# Klein-Gordon particles in Gödel-type Som-Raychaudhuri cosmic string
spacetime and the phenomenon of spacetime associated degeneracies

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

###### Abstract

Abstract: We argue that only exact, comprehensive, and explicit solutions for
the fundamental models, the Klein-Gordon (KG) oscillators and the KG-Coulomb,
would help to understand the effects of gravitational fields on the dynamics
of such quantum mechanical systems. In the current methodical proposal, the
effects of the gravitational fields generated by a Gödel-type Som-Raychaudhuri
(SR) cosmic string spacetime on KG-oscillators (KG-particles in general) are
studied and reported. In so doing, we revisit the KG-oscillators in a
topologically trivial Gödel-type spacetime background and use textbook
procedures to report its exact solution that covers even and odd parities.
Next, we discuss the drawbacks associated with the power series expansion
approach that implies the biconfluent Heun functions/polynomials solution. We,
therefore, recollect the so called pseudo perturbative shifted $\ell$
expansion technique (PSLET) as an alternative and more sophisticated
method/technique. Illustrative examples are used: (i) a KG-oscillator in a
topologically trivial Gödel-type spacetime, (ii) a quasi-free KG-oscillator in
Gödel SR-type cosmic string spacetime, (iii) a KG-Coulombic particle in Gödel
SR-type cosmic string spacetime at zero vorticity, and (iv) a massless KG-
particle in Gödel SR-type cosmic string spacetime in a Cornell-type Lorentz
scalar potential. The corresponding exact energies are obtained from the
zeroth (leading) order correction of PSLET, where all higher order correction
identically vanish. The comprehensive exactness of the reported solutions
manifestly suggest degeneracies associated with spacetime (STAD) phenomenon.

.

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

Keywords: Klein-Gordon (KG) particles, Gödel-type Som-Raychaudhuri cosmic
string spacetime, Spacetime associated degeneracies.

##  I Introduction

Topological defects in spacetime have stimulated intensive research studies in
quantum gravity. Amongst are, domain walls Vilenkin 1983 ; Vilenkin 1985 ,
cosmic string Kibble 1976 ; Hiscock 1985 , global monopole Barriola 1989  and
textures Cruz 2007 . The energy levels of relativistic/non-relativistic
quantum particles are shown to be affected by the gravitational fields
generated by different spacetime backgrounds with such topological defects,
not only in general relativity but also in the geometrical theory of defects
in condensed matter physics. In general relativity, for example, the Gödel
spacetime metric Godel 1949 , with an embedded cosmic string, introduces
itself as the first cosmological solution to Einstein’s equation with rotating
matter. Its compact form allowed analytical research studies of many physical
and mathematical systems in gravitational backgrounds with rotation and
causality violation. It has been shown that all spacetime homogeneous (ST-
homogeneous) Gödel-type metrics characterized by the vorticity $\Omega$ of
spacetime (a real number that represents uniform angular velocity or
rotational material, i.e, $\Omega=\pm\left|\Omega\right|$ R1 ) and a given
value of the parameter $\tilde{\mu};-\infty\leq\tilde{\mu}^{2}\leq\infty$, can
be transformed in cylindrical coordinates R1 ; Som 1968 ; Tiomno 1983 ;
Drukker 2004 ; Carvalho 2014 ; Tiomno 1992 ; Wang 2015 ; Ahmed 2018 ; Inzunza
2022 ; Montigny 2018 ; Sedaghatnia 2019  to

$ds^{2}=-\left(dt+\alpha\Omega\frac{\sinh^{2}\left(\tilde{\mu}\,r\right)}{\tilde{\mu}^{2}}d\varphi\right)^{2}+\alpha^{2}\frac{\sinh^{2}\left(2\,\tilde{\mu}\,r\right)}{4\,\tilde{\mu}^{2}}d\varphi^{2}+dr^{2}+dz^{2}.$
(1)

Where, for $\tilde{\mu}^{2}<0$ there is an infinite number of successive
causal and noncausal regions, and for $0\leq\tilde{\mu}^{2}<\Omega^{2}$ there
is one noncausal region for a given $r>r_{c}$. Here, $r_{c}$ is the critical
radius determined by $g_{\varphi\varphi}=0$ (e.g., R1 ) to imply

$\tanh\left(\tilde{\mu}\,r_{c}\right)=\frac{\tilde{\mu}\,}{\left|\Omega\right|}\Longrightarrow
r_{c}=\frac{1}{\tilde{\mu}\,}\text{arctanh}\left(\frac{\tilde{\mu}\,}{\left|\Omega\right|}\right).$

For $\tilde{\mu}^{2}=\Omega^{2}$ the Gödel spacetime metric (the first exact
Gödel solution of the Einstein equation describing a complete causal, ST-
homogeneous rotating universe Tiomno 1983 ) is recovered, and for
$\tilde{\mu}<\left|\Omega\right|$ there is a causal region (classified as
chronologically safe) with radius $r_{c}$ surrounded by the outer noncausal
space R1  Nevertheless, at the limit $\tilde{\mu}\longrightarrow 0$ of the
Gödel spacetime metric (1) we obtain the ST-homogeneous Som-Raychaudhuri (SR)
solution

$ds^{2}=-\left(dt+\alpha\,\Omega\,r^{2}d\varphi\right)^{2}+\alpha^{2}\,r^{2}d\varphi^{2}+dr^{2}+dz^{2}$
(2)

of the Einstein field equations Som 1968 ; R3 ; R4 ; R5 , in the presence of
cosmic string. Here, the disclination parameter $\alpha$ (curvature parameter)
admits the values $0<\alpha<1$ in general relativity (for cosmic strings with
positive curvature), $\alpha>1$ in the geometric theory of defects in
condensed matter (for a negative curvature), and $\Omega=0$ and $\alpha=1$
corresponds to Minkowski flat spacetime metric. Moreover, the covariant and
contravariant metric tensors associated with the Som-Raychaudhuri spacetime
are, respectively, given by

$g_{\mu\nu}=\left(\begin{tabular}[]{cccc}$-1\vskip 3.0pt plus 1.0pt minus
1.0pt$&$\,0\,$&$-\alpha\Omega r^{2}$&$\,0$\\\ $0$&$1\vskip 3.0pt plus 1.0pt
minus 1.0pt$&$0$&$0$\\\ $-\alpha\Omega
r^{2}$&$\,0$&$\,\alpha^{2}r^{2}\left(1-\Omega^{2}r^{2}\right)\,$&$0$\\\
$0$&$0$&$0$&$1$\end{tabular}\right)\Longleftrightarrow
g^{\mu\nu}=\left(\begin{tabular}[]{cccc}$\left(\Omega^{2}r^{2}-1\vskip 3.0pt
plus 1.0pt minus 1.0pt\right)$&$\,0\,$&$-\frac{\Omega}{\alpha}$&$0$\\\
$0$&$\,1\vskip 3.0pt plus 1.0pt minus 1.0pt$&$0$&$0$\\\
$-\frac{\Omega}{\alpha}$&$\,0$&$\,\frac{1}{\alpha^{2}r^{2}}\,$&$0$\\\
$0$&$0$&$0\,$&$\,1$\end{tabular}\right);\
\det\left(g\right)=-\alpha^{2}r^{2}.$ (3)

The objective of the current methodical proposal is to explore in a
comprehensive manner the effect of the gravitational field generated by the
Gödel-type cosmic string SR-spacetime (2) on the spectra of some Klein-Gordon
(KG) particles of fundamental quantum gravity nature, like the KG-oscillators
and KG-Coulomb models. Such fundamental models have been studied by different
authors (e.g., Carvalho 2014 ; Tiomno 1992 ; Wang 2015 ; Ahmed 2018 ; Inzunza
2022 ; Montigny 2018 ; Sedaghatnia 2019 ; Vitoria 2018 ; Neto 2020 ; Ahmed
2019 ) where the vorticity parameter $\Omega$ is assumed to take the form
$\Omega\geq 0$ and the energies are assumed to be positive $E=+\left|E\right|$
or $E=\pm\left|E\right|$ where all nodeless states are lost in the process
(excluding Hassanabadi et al Montigny 2018 ; Sedaghatnia 2019  who have
included all nodeless states with the radial quantum number $n_{r}=0$).
However, in Duarte de Oliveira et al R1 , $\Omega$ is a real number that
denotes uniform angular velocity or rotation of material (i.e.,
$\Omega=\pm\left|\Omega\right|$). In the current methodical proposal, we
discuss KG-particles’ spectral properties under the influence of the
gravitational field generated by the Gödel-type cosmic string SR-spacetime (2)
taking into account $E=\pm\left|E\right|$, $\Omega=\pm\left|\Omega\right|$,
and including nodeless states. In so doing, nevertheless, it is unavoidably
inviting to reconsider and report on the KG-oscillators in a topologically
trivial Gödel-type spacetime metric, used by Ahmed Ahmed 2018 ,

$ds^{2}=-dt^{2}+dx^{2}+\left(1-\alpha^{2}x^{2}\right)dy^{2}-2\alpha
xdtdy+dz^{2},$ (4)

where $H\left(x\right)=\alpha x$ and $D\left(x\right)=1$ in the general Gödel
metric Godel 1949 ; Tiomno 1983

$ds^{2}=-\left(dt+H\left(x\right)dy\right)^{2}+dx^{2}+D\left(x\right)^{2}dy^{2}+dz^{2},$

using the coordinates
$\left(x^{0}=t,x^{1}=x,x^{2}=y,x^{3}=z\,;\;x,y,z\in\left(-\infty,\infty\right)\right)$.
Where, the covariant and contravariant elements of the spacetime metric
tensors are given by

$\displaystyle g_{00}$ $\displaystyle=$
$\displaystyle-1,\,g_{11}=g_{33}=1,\,g_{02}=g^{20}=-\alpha
x,\,g_{22}=1-\alpha^{2}x^{2},$ $\displaystyle\,g^{00}$ $\displaystyle=$
$\displaystyle\alpha^{2}x^{2}-1,\,g^{11}=g^{22}=g^{33}=1,\text{
}\det\left(g\right)=-1.$ (5)

The organization of our methodical proposal is in order. We start with the KG-
oscillators in a topologically trivial Gödel-type spacetime (4) considered by
Ahmed Ahmed 2018 , who has reported only the positive energy states and missed
not only the ground state but also the negative energy states. For the sake of
scientific correctness, we give this problem a proper textbook quantum
mechanical treatment Flugge 1974  and report general forms of the spectra, in
section 2, that include all positive and negative energy states. In section 3,
we discuss KG-particles in Gödel SR-type cosmic string spacetime in a Cornnel-
type Lorentz scalar potentials and show the disadvantages of using a power
series (Frobinous method) expansion method that leads to biconfluent Heun
functions/polynomials. In section 4, we recollect, as a powerful alternative
method, the so called pseudo-perturbative shifted-$\ell$ expansion technique
(PSLET), to solve for Schrödinger equation (a common form that relativistic
wave equations collapse into) in $D$-dimensions Herschbach 1993 ; Imbo 1983 ;
Imbo 1984 ; Mustafa 1998 ; Mustafa 2000 ; Mustafa 1993 ; Mustafa 1994 ;
Mustafa 1991  to serves our current methodical proposal. PSLET has shown, with
a brut force evidence through comparison with numerical integration
techniques, to provide very highly accurate results (even with its non-
Hermitian $\mathcal{PT}$-symmetric version by Mustafa and Znojil Mustafa 2002
; Znojil 2002 ). We use some illustrative examples, in section 5, that include
a KG-oscillator (of section 2) in a topologically trivial Gödel-type spacetime
(4), a quasi-free KG-oscillator in Gödel SR-type cosmic string spacetime (2),
a KG-Coulombic particle in Gödel SR-type cosmic string spacetime (2) at zero
vorticity (i.e., $\Omega=0$), and a massless KG-particles in Gödel SR-type
cosmic string spacetime in a Cornell-type Lorentz scalar potential Quigg 1979
. We conclude in section 6.

##  II KG-oscillators in a topologically trivial Gödel-type spacetime
background

###  II.1 KG-oscillator I as a topologically trivial Gödel-type spacetime
byproduct:

In this section we recollect the topologically trivial Gödel-type spacetime
metric (4), along with the metric tensor elements in (5) and
$\det\left(g\right)=-1$, used by Ahmed Ahmed 2018 . Then, a KG-particle of
rest mass energy $m$ (denoting $mc^{2}$ in $c=\hbar=1$ units) in such a Gödel-
type spacetime is described by the KG-equation

$\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Psi\right)=m^{2}\Psi.$
(6)

Consequently, with

$\Psi\left(t,x,y,z\right)=\exp\left(i\left[k_{y}y+k_{z}z-Et\right]\right)\phi\left(x\right),$
(7)

it yields

$\phi^{\prime\prime}\left(x\right)-\left(\gamma^{2}x^{2}+2\,\gamma
k_{y}\,x\right)\,\phi\left(x\right)=\grave{\beta}\,\phi\left(x\right);\,\,\grave{\beta}=m^{2}+k_{y}^{2}+k_{z}^{2}-E^{2},\;\gamma^{2}=\alpha^{2}E^{2}.$
(8)

At this point, one should notice that $\gamma=\pm\alpha\left|E\right|$ as
mandated by the very nature of the Gödel-type spacetime in (4). Obviously,
this equation resembles the well known one-dimensional Schrödinger equation
for the so called shifted-oscillator (but does not inherit its parametric
characterizations) that can be treated, in a straightforward textbook manner,
by rewriting (8) as

$\phi^{\prime\prime}\left(x\right)-\left(\gamma
x+k_{y}\right)^{2}\,\phi\left(x\right)=\tilde{\beta}\,\phi\left(x\right)\,;\,\,\tilde{\beta}=\grave{\beta}-k_{y}^{2}=m^{2}+k_{z}^{2}-E^{2}.$
(9)

One would then use the change of variable $u=\gamma x+k_{y}$ to obtain

$\phi^{\prime\prime}\left(u\right)-\tilde{\gamma}^{2}\,u^{2}\,\phi\left(u\right)+\lambda\,\phi\left(u\right)=0\,;\,\,\lambda=-\frac{\tilde{\beta}}{\gamma^{2}}\,,\tilde{\gamma}^{2}=\frac{1}{\gamma^{2}}\,.$
(10)

This is the one-dimensional Schrödinger-like harmonic oscillator form that
admits exact textbook solution with the eigenvalues and eigenfunctions,
respectively, given by

$\lambda_{n}=\left|\tilde{\gamma}\right|\left(2n+1\right),\text{
and}\;\phi\left(u\right)=\mathcal{N}_{n}\,e^{-\frac{\left|\tilde{\gamma}\right|\,u^{2}}{2}}\,H_{n}\left(\sqrt{\left|\tilde{\gamma}\right|}\,u\right);\;n=0,1,2,\cdots,$
(11)

where $H_{n}\left(\sqrt{\left|\tilde{\gamma}\right|}\,u\right)$ are the
Hermite polynomials and $\mathcal{N}_{n}$ are the normalization constants. The
finiteness and square integrability of the wave function enforces the quantum
mechanical condition that
$\left|\tilde{\gamma}\right|=\pm\tilde{\gamma}_{\pm}\geq 0$ (i.e.,
$\tilde{\gamma}=\tilde{\gamma}_{\pm};\;\tilde{\gamma}_{+}=1/\alpha E_{+}$,
$\tilde{\gamma}_{-}=1/\alpha
E_{-}\Longrightarrow\left|\tilde{\gamma}\right|=\pm\tilde{\gamma}_{\pm}$,
where $E_{\pm}=\pm\left|E\right|$). Consequently, with the parametric
substitutions above (8), one obtains

$\frac{m^{2}+k_{z}^{2}-E_{\pm}^{2}}{\alpha^{2}E_{\pm}^{2}}=-\left|\tilde{\gamma}\right|\left(2n+1\right)\Rightarrow
E_{\pm}^{2}\mp\alpha
E_{\pm}\left(2n+1\right)-\left(m^{2}+k_{z}^{2}\right)=0;\;n=0,1,2,\cdots.$
(12)

Hence,

$E_{n}=E_{\pm}=\pm\alpha\left(n+\frac{1}{2}\right)\pm\sqrt{\alpha^{2}\left(n+\frac{1}{2}\right)^{2}+m^{2}+k_{z}^{2}}.$
(13)

Which in turn implies

$E_{+}=+\alpha\left(n+\frac{1}{2}\right)+\sqrt{\alpha^{2}\left(n+\frac{1}{2}\right)^{2}+m^{2}+k_{z}^{2}}$
(14)

as the positive energy solution, and

$E_{-}=-\alpha\left(n+\frac{1}{2}\right)-\sqrt{\alpha^{2}\left(n+\frac{1}{2}\right)^{2}+m^{2}+k_{z}^{2}}$
(15)

for the negative energy solution. Notably, unlike the solution reported by
Ahmed Ahmed 2018  (see equation (58) in Ahmed 2018 ), the textbook solution
above includes both even and odd parities, positive and negative energies, as
well as the ground state Flugge 1974 .

###  II.2 KG-oscillator II as a topologically trivial Gödel-type spacetime and
momentum operator deformation byproduct

Using similar recipe as that of Moshinsky and Szczepaniak Moshinsky 1989  and
Mirza and Mohadesi Mirza 2004  for the Dirac and KG oscillators, we redefine
the momentum operator as

$p_{\mu}\longrightarrow
p_{\mu}+i\eta\chi_{\mu};\;\chi_{\mu}=\left(0,x,0,0\right),$ (16)

where the dimensions of the second term suits the dimensions of the momentum
operator $p_{\mu}$ and here we consider $\eta$ to have positive and/or
negative values. Then the KG-equation reads

$\frac{1}{\sqrt{-g}}\left(\partial_{\mu}+\eta\chi_{\mu}\right)\sqrt{-g}g^{\mu\nu}\left(\partial_{\nu}-\eta\chi_{\nu}\right)\Psi=m^{2}\Psi.$
(17)

At this point, one may use $\eta=m\omega\geq 0$ (with $m$ denoting rest mass
of the KG-particle) to recover the traditionally used values as in (e.g.,
Ahmed 2018 ; Vitoria 2018 ; Neto 2020 ; Ahmed 2019 ; Moshinsky 1989 ; Mirza
2004  and other related references cited therein). However, we shall use a
general parameter $\eta$ and allow it to take positive and/or negative values.
Yet, in this case we avoid eminent confusion and inconsistency between $m^{2}$
(denoting $m^{2}c^{2}=mmc^{2}$, rest mass multiplied by rest mass energy) on
the R.H.S. and the rest mass of the particle on the L.H.S. of the KG-equation
(17) for $\eta=m\omega$. This point is made clear by Moshinsky and Szczepaniak
Moshinsky 1989  and Mirza and Mohadesi Mirza 2004  while dealing with the
Dirac and KG oscillators. We therefore stick with our assumption and use the
spacetime metric tensor elements in (5), to recast (17) as

$\phi^{\prime\prime}\left(x\right)-\left(\tilde{\omega}^{2}x^{2}+2\,\gamma
k_{y}\,x\right)\,\phi\left(x\right)=\zeta\,\phi\left(x\right);\,\,\zeta=m^{2}+k_{y}^{2}+k_{z}^{2}-E^{2}+\eta,\;\tilde{\omega}^{2}=\gamma^{2}+\eta^{2},\;\gamma=\alpha
E.$ (18)

Hereby, one should be aware that $\eta^{2}x^{2}$ would take the form of
$\eta^{2}x^{2}=m\left(m\omega^{2}x^{2}\right)$ as in Moshinsky 1989 ; Mirza
2004  when the speed of light is kept intact in Dirac and/or KG equations
(i.e., consistent with $m^{2}c^{2}=m\left(mc^{2}\right)$ on the R.H.S. of
(18)). The source of confusion is therefore clear and should be avoided.
Moreover, equation (18) admits solution in the form of hypergeometric
functions Flugge 1974  given by

$\,\phi\left(x\right)=e^{-\frac{\left|\tilde{\omega}\right|x^{2}}{2}-\frac{\delta
x}{\left|\tilde{\omega}\right|}}\left[A\,\,_{1}F_{1}\left(\tilde{a},\frac{1}{2},\frac{\left(\tilde{\omega}^{2}x+\delta\right)^{2}}{\left|\tilde{\omega}\right|^{3}}\right)+B\,\,\left(\tilde{\omega}^{2}x+\delta\right)\,_{1}F_{1}\left(\tilde{b},\frac{3}{2},\frac{\left(\tilde{\omega}^{2}x+\delta\right)^{2}}{\left|\tilde{\omega}\right|^{3}}\right)\right],$
(19)

where $\delta=\gamma k_{y}=\alpha Ek_{y}$, $\left|\tilde{\omega}\right|\geq 0$
to secure finiteness and square integrability of the wave function, and

$\tilde{b}=\frac{3}{4}-\frac{\delta^{2}}{4\left|\tilde{\omega}\right|^{3}}+\frac{\zeta}{4\left|\tilde{\omega}\right|}=\tilde{a}+\frac{1}{2}.$
(20)

Again, the condition for the hypergeometric functions to become polynomials of
degree $n\geq 0$ suggests that $a=-n$ and $b=a+1/2=-n$. This would imply that

$\tilde{a}=\frac{1}{4}-\frac{\delta^{2}}{4\left|\tilde{\omega}\right|^{3}}+\frac{\zeta}{4\left|\tilde{\omega}\right|}=-n\Longrightarrow\zeta_{2n}=-\left|\tilde{\omega}\right|\left(4n+1\right)+\frac{\delta^{2}}{\tilde{\omega}^{2}},$
(21)

for even parity, and

$\tilde{b}=-n\Longrightarrow\zeta_{2n+1}=-\left|\tilde{\omega}\right|\left(4n+3\right)+\frac{\delta^{2}}{\tilde{\omega}^{2}}.$
(22)

for odd parity solutions. The two results (21) and (22) would reduce to a
common form

$\zeta_{n}=-\left|\tilde{\omega}\right|\,\left(2n+1\right)+\frac{\delta^{2}}{\tilde{\omega}^{2}};\;\left|\tilde{\omega}\right|=\pm\alpha
E_{\pm}\sqrt{1+\frac{\eta^{2}}{\alpha^{2}E_{\pm}^{2}}}\geq 0,$ (23)

to imply that

$E_{\pm}^{2}-\left(m^{2}+k_{y}^{2}+k_{z}^{2}+\eta\right)=\pm\alpha
E_{\pm}\sqrt{1+\frac{\eta^{2}}{\alpha^{2}E_{\pm}^{2}}}\,\left(2n+1\right)-\frac{\alpha^{2}E_{\pm}^{2}k_{y}^{2}}{\alpha^{2}E_{\pm}^{2}+\eta^{2}}.$
(24)

Obviously, this result collapses into that in (13) for $\eta=0$, where the
effect of even and odd parities is reflected on the related spectrum.

##  III KG-particles in a 4-vector and scalar Lorentz potentials in Gödel SR-
type cosmic string spacetime background

The KG-equation for a spin-0 particle in a 4-vector $A_{\mu}$ and a radial
scalar $S\left(r\right)$ potentials in the Gödel SR-type spacetime background
(2) is given by

$\frac{1}{\sqrt{-g}}D_{\mu}\left(\sqrt{-g}g^{\mu\nu}D_{\nu}\Psi\right)=\left(m+S(r)\right)^{2}\Psi,$
(25)

where the gauge-covariant derivative is given by
$D_{\mu}=\partial_{\mu}-ieA_{\mu}$. This would in turn allow us to cast
equation (25) as

$\left\\{-D_{t}^{2}+\frac{1}{r}\,D_{r}\left(r\,D_{r}\right)+\left(\Omega\,rD_{t}-\frac{1}{\alpha
r}D_{\varphi}\right)^{2}+D_{z}^{2}-\left(m+S(r)\right)^{2}\right\\}\Psi=0.$
(26)

At this point, we may now define the corresponding gauge-covariant derivatives
so that

$D_{t}=\partial_{t}-ieA_{t}=\partial_{t}-iV\left(r\right),$ | $D_{r}=\partial_{r},$ | $D_{\varphi}=\partial_{\varphi}-ieA_{\varphi},$ | $D_{z}=\partial_{z},$
---|---|---|---
(27)

where, $V\left(r\right)=eA_{t}$ is the Lorentz 4-vector potential (i.e.,
transforms within the 4-vector potential $A_{\mu}$), $S\left(r\right)$ is the
Lorentz scalar potential (i.e., transforms like the rest mass energy
$m\longrightarrow m+S\left(r\right)$), and $eA_{\varphi}$ may include both
magnetic and Aharonov-Bohm flux fields effects Mustafa3 2021 ; Mustafa
Algadhi1 2020 . We now use the assumption that

$\Psi\left(t,r,\varphi,z\right)=\exp\left(i\left[\ell\varphi+k_{z}z-Et\right]\right)\psi\left(r\right)=\exp\left(i\left[\ell\varphi+k_{z}z-Et\right]\right)\frac{R\left(r\right)}{\sqrt{r}}$
(28)

in (26) to obtain

$\left\\{\partial_{r}^{2}+\frac{1}{4r^{2}}+\left(E+V\left(r\right)\right)^{2}-\left[\Omega
r\left(E+V\left(r\right)\right)+\frac{\left(\ell-eA_{\varphi}\right)}{\alpha
r}\right]^{2}-k_{z}^{2}-\left(m+S(r)\right)^{2}\right\\}R\left(r\right)=0,$
(29)

This result would describe KG-particles in Gödel SR-type cosmic string
spacetime background.

In what follows, we shall be interested in a set of KG-particles in Gödel SR-
type cosmic string spacetime under the influence of $A_{\varphi}=0$ (i.e., no
magnetic or Aharonov-Bohm flux fields effects), and no Lorentz 4-vector
potential (i.e., $V\left(r\right)=0$). Under such settings, the KG-equation
(29) reduces to

$\left\\{\partial_{r}^{2}-\frac{\left(\tilde{\ell}^{2}-1/4\right)}{r^{2}}-\Omega^{2}E^{2}r^{2}-2mS\left(r\right)-S(r)^{2}+\tilde{\lambda}\right\\}R\left(r\right)=0;\;\tilde{\ell}=\frac{\ell}{\alpha},\,\tilde{\lambda}=E^{2}-2\,\tilde{\ell}\,\Omega\,E-k_{z}^{2}-m^{2}.$
(30)

Obviously, this equation resembles the one-dimensional form of the two-
dimensional radial Schrödinger equation. Which, with a Cornell-type Lorentz
scalar potential $S\left(r\right)=ar+b/r$ would result

$\left\\{\partial_{r}^{2}-\frac{\left(\mathcal{L}^{2}-1/4\right)}{r^{2}}-\tilde{\Omega}^{2}r^{2}-2mar-\frac{2mb}{r}+\Lambda\right\\}R\left(r\right)=0,$
(31)

where

$\tilde{\Omega}^{2}=\Omega^{2}E^{2}+a^{2},\;\mathcal{L}^{2}=\tilde{\ell}^{2}+b^{2},\,\Lambda=E^{2}-2\,\tilde{\ell}\,\Omega\,E-\left(k_{z}^{2}+m^{2}+2ab\right).$
(32)

Equation (31) is known to admit, using power series method, a solution in the
form of biconfluent Heun functions

$R\left(r\right)=.C\,r^{\left|\mathcal{L}\right|+1/2}\,\exp\left(-\frac{\left|\tilde{\Omega}\right|}{2}r^{2}-\frac{am}{\left|\tilde{\Omega}\right|}r\right)\,H_{B}\left(\grave{\alpha},\grave{\beta},\grave{\gamma},\grave{\delta},\sqrt{\left|\tilde{\Omega}\right|}r\right).$
(33)

where
$\grave{\alpha}=2\left|\mathcal{L}\right|,\,\grave{\beta}=\frac{2am}{\left|\tilde{\Omega}\right|^{3/2}},\,\grave{\gamma}=\frac{a^{2}m^{2}+\Lambda\tilde{\Omega}^{2}}{\left|\tilde{\Omega}\right|^{3}},$
and $\delta=\frac{4mb}{\sqrt{\left|\tilde{\Omega}\right|}}$. For the case
$a=0=b$, this solution implies

$\displaystyle R\left(r\right)$ $\displaystyle=$
$\displaystyle.C\,r^{\left|\tilde{\ell}\right|+1/2}\,\exp\left(-\frac{\left|\Omega
E\right|}{2}r^{2}\right)\,H_{B}\left(2\left|\tilde{\ell}\right|,0,\frac{\Lambda}{\left|\Omega
E\right|},0,\sqrt{\left|\Omega E\right|}r\right)$ (34) $\displaystyle=$
$\displaystyle.C\,r^{\left|\tilde{\ell}\right|+1/2}\,\exp\left(-\frac{\left|\Omega
E\right|}{2}r^{2}\right)\,_{1}F_{1}\left(\left[\frac{1}{2}+\frac{\grave{\alpha}}{4}-\frac{\grave{\gamma}}{4}\right],\left[1+\frac{\grave{\alpha}}{2}\right],\left|\Omega
E\right|r^{2}\right),$

where the condition that

$\frac{1}{2}+\frac{\grave{\alpha}}{4}-\frac{\grave{\gamma}}{4}=\frac{1}{2}+\frac{\left|\tilde{\ell}\right|}{2}-\frac{\Lambda}{4\left|\Omega
E\right|}=-n_{r}$ (35)

is required for the confluent hypergeometric function to become a polynomial
of degree $n_{r}\geq 0$ and to secure the finiteness and square integrability
of the wave function. This would immediately result, in terms of the
associated Laguerre polynomials
$\,L_{n_{r}}^{\left|\tilde{\ell}\right|}\left(z^{2}\right)$, that

$\Lambda=2\left|\Omega
E\right|\left(2n_{r}+\left|\tilde{\ell}\right|+1\right),\text{ and
}R\left(r\right)=.C\,r^{\left|\tilde{\ell}\right|+1/2}\,\exp\left(-\frac{\left|\Omega
E\right|}{2}r^{2}\right)\,L_{n_{r}}^{\left|\tilde{\ell}\right|}\left(\left|\Omega
E\right|r^{2}\right)$ (36)

as the exact eigen energies and eigen functions for the two dimensional
Schrödinger harmonic oscillator. Moreover, the same condition (35) should also
be imposed on the biconfluent Heun function (33) so that a biconfluent Heun
polynomial R2  of degree $n=2n_{r}\geq 0$ is obtained (c.f., e.g., Neto 2020 ;
Mustafa3 2021 ; Mustafa 2022 ; Mustafa2 2021 ) to imply

$\Lambda=2\left|\Omega
E\right|\,\sqrt{1+\frac{a^{2}}{\Omega^{2}E^{2}}}\left(2n_{r}+\left|\mathcal{L}\right|+1\right),$
(37)

and

$E^{2}-2\,\tilde{\ell}\,\Omega\,E-\left(k_{z}^{2}+m^{2}+2ab\right)=2\left|\Omega
E\right|\,\sqrt{1+\frac{a^{2}}{\Omega^{2}E^{2}}}\left(2n_{r}+\left|\mathcal{L}\right|+1\right).$
(38)

Where the detailed analysis of such a quadratic equation are given in section
5-B, taking into account that

$\left|\Omega E\right|=\left\\{\begin{tabular}[]{l}$+\Omega_{\pm}E_{\pm}\vskip
3.0pt plus 1.0pt minus 1.0pt$\\\
$-\Omega_{\mp}E_{\pm}$\end{tabular}\right\\}\geq
0;\;E_{\pm}=\pm\left|E\right|,\;\Omega_{\pm}=\pm\left|\Omega\right|,$ (39)

At this point, the truncation order $n$ of a power series (biconfluent Heun
function here) does not make it a valid quantum number, but rather it should
be correlated to the well known quantum numbers as done above (i.e.,
$n=2n_{r}\geq 0$). Hereby, one should give credentials for Neto and co-workers
Neto 2020  as they resemble a group of the very few who have correctly used
condition (35) on the biconfluent Heun polynomials. Only under such condition
that the result of (38) would recover those of (36) for $a=0=b$. Obviously,
however, for the case $a=0=\Omega$ the solution in (38) collapses into a free
relativistic particle eigenvalues

$E^{2}-\left(k_{z}^{2}+m^{2}\right)=0,$ (40)

although we have an effective two-dimensional Coulomb problem (i.e., a KG-
Coulombic particle). This is a clear drawback of the Heun power series
(Frobinous method) expansion approach. Moreover, if condition (35) is not
implemented then the biconfluent Heun polynomial solution would fail to
address the exact solution for the KG-oscillator for $a=0=b$. One should,
therefore, search for an alternative method to deal with such a case in (31).

As a more reliable alternative method, to the power series (Frobinous method)
expansion, we recollect the so called pseudo-perturbative shifted-$\ell$
expansion technique (PSLET) for the $D$-dimensional radial Schrödinger-type
equation and solve for different KG-particles’ settings in the Gödel SR-type
cosmic string spacetime backgrounds of (31). In order to make our proposal
self contained, we, in short, discuss PSLET in the following section. We shall
also use Ahmed’s model (10) Ahmed 2018  as one of the illustrative examples to
be used in the current methodical proposal.

##  IV PSLET for the $D$-dimensional Schrödinger-type equation

Let us recollect that the $D$-dimensional radial Schrödinger equation
($\hbar=2m=1$ units are used in this section) reads

$\left\\{-\partial_{r}^{2}+\frac{\ell_{D}\left(\ell_{D}+1\right)}{r^{2}}+V\left(r\right)\right\\}\Phi_{k,\ell}\left(r\right)=E_{k,\ell}\Phi_{k,\ell}\left(r\right),$
(41)

where $\ell_{D}=\ell+\left(D-3\right)/2$ to incorporate interdimensional
degeneracies associated with the isomorphism between angular momentum and the
dimensionality $D$. Hereby, one should notice that for $D=1$,
$\ell_{1}=\ell-1$ and hence
$\ell_{D}\left(\ell_{D}+1\right)=\ell\left(\ell-1\right)=0$ for $\ell=0$ and
$\ell=1$ to denote even and odd parity, respectively, for the one-dimensional
case with $r=x\in\left(-\infty,\infty\right)$. In the two dimensional case
with radially cylindrical symmetry,
$r=\sqrt{x^{2}+y^{2}}\in\left(0,\infty\right)$ and
$\ell_{2}=\left|\ell\right|-1/2$ with $\ell$ denoting the magnetic quantum
number. Moreover, in the three-dimensional radially spherical symmetry,
$r=\sqrt{x^{2}+y^{2}+z^{2}}\in\left(0,\infty\right)$ and $\ell_{3}=\ell$ where
$\ell$ denotes angular momentum quantum number. Since we are going to use
$1/\ell_{D}$ as a small perturbation expansion parameter at large-$\ell_{D}$
limit (i.e., pseudo-classical limit, e.g., Imbo 1983 ; Imbo 1984 ; Herschbach
1993 ; Mustafa 1998 ; Mustafa 2000 ; Mustafa 2002 ; Mustafa 1993 ; Mustafa
1994 ; Mustafa 1991 ) we avoid the trivial case for $\ell_{D}=0$ (for $D=3$
and $\ell=0$) and use a shift $\beta$ so that $\breve{\ell}=\ell_{D}-\beta$.
Under such settings, our Schrödinger equation in (41) reads

$\left\\{-\partial_{r}^{2}+\frac{\breve{\ell}^{2}+\breve{\ell}\,\left(2\beta+1\right)+\beta\left(\beta+1\right)}{r^{2}}+V\left(r\right)\right\\}\Phi_{k,\ell}\left(r\right)=E_{k,\ell}\Phi_{k,\ell}\left(r\right).$
(42)

For convenience, we now shift the origin of the coordinate system and use

$x=\frac{\breve{\ell}^{1/2}}{r_{\circ}}\left(r-r_{\circ}\right),$ (43)

and expand about $x=0$ to obtain

$\left(\frac{x}{\breve{\ell}^{1/2}}+1\right)^{-2}=1-\frac{2x}{\breve{\ell}^{1/2}}+\frac{3x^{2}}{\breve{\ell}}+\frac{4x^{3}}{\breve{\ell}^{3/2}}+\cdots,$
(44)
$V\left(x\left(r\right)\right)=\frac{\breve{\ell}^{2}}{Q}\left[V\left(r_{\circ}\right)+V^{\prime}\left(r_{\circ}\right)\frac{r_{\circ}x}{\breve{\ell}^{1/2}}+V^{\prime\prime}\left(r_{\circ}\right)\frac{r_{\circ}^{2}x^{2}}{2\breve{\ell}}+V^{\prime\prime\prime}\left(r_{\circ}\right)\frac{r_{\circ}^{3}x^{3}}{6\breve{\ell}^{3/2}}+\cdots.\right].$
(45)

Moreover, let us expand the energy so that

$E_{k,\ell}=\frac{\breve{\ell}^{2}}{Q}\left[E_{0}+\frac{E_{1}}{\breve{\ell}}+\frac{E_{2}}{\breve{\ell}^{2}}+\frac{E_{3}}{\breve{\ell}^{3}}+\cdots,\right]$
(46)

where $Q$ is a constant parameter that scales the potential at
large-$\ell_{D}$ limit and is set equal to $\breve{\ell}^{2}$ at the end of
the calculation. Consequently, we may now re-write (42) as

$\displaystyle\left\\{-\partial_{x}^{2}+\left[\breve{\ell}+\left(2\beta+1\right)+\frac{\beta\left(\beta+1\right)}{\breve{\ell}}\right]\left(1-\frac{2x}{\breve{\ell}^{1/2}}+\frac{3x^{2}}{\breve{\ell}}+\frac{4x^{3}}{\breve{\ell}^{3/2}}+\cdots\right)\right.$
$\displaystyle\left.+\frac{r_{\circ}^{2}\,\breve{\ell}}{Q}\left[V\left(r_{\circ}\right)+V^{\prime}\left(r_{\circ}\right)\frac{r_{\circ}x}{\breve{\ell}^{1/2}}+V^{\prime\prime}\left(r_{\circ}\right)\frac{r_{\circ}^{2}x^{2}}{2\breve{\ell}}+\cdots\right]\right\\}\Phi_{k}\left(x\left(r\right)\right)=\xi_{k}\Phi_{k}\left(x\left(r\right)\right)$
(47)

with

$\xi_{k}=\frac{r_{\circ}^{2}\,\breve{\ell}}{Q}\left[E_{0}+\frac{E_{1}}{\breve{\ell}}+\frac{E_{2}}{\breve{\ell}^{2}}+\frac{E_{3}}{\breve{\ell}^{3}}+\cdots\right].$
(48)

This equation is to be compared with that of the one-dimensional anharmonic
oscillator

$\left[-\partial_{x}^{2}+\frac{1}{4}\omega^{2}x^{2}+\varepsilon_{\circ}+P\left(x\right)\right]\chi_{k}\left(x\right)=\lambda_{k}\chi_{k}\left(x\right)$
(49)

with $P\left(x\right)$ is a perturbation given by

$\displaystyle P\left(x\right)$ $\displaystyle=$
$\displaystyle\breve{\ell}^{-1/2}\left[\varepsilon_{1}x+\varepsilon_{3}\,x^{3}\right]+\breve{\ell}^{-1}\left[\varepsilon_{2}x^{2}+\varepsilon_{4}\,x^{4}\right]$
(50)
$\displaystyle+\breve{\ell}^{-3/2}\left[\delta_{1}x+\delta_{3}\,x^{3}+\delta_{5}\,x^{5}\right]++\breve{\ell}^{-2}\left[\delta_{2}x^{2}+\delta_{4}\,x^{4}+\delta_{6}\,x^{6}\right].$

It is obvious that equation (49) represents the one-dimensional Schrödinger
anharmonic oscillator that has been readily discussed by Imbo et al. Imbo 1983
; Imbo 1984  and by Mustafa and Barakat Mustafa 1998  and reported to admit
the eigenvalues

$\lambda_{k}=\varepsilon_{0}+\left(k+\frac{1}{2}\right)\,\omega+\frac{\alpha_{1}}{\breve{\ell}}+\frac{\alpha_{2}}{\breve{\ell}^{2}}.$
(51)

where $\alpha_{1}$ and $\alpha_{2}$ are given in the Appendix, and
$\left(k+\frac{1}{2}\right)\,\omega$ are the energies of the unperturbed one-
dimensional Schrödinger oscillator with $k=0,1,2,\cdots$ that denotes the
number of nodes in the wave function (for $D=2,3$ it denotes the radial
quantum number or if you wish, the number of nodes in the wave function). A
comparison between (47) and (49) would imply that

$\lambda_{k}=\breve{\ell}\left(1+\frac{r_{\circ}^{2}}{Q}V\left(r_{\circ}\right)\right)+\left[2\beta+1+\left(k+\frac{1}{2}\right)\,\omega\right]+\frac{1}{\breve{\ell}}\left[\beta\left(\beta+1\right)+\alpha_{1}\right]+\frac{\alpha_{2}}{\breve{\ell}^{2}}+\cdots.$
(52)

We may now simply compare (48) with (52) to obtain

$E_{0}=\frac{Q}{r_{\circ}^{2}}+V\left(r_{\circ}\right),\;E_{1}=\frac{Q}{r_{\circ}^{2}}\left[2\beta+1+\left(k+\frac{1}{2}\right)\,\omega\right],\;E_{2}=\frac{Q}{r_{\circ}^{2}}\left[\beta\left(\beta+1\right)+\alpha_{1}\right],\;E_{3}=\frac{Q}{r_{\circ}^{2}}\alpha_{2},$
(53)

where $r_{\circ}$ is chosen to minimize the zeroth order term so that it
satisfies the relations

$\frac{dE_{0}}{dr_{\circ}}=0,\;\frac{d^{2}E_{0}}{dr_{\circ}^{2}}>0\Longrightarrow\left(\ell_{D}-\beta\right)=\sqrt{\frac{r_{\circ}^{3}V^{\prime}\left(r_{\circ}\right)}{2}}.$
(54)

and the shifting parameter $\beta$ is determined through the requirement that
the first order term vanishes. This would result in

$E_{1}=0\Longrightarrow\beta=-\frac{1}{2}\left(k+\frac{1}{2}\right)\,\omega-\frac{1}{2},$
(55)

with

$\omega=2\sqrt{3+\frac{r_{\circ}V^{\prime\prime}\left(r_{\circ}\right)}{V^{\prime}\left(r_{\circ}\right)}.}$
(56)

One would therefore recast our anharmonic oscillator energies of (46) as

$E_{k}=E_{0}+\frac{1}{r_{\circ}^{2}}\left[\beta\left(\beta+1\right)+\alpha_{1}\right]+\frac{\alpha_{2}}{\breve{\ell}\,r_{\circ}^{2}}.$
(57)

Now we recall the corresponding Schrödinger potential for our KG-particles in
the Gödel SR-type cosmic string spacetime with a Cornnell-type Lorentz scalar
interaction of (31). This would result, using (56),(55) and (54), that

$\omega=2\,\sqrt{\frac{4\,\tilde{\Omega}^{2}r_{\circ}^{3}+3amr_{\circ}^{2}-bm}{\tilde{\Omega}^{2}r_{\circ}^{3}+amr_{\circ}^{2}-bm}},\;\beta=-\frac{1}{2}\left(k+\frac{1}{2}\right)\,\omega-\frac{1}{2},\;\tilde{\Omega}=\Omega
E$ (58)

and

$\breve{\ell}=\left(\ell_{D}-\beta\right)=\sqrt{\tilde{\Omega}^{2}r_{\circ}^{4}+amr_{\circ}^{4}-bmr_{\circ}^{2}};\;\ell_{D}=\ell+\left(D-3\right)/2.$
(59)

Various especial cases of such settings are given in the following
illustrative examples.

##  V Illustrative examples

In this section, we use PSLET and consider the following illustrative
examples.

###  V.1 KG-oscillator I as a topologically trivial Gödel-type spacetime
byproduct of (10)

We hereby recall the KG-oscillator I as a trivial Gödel-type spacetime of
section 2 described by (10) as

$\phi^{\prime\prime}\left(u\right)-\tilde{\gamma}^{2}\,u^{2}\,\phi\left(u\right)=-\lambda\,\phi\left(u\right)\,;\,\,\lambda=-\frac{\tilde{\beta}}{\gamma^{2}}\,,\tilde{\gamma}^{2}=\frac{1}{\gamma^{2}}\,.u=\gamma
x+k_{y}.$ (60)

Obviously, the central repulsive/attractive core
$\ell_{D}\left(\ell_{D}+1\right)/$ $r^{2}$ in the $D$-dimensional Schrödinger
equation (41) must vanish to resemble the one dimensional form of Schrödinger
equation (60). In this case,
$D=1\Longrightarrow\ell_{D}\left(\ell_{D}+1\right)=\ell\left(\ell-1\right)=0$
for $\ell=0$ and $\ell=1$ to denote even and odd parities
$\left(-1\right)^{\ell}$, respectively. Moreover, $u\equiv
r\in\left(-\infty,\infty\right)$ and $u_{\circ}$ is determined by (54) to read
$u_{\circ}=\sqrt{\breve{\ell}/\left|\tilde{\gamma}\right|}\in\mathbb{R};\;\breve{\ell}=\ell_{D}-\beta=\ell-1-\beta$.
This would in turn imply that $\omega=4$ by (56) and $\beta=-2k-3/2$ by (55).
Consequently, equations (53) would yield the zeroth-order correction

$E_{0}=2\left|\tilde{\gamma}\right|\left(2k+\ell+\frac{1}{2}\right)=\left\\{\begin{tabular}[]{ll}$2\left|\tilde{\gamma}\right|\left(2k+\frac{1}{2}\right)$&;
for even parity $\ell=0$ \vskip 6.0pt plus 2.0pt minus 2.0pt\\\
$2\left|\tilde{\gamma}\right|\left(2k+\frac{3}{2}\right)$&; for odd parity
$\ell=1$\end{tabular}\right\\},$ (61)

which results in a common form

$E_{0}=2\left|\tilde{\gamma}\right|\left(n+\frac{1}{2}\right);\;\,n=0,1,2,\cdots,$
(62)

with $E_{1},E_{2},$ and $E_{3}$ identically vanish. Therefore, the energies
associated with the shifted Schrödinger oscillator are

$\lambda_{n}=2\left|\tilde{\gamma}\right|\left(n+\frac{1}{2}\right);\;n=0,1,2,\cdots.$
(63)

Consequently, the KG-oscillator I above admits the exact energies reported in
(13) and goes through the same procedure discussed in section 2-A.

###  V.2 Quasi-free KG-oscillator in Gödel SR-type cosmic string spacetime (2)

Next, we consider a KG-particle in Gödel SR-type cosmic string spacetime
background described by (29) with the assumption that
$V\left(r\right)=S\left(r\right)=0$ (no interaction potential is involved,
hence the notion ”quasi-free KG-particle” is introduced). In this case one
would rewrite (29) as

$\left\\{\partial_{r}^{2}-\frac{\left(\tilde{\ell}^{2}-1/4\right)}{r^{2}}-\tilde{\Omega}^{2}r^{2}+\tilde{\lambda}\right\\}R\left(r\right)=0,$
(64)

where

$\tilde{\lambda}=E^{2}-2\,\tilde{\ell}\,\Omega\,E-\left(k_{z}^{2}+m^{2}\right)\text{;
\ }\tilde{\ell}=\frac{\ell}{\alpha},\;\tilde{\Omega}^{2}=\Omega^{2}E^{2}.$
(65)

Which resembles the two-dimensional radial Schrödinger-oscillator with an
irrational angular frequency $\tilde{\Omega}=\pm\left|\Omega E\right|$ (now
angular velocity). Under such settings, our PSLET implies that for
$D=2\Longrightarrow\ell_{D}=\left|\tilde{\ell}\right|-1/2$ (with
$\tilde{\ell}$ denoting irrational magnetic quantum number and $\tilde{\ell}$
replaces our $\ell$ of PSLET in (41)), $\mathbb{R}\ni
r\in\left(0,\infty\right)$ and $r_{\circ}$ is determined by (54) to read
$r_{\circ}=\sqrt{\breve{\ell}/\left|\tilde{\Omega}\right|}\in\mathbb{R}$ with
$\,\omega=4$ , $\beta=-2k-3/2$ by (58), and
$\breve{\ell}=\ell_{D}-\beta=2k+\left|\tilde{\ell}\right|+1$ by (59).
Consequently, equations (53) would yield the zeroth-order correction

$E_{0}=2\left|\Omega E\right|\left(2k+\left|\tilde{\ell}\right|+1\right).$
(66)

One should be aware that
$r_{\circ}=\sqrt{\breve{\ell}/\left|\tilde{\Omega}\right|}\in\mathbb{R}$ is
manifestly a condition that secures the finiteness and square integrability
wave function, and $E_{1},E_{2},$ and $E_{3}$ identically vanish.

Therefore, the energies associated with quasi-free KG-oscillator in Gödel SR-
type cosmic string spacetime are given by (65) as

$E^{2}-2\,\tilde{\ell}\,\Omega\,E-\left(k_{z}^{2}+m^{2}\right)=2\left|\Omega
E\right|\left(2k+\left|\tilde{\ell}\right|+1\right);\;\left|\Omega
E\right|=\left\\{\begin{tabular}[]{l}$+\Omega_{\pm}E_{\pm}\vskip 3.0pt plus
1.0pt minus 1.0pt$\\\ $-\Omega_{\mp}E_{\pm}$\end{tabular}\right\\}\geq 0.$
(67)

This would allow us to obtain

$\begin{tabular}[]{ll}$E_{\pm,1}=\pm\left|\Omega\right|\,\grave{n}_{+}\pm\,\sqrt{\Omega^{2}\,\grave{n}_{+}^{2}+k_{z}^{2}+m^{2}};\,\grave{n}_{+}=2k+\left|\tilde{\ell}\right|+\tilde{\ell}+1,$&for
$\left|\Omega E\right|=+\Omega_{\pm}E_{\pm}\vskip 3.0pt plus 1.0pt minus
1.0pt$\\\
$E_{\pm,2}=\pm\left|\Omega\right|\,\grave{n}_{-}\pm\,\sqrt{\Omega^{2}\,\grave{n}_{-}^{2}+k_{z}^{2}+m^{2}};\,\grave{n}_{-}=2k+\left|\tilde{\ell}\right|-\tilde{\ell}+1,$&for
$\left|\Omega E\right|=-\Omega_{\mp}E_{\pm}$\end{tabular}.$ (68)

However, we may rearrange such energies so that

$\begin{tabular}[]{ll}$E_{k,\ell,\pm}^{\left(+\right)}=\pm\left|\Omega\right|\,\grave{n}_{\pm}\pm\,\sqrt{\Omega^{2}\,\grave{n}_{\pm}^{2}+k_{z}^{2}+m^{2}}\Rightarrow\left\\{\begin{tabular}[]{l}$E_{k,\ell,+}^{\left(+\right)}=+\left|\Omega\right|\,\grave{n}_{+}+\sqrt{\Omega^{2}\,\grave{n}_{+}^{2}+k_{z}^{2}+m^{2}}$\\\
$E_{k,\ell,-}^{\left(+\right)}=-\left|\Omega\right|\,\grave{n}_{-}-\,\sqrt{\Omega^{2}\,\grave{n}_{-}^{2}+k_{z}^{2}+m^{2}}$\end{tabular}\right\\};\vskip
6.0pt plus 2.0pt minus 2.0pt\,$&for $\Omega=+\left|\Omega\right|$\\\
$E_{k,\ell,\pm}^{\left(-\right)}=\pm\left|\Omega\right|\,\grave{n}_{\mp}\pm\,\sqrt{\Omega^{2}\,\grave{n}_{\mp}^{2}+k_{z}^{2}+m^{2}}\Rightarrow\left\\{\begin{tabular}[]{l}$E_{k,\ell,+}^{\left(-\right)}=+\left|\Omega\right|\,\grave{n}_{-}+\sqrt{\Omega^{2}\,\grave{n}_{-}^{2}+k_{z}^{2}+m^{2}}$\\\
$E_{k,\ell,-}^{\left(-\right)}=-\left|\Omega\right|\,\grave{n}_{+}-\,\sqrt{\Omega^{2}\,\grave{n}_{+}^{2}+k_{z}^{2}+m^{2}}$\end{tabular}\right\\};$&for
$\Omega=-\left|\Omega\right|$\end{tabular},$ (69)

where in $E_{k,\ell,\pm}^{\left(+\right)}$ and
$E_{k,\ell,\pm}^{\left(-\right)}$ the superscripts $\left(+\right)$ and
$\left(-\right)$ identify states with positive and negative vorticities,
$\Omega=\Omega_{+}=+\left|\Omega\right|$ and
$\Omega=\Omega_{-}=-\left|\Omega\right|$, respectively. Moreover, one should
notice that
$\,\grave{n}_{\pm}\left(\tilde{\ell}=-\left|\tilde{\ell}\right|\right)=\grave{n}_{\mp}\left(\tilde{\ell}=\left|\tilde{\ell}\right|\right)$
and consequently
$E_{k,\ell,\pm}^{\left(+\right)}\left(\tilde{\ell}=-\left|\tilde{\ell}\right|\right)=E_{k,\ell,\pm}^{\left(-\right)}\left(\tilde{\ell}=\left|\tilde{\ell}\right|\right)$
indicating spacetime-type associated degeneracies (STAD). Moreover, when this
result is compared with that of Carvalho et al Carvalho 2014  (i.e., equation
(14) in Carvalho 2014 ) we observe that not only the negative energies are
missed but also the STADs discussed above. Their results should, therefore, be
generalized into those reported in (69). So should be the case with the
results reported on the linear confinement of a scalar particle in Gödel-type
space-time by Vitória Vitoria 2018  (their Eq. (23)) and in the related
comment by Neto et al Neto 2020  (their Eq. (18)) for zero linear confinement.
That would also include the results reported by Ahmed Ahmed 2020 , where he
also missed all nodeless (i.e., states with $n_{r}=0$) states.

###  V.3 KG-Coulombic particle in Gödel SR-type cosmic string spacetime (2) at
zero vorticity, $\Omega=0.$

Now, we consider a KG-particle in a Gödel SR-type cosmic string spacetime (2)
at zero vorticity, $\Omega=0$, and subjected to a Lorentz scalar potential
$S\left(r\right)=\beta_{2}/r$ in (29) to obtain

$\left\\{\partial_{r}^{2}-\frac{\left(\mathcal{L}^{2}-1/4\right)}{r^{2}}-\frac{\tilde{\beta}_{2}}{r}+\tilde{\lambda}\right\\}R\left(r\right)=0,$
(70)

where

$\tilde{\lambda}=E^{2}-\left(k_{z}^{2}+m^{2}\right)\text{; \
}\tilde{\ell}=\frac{\ell}{\alpha},\;\mathcal{L}^{2}=\tilde{\ell}^{2}+\beta_{2}^{2},\;\tilde{\beta}_{2}=2m\beta_{2}.$
(71)

In this case, our PSLET recipe above implies that
$D=2\Longrightarrow\ell_{D}=\left|\mathcal{L}\right|-1/2$ (with $\mathcal{L}$
now denotes irrational magnetic quantum number and $\mathcal{L}$ replaces our
$\ell$ of PSLET in (41)), $r\in\left(0,\infty\right)$ and $r_{\circ}$ is
determined by (59) to read $r_{\circ}=2\breve{\ell}^{2}/\tilde{\beta}_{2}$
with $\omega=2$, $\beta=-k-1$ by (58), and
$\breve{\ell}=\ell_{D}-\beta=k+\left|\mathcal{L}\right|+1/2$. Consequently,
equations (53) would yield

$E_{0}=-\frac{\tilde{\beta}_{2}^{2}}{4\left(k+\left|\mathcal{L}\right|+1/2\right)^{2}}$
(72)

with $E_{1},E_{2},$ and $E_{3}$ identically vanish. Therefore, the energies
associated with KG-Coulombic particle in Gödel SR-type cosmic string spacetime
are given by

$E_{k,\ell}=\pm\sqrt{k_{z}^{2}+m^{2}-\frac{m^{2}\beta_{2}^{2}}{\left(k+\left|\mathcal{L}\right|+1/2\right)^{2}}}.$
(73)

One should notice that the result of the Heun polynomials solution for the KG-
Coulomb in (40) for the case $a=0=\Omega$ and $b=\beta_{2}$ does not match the
exact solution (73), where the effect of the Coulombic parameter $b=\beta_{2}$
have no contribution in (40).

###  V.4 Massless KG-particles in Gödel SR-type cosmic string spacetime in a
Cornell-type Lorentz scalar potential

We hereby use the massless KG equation (30) with a Cornell-type Lorentz scalar
potential $S\left(r\right)=\beta_{1}r+\beta_{2}/r$ to obtain

$\left\\{\partial_{r}^{2}-\frac{\left(\mathcal{L}^{2}-1/4\right)}{r^{2}}-\tilde{\Omega}^{2}r^{2}+\tilde{\lambda}\right\\}R\left(r\right)=0,.$
(74)
$\tilde{\Omega}^{2}=\Omega^{2}\,E^{2}+\beta_{1}^{2},\;\mathcal{L}^{2}=\tilde{\ell}^{2}+\beta_{2}^{2},\;\tilde{\ell}=\frac{\ell}{\alpha},\,\,\tilde{\lambda}=E^{2}-2\,\tilde{\ell}\,\Omega\,E-k_{z}^{2}-2\beta_{1}\beta_{2}.$
(75)

Then, our PSLET recipe implies that
$D=2\Longrightarrow\ell_{D}=\left|\mathcal{L}\right|-1/2$ (with $\mathcal{L}$
denoting irrational magnetic quantum number and $\mathcal{L}$ replaces our
$\ell$ of PSLET in (41)), where $r_{\circ}$ is determined by (59) to read
$r_{\circ}=\sqrt{\breve{\ell}/\left|\tilde{\Omega}\right|}\in\mathbb{R}$ ,
$\,\omega=4$ and $\beta=-\left(2k+3/2\right)$ by (58) to imply
$\breve{\ell}=\ell_{D}-\beta=2k+\left|\mathcal{L}\right|+1$. Under such
settings, one obtains

$E_{0}=2\,\left|\Omega
E\right|\sqrt{1+\frac{\beta_{1}^{2}}{\Omega^{2}\,E^{2}}}\,\left(2k+\left|\mathcal{L}\right|+1\right)\Longrightarrow\tilde{\lambda}=2\,\left|\Omega
E\right|\sqrt{1+\frac{\beta_{1}^{2}}{\Omega^{2}\,E^{2}}}\,\left(2k+\left|\mathcal{L}\right|+1\right),$
(76)

where $E_{2}=0,$ $E_{3}=0$, and $\left|\tilde{\Omega}\right|=\,\left|\Omega
E\right|\sqrt{1+\beta_{1}^{2}/\Omega^{2}\,E^{2}}\geq 0$ is used. Consequently,
we get

$\tilde{\lambda}=E^{2}-2\tilde{\ell}\Omega
E-\left(k_{z}^{2}+2\beta_{1}\beta_{2}\right)=2\,\left|\Omega
E\right|\sqrt{1+\frac{\beta_{1}^{2}}{\Omega^{2}\,E^{2}}}\,\left(2k+\left|\mathcal{L}\right|+1\right)$
(77)

to yield

$E_{\pm}^{2}-2\tilde{\ell}\,\Omega_{\pm}\,E_{\pm}-\left(k_{z}^{2}+2\beta_{1}\beta_{2}\right)=2\Omega_{\pm}\,E_{\pm}\sqrt{1+\frac{\beta_{1}^{2}}{\Omega^{2}\,E^{2}}}\,\left(2k+\left|\mathcal{L}\right|+1\right)$
(78)

for $\left|\Omega E\right|=\Omega_{\pm}\,E_{\pm}$, and

$E_{\pm}^{2}-2\tilde{\ell}\,\Omega_{\mp}\,E_{\pm}-\left(k_{z}^{2}+2\beta_{1}\beta_{2}\right)=-2\Omega_{\mp}\,E_{\pm}\sqrt{1+\frac{\beta_{1}^{2}}{\Omega^{2}\,E^{2}}}\,\left(2k+\left|\mathcal{L}\right|+1\right).$
(79)

for $\left|\Omega E\right|=-\Omega_{\mp}\,E_{\pm}$ . However, the two results
above would eventually imply that

$E_{\pm}^{2}\mp
2\,\left|\Omega\right|\,E_{\pm}N_{\pm}-\left(k_{z}^{2}+2\beta_{1}\beta_{2}\right)=0\Rightarrow
E_{k,l,\pm}=\pm\left|\Omega\right|N_{\pm}\pm\sqrt{\Omega^{2}N_{\pm}^{2}+k_{z}^{2}+2\beta_{1}\beta_{2}},\text{
}$ (80)

to read

$E_{k,l,+}^{\left(+\right)}=+\left|\Omega\right|N_{+}+\sqrt{\Omega^{2}N_{+}^{2}+k_{z}^{2}+2\beta_{1}\beta_{2}}\text{
\ \ and \
}E_{k,l,-}^{\left(+\right)}=-\left|\Omega\right|N_{-}-\sqrt{\Omega^{2}N_{-}^{2}+k_{z}^{2}+2\beta_{1}\beta_{2}}$
(81)

for $\Omega=\Omega_{+}=+\left|\Omega\right|$, and

$E_{\pm}^{2}\mp
2\,\left|\Omega\right|\,E_{\pm}N_{\mp}-\left(k_{z}^{2}+2\beta_{1}\beta_{2}\right)=0\Rightarrow
E_{k,l,\pm}=\pm\left|\Omega\right|N_{\mp}\pm\sqrt{\Omega^{2}N_{\mp}^{2}+k_{z}^{2}+2\beta_{1}\beta_{2}},$
(82)

to read

$E_{k,l,+}^{\left(-\right)}=+\left|\Omega\right|N_{-}+\sqrt{\Omega^{2}N_{-}^{2}+k_{z}^{2}+2\beta_{1}\beta_{2}}\text{
\ \ and \
}E_{k,l,-}^{\left(-\right)}=-\left|\Omega\right|N_{+}-\sqrt{\Omega^{2}N_{+}^{2}+k_{z}^{2}+2\beta_{1}\beta_{2}},$
(83)

for $\Omega=\Omega_{-}=-\left|\Omega\right|$, where

$N_{\pm}=\sqrt{1+\frac{\beta_{1}^{2}}{\Omega^{2}\,E^{2}}}\,\left(2k+\left|\mathcal{L}\right|+1\right)\pm\tilde{\ell}.$
(84)

Again, one observes that
$\,N_{\pm}\left(\tilde{\ell}=-\left|\tilde{\ell}\right|\right)=N_{\mp}\left(\tilde{\ell}=\left|\tilde{\ell}\right|\right)$
and consequently
$E_{k,\ell,\pm}^{\left(+\right)}\left(\tilde{\ell}=-\left|\tilde{\ell}\right|\right)=E_{k,\ell,\pm}^{\left(-\right)}\left(\tilde{\ell}=\left|\tilde{\ell}\right|\right)$
to represent spacetime-type associated degeneracies (STAD).

##  VI Concluding remarks

In this paper, we started with KG-oscillators in a topologically trivial
Gödel-type spacetime background. A KG-oscillator I as a topologically trivial
Gödel-type spacetime byproduct, and a KG-oscillator II as a topologically
trivial Gödel-type spacetime plus momentum operator deformation byproduct. We
have used simple and straightforward textbook procedures and reported, for the
sake of scientific correctness, their exact relativistic energies in (14),
(15), and (24). This would make the results reported by Ahmed 2018  on KG-
oscillators in topologically trivial Gödel-type spacetime (4) only partially
correct and consequently should be generalized into those reported in (24).
Next, we have discussed the drawbacks associated with the power series
(Frobinous method) expansion approach that implied the biconfluent Heun
functions/polynomials solution (33). We have shown that for the effective
interaction potential $V_{eff}\left(r\right)=$
$\tilde{\Omega}^{2}r^{2}+2mar+2mb/r$ in the Schrödinger-like quantum model
(31), the biconfluent Heun polynomial solution (33) cease to be able to
address the especial case of KG-Coulombic particle when $a=0=\Omega$.
Consequently, the biconfluent Heun polynomial solution (33) can only be
classified as a conditionally exact solution that may address the KG-
oscillator (36), when $a=0=b$, but tragically fails to address the KG-Coulomb
case when $a=0=\Omega$.

As an alternative method/technique, we have recollected, in short, the so
called pseudo perturbative shifted $\ell$ expansion technique PSLET (e.g.,
Imbo 1983 ; Imbo 1984 ; Mustafa 1993 ; Mustafa 1994 ; Mustafa 1991 ; Mustafa
1998 ; Mustafa 2000 ; Mustafa 2002 ; Znojil 2002 ) to solve for the
$D$-dimensional Schrödinger-type equations (which is a common form the
relativistic wave equations collapse into, like (60), (31), etc). Some
illustrative examples on the straightforward/simplistic applicability of PSLET
procedure are used. Amongst, a KG-oscillator as a topologically trivial Gödel-
type spacetime byproduct of (10) that collapses into the one dimensional form
of Schrödinger equation, and three examples that resemble the radial two-
dimensional form of Schrödinger equation: a Quasi-free KG-oscillator in Gödel
SR-type cosmic string spacetime (2), a KG-Coulombic particle in Gödel SR-type
cosmic string spacetime (2) at zero vorticity $\Omega=0$, and a massless KG-
particle in Gödel SR-type cosmic string spacetime in a Cornell-type Lorentz
scalar potential. In all of our illustrative examples above, the exact
energies for the harmonic oscillator (i.e., (60), (64), and (74)) like and
Coulomb like (70) interactions are obtained from the zeroth (leading) order
correction, where all higher order correction identically vanish. In
connection with the effective potential $V_{eff}\left(r\right)=$
$\tilde{\Omega}^{2}r^{2}+2mar+2mb/r$ in the Schrödinger-like quantum model
(31), nevertheless, one would need to numerically solve for $\omega$ and
$r_{\circ}$ in (58) and (59) for each parametric value involved in the Gödel
SR-type cosmic string spacetime metric (2). In this case, although tedious,
one would work out the corresponding eigenvalues for such a quantum mechanical
problem. The accuracy of PSLET is shown to be very satisfactory and reliable
for similar Schrödinger-like models Mustafa 1998 ; Mustafa 2000 ; Mustafa 1993
; Mustafa 1994 ; Mustafa 2002 ; Znojil 2002 . This issue already lies far
beyond the scope of the current proposal.

Finally, we believe that the effects of gravitational fields introduced by
different spacetime structures can be clarified if only if the fundamental
models like the KG-oscillators and KG-Coulomb are comprehensively and
explicitly reported. To the best of our knowledge, the results of the current
methodical proposal have never been published elsewhere.

##  VII Appendix: Explicit forms of $\alpha_{1}$ and $\alpha_{2}$

In this section we give the explicit forms of $\alpha_{1}$ and $\alpha_{2}$ as
given by Imbo 1984 ; Mustafa 1998 ,

$\alpha_{1}=\left[\left(1+2k\right)e_{2}+3\left(1+2k+2k^{2}\right)e_{4}\right]-\frac{1}{\omega}\left[e_{1}^{2}+6\left(1+2k\right)e_{1}e_{3}+\left(11+30k+30k^{2}\right)e_{3}^{2}\right],$
(85) $\displaystyle\alpha_{2}$ $\displaystyle=$
$\displaystyle\left[\left(1+2k\right)d_{2}+3\left(1+2k+2k^{2}\right)d_{4}+5\left(3+8k+6k^{2}+4k^{3}\right)d_{6}\right]$
(86)
$\displaystyle-\frac{1}{\omega}\left[\left(1+2k\right)e_{2}^{2}+12\left(1+2k+2k^{2}\right)e_{2}e_{4}+2e_{1}d_{1}+2\left(21+59k+51k^{2}+34k^{3}\right)e_{4}^{2}\right.$
$\displaystyle\left.+6\left(1+2k\right)e_{1}d_{3}+30\left(1+2k+2k^{2}\right)e_{1}d_{5}+6\left(1+2k\right)e_{3}d_{1}+2\left(11+30k+30k^{2}\right)e_{3}d_{3}\right.$
$\displaystyle\left.+10\left(13+40k+42k^{2}+28k^{3}\right)e_{3}d_{5}\right]+\frac{1}{\omega^{2}}\left[4e_{1}^{2}e_{2}+36\left(1+2k\right)e_{1}e_{2}e_{3}\right.$
$\displaystyle\left.+8\left(11+30k+30k^{2}\right)e_{2}e_{3}^{2}+24\left(1+2k\right)e_{1}^{2}e_{4}+8\left(31+78k+78k^{2}\right)e_{1}e_{2}e_{4}\right.$
$\displaystyle\left.+12\left(57+189k+225k^{2}+150k^{3}\right)e_{3}^{2}e_{4}\right]-\frac{1}{\omega^{3}}\left[8e_{1}^{3}e_{3}+108\left(1+2k\right)e_{1}^{2}e_{3}^{2}\right.$
$\displaystyle\left.+48\left(11+30k+30k^{2}\right)e_{1}e_{3}^{3}+30\left(31+109k+141k^{2}+94k^{3}\right)e_{3}^{4}\right],$

with

$e_{j}=\varepsilon_{j}/\omega^{j/2}\,;\;d_{i}=\delta_{i}/\omega^{i/2}\;;\;j=1,2,3,4,\;i=1,2,3,4,5,6,$ (87) $\varepsilon_{1}=-4\beta,$ | $\varepsilon_{2}=6\beta,\;$ | $\varepsilon_{3}=-4+r_{\circ}^{5}V^{\prime\prime\prime}\left(r_{\circ}\right)/6Q,\;$ | $\varepsilon_{4}=5+r_{\circ}^{6}V^{\prime\prime\prime\prime}\left(r_{\circ}\right)/24Q,$
---|---|---|---
(88) $\delta_{1}=-2\left(\beta^{2}-1/4\right),\;\delta_{2}=3\left(\beta^{2}-1/4\right),\;$ | $\delta_{3}=-8\beta\vskip 6.0pt plus 2.0pt minus 2.0pt,\;\delta_{4}=10\beta,$
---|---
$\delta_{5}=-6+r_{\circ}^{7}V^{{}^{\prime\prime\prime\prime\prime}}\left(r_{\circ}\right)/120Q,$ | $\delta_{6}=7+r_{\circ}^{8}V^{{}^{\prime\prime\prime\prime\prime\prime}}\left(r_{\circ}\right)/120Q.$
(89)

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