}\,\eta\,r}{2\tilde{\omega}}\right)\,H_{B}\left(2|\tilde{\beta}|,\,\frac{\tilde{a}\eta}{\tilde{\omega}^{3/2}},\,\frac{\tilde{a}^{2}\eta^{2}+4\mathcal{E}\tilde{\omega}^{2}}{4\,\tilde{\omega}^{3}},\,\frac{2\,\tilde{b}}{\sqrt{\tilde{\omega}}},\,\sqrt{\tilde{\omega}}\,r\right).$
(27)

To retrieve the two-dimensional oscillator eigenvalues
$\tilde{\lambda}=\tilde{\mathcal{E}}$, one may set $\tilde{a}=\tilde{b}=0$ in
(24) to obtain

$U(r)=\mathcal{N}\,r^{|\tilde{\beta}|+1/2}\,exp\left(-\frac{1}{2}\tilde{\omega}\,r^{2}\right)\,H_{B}\left(2|\tilde{\beta}|,\,0,\,\frac{\tilde{\mathcal{E}}}{\tilde{\omega}},0,\,\sqrt{\tilde{\omega}}\,r\right).$
(28)

In this case, we again use the Kummer relation Ron 1995

$H_{B}\left(2|\tilde{\beta}|,\,0,\frac{\tilde{\mathcal{E}}}{\tilde{\omega}},0,\,\sqrt{\tilde{\omega}}\,r\right)=\,_{1}F_{1}\left(\left[\frac{2\,|\tilde{\beta}|\,\tilde{\omega}-\tilde{\mathcal{E}}+2\,\tilde{\omega}^{2}}{4\,\tilde{\omega}}\right],\,\left[1+|\tilde{\beta}|\right],\,\tilde{\omega}\,r\right)$
(29)

to imply that

$\frac{2\,|\tilde{\beta}|\,\tilde{\omega}-\tilde{\mathcal{E}}+2\,\tilde{\omega}^{2}}{4\,\tilde{\omega}}=\,-n_{r}\Rightarrow\,\tilde{\mathcal{E}}=2\,\tilde{\omega}\left(2\,n_{r}+|\tilde{\beta}|+1\right)$
(30)

This would give us the relation between $\tilde{\alpha}$ and $\gamma$ in the
biconfluent Heun function $H_{B}(\tilde{\alpha},\beta,\gamma,\delta,z)$ as
$\gamma=2(2n_{r}+1)+\tilde{\alpha}$, so that a finite biconfluent Heun
polynomial of degree $2n_{r}\geq 0$ (i.e., of degree $n_{r}\geq 0$ ) is
obtained. Consequently, for our biconfluent Heun polynomial in (27) we get

$\frac{\tilde{a}^{2}\eta^{2}+4\mathcal{E}\tilde{\omega}^{2}}{4\,\tilde{\omega}^{3}}=2(2n_{r}+1)+2|\tilde{\beta}|\Rightarrow\mathcal{E}=2\,\tilde{\omega}\left(2\,n_{r}+|\tilde{\beta}|+1\right)-\frac{\tilde{a}^{2}\eta^{2}}{4\,\tilde{\omega}^{2}}.$
(31)

Which would result, along with (25) and (26), the energy eigenvalues as

$E^{2}=2\,\tilde{\omega}\left(2\,n_{r}+|\tilde{\beta}|+1\right)+k_{z}^{2}+Q^{2}+m^{2}+2\,{\omega}\,\tilde{\gamma}+\frac{b^{2}}{16}+\frac{\sigma+4}{2}\,\eta-\frac{\tilde{a}^{2}\eta^{2}}{4\,\tilde{\omega}^{2}}$
(32)

One should notice that this result collapses into that in (18) and (19) for
$m(r)=1$ for $\sigma=0=b$ and $a=1$ of (22) as a natural tendency of the
solution of the more general problem at hand. Therefore, the parametric
settings adopted above, for our biconfluent Heun polynomial of (27) are
reasonable/consistent, and mathematically and/or quantum mechanically
acceptable settings.

Figure 3:  For $m=k_{z}=Q=\Phi=B_{\circ}=\eta=1$ we show the effect of the PDM
in (22) we plot the energy levels $E_{n_{r},\ell}$ so that in (a)
$E_{n_{r},0}$ with $\alpha=0.5$ and $b=1$ for $n_{r}=0,1,2,3,4$ at different
values of the PDM parameter $\sigma$, (b) $E_{n_{r},0}$ with $\alpha=0.5$ and
$\sigma=1$ for $n_{r}=0,1,2,3,4$ at different values of $b$, and $\ell=0,\pm
1,\pm 2$. For $m=k_{z}=Q=B_{\circ}=\eta=1$, $\Phi=0$ we plot in (c)
$E_{n_{r},\ell}$ with $\sigma=b=1$ for $n_{r}=0,2$ and $\ell=0,\pm 1,\pm 2$ at
different values of $\alpha$.

In Fig.3, we show the energy levels (32) corresponding to the PDM of (22).
Where we observe energy levels clustering as the PDM parameter $|\sigma|$
grows up from zero (for a fixed value of the PDM parameter $b$) in Fig. 3(a),
and as the PDM parameter $|b|$ grows up from zero (for a fixed value of
$\sigma$) in Fig. 3(b). In Fig. 3(c), similar trends of clustering, batching-
up, and energy levels crossings (as in 1(b) and 1(c)) are observed.

##  V A confined PDM KG-oscillator-III in cosmic string spacetime within KKT

Let us now consider the PDM KG-oscillator-III, with

$m(r)=\tilde{A}\,e^{2\,\Omega\,r^{2}}$ (33)

of (14), subjected to a Cornell-type confinement $S(r)=A\,r+B/r$. In this
case, equation (13) yields $M(r)=-(\Omega^{2}\,r^{2}+2\,\Omega)$ and equation
(12) reads

$U^{\prime\prime}\left(r\right)+\left[\tilde{\lambda}_{1}-\frac{\left({\tilde{\gamma}_{1}}^{2}-1/4\right)}{r^{2}}-{\tilde{\omega}_{1}}^{2}\,r^{2}-2\,m\,A\,r-\frac{2\,B}{r}\right]\,U\left(r\right)=0,$
(34)

where $\tilde{\lambda}_{1}=\lambda-2\,(\eta+\Omega)-2AB$,
${\tilde{\omega}_{1}}^{2}=\omega^{2}+(\eta^{2}+\Omega^{2})+A^{2}$, and
${\tilde{\gamma}_{1}}^{2}=\tilde{\gamma}^{2}+B^{2}$. This differential
equation admits a solution in the form of biconfluent Heun function given by

$U(r)=\mathcal{N}\,r^{|{\tilde{\gamma}_{1}}|+1/2}\,exp\left(-\frac{1}{2}{\tilde{\omega}_{1}}\,r^{2}+\frac{m\,A\,r}{{\tilde{\omega}_{1}}}\right)\,H_{B}\left(2|{\tilde{\gamma}_{1}}|,\,\frac{2\,m\,A}{{\tilde{\omega}_{1}}^{3/2}},\,\frac{m^{2}A^{2}+\tilde{\lambda}_{1}\,{\tilde{\omega}_{1}}^{2}}{{\tilde{\omega}_{1}}^{3}},\,\frac{4\,m\,B}{\sqrt{{\tilde{\omega}_{1}}}},\,\sqrt{{\tilde{\omega}_{1}}}\,r\right).$
(35)

In the above section, we have readily asserted that the relation between
$\tilde{\alpha}$ and $\gamma$ in the biconfluent Heun function
$H_{B}(\tilde{\alpha},\beta,\gamma,\delta,z)$ is given by
$\gamma=2(2n_{r}+1)+\tilde{\alpha}$. This would result in a biconfluent Heun
polynomial of degree $2n_{r}\geq 0$ (i.e., of degree $n_{r}\geq 0$ ) to secure
the finiteness of the wave function. Consequently, for our biconfluent Heun
polynomial in (35) we get

$\frac{m^{2}A^{2}+\tilde{\lambda}_{1}\,{\tilde{\omega}_{1}}^{2}}{{\tilde{\omega}_{1}}^{3}}=2(2n_{r}+1)+2|{\tilde{\gamma}_{1}}|\Rightarrow\tilde{\lambda}_{1}=2\,{\tilde{\omega}_{1}}\left(2\,n_{r}+|{\tilde{\gamma}_{1}}|+1\right)-\frac{m^{2}A^{2}}{{\tilde{\omega}_{1}}^{2}}.$
(36)

Which would result

$E^{2}=2\,{\tilde{\omega}_{1}}\left(2\,n_{r}+|{\tilde{\gamma}_{1}}|+1\right)+2\,{\omega}\,\tilde{\gamma}+k_{z}^{2}+Q^{2}+m^{2}+2\,[\eta+\Omega]+2\,A\,B-\frac{m^{2}A^{2}}{{\tilde{\omega}_{1}}^{2}}.$
(37)

Figure 4:  For $m=k_{z}=Q=\Phi=B_{\circ}=\eta=A=B=1$ and $\alpha=0.5$ we show
the effect of the PDM parameter $\Omega$ in (33) on the energy levels
$E_{n_{r},\ell}$. We plot in (a) $E_{n_{r},0}$ states for $\ell=0$ and and
$n_{r}=0,1,2,3,4$ at different values of the PDM parameter $\Omega$, (b)
$E_{n_{r},\ell}$ for $\ell=0,\pm 1,\pm 2$ and $n_{r}=0,3$ at different values
of $\Omega$, and (c) $E_{n_{r},\ell}$ with $\Omega=1$ for $n_{r}=0,2$ and
$\ell=0,\pm 1,\pm 2$ at different values of $\alpha$.

In Figure 4, we show the effect of the PDM parameter $\Omega$ on the energy
levels, which are readily quasi-Landau levels. In Fig. 4(a) we see that the
energy gap increases more rapidly for $\Omega>0$ than that for $\Omega<0$.
This is because of the contribution of the sixth term in (37). In Fig 4(b), we
observe energy levels crossing because of the irrational magnetic quantum
number $\tilde{\gamma}$ involved in the first and second terms of (37), where
the PDM parameter $\Omega$ is only indulged in the first term as $\Omega^{2}$
(consequently, positive or negative $\Omega$ have the same effect). In Fig.
4(c), we again observe the same trend of behaviour as that readily discussed
for Fig. 1(b).

##  VI $\mathcal{PT}$-symmetric PDM KG-particle in cosmic string spacetime
within KKT

The first interest in non-Hermitian Hamiltonians date back to Caliceti et al
Caliceti 1980  which offered the first rigorous explanation why the spectrum
in non-Hermitian model may be real and discrete. This has initiated extensive
discussions (cf., e.g., Buslaev 1993 ) which eventually resulted in the
proposal of $\mathcal{PT}$-symmetric quantum mechanics of Bender and Boettcher
Bender 1998 . Where the key idea lies in the existence of the real spectrum
need not necessarily be attributed to the Hermiticity of the Hamiltonian. The
textbook Hermiticity assumption $H=H^{\dagger}$ is replaced by the mere
$\mathcal{PT}$-symmetry $H=H^{\ddagger}=\mathcal{PT}H\mathcal{PT}$ (for more
details on this issue the reader may refer to Znojil Levai 2000 ; Mustafa
Znojil 2002  and references cited therein). Hereby, $\mathcal{P}$ denotes
space reflection: $\mathcal{PT}x\,\mathcal{PT}\rightarrow-x$, $\mathcal{T}$
mimics the time-reversal: $\mathcal{PT}i\,\mathcal{PT}\rightarrow-i$. Under
such non-Hermitian settings, we use a $\mathcal{PT}$-symmetric PDM so that

$m(r)=a\,e^{4\,i\tilde{A}\,r}.$ (38)

This would, in turn, imply that equation (13), with $\eta=0$ now reads

$M(r)={\tilde{A}}^{2}-\frac{i\,\tilde{A}}{r}.$ (39)

Under such settings and equation (12) yields

Figure 5:  For $m=k_{z}=Q=\Phi=\tilde{A}=1$ we plot $E_{n_{r},\,\ell,\,q}$
against $\alpha$ (a) for $n_{r}=0,1,2,3$, $\ell=0$, and with even quasi-parity
$q=+1$, (b) for $n_{r}=0,1,3,\,\ell=-1,\pm 2,\pm 3$ , and with even quasi-
parity $q=+1$, and (c) for $n_{r}=0,1,2,4,\,\ell=1$, and $q=\pm 1$.

$U^{\prime\prime}\left(r\right)+\left[\tilde{\mathcal{E}}-\frac{\left(\tilde{\gamma}^{2}-1/4\right)}{r^{2}}-\frac{i\,\tilde{A}}{r}\right]\,U\left(r\right)=0,$
(40)

where, $\tilde{\mathcal{E}}=E^{2}-Q^{2}-k_{z}^{2}-m^{2}+{\tilde{A}}^{2}$,
$\tilde{\gamma}^{2}=\left(\frac{\ell}{\alpha}+\frac{\Phi
Q}{2\,\alpha\,\pi}\right)^{2}$ and our PDM KG-Coulombic particle is only under
Aharonov-Bohm flux field and no magnetic field is involved in this process
(i.,e., $B_{\circ}=0$). This Schrödinger-like Hamiltonian (i.e.,
$\hat{H}U(r)=\tilde{\mathcal{E}}U(r)$) is $\mathcal{PT}$\- symmetric by
structure and admits real eigenvalues $\tilde{\mathcal{E}}\in\mathbb{R}$. In
fact, this equation (40) represents the so called $\mathcal{PT}$-symmetric
Schrödinger-Coulomb problem. It has been thoroughly discussed by Znojil and
Lévai Znojil Levai 2000  and later on by Mustafa and Znojil Mustafa Znojil
2002  using the so called $\mathcal{PT}$-symmetric pseudo-perturbation
shifted-$\ell$ expansion quasi-perturbation recipe. Where, it has been shown,
in Znojil Levai 2000 ; Mustafa Znojil 2002 , that if the central-like
repulsive/attractive core, in Mustafa Znojil 2002 , is replaced through the
transformation $\ell_{d}(\ell_{d}+1)\rightarrow\tilde{\gamma}^{2}-1/4$ , i.e.
$\ell_{d}=q\,|\tilde{\gamma}|-1/2$, with $q=\pm 1$ denoting quasi-parity, then
the reported exact eigenvalues $\tilde{\mathcal{E}}$ would read

$\tilde{\mathcal{E}}=\frac{{\tilde{A}}^{2}}{4\,(n_{r}-\ell_{d})^{2}}\Rightarrow\tilde{\mathcal{E}}=\frac{{\tilde{A}}^{2}}{(2\,n_{r}-2\,q\,|\tilde{\gamma}|+1)^{2}}.$
(41)

At this point, one should notice that the degeneracy associated with the
regular Hermitian Coulomb eigenvalues
$\tilde{\mathcal{E}}=-{\tilde{A}}^{2}/(2\,n^{2});\,n=n_{r}+|\ell|+1/2=1,\,2,\cdots$,
is now lifted upon the complexification of, say, the dielectric constant
embedded in $\tilde{A}$. Hence, the energies for our $\mathcal{PT}$-symmetric
KG-Coulombic particle in cosmic string spacetime within KKT are given by

$E^{2}=\frac{{\tilde{A}}^{2}}{(2\,n_{r}-2\,q\,|\tilde{\gamma}|+1)^{2}}-{\tilde{A}}^{2}+Q^{2}+k_{z}^{2}+m^{2},$
(42)

and the radial wave functions Znojil Levai 2000  are given by

$U\left(r\right)\sim
r^{\frac{1}{2}-q\,\left|\tilde{\gamma}\right|}\exp\left(i\,{\tilde{A}}^{2}\,r\right)L_{n_{r}}^{-2\,q\,\left|\tilde{\gamma}\right|}\left(-2\,i\,{\tilde{A}}^{2}\,r\right)\Longrightarrow
R\left(r\right)\sim
r^{-q\,\left|\tilde{\gamma}\right|}\exp\left(i\,{\tilde{A}}^{2}\,r\right)L_{n_{r}}^{-2\,q\,\left|\tilde{\gamma}\right|}\left(-2\,i\,{\tilde{A}}^{2}\,r\right),$
(43)

Moreover, the energies in (42) suggest the phenomenon of flown away states at
$n_{r}=q\,|\tilde{\gamma}|-1/2\geq 0$ (i.e., for each $n_{r}$-state there is
an $\ell_{d}=q\,|\tilde{\gamma}|-1/2$ state to fly away). Under such
condition, only states with even quasi-parity, $q=+1$, are destined to fly
away at some specific values of the curvature parameter $\alpha$. For more
details on the issue of flown-away states and thorough discussion on the
$\mathcal{PT}$-symmetric Coulomb problem the reader is advised to refer to
Znojil Levai 2000 ; Mustafa Znojil 2002  and related references cited therein.

In Figure 5, we show the energy levels $E_{n_{r},\,\ell,\,q}$ against
$\alpha$. For states with even quasi-parity, we observe energy levels
crossings (in 5(a) for different $n_{r}$ and $\ell=0$, and in 5(b) for
different $n_{r}$ and different $\ell$. Whereas, in Fig.5(c) we observe energy
levels crossings between states with even and odd quasi-parities. That is, a
state with odd quasi-parity $n_{r}^{\prime}$ crosses with a state with even
parity $n_{r}$ when the relation
$\alpha\,(n_{r}-n_{r}^{\prime})=2\,|\ell+\frac{\Phi\,Q}{2\,\pi}|$ is
satisfied.

##  VII Concluding remarks

In the current methodical proposal, we have introduced the effective PDM
setting for the KG-oscillators (in particular and KG-particles in general) in
cosmic string spacetime in magnetic and Aharonov-Bohm flux fields within the
Kaluza-Klein theory. The effective PDM is introduced as a defect/deformation
in the momentum operator analogous to effective PDM in Schrödinger theory. We
have reported, in section 2, two new PDM KG-oscillators among four feasible
KG-oscillators in the cosmic string spacetime within KKT. Therein, we have
shown that the KG-oscillator-I by Mirza’s et al Mirza 2004  turns out to be
nothings but a byproduct of the PDM setting in KG-oscillator-III of (14).
Effectively, the topological defect in the momentum operator suggested by
Mirza et al. may very well be explained as an effective topological defect
introduced by effective PDM settings. Moreover, we have observed that square
the energies in (18) (i.e., $E^{2}$) resembles the Landau-type energies-
squared with an irrational magnetic quantum number $\tilde{\gamma}$ that
indulges within the Aharonov-Bohm flux field effect. We may argue, therefore,
that the additional compact dimension of the Kaluza-Klein theory offers quasi-
Landau energy levels (e.g., Geusa 2001 ; Bezerra 2019 ; Zeinab 2020 ). Hereby,
the behaviour of the energy levels $E_{n_{r},\ell}$ is found to follow two
different trends of clustering and batching up for the curvature parameter
$\alpha\approx 0$ and $\alpha\gg 1$. This would necessarily suggest that
energy levels crossings are unavoidable in the process (documented in Fig.s
1(b), 1(c), 2(b), 2(c), 3(c), 4(b), and 4(c)). Whereas, at $\alpha=1$ (i.e.,
Minkowski flat spacetime) the energy levels maintain their regular quasi-
Landau-type structure.

Next, we have discussed, in section 4, a mixed power-law and exponential type
PDM model that resulted in a pseudo-confined PDM KG-oscillator in cosmic
string spacetime within KKT. The effective confining potential is introduced
as a byproduct of the PDM settings of the KG-oscillator. In this case, the PDM
KG-oscillators are shown to be confined in their own PDM manifested effective
interaction potential force field, hence the notion ”pseudo-confined PDM KG-
oscillators” is unavoidably inviting. We have considered, in section 5, a
confined PDM KG-oscillator-III with the PDM (33) which could basically replace
Mirza et al’s recipe Mirza 2004  with $\Omega=\eta$ in (34). Finally, we have
reported on a non-Hermitian $\mathcal{PT}$-symmetric PDM-Coulombic KG-particle
model in cosmic string spacetime within KKT. Some unusual quantum mechanical
properties such as flown away states are observed. The detailed discussions on
this issue may be found in (cf., e.g., Bender 1998 ; Znojil Levai 2000 ;
Mustafa Znojil 2002 ). The current methodical proposal should, therefore, be
considered as a new generalization where its applicability covers not only PDM
KG-oscillators (documented in sections 3, 4, and 5) but also PDM KG-particles
(documented in section 6) in cosmic string spacetime or in the geometric
theory of topological defects in condensed matter. To the best of our
knowledge, the current methodical proposal did not appear elsewhere.

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# Improving Classification Neural Networks by using Absolute activation
function (MNIST/LeNET-5 example)

Oleg I.Berngardt

###### Abstract

The paper discusses the use of the Absolute activation function in
classification neural networks. An examples are shown of using this activation
function in simple and more complex problems. Using as a baseline LeNet-5
network for solving the MNIST problem, the efficiency of Absolute activation
function is shown in comparison with the use of Tanh, ReLU and SeLU
activations. It is shown that in deep networks Absolute activation does not
cause vanishing and exploding gradients, and therefore Absolute activation can
be used in both simple and deep neural networks. Due to high volatility of
training networks with Absolute activation, a special modification of ADAM
training algorithm is used, that estimates lower bound of accuracy at any test
dataset using validation dataset analysis at each training epoch, and uses
this value to stop/decrease learning rate, and reinitializes ADAM algorithm
between these steps. It is shown that solving the MNIST problem with the
LeNet-like architectures based on Absolute activation allows to significantly
reduce the number of trained parameters in the neural network with improving
the prediction accuracy.

##  1 Introduction and basic idea

Many tasks have recently been effectively solved by neural networks. Deep
neural networks are the most actively studied, the intensive progress in their
development is associated with successful solution of the problem of vanishing
and exploding gradients in deep networks.

In practical use, the problem often arises of constructing from already
trained deep network a smaller network allowing one to successfully solve the
problem with similar quility[Hinton et al.(2015)]. The main theoretical basis
for the possibility of creating small networks are the theorems of Kolmogorov-
Arnold [Kolmogorov(1957), Arnold(1963)], Cybenko [Cybenko(1989)], Funahashi
[Funahashi(1989)], and Hornik [Hornik et al.(1989), Hornik(1991)] (also known
as the universal approximation theorem) .

In accordance with the Kolmogorov-Arnold theorem, the approximation of an
unknown function of N variables can be represented as:

$f(\overrightarrow{x})=\sum_{q=0}^{2N}\Phi_{q}\left(\sum_{p=1}^{N}\phi_{q,p}\left(x_{p}\right)\right)$
(1)

The theorem is not constructive, and proves the existence of optimal functions
of one argument $\Phi_{q}\left(x\right),\phi_{q,p}\left(x\right)$ that provide
such an approximation, but does not specify their specific shape.

The theorems [Cybenko(1989), Funahashi(1989), Hornik et al.(1989),
Hornik(1991), Sonoda and Murata(2017)] specify the types of these functions
and prove the possibility of using various functions
$\Phi_{q}\left(x\right),\phi_{q,p}\left(x\right)$. Thus, theoretically, most
of the practical problems should be solved by a neural network with one hidden
layer by choosing correct functions
$\Phi_{q}\left(x\right),\phi_{q,p}\left(x\right)$.

Traditionally, however, for large input vectors this hidden layer size and
number of unknown functions becomes as huge as $O(N^{2})$, where N is
dimension of input vector. Therefore the solution of many problems is reduced
to the construction of neural networks with a larger number of hidden layers,
and smaller number of neurons in each layer (’thin networks’), that
efficiently decrease number of unknown (trained) parameters. However, there
are still problems in which the use of a large number of hidden layers is not
required, and the problem can be solved with a simpler network.

The simplest form of the function $\phi_{q,p}$ is linear:

$\phi_{q,p}(x)=A_{p,q}x+B_{p,q}$ (2)

and the $\Phi_{q}$ function is the neuron activation function. This allows one
to build neural networks with one hidden layer to solve some problems by
choosing corresponding activation function and fitting coefficients
$A_{p,q},B_{p,q}$.

A difficult task is to choose the function $\Phi_{q}$ \- the activation
function of the hidden layer. The theorem [Hornik(1991)] proves that as this
function one can use a sufficiently arbitrary function - continuous and
bounded. However even non-bounded functions can be used, for example widely
used ReLU [Agarap(2018)], that satisfy less strong conditions [Sonoda and
Murata(2017)] and also could be used in universal approximators(networks).

The only problem is to find the coefficients $A_{p,q},B_{p,q}$. In modern time
this problem is solved by gradient descent methods. Modern experience in
building deep neural networks shows that in order to avoid vanishing and
exploding gradients [Bengio et al.(1994)] when searching for optimal network
coefficients using the gradient descent method, only some activation functions
are beneficial. This greatly limits the activation functions suitable for
building deep neural networks. For optimal activation functions, the
derivative of the function should be as close as possible to 1. This is one of
the reasons for replacing Sigmoid (logistic function) with Tanh (hyperbolic
tangent) in the LeNet[Lecun et al.(1998)] network, replacing Tanh with ReLU
(Rectified Linear Unit) in the AlexNet[Krizhevsky et al.(2017)] network, and
the emergence of Residual blocks and ResNet[He et al.(2015)] networks, that
are widely used today.

Therefore, the requirement that the derivative of the activation function be
close to 1 is a very important requirement when choosing activation functions
for deep neural networks. An ideal function that satisfies this condition is
linear. This function is not useful, since it will not allow effective
building deep networks - the superposition of layers with linear activation
functions is equivalent to a single layer. Thus, we face with a contradiction,
which is usually resolved by using the ReLU and similar right-hand-linear
functions (SeLU, SWISH, MISH, APTx, etc.) or Residual blocks.

However, this contradiction is apparent: in order for the gradient to do not
vanish or explode, a slightly different condition should be met. Namely, not
the derivative, but the modulus of the derivative of the activation function
must be equal to 1. This property is possessed not only by a linear function,
but also by many other real-valued functions whose derivative modulus are
equal to 1 almost everywhere. We can classify these functions by the number of
points where their differential is not defined. The simplest function with
none of these points is linear, and the function with single point is absolute
value (module). So we can use absolute value function as an activation
function (Absolute activation function, Abs), and $\Phi_{q}$ becomes:

$\Phi_{q}(y)=W_{q}|y|$ (3)

It should be noted that most of the popular activation functions (Sigmoid,
Tanh, ReLU, SeLU, LeakyReLU, etc.) are monotonically nondecreasing. Absolute
activation function Abs differs significantly from them - it is nonmonotone,
but continuous, like SWISH, MISH and APTx functions[Kumar(2022)]. The Absolute
activation function already mentioned by researchers [Karnewar(2018), Apicella
et al.(2021)], but its efficiency has not been demonstrated in details. The
Absolute activation function satisfy conditions of [Sonoda and Murata(2017)]
and therefore could be used in universal approximators (neural networks).

A neural network with one hidden layer with an N-dimensional input vector and
Absolute activation will implement the operation:

$f(\overrightarrow{x})=\sum_{q=0}^{2N}W_{q}\left|\sum_{p=1}^{N}\left(A_{p,q}x_{p}+B_{p,q}\right)\right|$
(4)

It can be seen from the eq.(4) that the network is implemented by two layers
of neurons - the hidden fully connected layer with an Absolute activation
function (Abs, $\varphi(y)=|y|$) and with the number of neurons 2N+1, where N
is the dimension of the input vector. The decision layer is standard for the
given problem: with a linear activation function in the case of solving the
approximation problem, as in (4) or with a Softmax function in the case of
solving the classification problem:

$f_{c}(\overrightarrow{x})=Softmax_{c}\left(\sum_{q=0}^{2N}W_{q,c}\left|\sum_{p=1}^{N}\left(A_{p,q}x_{p}+B_{p,q}\right)\right|\right)$
(5)

where c is the class number and $W_{q}$ are the coefficients of the decision
layer.

##  2 Simplest 2D classification problems

Let us demonstrate the possibility of using the Absolute activation function
for effective solving simple classification problems. Let us compare three
networks: a fully-connected network with single hidden layer with ReLU
activations, a fully-connected network with two hidden layers with ReLU
activations, and a fully-connected network with single hidden layer with Abs
activations. We will use two-dimensional input vectors (N=2). The three
synthetic problems of two-class classification were solved: a) linear
separation, b) “cross” type separation, c) circular area separation. These
tasks are shown in Fig.1.

In accordance with (4) and the Kolmogorov-Arnold theorem (1), we choose the
number of neurons in each layer equal to 2N+1=5. The number of training epochs
is 1000, loss function is cross-entropy, the optimum search method is ADAM
[Kingma and Ba(2014)] with learning rate $10^{-3}$. The first network has two
hidden layers and 57 unknown parameters, the second and third networks have
one hidden layer and 27 unknown parameters.

The architectures of the studied networks and training processes are shown in
Fig.2. Fig.2 also shows the results of training the networks on these
synthetic test datasets (A-C).

Figure 1: Synthetic datasets for the simplest classification: A) linear
separation; B) “cross” separation; C) circular area Figure 2: The architecture
of the studied simple networks: a) two hidden layers with ReLU activations, b)
single hidden layer with ReLU activations c) single hidden layer with Abs
activations. Results of training three networks on datasets (A-C, as shown in
Fig.1). For each dataset, the dependence of the loss function on the training
epoch on the training and validation datasets is shown.

It should be noted that the solution of datasets (B-C) by a network with
single hidden ReLU layer looks unstable. In the most cases it predicts
accurate classification, but in some cases - an inaccurate one (see Fig.2B for
datasets B and C). Experiments shows, that for dataset B inaccurate training
(final accuracy is less than 0.8) is about 9% of all cases, for dataset C
inaccurate training is about 16% of all cases. For two hidden layer ReLU
network inaccurate training probability is about 3% of B-C cases (possibly due
to small number of training epochs), for single hidden layer Abs network there
are no inaccurate runs.

This demonstrates a dependence of the single fully-connected hidden layer ReLU
network on the initial conditions (initial coefficients values), and weaker
dependence of the single fully-connected hidden layer Abs network results on
initial conditions.

Fig.2 shows that the neural network with single hidden layer and Abs
activations is no less efficient than the network with single ReLU hidden
layer, but less efficient than the network with two ReLU hidden layers. At the
same time, it has the same number of free parameters as the network with one
hidden ReLU layer and is less dependent on initial conditions during training.
So from the first view Abs activation function can be effectively used in
classification networks and may improve their efficiency. Let us study the use
of Absolute activation in more complex classification problems.

##  3 LeNet-kind solution of MNIST problem

###  3.1 Preliminary analysis

Let us consider the solution of the MNIST problem - the task of classifying
handwritten digital characters into 10 classes[Simard et al.(2003)].

When solving the MNIST problem, the LeNet-5[Lecun et al.(1998)] network was
frequently used as a baseline solution, which is quite compact and fast. To
demonstrate the effectiveness of Absolute activation, the network was analyzed
in two versions - in its standard variant (with Tanh activation functions,
LeNet-5) and its modernization, obtained from LeNet-5 by replacing all the
activation functions by Abs (referred as LeNet+Abs).

Both networks have 368,426 free parameters, network architectures are similar
and shown in Fig.3. Significant differences are highlighted in bold. There are
also more efficient networks exist for solving the MNIST problem, for example
listed in [Apicella et al.(2021)], but usualy they contain a larger number of
free parameters (for example as much as 2,888,000, as in [Simard et
al.(2003)]). Therefore, to demonstrate the effectiveness of the Absolute
activation function, we will use a relatively simple LeNet-5 as baseline
solution.

Figure 3: Tested network architectures: A) standard LeNet-5; B) LeNet+Abs
(with Absolute activation functions). The results of training the two
networks, from top to bottom: loss on the test and validation datasets,
accuracy (accuracy) on the training and validation datasets.

Fig.3 shows the dependencies of the loss function and accuracy of both
networks after 100 training epochs on identical MNIST training and validation
datasets (80% training data followed by 20% of validation data). Optimization
algorithm is ADAM with batchsize 128 and learning rate $10^{-3}$.

It can be seen from Fig.3 that the resulting accuracy of the LeNet+Abs network
looks higher than the accuracy of the standard LeNet-5, and with an accurate
training stop, it should provide the better accuracy and the smaller value of
the loss function on the validation dataset, than standard LeNet-5 does. Thus,
using the Abs activation function in this problem at first view is more
profitable than the use of Tanh activations due to the greater accuracy
achieved.

###  3.2 Training process details

It can be seen from the Fig.3 that the dependence of the LeNet+Abs accuracy
and loss function on the training epoch is not smooth, but much more volatile
than those for LeNet-5. There seems to be two reasons for this. First, when
training the network, by default, the epoch accuracy is calculated over the
last batch, i.e. not over the entire training dataset. The second is that the
derivative of the Absolute activation function, indicating the direction of
the gradient, is discontinuous at zero argument, so small changes in the
argument can lead to sharp changes in the gradient, so the ADAM algorithm does
not provide a smooth enough descent, and one should use lower learning rates,
slowing down training process.

To find the optimally trained model we should find the epoch with maximal
accuracy at unknown test dataset. We do not use the test dataset at training,
so we may only estimate this accuracy. To prevent overtraining, we should not
use train dataset information, and only validation dataset could be used.

Optimally trained network should provide good accuracy exceeding a given limit
at almost any test dataset. After the network trained this minimal accuracy
bound should be as high as possible.

To estimate the lower bound of accuracy at any test dataset we should know the
distribution of network accuracy at different validation datasets. We have
only single validation dataset, so the lower bound could be estimated by
bootstrap algorithm [Efron(1979)]. But for using bootstrap we should
accurately choose significance level, that defines how low the lower bound
over test dataset should be below the mean validation accuracy. As shows
experiments, more accurate results can be obtained when we just choose minimal
accuracy over two halves of validation dataset:

$ACC_{expected@test}=\min(ACC(ValidationData1),ACC(ValidationData2))$ (6)

Another training problem is choosing a right learning rate for training
process to be fast and accurate. We reach this by by step-by-step decrease
learning rate with reinitializaing ADAM between steps. The ADAM algorithm is
known to be very affective, but have hidden parameters, optimized during
training. Reinitialization resets these hidden parameters to their initial
values. Training process at given learning rate stops if expected accuracy
lower bound has not increased for 10 epoches since epoch of previous increase.
During training we use learning rates stepping from $10^{-3}$ downto $10^{-6}$
with 10 times decrease between steps. This approach is close to well known
ReduceLROnPlateau algorithm with patience 10, monitoring of
$ACC_{expected@test}$ and reinitializing ADAM before reduction.

The training algorithm is shown as Algorithm 21

Algorithm 1 Training procedure

procedure CallAfterEachTrainingEpoch($Model,EpochNumber,ValidationDataset$)

$ExpectedAccuracy\leftarrow ACC_{expected@test}(ValidationDataset)$

if $ExpectedAccuracy>BestAccuracy$ then

$BestAccuracy\leftarrow ExpectedAccuracy$

$BestEpoch\leftarrow EpochNumber$

SaveModel(Model)

end if

if $EpochNumber>BestEpoch+10$ then

StopTrainingModel()

end if

end procedure

for $LR\in[10^{-3}..10^{-6},step:\times 0.1]$ do

$ADAM\leftarrow InitializeADAM(LearningRate=LR)$

if $LR<10^{-3}$ then

$Model\leftarrow RestoreModelFromSaved()$

else

$Model\leftarrow InitModel()$

end if

$BestEpoch\leftarrow 0,BestAccuracy\leftarrow 0$

$TrainModelUntilStop(Model,ADAM,TrainDataset,ValidationDataset)$

end for

The Table 1 shows the accuracy achieved and total epoch number during training
of the standard LeNet-5 model and the LeNet+Abs model (both marked as ”Not
degraded” in the Table 1). The table shows that the accuracy of LeNet+Abs
exceeds LeNet-5 accuracy, and the number of errors on the test dataset has
fallen from 1.14% (for LeNet-5) to 0.56% (for LeNet+Abs) - i.e. almost 2
times. Also the found error level 0.56% for LeNet+Abs looks better than the
lowest error level 0.7% reported in [Lecun et al.(1998)] even for boosted
variants of LeNet. One can also see that training of LeNet+Abs slower than
training LeNet-5 for about 1.5 times (100 training epochs vs. 68
correspondingly).

###  3.3 Gradient vanishing robustness

It is known that an important problem in training deep and recurrent networks
is gradient vanishing and explosion - the loss of the network’s ability to
train layers with an increase of their number [Bengio et al.(1994)]. Recently,
Residual blocks[He et al.(2015)] have been found to solve this problem. Let us
demonstrate that the Absolute activation function is resistant to the gradient
vanishing and explosion effect, not worse than popular ReLU and SeLU
functions.

To show this, inside each of the networks (LeNet-5 and LeNet+Abs) 20
intermediate (disturbing) layers were placed (as the last hidden layers),
complicating training the network by vanishing gradient. To LeNet-5 and
LeNet+Abs architectures a 20 layers with Tanh, ReLU, SeSU, Abs activations
were added, producing 8 final architures: Lenet-5+20DTanh, Lenet-5+20DReLU,
Lenet-5+20DSeLU, Lenet-5+20DAbs, Lenet+Abs+20DTanh, Lenet+Abs+20DReLU,
Lenet+Abs+20DSeLU, Lenet+Abs+20DAbs. Examples of these architectures are shown
in Fig.4.

All 8 resulting (deep) networks have 511226 free coefficients, and about 143
thousand of them are from the last (disturbing) layers, and these networks
differ only by activation functions in layers.

The training results are shown in Table 1.

Figure 4: Degraded versions of LeNet-5 and LeNet+Abs networks to test the
effect of gradient vanisning and explosion effect: LeNet-5+20DTanh (A),
LeNet+Abs+20DAbs (B) , LeNet-5+20Drelu (C), LeNet+Abs+20Drelu (D).
Architecture and quality of training.

It can be seen from the Table 1 that adding 20 more hidden layers with Abs
activation at the end makes it still possible to train LeNet+Abs+20DAbs,
although with somewhat lower accuracy than in the absence of these layers -
LeNet+Abs network (99.05% vs. 99.44% correspondingly). This confirms with our
theoretical expectation that the Absolute activation function does not cause
significant gradient vanishing or explosion, and therefore can be effectively
used in both simple and deep neural networks. It can be seen from Table 1 that
the gradient vanishing/explosion it causes is better than that of ReLU, and
that Abs as good as SeLU activation. This allows to use Abs in deep networks.

Table 1: Accuracy (%) of various networks - standard LeNet-5 and LeNet+Abs, degraded by the last 20 hidden fully connected layers with different activation functions. *) marks the cases when learning rate $10^{-3}$ is not enought to start training and we start from $10^{-4}$ instead. Bootstrap confidential interval over test dataset at standard confidence level 0.95 is also shown. Network | Accuracy | Trained Epoches | Conf.interval
---|---|---|---
LeNet-5 architecture
Not degraded | 98.86 | 68 | (98.65, 99.07)
+20DTanh | 98.41 | 80 | (98.17,98.66)
+20DReLU | 98.32 | 129 | (98.07,98.57)
+20DSeLU | 98.75 | 80 | (98.53,98.97)
+20DAbs(*) | 98.71 | 66 | (98.49,98.93)
LeNet+Abs architecture
Not degraded | 99.44 | 100 | (99.30,99.59)
+20DTanh(*) | 97.51 | 80 | (96.90,97.54)
+20DReLU | 98.92 | 118 | (98.72,99.12)
+20DSeLU | 99.03 | 98 | (98.84,99.22)
+20DAbs | 99.05 | 112 | (98.86,99.24)

###  3.4 Reducing the LeNet-5 network size without loss of accuracy

The good stability of the Absolute activation function against gradient
vanishing suggests that the original LeNet-5 neural network could be too
complex for its reported accuracy. Let us reduce its size without loosing the
accuracy. The most obvious way is to remove the last hidden convolutional
layer from LeNet+Abs. This resulting network referred as SmallLeNet+Abs
network and shown in Fig.5A. The TinyLeNet+Abs model is made from the
SmallLeNet+Abs model by reducing the number of convolutions in the second
convolution layer from 16 to 3. This new architecture is also shown in Fig.5B.
Bold marks important changes.

Figure 5: Smaller models A)SmallLeNet+Abs; B)TinyLeNet+Abs;
C)LeNet+Conv120+Abs; D)LeNet+Conv64+Conv120+Abs. Bold highlights the main
difference from the previous model.

The Table 2 shows the accuracies achieved by these networks using training
Algorithm 21. In addition, the number of trained model coefficients, total
trained epoches, and bootstrap confidence interval on the test dataset are
shown (with a standard significance level of 0.95).

The Table 2 shows that the best accuracy on the test dataset is provided by
the LeNet+Abs model. The SmallLeNet+Abs model with 52 thousand trained
parameters, and TinyLeNet+Abs model with 10.6 thousand trained parameters also
exceeds the initial LeNet-5 (with 368 thousand parameters) accuracy in most
cases.

More accurate network is extended LeNet architecture - Lenet+Conv120+Abs,
created by adding Conv120 and AveragePooling layers to LeNet (it has 76226
parameters and is shown in Fig.5C), provides the even better accuracy than
LeNet+Abs architecture - 99.51%, but having 4.8 times less parameters.

Thus, the replacement of Tanh activation functions with Abs makes it possible
to reduce the size of original LeNet-5 model by more than 4.8 and 7 times with
an improvement of its accuracy (LeNet+Conv120+Abs and SmallLeNet+Abs variants
correspondingly), and providing better accuracy than ReLU and SeLU variants.
It can be seen from the bootstrap confidence intervals that even TinyLeNet+Abs
model with the number of parameters 35 times less than the number of LeNet-5
parameters provides an accuracy that not worse than the accuracy of original
LeNet-5 with Tanh, ReLU, or SeLU activations.

Table 2: The best accuracy obtained on test dataset for different architectures with different activation functions. The number of trained model parameters and the bootstrap confidence interval (standard confidence level 0.95) over the test dataset. Bold marks best activation function for given architecture. $ACC_{expected@test}$ according to eq.6.  | Accuracy | Trained epoches |  Conf.interval |  # coefs
---|---|---|---|---
LeNet architecture |  368426
+Tanh (LeNet-5) | 98.86 | 68 |  (98.65, 99.07) |  368426
\+ ReLU | 99.13 | 72 |  (98.95,99.31) |  -
\+ SeLU | 99.03 | 62 |  (98.84,99.23) |  -
\+ Abs | 99.44 | 100 |  (99.30,99.58) |  -
SmallLeNet architecture |  51890
\+ Tanh | 99.03 | 64 |  (98.84, 99.22) |  51890
\+ Relu | 99.10 | 88 |  (98.91, 99.28) |  -
\+ Selu | 99.10 | 59 |  (98.92, 99.28) |  -
\+ Abs | 99.28 | 116 |  (99.12, 99.44) |  -
TinyLeNet architecture |  10615
\+ Tanh | 98.76 | 75 |  (98.54, 98.97) |  10615
\+ Relu | 98.94 | 71 |  (98.74, 99.14) |  -
\+ Selu | 98.65 | 74 |  (98.42, 98.88) |  -
\+ Abs | 99.08 | 87 |  (98.90, 99.27) |  -
LeNet+Conv120 architecture |  76226
\+ Tanh | 99.09 | 72 |  (98.91, 99.27) |  -
\+ Relu | 99.28 | 80 |  (99.12, 99.45) |  -
\+ Selu | 99.26 | 65 |  (99.09, 99.43) |  -
\+ Abs | 99.51 | 103 |  (99.38, 99.65) |  -

As we have shown, even standard LeNet-5 with Abs activation (LeNet+Abs) can
provide very good accuracy (99.44%) comparable with top solutions for fixed
shape activation functions [Apicella et al.(2021)]. Extended LeNet versions
(Lenet+Conv120+Abs) have accuracy up to (99.51%), very close to top solutions
for fixed shape activation functions [Apicella et al.(2021)], and having less
than 77,000 trained parameters.

##  4 More efficient accuracy predictions for training

As we have shown above, the networks with Abs activation looks better than the
networks with ReLU, SeLU and Tanh activations. But there are two problems
arise.

Problem 1. The stabuility of training the network. We use random initial
network parameters, no regularization, so the found loss function minimum is
not global one. We need to check how the initializing and training affect the
trained network accuracy.

Problem 2. The stop condition. As a stop condition we use prediction of
network accuracy at unknown test dataset over validation dataset. It is
obvious that different prediction algorithms lead to different network
accuracy.

To study these problems we created 3 ensembles for LeNet+Conv120 network
architecture (LeNet+Conv120+Abs, LeNet+Conv120+ReLU, LeNet+Conv120+SeLU,
LeNet+Conv120+Tanh), that differs only by activation function (Abs, ReLU,
SeLU, Tanh). Each of 3 ensembles is trained by using different model for
accuracy at test dataset prediction over validation dataset (
$ACC_{expected@test}$ in Algorithm 21 ):

$ACC_{expected@test,1}=\min(ACC(ValidationData1),ACC(ValidationData2))$ (7)

$ACC_{expected@test,2}=MEAN_{bootstrap}(ValidationData)-STD_{bootstrap}(ValidationData)$
(8)

$ACC_{expected@test,3}=ACC_{expected@test,1}-STD_{bootstrap}(ValidationData)$
(9)

The first prediction method is discussed earlier with Algorithm 21, the second
one is based on bootstrap estimate of lower bound of prediction interval over
validation dataset with significance level 0.68, the last one is combination
of the first and the second, lowering the expected bound.

Each ensemble of trained networks were used to produce 2 values, shown in
Table 3. The first one is the accuracy over the test dataset by majority
ensemble voting, when the predicted label is found as the label predicted by
maximal number of networks (marked as M.V. - majority voting). Another measure
is prediction interval over the test dataset over the ensemble of trained
networks (marked as C.I. - confidence interval).

For different purposes different M.V. and C.I. are required. For example, for
majority ensemble voting the networks should be overtrained and M.V. should be
as high as possible. As shown in Table 3, for Abs network the best M.V. is
provided by use $ACC_{expected@test,1}$ or $ACC_{expected@test,3}$ accuracy
prediction variant during train.

When only single network is planned for prediction, we need the highest
confidential interval. In this case using $ACC_{expected@test,3}$ prediction
variant is preferable. For both cases using $ACC_{expected@test,3}$ accuracy
prediction for training the LeNet+Conv120+Abs is preferable.

To improve the result, a LeNet+Conv64+Conv120+Abs were made - it is wider and
deeper LeNet-like network, having more convolutions at top layer, than
LeNet+Conv120+Abs has. Its architecture is shown in Fig.5D.

The model has 227474 parameters, that is also smaller than number of
parameters in original LeNet-5 network, and provides better accuracy than
LeNet+Conv120+Abs. It’s ensemble training using $ACC_{expected@test,3}$ test
dataset accuracy prediction shows its increased accuracy, also exceding its
ReLU, SeLU and Tanh varaints (shown in Table 3).

Table 3:  Accuracy results for ensemble of from 20 training runs, produced 20 trained variants of LeNet+Conv120 and LeNet+Conv64+Conv120 networks, with different algorithms for predicting accuracy at test dataset from the validation dataset during training, eqs.7-9. M.V. rows - majority voting ensemble accuracy over 20 networks at test dataset, C.I. rows - accuracy limits over the ensemble at test dataset. Best results marked by bold. | $ACC_{expected@test,1}$ | $ACC_{expected@test,2}$ | $ACC_{expected@test,3}$
---|---|---|---
Lenet+Conv120 architecture
+Tanh C.I. | [98.95,99.20] | [98.93,99.21] | [99.03,99.22]
+Tanh M.V. | 99.33 | 99.32 | 99.36
+ReLU C.I. | [99.12,99.38] | [99.16,99.37] | [99.13,99.37]
+ReLU M.V. | 99.52 | 99.52 | 99.48
+SeLU C.I. | [99.19,99.36] | [99.17,99.39] | [99.20,99.35]
+SeLU M.V. | 99.54 | 99.55 | 99.54
+Abs C.I. | [99.39,99.56] | [99.35,99.57] | [99.41,99.59]
+Abs M.V. | 99.64 | 99.61 | 99.64
Lenet+Conv64+Conv120 architecture
+Tanh C.I. |  |  | [98.87,99.15]
+Tanh M.V. |  |  | 99.37
+ReLU C.I. |  |  | [99.24,99.50]
+ReLU M.V. |  |  | 99.55
+SeLU C.I. |  |  | [99.29,99.44]
+SeLU M.V. |  |  | 99.56
+Abs C.I. |  |  | [99.45,99.60]
+Abs M.V. |  |  | 99.64

##  5 Conclusion

In the paper we discuss the Absolute activation function for classification
neural networks. An examples of using this activation function in simple and
complex classifying problems are presented. In solving the MNIST problem with
the LeNet-5 network, the efficiency of Abs is shown in comparison with Tanh,
ReLU and SeLU activations. It allows to reach 99.44% accuracy at standard
LeNet-5 network by only changing all activations into Abs. It is shown that
its use practically does not lead to gradient vanishing/explosion, and
therefore Absolute activation can be used in both small and deep neural
networks. It is shown that in solving the MNIST problem with the LeNet-5
architecture, the use of Absolute activation helps to significantly reduce the
size of the neural network (up to 2 orders by number of trained parameters)
and improve the accuracy of the solution. The mean reached accuracy could be
about 99.51% ([99.41%..99.59%] at different trained varaints) at train dataset
with decreasing the network size from 368 thousands to 76 thousands
coeficients(LeNet+Conv120+Abs) and about 99.53% ([99.45%..99.60%] at different
trained varaints) at train dataset with decreasing the network size from 368
thousands to 227 thousands coeficients(LeNet+Conv64+Conv120+Abs). Therefore
these networks with Absolute activations could be close to the best solutions
accuracy for fixed shape activation functions, reached by more complicated
networks [Apicella et al.(2021)].

It is demonstrated that training curve (accuracy vs. training epoch) when
using Abs activation is not smooth, so one should use more complex technique
for changing learning ratio and stop training. We doing this by estimating the
lower bound of network accuracy at any test dataset using accuracy calculated
over the halves of validating dataset and by bootstrap method and combining
these predictions.

Thus, the high efficiency of the Absolute activation function has been
demonstrated, which makes it possible to improve the accuracy of current
classification neural networks even without changing their architecture.

## Acknowledgements

The data analysis was performed in part on the equipment of the Bioinformatics
Shared Access Center, the Federal Research Center Institute of Cytology and
Genetics of Siberian Branch of the Russian Academy of Sciences (ICG SB RAS).
The work has been done under financial support of the Ministry of Science and
Higher Education of the Russian Federation (Subsidy No.075-GZ/C3569/278).

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