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# User Power Measurement Based IRS Channel Estimation via Single-Layer Neural
Network

He Sun1, Weidong Mei4, Lipeng Zhu1, and Rui Zhang231

This paper has been accepted by GLOBECOM 2023. 1Department of Electrical and
Computer Engineering, National University of Singapore, Singapore 117583.
2School of Science and Engineering, Shenzhen Research Institute of Big Data.
3Chinese University of Hong Kong, Shenzhen, Guangdong, China 518172.
4National Key Laboratory of Science and Technology on Communications,
University of Electronic Science and Technology of China, China 611731.
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

One main challenge for implementing intelligent reflecting surface (IRS) aided
communications lies in the difficulty to obtain the channel knowledge for the
base station (BS)-IRS-user cascaded links, which is needed to design high-
performance IRS reflection in practice. Traditional methods for estimating IRS
cascaded channels are usually based on the additional pilot signals received
at the BS/users, which increase the system training overhead and also may not
be compatible with the current communication protocols. To tackle this
challenge, we propose in this paper a new single-layer neural network
(NN)-enabled IRS channel estimation method based on only the knowledge of
users’ individual received signal power measurements corresponding to
different IRS random training reflections, which are easily accessible in
current wireless systems. To evaluate the effectiveness of the proposed
channel estimation method, we design the IRS reflection for data transmission
based on the estimated cascaded channels in an IRS-aided multiuser
communication system. Numerical results show that the proposed IRS channel
estimation and reflection design can significantly improve the minimum
received signal-to-noise ratio (SNR) among all users, as compared to existing
power measurement based designs.

##  I Introduction

Intelligent reflecting surface (IRS) has recently emerged as a candidate
technology for the future six-generation (6G) wireless communication systems
due to its capability of realizing smart and reconfigurable propagation
environment cost-effectively [1]. Specifically, an IRS consists of a large
number of passive reflecting elements with independently tunable reflection
coefficients, which can be jointly designed to alter the phase and/or
amplitude of its incident signal to achieve high performance passive
beamforming for various purposes, such as signal boosting, interference
cancellation, target sensing, etc[1, 2, 3].

To this end, IRS passive beamforming or in general passive reflection should
be properly designed. In the existing literature, there are two main
approaches for IRS passive beamforming design, which are based on channel
estimation pilots and user signal power measurements, respectively. In the
former approach, the cascaded base station (BS)-IRS-user/user-IRS-BS channels
are first estimated based on the downlink/uplink pilots received at the
users/BS with time-varying IRS training reflections, and then the IRS
reflection for data transmission is optimized based on the estimated IRS
cascaded channels[4, 5, 6, 7]. Alternatively, the authors in [8] proposed to
train a deep neural network (NN) to directly learn the mapping from the
received pilot signals to the optimal IRS reflection. However, the above
pilot-based designs require additional training pilots for IRS channel
estimation or NN training, which not only increases the system training
overhead but also may not be compatible with the current cellular transmission
protocols that cater to the user-BS direct channel (without IRS) estimation
only. To efficiently integrate IRS into current wireless systems without the
need of changing their protocols, the latter approach designs IRS reflection
for data transmission based on the received (pilot or data) signal power
measurements at each user’s receiver with time-varying IRS reflections, which
can be easily obtained in existing wireless systems. For example, passive beam
training for IRS-aided millimeter-wave (mmWave) systems[9, 10] and conditional
sample mean (CSM)-based IRS reflection for IRS-aided sub-6 GHz systems[11]
have been proposed. In particular, it was shown in [11] that in the single-
user case, the CSM method can achieve an IRS passive beamforming gain in the
order of the number of IRS reflecting elements, which is identical to that
under perfect channel state information (CSI)[1]. However, the number of
random IRS reflections needed for CSM to obtain sufficient user power
measurement samples is very large (hundreds or even thousands) for even the
single-user case, which still results in high implementation overhead and
large training delay. The fundamental reason for CSM’s low efficiency lies in
its lack of IRS channel information extraction from the users’ power
measurements.

Figure 1: IRS-aided multicasting with users’ power measurements.

In this paper, we propose a new IRS cascaded channel estimation and IRS
reflection design method based on users’ power measurements similar to CSM in
[11]. However, different from CSM, we first estimate the IRS cascaded channels
based on user power measurements and then design the IRS reflection for data
transmission based on the estimated channels. This thus overcomes the
aforementioned inefficacy of CSM due to the lack of channel information
extraction. In particular, our proposed IRS channel estimation method based on
user power measurements leverages a simple single-layer NN formulation.
Specifically, we first reveal that for any given IRS reflection, the received
signal power at each user can be equivalently modeled as the output of a
single-layer NN, with its weights corresponding to the coefficients of the
cascaded BS-IRS-user channel. Inspired by this, we optimize the weights of the
single-layer NN to minimize the mean squared error (MSE) between its output
and each user’s power measurement via the stochastic gradient descent method,
thereby estimating the cascaded BS-IRS-user channel. Next, to evaluate the
effectiveness of the proposed channel estimation method, we design the IRS
reflection for data transmission based on the estimated cascaded channels in
an IRS-aided multiuser multicast communication system, as shown in Fig. 1. We
aim to optimize the IRS reflection to maximize the minimum received signal-to-
noise ratio (SNR) among all users and solve this problem efficiently by
applying various optimization techniques. Numerical results show that the
proposed IRS channel estimation and IRS reflection design can yield much
better performance than existing user power measurement based schemes such as
CSM.

_Notations_ : Scalars, vectors and matrices are denoted by lower/upper case,
boldface lower case and boldface upper case letters, respectively. For any
scalar/vector/matrix, $(\cdot)^{*}$, $(\cdot)^{T}$ and $(\cdot)^{H}$
respectively denote its conjugate, transpose and conjugate transpose.
$\mathbb{C}^{n\times m}$ and $\mathbb{R}^{n\times m}$ denote the sets of
$n\times m$ complex and real matrices, respectively. $\parallel\cdot\parallel$
denotes the Euclidean norm of a vector, and $\mid\cdot\mid$ denotes the
cardinality of a set or the amplitude of a complex scalar. $j=\sqrt{-1}$
denotes the imaginary unit. $\rm Re(\cdot)$ and $\rm Im(\cdot)$ denote the
real and imaginary parts of a complex vector/number, respectively.
$\boldsymbol{V}\succeq\textbf{0}$ indicates that $\boldsymbol{V}$ is a
positive semidefinite matrix. $\text{Tr}(\cdot)$ denotes the trace of a
matrix. The distribution of a circularly symmetric complex Gaussian (CSCG)
random variable with zero mean and covariance $\sigma^{2}$ is denoted by
$\mathcal{CN}(0,\sigma^{2})$.

##  II System Model and Problem Formulation

As shown in Fig. 1, we consider an IRS-aided multicast communication system,
where a single-antenna BS (or multi-antenna BS with fixed downlink precoding)
transmits a common message to $K$ single-antenna users (or independent
messages to different users over orthogonal frequency bands), with the help of
an IRS consisting of $N$ reflecting elements. It is assumed that there is a
central controller in the system (the BS or another dedicated unit) which can
collect the users’ received signal power measurements and thereby optimize the
IRS passive reflection. Let $U_{k}$ denote the $k$-th user,
$k\in{\cal{K}}\triangleq{\\{1,2,...,K\\}}$. In this paper, we consider quasi-
static block-fading channels and focus on a given fading block, during which
all the channels involved are assumed to be constant. The baseband equivalent
channel from the BS to the IRS, that from the BS to $U_{k}$, and that from the
IRS to $U_{k}$ are denoted by $\boldsymbol{h}_{BI}\in{\mathbb{C}}^{N\times
1}$, $h_{B{U_{k}}}\in{\mathbb{{C}}}$ and
$\boldsymbol{h}_{I{U_{k}}}^{H}\in{\mathbb{C}}^{1\times N}$, respectively. Let
$\mathbf{\Theta}=\text{diag}(e^{j{\theta}_{1}},...,e^{j{\theta}_{N}})$ denote
the reflection matrix of the IRS, where ${\theta}_{i}$ denotes the phase shift
of its $i$-th reflecting element, $1\leq i\leq N$. Due to the hardware
constraints, we consider that the phase shift of each reflecting element can
only take a finite number of discrete values in the set
$\Phi_{\alpha}=\\{\omega,{2\omega},{3\omega},...,2^{\alpha}{\omega}\\}$, where
$\alpha$ is the number of bits used to uniformly quantize the continuous phase
shift in $(0,2\pi]$, and $\omega=\frac{2\pi}{2^{\alpha}}$[12]. Let $P$ denote
the transmit power of the BS. The effective channel from the BS to $U_{k}$ is
expressed as

$g_{k}=\sqrt{P}\left({h}_{B{U_{k}}}+{\boldsymbol{h}_{I{U_{k}}}^{H}}\mathbf{\Theta}\boldsymbol{h}_{BI}\right),\
k\in{\mathcal{K}},$ (1)

where we have incorporated the effect of the BS transmit power $P$ into the
BS-$U_{k}$ effective channel, since it may be practically unknown to the
central controller. Let
${\bar{\boldsymbol{v}}}^{H}=\left[e^{j{\theta}_{1}},...,e^{j{\theta}_{N}}\right]$
denote the passive reflection of the IRS, and
$\bar{\boldsymbol{h}}_{k}={\text{diag}}({\boldsymbol{h}_{I{U_{k}}}^{H}}){\boldsymbol{h}_{BI}}$
denote the cascaded BS-IRS-$U_{k}$ channel. As such, the channel in (1) can be
simplified as

$g_{k}=\sqrt{P}{{h}}_{B{U_{k}}}+\sqrt{P}\bar{\boldsymbol{v}}^{H}\bar{\boldsymbol{h}}_{k},\
k\in{\mathcal{K}}.$ (2)

By extending the IRS passive reflection vector into
$\boldsymbol{v}^{H}=\left[1,\bar{\boldsymbol{v}}^{H}\right]$ and stacking the
direct and cascaded BS-$U_{k}$ channels into
$\boldsymbol{h}_{k}^{H}=\sqrt{P}\left[{{h}_{B{U_{k}}}^{*}},\bar{\boldsymbol{h}}_{k}^{H}\right]$,
the baseband equivalent channel in (2) can be further simplified as

$g_{k}=\boldsymbol{v}^{H}{\boldsymbol{h}_{k}},\ k\in{\mathcal{K}}.$ (3)

Let $s\in\mathbb{C}$ denote the transmitted symbol (pilot or data) at the BS
with $\left|s\right|^{2}=1$. Hence, the received signal at $U_{k}$ is given by

$y_{k}={g_{k}}{s}+\emph{n}_{k},\ k\in{\mathcal{K}},$ (4)

where $n_{k}\sim\mathcal{CN}(0,\sigma^{2})$ denotes the complex additive white
Gaussian noise (AWGN) at $U_{k}$ with power ${\sigma}^{2}$. Accordingly, the
received SNR at $U_{k}$ is

$\text{SNR}_{k}=\frac{{\left|g_{k}\right|}^{2}}{{\sigma}^{2}}=\frac{{\boldsymbol{v}}^{H}{\boldsymbol{G}_{k}}{\boldsymbol{v}}}{{\sigma}^{2}},\
k\in{\mathcal{K}},$ (5)

where $\boldsymbol{G}_{k}={\boldsymbol{h}_{k}}\boldsymbol{h}_{k}^{H}$ denotes
the covariance matrix of $\boldsymbol{h}_{k}$. In this paper, we aim to
optimize the IRS passive reflection to maximize the minimum received SNR among
all $K$ users. The associated optimization problem is thus formulated as

(P1): $\displaystyle\max_{\boldsymbol{v}}\ \min_{{{k\in\mathcal{K}}}}\
{\boldsymbol{v}}^{H}{\boldsymbol{G}_{k}}{\boldsymbol{v}}$ (6a)
$\displaystyle\text{s.t.}\ \ \
{\theta}_{i}\in\Phi_{\alpha},i=1,...,N,\vspace{-3pt}$ (6b)

where the constant scalar $1/\sigma^{2}$ is omitted in (6a). Problem (P1) is a
discrete optimization problem and can be solved by a variety of methods (see
e.g., [12, 13]), given that the perfect channel knowledge (i.e.,
$\boldsymbol{h}_{k}$ or $\boldsymbol{G}_{k}$, $k\in{\mathcal{K}}$) is
available. However, perfect CSI is difficult to acquire in practice for IRS
cascaded channels [1]. Thus, in this paper, we propose an efficient single-
layer NN-enabled method to estimate $\boldsymbol{h}_{k}$, $k\in{\mathcal{K}}$,
based on the users’ individual power measurements and then optimize
$\boldsymbol{v}$ via solving (P1), as detailed in the next section.

##  III Single-Layer NN-enabled IRS Channel Estimation and IRS Reflection
Optimization

###  III-A User Power Measurement

In current wireless systems (e.g., cellular and WiFi), user terminals are able
to measure the average power of their received signals, referred to as the
reference signal received power (RSRP). By leveraging this capability, we
propose to estimate the IRS cascaded channels based on users’ power
measurements (to be presented in the next subsection). Specifically, we assume
that the IRS applies randomly generated phase shifts of its reflecting
elements (subject to (6b)) to reflect the BS’s signals to all $K$ users
simultaneously. Let $M$ and ${\boldsymbol{v}}_{m}$,
$m\in{\cal{M}}\triangleq\left\\{1,2,...,M\right\\}$, denote the number of
random reflection sets generated and its $m$-th reflection set, respectively.
In the meanwhile, the users independently measure the power of their received
signals corresponding to each IRS reflection set and send the results to the
central controller (see Fig.1). For each reflection set, we consider that
$U_{k}$ takes $Q$ samples of its received signal to calculate the RSRP based
on them. Note that in practice, it usually holds that $Q\gg 1$ since the IRS’s
reflection switching rate is usually much lower than the symbol rate of each
user. Thus, the power measurement of $U_{k}$ under the IRS’s $m$-th reflection
set is given by

${\bar{p}}_{k}(\boldsymbol{v}_{m})=\frac{1}{Q}\sum_{q=1}^{Q}{\left|{g_{k,m}}{s}+\emph{n}_{k}(q)\right|^{2}},\
k\in{\mathcal{K}},\ m\in{\cal{M}},\vspace{-3pt}$ (7)

where $g_{k,m}=\boldsymbol{v}_{m}^{H}\boldsymbol{h}_{k}$ and
$\emph{n}_{k}(q)\sim\mathcal{CN}(0,\sigma^{2})$ denotes the $q$-th sampled
AWGN at $U_{k}$. Let
$\mathcal{P}_{k}=\left[{\bar{p}}_{k}(\boldsymbol{v}_{1}),{\bar{p}}_{k}(\boldsymbol{v}_{2}),...,{\bar{p}}_{k}(\boldsymbol{v}_{M})\right]$
denote the collection of $U_{k}$’s received signal power measurements under
the $M$ reflection sets of the IRS. After the above power measurements, each
user $U_{k}$ reports $\mathcal{P}_{k}$ to the central controller, which
estimates ${\boldsymbol{h}}_{k},k\in{\cal{K}}$ as presented next.

###  III-B NN-enabled Channel Estimation

For any given IRS reflection set $\boldsymbol{v}$, the desired signal power at
each user $U_{k}$ is given by

$p_{k}(\boldsymbol{v})={\left|{\boldsymbol{v}^{H}{\boldsymbol{h}_{k}}}\right|}^{2}.$
(8)

It is worth mentioning that if the number of samples $Q$ is sufficiently
large, we have
${\bar{p}}_{k}(\boldsymbol{v})\approx{p}_{k}(\boldsymbol{v})+\sigma^{2}$.
Thus, we aim to estimate $\boldsymbol{h}_{k},k\in{\cal{K}}$ based on
${\bar{p}}_{k}(\boldsymbol{v}_{m}),m\in{\cal{M}}$. To this end, note that (8)
can be modeled as a single-layer NN, explained as follows. In particular, this
NN takes the reflection pattern $\boldsymbol{v}$ and the cascaded channel
${\boldsymbol{h}_{k}}$ as its input and weights, respectively, while the
nonlinear activation function at the output layer is the squared amplitude of
${\boldsymbol{v}^{H}{\boldsymbol{h}_{k}}}$, as given in (8). However, as both
$\boldsymbol{v}$ and ${\boldsymbol{h}_{k}}$ are complex numbers in general,
such a single-layer NN requires the implementation in the complex domain. To
avoid this issue, we express (8) equivalently in the real domain as

$p_{k}(\boldsymbol{v})={\left|{\boldsymbol{v}^{H}{\boldsymbol{h}_{k}}}\right|}^{2}=\left\|{\boldsymbol{x}^{T}\boldsymbol{R}_{k}}\right\|^{2},$
(9)

where $\boldsymbol{x}$ consists of the real and imaginary parts of
$\boldsymbol{v}$, i.e.,
$\boldsymbol{x}^{T}=\left[{\begin{array}[]{*{20}{c}}{{\mathop{\rm
Re}\nolimits}\left({{\boldsymbol{v}^{T}}}\right)},&{{\mathop{\rm
Im}\nolimits}\left({{\boldsymbol{v}^{T}}}\right)}\end{array}}\right]$, and
$\boldsymbol{R}_{k}$ denotes the real-valued cascaded channel, i.e.,

$\boldsymbol{R}_{k}=\left[{\begin{array}[]{*{20}{c}}{{{\mathop{\rm
Re}\nolimits}\left(\boldsymbol{h}_{k}\right)}}&{{{{\mathop{\rm
Im}\nolimits}\left(\boldsymbol{h}_{k}\right)}}}\\\ {{{{\mathop{\rm
Im}\nolimits}\left(\boldsymbol{h}_{k}\right)}}}&{{-{{\mathop{\rm
Re}\nolimits}\left(\boldsymbol{h}_{k}\right)}}}\end{array}}\right]\in\mathbb{R}^{{(2N+2)}\times
2}.$ (10)

Based on (9), we can construct an equivalent single-layer NN to (8) in the
real-number domain. Specifically, as shown in Fig. 2, the input of this
single-layer NN is $\boldsymbol{x}$. Let $W_{k,i,j}$ denote the weight of the
edge from the $i$-th input to the $j$-th neuron in the hidden layer, with
$i=1,2,...,2N+2$ and $j=1,2$. The two neurons at the hidden layer of this NN
are given by

$\left[{\begin{array}[]{*{20}{c}}{a_{k}}\\\
{b_{k}}\end{array}}\right]^{T}=\boldsymbol{x}^{T}\boldsymbol{W}_{k},$ (11)

where $\boldsymbol{W}_{k}\in\mathbb{R}^{(2N+2)\times 2}$ denotes the weight
matrix of this NN, with $W_{k,i,j}$ being its entry in the $i$-th row and the
$j$-th column.

Finally, the activation function at the output layer is given by the squared
norm of (11), and the output of this NN is

${{\hat{p}}_{k}}(\boldsymbol{v})=a_{k}^{2}+b_{k}^{2}=\left\|{\boldsymbol{x}^{T}\boldsymbol{W}_{k}}\right\|^{2}.$
(12)

Figure 2: Single-layer NN architecture for $U_{k}$.

By comparing (12) with (9), it is noted that this real-valued NN can imitate
the received signal power by $U_{k}$. In particular, if
${\boldsymbol{R}}_{k}={\boldsymbol{W}}_{k}$, we have
$\hat{p}_{k}(\boldsymbol{v})=p_{k}(\boldsymbol{v})$. Motivated by this, we
propose to recover ${\boldsymbol{R}}_{k}$ (and ${\boldsymbol{h}}_{k}$) by
estimating the weight matrix ${\boldsymbol{W}}_{k}$ via training this single-
layer NN. To this end, we consider that ${\boldsymbol{W}}_{k}$ takes a similar
form to ${\boldsymbol{R}}_{k}$ in (10), i.e.,

${\boldsymbol{W}}_{k}=\left[{\begin{array}[]{*{20}{c}}{{{\boldsymbol{w}_{1,k}}}}&{{{\boldsymbol{w}}_{2,k}}}\\\
{{{\boldsymbol{w}_{2,k}}}}&{{-{\boldsymbol{w}_{1,k}}}}\end{array}}\right],$
(13)

where ${\boldsymbol{w}}_{1,k}\in\mathbb{R}^{N+1}$ and
${\boldsymbol{w}}_{2,k}\in\mathbb{R}^{N+1}$ correspond to ${{{\mathop{\rm
Re}\nolimits}\left(\boldsymbol{h}_{k}\right)}}$ and ${{{\mathop{\rm
Im}\nolimits}\left(\boldsymbol{h}_{k}\right)}}$ in (10), respectively.

With (13), we present the following lemma.

###### Lemma 1

If

$\|\boldsymbol{x}^{T}{\boldsymbol{W}}_{k}\|^{2}=\|{\boldsymbol{x}^{T}\boldsymbol{R}_{k}}\|^{2}$
(14)

holds for any $\boldsymbol{x}\in\mathbb{R}^{2N+2}$, we have
${\boldsymbol{h}}_{k}={\boldsymbol{w}}_{k}e^{j\phi_{k}}$, where
${\boldsymbol{w}}_{k}={{\boldsymbol{w}}_{1,k}}+j{\boldsymbol{w}}_{2,k}$,
$\phi_{k}\in[0,2\pi)$ denotes an arbitrary phase.

###### Proof:

By substituting (13) into the left-hand side of (14), we have

$\|\boldsymbol{x}^{T}{\boldsymbol{W}}_{k}\|^{2}={\left|{\boldsymbol{v}^{H}{\boldsymbol{w}}_{k}}\right|}^{2}={\boldsymbol{v}^{H}{\boldsymbol{w}}_{k}}{{\boldsymbol{w}}_{k}^{H}\boldsymbol{v}}.$
(15)

Next, by substituting (9) and (15) into (14), we have

${\boldsymbol{v}^{H}{\boldsymbol{w}}_{k}}{{\boldsymbol{w}}_{k}^{H}\boldsymbol{v}}={\left|{\boldsymbol{v}^{H}{\boldsymbol{h}_{k}}}\right|}^{2}={{\boldsymbol{v}}^{H}{{{{\boldsymbol{h}}}}_{k}{{{\boldsymbol{h}}}}_{k}^{H}}{\boldsymbol{v}}},\
\forall\boldsymbol{v}\in{\mathbb{C}}^{N+1},$ (16)

which implies that

${\boldsymbol{v}}^{H}\left({{{\boldsymbol{h}}}}_{k}{{{\boldsymbol{h}}}}_{k}^{H}-{\boldsymbol{w}}_{k}{\boldsymbol{w}}_{k}^{H}\right){\boldsymbol{v}}=0,\
\forall{\boldsymbol{v}}\in{\mathbb{C}}^{N+1}.$ (17)

For (17) to hold for any ${\boldsymbol{v}}\in{\mathbb{C}}^{N+1}$, it should be
satisfied that
${{{\boldsymbol{h}}}}_{k}{{{\boldsymbol{h}}}}_{k}^{H}={\boldsymbol{w}}_{k}{\boldsymbol{w}}_{k}^{H}$.
As such, we have ${\boldsymbol{h}}_{k}={\boldsymbol{w}}_{k}e^{j\phi_{k}}$. The
proof is thus completed. ∎

It follows from Lemma 1 that we can estimate ${\boldsymbol{h}}_{k}$ by
training the single-layer NN in Fig. 2 to estimate ${\boldsymbol{W}}_{k}$
first. Although we cannot derive the exact ${\boldsymbol{h}}_{k}$ due to the
presence of the unknown phase $\phi_{k}$, the objective function of (P1) only
depends on the channel covariance matrix ${\boldsymbol{G}}_{k}$, and we have
${\boldsymbol{G}}_{k}={{{\boldsymbol{h}}}}_{k}{{{\boldsymbol{h}}}}_{k}^{H}={\boldsymbol{w}}_{k}{\boldsymbol{w}}_{k}^{H},k\in{\cal{K}}$.
As such, the unknown common phase does not affect the objective function of
(P1). It should also be mentioned that Lemma 1 requires that (14) holds for
any ${\boldsymbol{x}}\in{\mathbb{R}}^{2N+2}$ or
${\boldsymbol{v}}\in{\mathbb{C}}^{N+1}$. However, due to (6b), the discrete
IRS passive reflection set can only take a finite number of values in a
subspace of ${\mathbb{C}}^{N+1}$, and
${\boldsymbol{h}}_{k}={\boldsymbol{w}}_{k}e^{j\phi_{k}}$ may not always hold
in such a subspace. Nonetheless, the proposed design is still effective, as
will be explained in Remark 1 later.

To estimate ${{{\boldsymbol{w}}}}_{k}$ or ${{{\boldsymbol{W}}}}_{k}$, we can
train the NN in Fig. 2 by using the stochastic gradient descent method to
minimize the MSE between its output and the training data. In particular, we
can make full use of each user’s power measurements, i.e., ${\cal
P}_{k},k\in{\cal K}$, as the training data. Specifically, we divide them into
two data sets, namely, the training set and validation set. The training set
consists of $M_{0}$ $(M_{0}<M)$ entries of ${\cal P}_{k}$, while the remaining
$M-M_{0}$ entries of ${\cal P}_{k}$ are used as the validation set to evaluate
the model fitting accuracy. Accordingly, the MSE for the training data is set
as the following loss function,

${\cal
L}_{\boldsymbol{W}_{k}}=\frac{1}{M_{0}}\sum_{m=1}^{M_{0}}{\left({\hat{{{p}}}}_{k}(\boldsymbol{v}_{m})-{{\bar{p}}_{k}}(\boldsymbol{v}_{m})\right)}^{2}.$
(18)

Given this loss function, we can use the backward propagation [14] to
iteratively update the NN weights. Specifically, with (13), the weight matrix
${\boldsymbol{W}_{k}}$ can be expressed by a vector
$\boldsymbol{\gamma}_{k}=\left[{{\boldsymbol{w}}_{1,k}}^{T},{\boldsymbol{w}}_{2,k}^{T}\right]^{T}\in\mathbb{R}^{2N+2}$.
Let $\boldsymbol{\gamma}_{k,t}$ denote the updated value of
$\boldsymbol{\gamma}_{k}$ after the $t$-th iteration. As such, the iteration
proceeds as

${\boldsymbol{\gamma}_{k,t+1}}={\boldsymbol{\gamma}_{k,t}}-\rho
F\left({\boldsymbol{\gamma}_{k,t}}\right),$ (19)

where $\rho>0$ denotes the learning rate, and
$F\left({\boldsymbol{\gamma}_{k}}\right)=\frac{\partial{{\cal
L}_{{\boldsymbol{W}}_{k}}}}{\partial{{{\boldsymbol{\gamma}}_{k}}}}$ denotes
the derivative of the loss function ${\cal L}_{\boldsymbol{W}_{k}}$ with
respect to $\boldsymbol{\gamma}_{k}$. Here,
$F\left({\boldsymbol{\gamma}_{k}}\right)$ can be calculated using the chain
rule,

$F\left(\boldsymbol{\gamma}_{k}\right)=\frac{\partial{{\cal
L}_{{\boldsymbol{W}}_{k}}}}{\partial{{{\hat{p}}_{k}}}}\left[\frac{\partial{{{\hat{p}}_{k}}}}{\partial{{{{a}}_{k}}}},\
\frac{\partial{{{\hat{p}}_{k}}}}{\partial{{{{b}}_{k}}}}\right]\left[\frac{\partial{{{{a}}_{k}}}}{\partial{{{{\boldsymbol{\gamma}}}_{k}}}},\
\frac{\partial{{{{b}}_{k}}}}{\partial{{{{\boldsymbol{\gamma}}}_{k}}}}\right]^{T},$
(20)

where $\frac{\partial{{\cal
L}_{{\boldsymbol{W}}_{k}}}}{\partial{{{\hat{p}}_{k}}}}$ can be calculated
based on (18), while the other four derivatives in (20) can be calculated
based on (11) as

$\displaystyle\frac{\partial{{{{a}}_{k}}}}{\partial{{{{\boldsymbol{\gamma}}}_{k}}}}$
$\displaystyle=\left[{1,{\cos\left({{\theta_{1}}}\right)},\\!\cdots\\!,{\cos\left({{\theta_{N}}}\right)},{0,-\sin\left({{\theta_{1}}}\right)},\\!\cdots\\!,{-\sin\left({{\theta_{{N}}}}\right)}}\right]^{T},$
$\displaystyle\frac{\partial{{{{b}}_{k}}}}{\partial{{{{\boldsymbol{\gamma}}}_{k}}}}$
$\displaystyle=\left[{0,\sin\left({{\theta_{1}}}\right)},\\!\cdots\\!,{\sin\left({{\theta_{N}}}\right)},{1,\cos\left({{\theta_{1}}}\right)},\\!\cdots\\!,{\cos\left({{\theta_{N}}}\right)}\right]^{T},$
$\displaystyle\frac{\partial{{{\hat{p}}_{k}}}}{\partial{{{{a}}_{k}}}}$
$\displaystyle=2{{{a}}_{k}},\ \text{and}\ \
\frac{\partial{{{\hat{p}}_{k}}}}{\partial{{{{b}}_{k}}}}=2{{b}}_{k}.$ (21)

The NN training process terminates after $Z$ rounds of iterations, and the
weight matrix of the NN is determined as

$\boldsymbol{W}^{\divideontimes}_{k}=\arg\min_{1\leq t\leq
Z}\left(\sum_{m=M_{0}+1}^{M}{(\hat{p}_{k,t}}(\boldsymbol{v}_{m})-{{\bar{p}}_{k}}(\boldsymbol{v}_{m}))^{2}\right),$
(22)

based on the validation set, where
${\hat{p}}_{k,t}(\boldsymbol{v}_{m})=\|\boldsymbol{x}_{m}^{T}{\boldsymbol{W}}_{k,t}\|^{2}$
denotes the output of the NN after the $t$-th iteration, and
$\boldsymbol{W}_{k,t}$ denotes the updated version of $\boldsymbol{W}_{k}$
after the $t$-th iteration. Based on the above, the complex-valued cascaded
channel can be estimated as
${\boldsymbol{w}}^{\divideontimes}_{k}={{\boldsymbol{w}}^{\divideontimes}_{1,k}}+j{\boldsymbol{w}}^{\divideontimes}_{2,k}$.

###### Remark 1

In the case with one-bit IRS phase shifts, i.e., $\alpha=1$, the cascaded
channel ${\boldsymbol{h}}_{k}$ may not be estimated as
${\boldsymbol{w}}_{k}e^{j\phi_{k}}$. This is because in this case, we have
${\boldsymbol{v}^{*}}={\boldsymbol{v}}$, which results in

${\boldsymbol{v}^{H}}{{{\boldsymbol{h}}}}_{k}{{{\boldsymbol{h}}}}_{k}^{H}\boldsymbol{v}={\boldsymbol{v}^{H}}{\boldsymbol{h}}_{k}^{*}{\boldsymbol{h}}_{k}^{T}\boldsymbol{v}.$
(23)

Based on (17), we may estimate ${\boldsymbol{h}}_{k}^{*}$ as
${\boldsymbol{w}}_{k}e^{j\phi_{k}}$, while the actual channel
${\boldsymbol{h}}_{k}$ should be estimated as
${\boldsymbol{w}}_{k}^{*}e^{-j\phi_{k}}$. However, this does not affect the
efficacy of the proposed design, since both estimations lead to the same
received signal power due to (23).

###  III-C IRS Reflection Optimization

After estimating ${\boldsymbol{h}}_{k},k\in{\cal{K}}$, we can substitute them
into (6a) and solve (P1) accordingly. Next, we present the optimal and
suboptimal algorithms to solve (P1) in the cases of $K=1$ and $K>1$,
respectively. First, if $K=1$, it has been shown in [15] that (P1) can be
optimally solved by applying a geometry-based method, and the details are thus
omitted.

However, if $K>1$, problem (P1) is generally difficult to be optimally solved.
Next, we consider combining the SDR technique[16] and the successive
refinement method[12] to solve it. First, let
${\boldsymbol{V}}=\boldsymbol{v}\boldsymbol{v}^{H}$ denote the covariance
matrix of $\boldsymbol{v}$, with $\boldsymbol{V}\succeq\textbf{0}$. Problem
(P1) can be equivalently reformulated as

(P2): $\displaystyle\max_{{\boldsymbol{V}}}\ \xi$ (24a)
$\displaystyle\text{s.t.}\ \ \
\text{Tr}({\hat{{\boldsymbol{G}}}_{k}}{\boldsymbol{V}})\geq\xi,\ \forall\ k,$
(24b) $\displaystyle\ \ \ \ \ \ \text{rank}({\boldsymbol{V}})=1,$ (24c)
$\displaystyle\ \ \ \ \ \ {\boldsymbol{V}}\succeq\textbf{0},$ (24d)
$\displaystyle\ \ \ \ \ \ \ {\theta}_{i}\in\Phi_{\alpha},\ i=1,...,N,$ (24e)

where $\xi$ is an auxiliary variable. Problem (P2) is still difficult to solve
due to the rank-one and discrete-phase constraints in (24c) and (24e),
respectively. Next, we relax both constraints and thereby transform (P2) into
a semidefinite programming (SDP) problem, which can be optimally solved by the
interior-point algorithm[17]. However, the obtained solution may not be rank-
one, and its entries may not satisfy the discrete constraint (24e). In this
case, we can apply the Gaussian randomization method jointly with the solution
quantization to construct a rank-one solution that satisfies (24e), denoted as
$\hat{\boldsymbol{v}}$. Based on the initial passive reflection
$\hat{\boldsymbol{v}}$, we successively refine $\theta_{i}$ by enumerating the
elements in $\Phi_{\alpha}$, with $\theta_{j},j\neq i,j=1,2,...,N$ being
fixed, until the convergence is reached.

###  III-D Complexity Analysis

In the proposed IRS channel estimation and IRS reflection design method, the
computational complexity is mainly due to the NN training procedures for
channel estimation and the passive reflection optimization for solving (P2).
In particular, the training complexity depends on the size of the NN
structure. In the NN for $U_{k},k\in{\cal K}$, as shown in Fig. 2, the number
of neurons is $2$, and the number of weights is $2N+2$, which entails the
complexity for all $K$ users in the order of ${\cal O}\left(KN\right)$[18].
Furthermore, in the passive reflection optimization, the SDR-based
initialization incurs the complexity of ${\cal O}\left((K+N)^{3.5}\right)$,
while the successive refinement incurs the complexity of ${\cal
O}\left(KN\right)$. Thus, the overall complexity of the proposed design is
dominated by the SDR, i.e., ${\cal O}\left((K+N)^{3.5}\right)$. In practice,
we can apply the successive refinement only for solving (P2) with affordable
performance loss (to be shown in Section IV via simulation), while the
complexity is decreased significantly to ${\cal O}\left(KN\right)$, thus
reducing the overall complexity to linear over $N$.

##  IV Numerical Results

###  IV-A Simulation Setup

Consider a three-dimensional Cartesian coordinate system in meter (m) with $K$
users, where the BS is deployed at $(50,-200,20)$, while the locations of all
users are randomly generated in a square area with the coordinates of its four
corner points given by $(0,0,0),(10,0,0),(10,10,0)$ and $(0,10,0)$,
respectively. The IRS is equipped with a uniform planar array (UPA) and
assumed to be parallel to the $y$-$z$ plane, with $N=N_{y}\times N_{z}$
reflecting elements, where $N_{y}$ and $N_{z}$ denote the numbers of
reflecting elements along the axes $y$ and $z$, respectively. We set
$N_{y}=N_{z}=8$ and half-wavelength spacing for the adjacent IRS reflecting
elements. The location of the reference point of the IRS is set as
$(-2,-1,0)$. Let $\beta_{0,k}$, $\beta_{1}$ and $\beta_{2,k}$ denote the path
loss (in dB) of the BS-$U_{k}$, BS-IRS and IRS-$U_{k}$ channels, respectively,
which are set to $\beta_{0,k}=33+37\text{log}_{10}(d_{0,k})$,
$\beta_{1}=30+20\text{log}_{10}(d_{1})$ and
$\beta_{2,k}=30+20\text{log}_{10}(d_{2,k})$, respectively, with $d_{0,k}$,
$d_{1}$ and $d_{2,k}$ denoting the distance from the BS to $U_{k}$, that from
the BS to the IRS, and that from the IRS to $U_{k}$. We assume Rayleigh fading
for the BS-$U_{k}$ channel, i.e.,
${{\emph{{h}}}_{B{U_{k}}}}=10^{-\beta_{0,k}/20}\zeta_{k}$, where $\zeta_{k}$
denotes the small-scale fading following $\mathcal{CN}(0,1)$. In addition, a
multipath channel model is assumed for the BS-IRS and IRS-$U_{k}$ channels,
and the BS-IRS channel is expressed as

${{{\boldsymbol{h}}}_{BI}}=\sqrt{\frac{\varepsilon_{BI}}{1+\varepsilon_{BI}}}{\boldsymbol{h}}_{LoS}+\sqrt{\frac{1}{1+\varepsilon_{BI}}}{\boldsymbol{h}}_{NLoS},$
(25)

where $\varepsilon_{BI}$ is the ratio of the line-of-sight (LoS) path power to
that of the non-LoS (NLoS) path. ${\boldsymbol{h}}_{LoS}$ and
${\boldsymbol{h}}_{NLoS}$ denote the LoS and NLoS components, which are
respectively given by

$\displaystyle{\boldsymbol{h}}_{LoS}\ $
$\displaystyle=10^{-\beta_{1}/20}{e^{\frac{-j2\pi{d_{1}}}{\lambda}}}{\boldsymbol{u}_{N}(\vartheta_{0},\varphi_{0})},$
(26a) $\displaystyle\boldsymbol{h}_{NLoS}$
$\displaystyle=\sqrt{\frac{1}{L}}\sum_{l=1}^{L}{{\kappa_{l}}\boldsymbol{u}_{N}(\vartheta_{l},\varphi_{l})},$
(26b)

where $\lambda$ denotes the wavelength. In (26), $L$ denotes the number of
NLoS multipath components, $\kappa_{l}$ denotes the amplitude of the $l$-th
multipath component following ${\cal{CN}}(0,10^{-\beta_{1}/10})$, and
$\boldsymbol{u}_{N}(\vartheta_{l},\varphi_{l})$ denotes the steering vector of
the $l$-th path from the BS to the IRS with $\vartheta_{l}\in[0,\pi]$ and
$\varphi_{l}\in[0,\pi]$ denoting the azimuth and the elevation angles of
arrival at the IRS in this path, respectively. In particular, let
$\boldsymbol{e}(\gamma,n)=[1,e^{-j\pi\gamma},e^{-j2\pi\gamma},...,e^{-j{(n-1)}\pi\gamma}]^{T}$
denote the steering vector function of a uniform linear array with $n$
elements and directional cosine $\gamma$. As such, we have
${\boldsymbol{u}_{N}(\vartheta_{l},\varphi_{l})}=\boldsymbol{e}({\rm
sin}(\vartheta_{l}){\rm sin}(\varphi_{l}),N_{y})\otimes\boldsymbol{e}({\rm
cos}(\vartheta_{l}),N_{z})$, where $\otimes$ denotes the Kronecker product.
The IRS-$U_{k}$ channel can be expressed similarly and we denote by
$\varepsilon_{IU_{k}}$ the ratio of its LoS path power to that of the NLoS
counterpart. We set $L=5$, $\varepsilon_{BI}=10$ and $\varepsilon_{IU_{k}}=1$.
The number of power measurements obtained by one user under each IRS
reflection set is $Q=10$. The transmit power is $P=30$ dBm, and the noise
power is $\sigma^{2}=-90$ dBm. All results are averaged over $10^{3}$
realizations of channels and user locations.

###  IV-B Benchmark Schemes

We adopt the CSM[11] and random-max sampling (RMS)[19] methods as benchmark
schemes, both of which design the IRS passive reflection for data transmission
based on the users’ power measurements, but without estimating the IRS
cascaded channels by further exploiting the power measurements. Specifically,
the RMS method sets the IRS reflection as the one that maximizes the minimum
received signal power among all users over $M$ random IRS reflection sets,
i.e.,

${\boldsymbol{v}}^{\text{RMS}}=\boldsymbol{v}_{m^{\star}},\ \ \ \text{with}\ \
\ {m^{\star}}=\arg{{\max_{m\in\mathcal{M}}\
{\min_{k\in\mathcal{K}}{{\bar{p}}_{k}(\boldsymbol{v}_{m})}}}}.$ (27)

Moreover, the CSM method first calculates the sample mean of the minimum power
measurement among all users conditioned on
$\theta_{i}=\psi,\psi\in\Phi_{\alpha}$, i.e.,

${{\mathbb{E}}}[p|\theta_{i}=\psi]=\frac{1}{\left|\mathcal{A}_{i}(\psi)\right|}\sum_{\boldsymbol{v}\in\mathcal{A}_{i}(\psi)}{\min_{k\in\mathcal{K}}{{\bar{p}}_{k}(\boldsymbol{v})}},$
(28)

where $\mathcal{A}_{i}(\psi)$ denotes a subset of the $M$ random reflection
sets with $\theta_{i}=\psi$, $i=1,2,...,N$. Finally, the phase shift of the
$i$-th reflecting element is set as

$\theta_{i}^{\text{CSM}}=\arg{\max_{\psi\in\Phi_{\alpha}}{{{\mathbb{E}}}[p|\theta_{i}=\psi]}},\
i=1,...,N.$ (29)

In addition, the IRS passive reflection design based on perfect CSI by solving
(P2) with ${\hat{\boldsymbol{G}}}_{k}$ replaced by
${{\boldsymbol{G}}}_{k},k\in{\cal{K}}$, is included as the performance upper
bound to evaluate the efficacy of the proposed scheme.

Figure 3: Received SNR versus the number of IRS reflection sets with $K=1$ and
$\alpha=1$. Figure 4: Received SNR versus the number of IRS reflection sets
with $K=1$ and $\alpha=2$.

###  IV-C Simulation Results

We evaluate the received SNR by different schemes in both single-user and
multiuser scenarios. For the proposed scheme, we first use the single-layer NN
to estimate ${{{\boldsymbol{h}}}}_{k},k\in{\cal{K}}$, and then apply the
geometry-based method and the SDR method to optimize the IRS passive
reflection in the single-user and multiuser scenarios, respectively (labeled
as “NN-GE” and “NN-SDR”), as presented in Section III-C. In addition, for both
scenarios, we also show the performance by directly applying the successive
refinement method, where the IRS passive reflection is initialized based on
RMS given in (27) (labeled as “NN-SR”).

First, Fig. 3 and Fig. 4 show the received SNR under different schemes in the
single-user case with the number of controlling bits for IRS phase shifts
$\alpha=1$ and $\alpha=2$, respectively. It is observed that both the NN-GE
and NN-SR methods significantly outperform the benchmark schemes by fully
exploiting the users’ power measurements for channel estimation. In
particular, with increasing $M$, the performance of our proposed scheme
quickly converges to the upper bound achievable with perfect CSI. Moreover,
the SNR performance improves by increasing $\alpha$ from 1 to 2, as expected,
thanks to the higher phase-shift resolution for both channel estimation and
reflection design. Furthermore, the small gap between the NN-SDR and NN-SR
schemes demonstrates that the IRS passive reflection can be more efficiently
optimized with linear complexity over $N$ if small performance loss is
tolerable. Next, Fig. 5 and Fig. 6 show the minimum received SNR among $K=5$
users with $\alpha=1$ and $\alpha=2$, respectively. Similar observations for
the single-user case can be made for the multiuser case. In addition, it is
observed that CSM performs worse than RMS in the multiuser case, due to its
inefficacy to adapt to more complex utility functions such as that given in
(28) when $K>1$.

Figure 5: Minimum received SNR versus the number of IRS reflection sets with
$K=5$ and $\alpha=1$.  Figure 6: Minimum received SNR versus the number of IRS
reflection sets with $K=5$ and $\alpha=2$.

##  V Conclusion

In this paper, we proposed a new IRS channel estimation method based on user
received signal power measurements with randomly generated IRS reflections, by
exploiting a simple single-layer NN formulation. Numerical results showed that
the IRS passive reflection design based on the estimated IRS channels can
significantly outperform the existing power measurement based schemes and
approach the optimal performance assuming perfect CSI with significantly
reduced power measurements in an IRS-aided multiuser communication system. The
proposed IRS channel estimation and reflection optimization approach can be
extended to other setups such as multi-antenna BS with adaptive precoding,
multi-antenna user receivers, multiple IRSs, which will be studied in future
work.

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# PDM Klein-Gordon particles in Gödel-type Som-Raychaudhuri cosmic string
spacetime background

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

###### Abstract

Abstract: In Gödel-type Som-Raychaudhuri (SR) cosmic string spacetime
background, we re-cycle the Klein-Gordon (KG) oscillators and report their
_correct_ exact solutions. We argue that the mathematical collapse of the KG-
equation into the 2-dimensional radial Schrödinger-like oscillator does not
yield that the parametric characterizations of one is inherited by the other.
The angular frequency (positive) in the Schrödinger case is replaced by an
_irrational_ frequency-like (positive and negative) in the KG-case. Inheriting
the Schrödinger oscillator’s parametric characterizations implies that at
least half of the spectra (the negative part) is lost in the process. We also
introduce KG-oscillators in pseudo-Gödel SR-type spacetime that admit
invariance and isospectrality with those in Gödel SR-type spacetime
background. We introduce position-dependent mass (PDM) settings to KG-
particles in 4-vector and scalar Lorentz potentials in magnetic field in the
Gödel SR-type spacetime background. Four illustrative examples of fundamental
nature are discussed and their exact or conditionally exact solvability are
reported. Amongst are, a PDM KG-Coulombic particle in 4-vector and scalar
Coulombic Lorentz potentials in magnetic field, a PDM KG-Coulombic particle in
4-vector and scalar Coulombic Lorentz potentials in magnetic field at zero
vorticity, a PDM KG-Coulombic particle in equally mixed 4-vector and scalar
Coulombic Lorentz potentials in magnetic field, and a quasi-free PDM KG-
oscillator. We also emphasis that the biconfluent Heun polynomial approach, to
the effective oscillator plus Cornell type potential, yields conditionally
exact solvability that paralyzes the solution from collapsing into that of
pure Coulombic one.

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

Keywords: Klein-Gordon (KG) particles, Gödel-type Som-Raychaudhuri cosmic
string spacetime, effective position-dependent mass (PDM), PDM-KG Coulombic
and oscillators, quasi-free KG-oscillators.

##  I Introduction

Early universe theories have predicted several topological structural defects
in spacetime that have inspired intensive research studies in quantum gravity,
such as domain wall Vilenkin 1983 ; Vilenkin 1985 , cosmic string Kibble 1976
; Hiscock 1985 , global monopole Barriola 1989  and textures Cruz 2007 . It
has been observed that the energy levels of relativistic and non-relativistic
quantum systems are infected by such topological defects, not only in general
relativity but also in the geometrical theory of topological defects in
condensed matter. For example, in general relativity the Gödel spacetime
metric Godel 1949 , with an embedded cosmic string, introduces itself as the
first cosmological solution with rotating matter. Its compact form allowed
analytical research studies of many physical and mathematical systems in
gravitational backgrounds with rotation and causality violation. Rebouças and
Tiomno Tiomno 1983  have examined the conditions for spacetime homogeneity of
the Riemannian manifold with a Gödel spacetime background and found that the
Gödel universe is homogeneous in spacetime (ST-homogeneous). They have shown
that all ST-homogeneous Gödel-type metrics characterized by the vorticity
$\Omega$ and a given value of the parameter $\tilde{\mu}$
($-\infty\leq\tilde{\mu}^{2}\leq\infty$) can be transformed in polar
coordinates Drukker 2004 ; Carvalho 2014 ; Tiomno1 1992 ; Wang 2015  to

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

Where for $\tilde{\mu}^{2}<0$ there is an infinite number of successive causal
and noncausal regions, for $0\leq\tilde{\mu}^{2}<\Omega^{2}$ there is one
noncausal region for a given $r>r_{c}$ ( $r_{c}$ is the critical radius), and
$\tilde{\mu}=\Omega$ corresponds to the Gödel spacetime metric (the first
exact Gödel solution of the Einstein equation describing a complete causal,
ST-homogeneous rotating universe Tiomno 1983 ). 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 . Where the disclination parameter
$\alpha$ 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 $\alpha=1$ corresponds to
Minkowski flat spacetime metric. Moreover, the covariant metric tensor
associated with the Som-Raychaudhuri spacetime is 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)\text{ };\text{ \
}\det\left(g\right)=-\alpha^{2}r^{2}.$ (3)

The Gödel SR- type ST-homogeneous metric (2) is the core of the current
methodical proposal.

On the other hand, effective position-dependent mass (PDM) settings in the von
Roos Schrödinger Hamiltonian von Roos  have attracted research interest over
the last few decads both in quantum and classical mechanics (e.g., M-L 1974 ;
Carinena Ranada Sant 2004 ; Mustafa 2019 ; Mustafa arXiv ; Mustafa Phys.Scr.
2020 ; Mustafa Habib 2007 ; Khlevniuk 2018 ; Mustafa 2015 ; dos Santos 2021 ;
Nabulsi1 2020 ; Quesne 2015 ; Tiwari 2013 ). Hereby, it has been asserted that
the PDM notion is a metaphoric manifestation of coordinate
transformation/deformation Mustafa Algadhi 2019 ; Mustafa 2020 ; Mustafa 2022
; Mustafa2 2021 ; Mustafa3 2021 ; Mustafa Algadhi1 2020 . This would, in turn,
effectively change the canonical momentum for PDM classical systems and the
momentum operator for PDM quantum systems. Namely, the PDM canonical momentum
reads $p\left(x\right)=m\left(x\right)\dot{x}$ and consequently negative the
gradient of the potential force field is no longer given by the time
derivative of the canonical momentum but rather related to the time derivative
of the pseudo-momentum (i.e., Noether momentum)
$\pi\left(x\right)=\sqrt{m\left(x\right)}\dot{x}$ Mustafa arXiv . Yet, in
quantum mechanics the PDM-momentum operator is shown to be given by
$\hat{p}_{j}\left(\mathbf{r}\right)=-i\left(\partial_{j}-\partial_{j}m\left(\mathbf{r}\right)/4m\left(\mathbf{r}\right)\right);\,j=1,2,3,$
and consequently yields, in its most simplistic one-dimensional form, the von
Roos von Roos  PDM kinetic energy operator
$\hat{T}=\left(\hat{p}_{x}\left(x\right)/\sqrt{m\left(x\right)}\right)^{2}=-m\left(x\right)^{-1/4}\partial_{x}m\left(x\right)^{-1/2}\partial_{x}m\left(x\right)^{-1/4}$,
which is known in the literature as Mustafa and Mazharimousavi’s Mustafa Habib
2007  ordering. Basically, in short, a quantum mechanical PDM particle should
be associated with the PDM-momentum operator both for relativistic and non-
relativistic PDM-quantum particles. Therefore, in the relativistic Dirac
and/or Klein-Gordon (KG) equations, the assumption that the rest mass energy
term $m\longrightarrow m+m\left(x\right)+S\left(x\right)$ does not introduce
PDM relativistic particles but rather introduces an amendment to the already
existing Lorentz scalar potential, i.e.,
$\tilde{S}\left(x\right)=m\left(x\right)+S\left(x\right)$, (c.f., e.g., ikot
2016 ; Ghabab 2016 ; Mustafa 2008 ; Mustafa 2007 ; Vitoria 2016  and
references cited therein). Very recently, however, we have introduced and
discussed PDM relativistic particles Mustafa 2022  in Gürses spacetime
background Gurses 1994 , in cosmic string spacetime in Kaluza-Klein theory
Mustafa3 2021  and in a cosmic spacetime with a cosmic space-like dislocation
Mustafa2 2021 . It would be, therefore, interesting to study PDM KG-particles
in the Gödel-type Som-Raychaudhuri (SR) spacetime background Tiomno1 1992 ;
Bruce 1993 ; Das 2008 ; Carvalho 2016 ; Garcia 2017 ; Ahmed1 2018 ; Lutfuoglu
2020 .

The organization of our paper is in order. In section 2, we recycle and
discuss a KG-oscillators in the Gödel SR-type cosmic string spacetime
background. Hereby, we pinpoint and emphasis (along with the analysis carried
out by Fernandez Fernandez 2021  on the related references cited therein, some
of which are cited here) that the KG-oscillators and the Schrödinger-
oscillator are two different quantum mechanical systems and should not be
confused with each other. The mathematical collapse of the KG equation into
Schrödinger-like oscillator does not mean that the parametric
characterizations are copied from one into the other. By recycling this
problem, we provide solid/correct interpretation grounds to be used in the
subsequent sections. We report, in section 3, a set of KG-oscillators in
pseudo-Gödel SR-type cosmic string spacetime that admit invariance and
isospectrality with the KG-oscillator discussed in section 2. We introduce, in
section 4, PDM-settings (metaphorically speaking) for KG-particles in 4-vector
and/or scalar Lorentz potentials in Gödel SR-type cosmic string spacetime
background and subjected to a magnetic field. Therein, we discuss four
illustrative examples of fundamental nature: (i) PDM-KG particles in a
4-vector and scalar Lorentz potentials and a magnetic field in the Gödel SR-
type cosmic string spacetime background, (ii) a PDM-KG Coulombic particle in
4-vector and scalar coulombic potentials and a magnetic field in Gödel SR-type
spacetime at zero vorticity, (iii) a PDM-KG Coulombic particle in equally
mixed 4-vector and scalar Coulombic Lorentz potentials and a magnetic field in
Gödel SR-type spacetime, and (iv) a quasi-free PDM KG-oscillator in Gödel SR-
type cosmic string spacetime background. Our concluding remarks are given in
section 5.

##  II Klein-Gordon oscillators in the Gödel SR-type cosmic string spacetime
background

The KG-equation describing a spin-0 particle of rest mass energy $m$ (denoting
$mc^{2}$ in $c=\hbar=1$ units) in the Gödel SR- type ST-homogeneous metric (2)
is given by

$\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Psi\right)=m^{2}\Psi.$
(4)

Which, in a straightforward manner, would result

$\left\\{-\partial_{t}^{2}+\left(\partial_{r}^{2}+\frac{1}{r}\partial_{r}\right)+\left(\Omega\,r\,\partial_{t}-\frac{1}{\alpha\,r}\partial_{\varphi}\right)^{2}+\partial_{z}^{2}-m^{2}\right\\}\Psi=0.$
(5)

The substitution of

$\Psi\left(t,r,\varphi,z\right)=\exp\left(i\left[\ell\varphi+k_{z}z-Et\right]\right)\psi\left(r\right)=\exp\left(i\left[\ell\varphi+k_{z}z-Et\right]\right)\frac{R\left(r\right)}{\sqrt{r}}$
(6)

would yield

$R^{\prime\prime}\left(r\right)+\left[\lambda-\frac{\left(\tilde{\ell}^{2}-1/4\right)}{r^{2}}-\tilde{\omega}^{2}\,r^{2}\right]R\left(r\right)=0,$
(7)

where

$\lambda=E^{2}-2\,\tilde{\ell}\,\Omega\,E-k_{z}^{2}-m^{2}\text{ ; \
}\tilde{\ell}=\frac{\ell}{\alpha};\text{ }\tilde{\omega}^{2}=\Omega^{2}E^{2}.$
(8)

It is obvious that this equation (7) represents a KG-oscillator (as a
manifestation of the Gödel SR-type spacetime structure) and resembles the one-
dimensional form of the two-dimensional radially symmetric Schrödinger-
oscillator. However, the parameter $\tilde{\omega}=\pm\left|\Omega E\right|$
should never be confused with the Schrödinger-oscillator frequency for it does
not admit the same parametric characterization..Therefore, this equation would
only mathematically inherit the textbook exact eigenvalues and eigenfunctions
of the Schrödinger-oscillator so that

$\lambda=2\tilde{\omega}\left(2n_{r}+\left|\tilde{\ell}\right|+1\right)$ (9)

and

$R\left(r\right)\sim
r^{\left|\tilde{\ell}\right|+1/2}\exp\left(-\frac{\tilde{\omega}r^{2}}{2}\right)L_{n_{r}}^{\left|\tilde{\ell}\right|}\left(\tilde{\omega}r^{2}\right),$
(10)

where $L_{n_{r}}^{\left|\tilde{\ell}\right|}\left(\tilde{\omega}r^{2}\right)$
are the associated Laguerre polynomials. Under such settings, one obtains

$E=\Omega\,\left(2n_{r}+\left|\tilde{\ell}\right|+\tilde{\ell}+1\right)\pm\sqrt{\Omega^{2}\,\left(2n_{r}+\left|\tilde{\ell}\right|+\tilde{\ell}+1\right)^{2}+m^{2}+k_{z}^{2}}.$
(11)

We clearly observe, hereby, that the vorticity parameter $\Omega$ plays a
magnetic field-like role of lifting the degeneracies associated with the
irrational magnetic quantum number
$\tilde{\ell}=\ell/\alpha=\pm\left|\tilde{\ell}\right|$ and results in quasi
Landau energy levels. Moreover, it should be noted here that for $\Omega=0$,
$k_{z}=0$ and $\alpha=1$, the rest mass energies for the one-dimensional KG-
particle are retrieved, with $k_{z}=0$, as a natural tendency of the more
general case at hand. However, when this result is compared with that of
Carvalho et al Carvalho 2014  (i.e., equation (14) in Carvalho 2014 ) we
observe that the negative energies are missed in their results and should be
corrected into those reported in (11) above. 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 Francisco 2020  (their Eq. (18)) for zero linear confinement.
That would also include the results reported by Ahmed Ahmed 2020 , where not
only half of the spectra (the negative energies) is missed but also all
nodeless (i.e., states with $n_{r}=0$) states.

##  III KG-oscillators in pseudo-Gödel SR-type cosmic string spacetime
admitting invariance and isospectrality

Let us consider that the Gödel SR-type spacetime metric $ds^{2}$ of (2) be
transformed so that

$ds^{2}\longrightarrow
d\,\tilde{s}^{2}=-\left(d\tilde{t}+\alpha\,\Omega\,\tilde{r}^{2}d\tilde{\varphi}\right)^{2}+\alpha^{2}\,\tilde{r}^{2}d\tilde{\varphi}^{2}+d\tilde{r}^{2}+d\tilde{z}^{2}$
(12)

where

$\begin{tabular}[]{llll}$d\tilde{t}=dt,d\tilde{r}=\sqrt{f\left(r\right)}dr,$&$\tilde{r}=\sqrt{Q\left(r\right)}r,$&$d\tilde{\varphi}=d\varphi,$&$d\tilde{z}=dz$\end{tabular},$
(13)

and hence

$\frac{d\tilde{r}}{dr}\Longrightarrow\sqrt{f(r)}=\sqrt{Q\left(r\right)}\left[1+\frac{Q^{\prime}\left(r\right)}{2Q\left(r\right)}r\right],$
(14)

to govern the correlation between $f\left(r\right)$ and $Q\left(r\right)$.
Under such transformation settings, a pseudo-Gödel SR-type spacetime metric is
manifestly introduced as

$d\,\tilde{s}^{2}=-\left(dt+\alpha\,\Omega\,g\left(r\right)d\varphi\right)^{2}+\alpha^{2}\,g\left(r\right)d\varphi^{2}+f\left(r\right)dr^{2}+dz^{2}$
(15)

Then the covariant metric tensor associated to such a pseudo-Gödel SR-type
spacetime is given by

$\tilde{g}_{\mu\nu}=\left(\begin{tabular}[]{cccc}$-1\vskip 3.0pt plus 1.0pt
minus 1.0pt$&$\,0\,$&$-\alpha\,\Omega g\left(r\right)$&$\,0$\\\
$0$&$f\left(r\right)\vskip 3.0pt plus 1.0pt minus 1.0pt$&$0$&$0$\\\
$-\alpha\,\Omega\,g\left(r\right)$&$\,0$&$\,\alpha^{2}\,g\left(r\right)\left(1-\Omega^{2}\,g\left(r\right)\right)\,$&$0$\\\
$0$&$0$&$0$&$1$\end{tabular}\right)\Longleftrightarrow\tilde{g}^{\mu\nu}=\left(\begin{tabular}[]{cccc}$\left(\Omega^{2}\,g\left(r\right)-1\vskip
3.0pt plus 1.0pt minus 1.0pt\right)$&$\,0\,$&$-\frac{\Omega}{\alpha}$&$0$\\\
$0$&$\,\frac{1\vskip 3.0pt plus 1.0pt minus
1.0pt}{f\left(r\right)}$&$0$&$0$\\\
$-\frac{\Omega}{\alpha}$&$\,0$&$\,\frac{1}{\alpha^{2}\,g\left(r\right)}\,$&$0$\\\
$0$&$0$&$0\,$&$\,1$\end{tabular}\right),$ (16)

where $g\left(r\right)=Q\left(r\right)r^{2}$ and
$\det\left(g\right)=-\alpha^{2}\,f\left(r\right)g\left(r\right)$. We may now
use similar assumption to that in (6) and define

$\Psi\left(t,\tilde{r},\varphi,z\right)=\exp\left(i\left[\ell\varphi+k_{z}z-Et\right]\right)\frac{R\left(\tilde{r}\right)}{\sqrt{\tilde{r}}};\text{
}\tilde{r}=\sqrt{g\left(r\right)}=\sqrt{Q\left(r\right)}r.$ (17)

Consequently, the KG-equation (4) would read

$\left[\frac{1}{\sqrt{g\left(r\right)f\left(r\right)}}\partial_{r}\left(\sqrt{\frac{g\left(r\right)}{f\left(r\right)}}\partial_{r}\right)+\lambda-\frac{\tilde{\ell}^{2}}{g\left(r\right)}-\tilde{\omega}^{2}g\left(r\right)\right]\frac{R\left(\tilde{r}\right)}{\sqrt{\tilde{r}}}=0,$
(18)

to imply, with $\tilde{r}=\sqrt{g\left(r\right)}=\sqrt{Q\left(r\right)}r$,
that

$\frac{d^{2}R\left(\tilde{r}\right)}{d\tilde{r}^{2}}+\left[\lambda-\frac{\left(\tilde{\ell}^{2}-1/4\right)}{\tilde{r}^{2}}-\tilde{\omega}^{2}\,\tilde{r}^{2}\right]R\left(\tilde{r}\right)=0.$
(19)

Obviously, this equation is invariant with (7) and shares the same
eigenvalues. We may now conclude that the two equations (7) and (19) are
invariant and isospectral. Moreover, our transformed metric in (15) is a
pseudo-Gödel SR-type spacetime metric under our transformation recipe in (13)
and (14). This would in effect introduce new pseudo-Gödel SR-type solutions
provided that $Q\left(r\right)$ and $f\left(r\right)$ are correlated by (14).
Yet, one should notice that for $Q\left(r\right)=1\Longleftrightarrow$
$f\left(r\right)=1$, our pseudo-Gödel SR-type spacetime metric (15) collapses
into that of Gödel SR-type in (2). This would in effect introduce a new set of
the Gödel SR-type spacetime metrics that may very well be classified as
pseudo-Gödel SR-type spacetime metrics that can be transformed back into the
Gödel SR-type spacetime metrics through the transformation in (14).

##  IV PDM-KG particles in a 4-vector and scalar Lorentz potentials and a
magnetic field in Gödel SR-type cosmic string spacetime background

The KG-equation for a spin-0 particle in a 4-vector potential $A_{\mu}$ in the
Gödel SR-type spacetime background (2) is given by

$\frac{1}{\sqrt{-g}}D_{\mu}\left(\sqrt{-g}g^{\mu\nu}D_{\nu}\Psi\right)=m^{2}\Psi,$
(20)

where the gauge-covariant derivative is given by
$D_{\mu}=\partial_{\mu}-ieA_{\mu}$. Moreover, in a recent paper, we have
asserted that the PDM-setting is a manifestation of coordinate
deformation/transformation Mustafa Algadhi 2019 ; Mustafa 2020 ; Mustafa 2022
; Mustafa2 2021 ; Mustafa3 2021  that yields to the introduction of the PDM-
momentum operator

$\mathbf{\hat{p}}\left(\mathbf{r}\right)=-i\left(\mathbf{\nabla-}\frac{\mathbf{\nabla}f\left(\mathbf{r}\right)}{4f\left(\mathbf{r}\right)}\right)\Longleftrightarrow
p_{j}=-i\left(\partial_{j}-\frac{\partial_{j}f\left(\mathbf{r}\right)}{4f\left(\mathbf{r}\right)}\right).$
(21)

This would, in effect, suggest that a PDM-quantum particle (metaphorically
speaking) is described by the PDM-momentum operator in (21) and may be
subjected to some interaction potential force fields. However, should one be
interested in invariant and isospectral systems, then our recipe in the
preceding section shall be followed (c.f., e.g., Mustafa 2020 ). In the
current section, however, we shall assume that we have only a
deformation/defect in the radial coordinate so that

$\partial_{\mu}\longrightarrow\partial_{\mu}+\mathcal{F}_{\mu};\mathcal{F}_{\mu}=\left(0,\mathcal{F}_{r},0,0\right),\mathcal{F}_{r}=\frac{f^{\prime}\left(r\right)}{4f\left(r\right)}$
(22)

(e.g., Mustafa 2022 ; Mustafa2 2021 ; Mustafa3 2021 ). This would in turn
allow us to cast equation (20) as

$\frac{1}{\sqrt{-g}}\tilde{D}_{\mu}\left(\sqrt{-g}g^{\mu\nu}\tilde{D}_{\nu}\Psi\right)=\left(m+S\left(r\right)\right)^{2}\Psi;\;\tilde{D}_{\mu}=\partial_{\mu}+\mathcal{F}_{\mu}-ieA_{\mu}.$
(23)

Consequently, one obtains

$\left\\{-D_{t}^{2}+\frac{1}{r}D_{r}\left(r\left[D_{r}-\mathcal{F}_{r}\right]\right)+\left(\Omega\,rD_{t}-\frac{1}{\alpha
r}D_{\varphi}\right)^{2}+D_{z}^{2}+\mathcal{F}_{r}\left(D_{r}-\mathcal{F}_{r}\right)-\left(m+S(r)\right)^{2}\right\\}\Psi=0.$
(24)

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

$D_{t}=\partial_{t}-ieA_{t}=\partial_{t}-iV\left(r\right),$ | $D_{r}=\partial_{r},$ | $D_{\varphi}=\partial_{\varphi}-ieA_{\varphi},$ | $D_{z}=\partial_{z},$
---|---|---|---
(25)

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 may now use the assumption of (6) in (24) and obtain

$\left\\{\partial_{r}^{2}+\frac{1}{4r^{2}}+\left(E+V\left(r\right)\right)^{2}-\left[\Omega
r\left(E+V\left(r\right)\right)+\frac{\left(\ell-eA_{\varphi}\right)}{\alpha
r}\right]^{2}-M\left(r\right)-k_{z}^{2}-\left(m+S(r)\right)^{2}\right\\}R\left(r\right)=0,$
(26)

where

$M\left(r\right)=-\frac{3}{16}\left(\frac{f^{\prime}\left(r\right)}{f\left(r\right)}\right)^{2}+\frac{1}{4}\frac{f^{\prime\prime}\left(r\right)}{f\left(r\right)}+\frac{f^{\prime}\left(r\right)}{4rf\left(r\right)}.$
(27)

This result would describe KG-particles in Gödel SR-type cosmic string
spacetime background within effective PDM introduced as a topological defect
in the coordinate system. In the following subsections, we discuss some
illustrative examples of fundamental nature.

###  IV.1 PDM-KG Coulomb particle in 4-vector and scalar Coulombic potentials
and a magnetic field in Gödel SR-type cosmic string spacetime background

Let us consider a PDM-KG particle with an effective PDM function (a
dimensionless scalar multiplier) given by

$f\left(r\right)=\exp\left(4\beta r\right),$ (28)

so that equations (22) and (27) yield

$\mathcal{F}_{r}=\frac{f^{\prime}\left(r\right)}{4f\left(r\right)}=\beta\Longleftrightarrow
M\left(r\right)=\beta^{2}+\frac{\beta}{r}.$ (29)

Moreover, for $V\left(r\right)=\beta_{1}/r$, $S\left(r\right)=\beta_{2}/r$,
and $A_{\varphi}=B_{\circ}r/2$ equation (26) would read

$\left\\{\partial_{r}^{2}-\frac{\left(\mathcal{L}^{2}-1/4\right)}{r^{2}}-\tilde{\omega}^{2}r^{2}-2E\Omega\left(\Omega\beta_{1}+\tilde{B}\right)r+\frac{\tilde{\beta}}{r}+\tilde{\lambda}\right\\}R\left(r\right)=0,$
(30)

where

$\tilde{\lambda}=E^{2}-k_{z}^{2}-m^{2}-\tilde{B}^{2}-\Omega^{2}\beta_{1}^{2}-\beta^{2}-2\,\tilde{\ell}\,\Omega\,E+2\Omega\tilde{B}\beta_{1}\text{
; \ }\tilde{\ell}=\frac{\ell}{\alpha},\tilde{B}=\frac{eB_{\circ}}{2\alpha},$
(31)

and

$\tilde{\beta}=-\beta-2m\beta_{2}+2E\beta_{1}+2\tilde{\ell}\tilde{B}-2\,\tilde{\ell}\,\Omega\beta_{1},\;\mathcal{L}^{2}=\tilde{\ell}^{2}-\beta_{1}^{2}+\beta_{2}^{2},\;\tilde{\omega}^{2}=\Omega^{2}E^{2}.$
(32)

The differential equation (30) at hand admits a solution in the form of
biconfluent Heun functions so that

$\psi\left(r\right)=\frac{R\left(r\right)}{\sqrt{r}}=\mathcal{N}\,r^{\left|\mathcal{L}\right|}\,e^{-\left(\frac{\tilde{\omega}r^{2}}{2}+\Omega\beta_{1}r+\tilde{B}r\right)}\,H_{B}\left(\alpha^{\prime},\beta^{\prime},\gamma^{\prime},\delta^{\prime},\sqrt{\tilde{\omega}}r\right),$
(33)

where

$\alpha^{\prime}=2\left|\mathcal{L}\right|\,,\,\beta^{\prime}=\frac{2\left(\Omega\beta_{1}+\tilde{B}\right)}{\sqrt{\tilde{\omega}}}\,,\,\gamma^{\prime}=\frac{\tilde{\lambda}+\left(\Omega\beta_{1}+\tilde{B}\right)^{2}}{\tilde{\omega}}\,,\delta^{\prime}=-\frac{2\tilde{\beta}}{\sqrt{\tilde{\omega}}}.$
(34)

This would result in a biconfluent Heun polynomial of degree
$n^{\prime}=2n_{r}\geq 0$ (where $n_{r}$ denotes the radial quantum number)
when $\gamma^{\prime}=2\left(2n_{r}+1\right)+\alpha^{\prime}$ and consequently
yields

$E=\,\Omega\,\left(2n_{r}+\left|\mathcal{L}\right|+\,\tilde{\ell}\,+1)\right)\pm\sqrt{\,\Omega^{2}\,\left(2n_{r}+\left|\mathcal{L}\right|+\,\tilde{\ell}\,+1)\right)^{2}+m^{2}+k_{z}^{2}+\beta^{2}-4\Omega\beta_{1}\tilde{B}}.$
(35)

This condition would in turn classify the solution as a conditionally exact
solution, therefore (c.f., e.g., Fernandez 2021 ; Mustafa 2022 ; Mustafa3 2021
for more details). Nevertheless, the choice, $n^{\prime}=2n_{r}\geq 0$, is
manifested by the requirement that the eigenvalues as well as the
eigenfunctions should naturally retrieve those of the KG-oscillator when
$\beta=\beta_{1}=\beta_{2}=\tilde{B}=0$ (e.g., Mustafa 2022 ). However, one
would notice that for the case where the vorticity $\Omega=0$, the
differential equation in (30) effectively reduces to that of the radial
Schrödinger Coulomb problem. However, neither the reported biconfluent Heun
polynomial in (33) nor the eigenvalues in (35) collapse into the corresponding
known Schrödinger-like KG-Coulombic solution. One has therefore to restart
from the beginning and work out the related solution as in the sequel
subsection.

In fact, Fernández Fernandez 2021  has provided through and comprehensive
details on the conditional exact and/or numerical exact solvability of the
corresponding one-dimensional Schrödinger model

$\left\\{\partial_{r}^{2}-\frac{\left(\mathcal{L}^{2}-1/4\right)}{r^{2}}-Ar^{2}-Br+\frac{C}{r}+E\right\\}R\left(r\right)=0.$
(36)

Fernández work, along with the arguments reported in the model above, provide
a brut-force evidence on not only the conditional exactness of the biconfluent
Heun polynomials solution but also on the lack of the correct convergence of
these polynomials into the pure Coulombic (when $A=0=B$ in (36)) spectral
problem.

###  IV.2 A PDM-KG Coulombic particle in 4-vector and scalar Coulombic
potentials and magnetic field in Gödel SR-type spacetime at zero vorticity

For such a model, we choose to work with $\Omega=0$ (no vorticity) so that
equation (30) reads

$\left\\{\partial_{r}^{2}-\frac{\left(\mathcal{L}^{2}-1/4\right)}{r^{2}}+\frac{\tilde{\beta}}{r}+\tilde{\lambda}\right\\}R\left(r\right)=0.$
(37)

Which resembles the two-dimensional (2D) Coulombic problem that admits exact
solution in the form of Whittaker and confluent hypergeometric functions as

$R\left(r\right)\sim\mathcal{\,}W_{M}\left(-\frac{i\tilde{\beta}}{2\sqrt{\tilde{\lambda}}},\left|\mathcal{L}\right|,2i\sqrt{\tilde{\lambda}}r\right)\sim
r^{\left|\mathcal{L}\right|+1/2}\exp\left(i\sqrt{\tilde{\lambda}}r\right)\,_{1}F_{1}\left(\left[\frac{1}{2}+\left|\mathcal{L}\right|+\frac{i\tilde{\beta}}{2\sqrt{\tilde{\lambda}}}\right],\left[1+2\left|\mathcal{L}\right|\right],2i\sqrt{\tilde{\lambda}}r\right).$
(38)

Consequently, a finite confluent hypergeometric polynomial of order $n_{r}$ is
obtained when

$\frac{1}{2}+\left|\mathcal{L}\right|+\frac{i\tilde{\beta}}{2\sqrt{\tilde{\lambda}}}=-n_{r}\Longleftrightarrow\sqrt{\tilde{\lambda}}=-\frac{i\tilde{\beta}}{2n_{r}+2\left|\mathcal{L}\right|+1}\Longleftrightarrow\tilde{\lambda}=-\frac{\tilde{\beta}^{2}}{\left(2n_{r}+2\left|\mathcal{L}\right|+1\right)^{2}}.$
(39)

In fact, $\tilde{\lambda}$ in (39) represents the exact eigenvalues of the
2D-Schrödinger Coulomb problem and the exact eigenfunctions are given by

$\psi\left(r\right)=\frac{R\left(r\right)}{\sqrt{r}}=\mathcal{N\,}r^{\left|\mathcal{L}\right|}\exp\left(i\sqrt{\tilde{\lambda}}r\right)\,_{1}F_{1}\left(-n_{r},\left[1+2\left|\mathcal{L}\right|\right],\Lambda
r\right);\,\Lambda=\frac{\tilde{\beta}}{2n_{r}+2\left|\mathcal{L}\right|+1},$
(40)

where $n_{r}=0,1,2,\cdots$ is the radial quantum number.

Under such settings, equation (31) along with (39) would imply, with
$\tilde{n}=n_{r}+\left|\mathcal{L}\right|+1/2$, that

$E^{2}-k_{z}^{2}-m^{2}-\tilde{B}^{2}-\beta^{2}=-\frac{\left(-\beta-2m\beta_{2}+2E\beta_{1}+2\tilde{\ell}\tilde{B}\right)^{2}}{\left(2\tilde{n}\right)^{2}}\text{,}$
(41)

where $\tilde{\ell}=\ell/\alpha,\tilde{B}=eB_{\circ}/2\alpha,$, and
$\mathcal{L}^{2}=\tilde{\ell}^{2}-\beta_{1}^{2}+\beta_{2}^{2}$. This result
can be simplified into

$E=\frac{-\beta_{1}\Lambda_{2}\pm\sqrt{4\tilde{n}^{2}\Lambda_{1}\left(\tilde{n}^{2}+\beta_{1}^{2}\right)-\tilde{n}^{2}\Lambda_{2}^{2}}}{2\left(\tilde{n}^{2}+\beta_{1}^{2}\right)},$
(42)

where

$\Lambda_{1}=k_{z}^{2}+m^{2}+\tilde{B}^{2}+\beta^{2};\;\Lambda_{2}=-\beta-2m\beta_{2}+2\tilde{\ell}\tilde{B}.$
(43)

Obviously, the term $2\tilde{\ell}\tilde{B}$ (i.e., the magnetic field effect)
in $\Lambda_{2}$ removes the degeneracies associated with
$\left|\mathcal{L}\right|$ in the new irrational quantum number
$\tilde{n}=n_{r}+\left|\mathcal{L}\right|+1/2$ of (42). Moreover, one should
notice that the PDM settings of (28) and (29) have provided a Coulomb-like
interaction as a byproduct of its own.

###  IV.3 A PDM-KG Coulombic particle in equally mixed 4-vector and scalar
Lorentz potentials and a magnetic field in Gödel SR-type spacetime

We now consider a PDM particle described by (28) in Gödel SR-type spacetime in
a magnetic field along with equally mixed 4-vector and scalar Coulombic
Lorentz potentials and a magnetic field, where
$V\left(r\right)=S\left(r\right)=\beta_{\circ}/r:\,$
$\beta_{1}=\beta_{2}=\beta_{\circ}$. In this case, equation (30) reads

$\left\\{\partial_{r}^{2}-\frac{\left(\,\tilde{\ell}\,^{2}-1/4\right)}{r^{2}}-\tilde{\omega}^{2}r^{2}-2E\Omega\left(\Omega\beta_{\circ}+\tilde{B}\right)r+\frac{\tilde{\beta}}{r}+\tilde{\lambda}\right\\}R\left(r\right)=0,$
(44)

where

$\tilde{\lambda}=E^{2}-k_{z}^{2}-m^{2}-\tilde{B}^{2}-\Omega^{2}\beta_{\circ}^{2}-\beta^{2}-2\,\tilde{\ell}\,\Omega\,E+2\Omega\tilde{B}\beta_{\circ}\text{
; \ }\tilde{\ell}=\frac{\ell}{\alpha},\tilde{B}=\frac{eB_{\circ}}{2\alpha},$
(45)

and

$\tilde{\beta}=-\beta-2m\beta_{\circ}+2E\beta_{\circ}+2\tilde{\ell}\tilde{B}-2\,\tilde{\ell}\,\Omega\beta_{\circ},\;\tilde{\omega}^{2}=\Omega^{2}E^{2}.$
(46)

Which admits conditionally exact solution in the form of biconfluent Heun
polynomials so that

$\psi\left(r\right)=\frac{R\left(r\right)}{\sqrt{r}}=\mathcal{N}\,r^{\left|\tilde{\ell}\,\right|}\,e^{-\left(\frac{\tilde{\omega}r^{2}}{2}+\Omega\beta_{\circ}r+\tilde{B}r\right)}\,H_{B}\left(\alpha^{\prime},\beta^{\prime},\gamma^{\prime},\delta^{\prime},\sqrt{\tilde{\omega}}r\right),$
(47)

where

$\alpha^{\prime}=2\left|\tilde{\ell}\,\right|\,,\,\beta^{\prime}=\frac{2\left(\Omega\beta_{\circ}+\tilde{B}\right)}{\sqrt{\tilde{\omega}}}\,,\,\gamma^{\prime}=\frac{\tilde{\lambda}+\left(\Omega\beta_{\circ}+\tilde{B}\right)^{2}}{\tilde{\omega}}\,,\delta^{\prime}=-\frac{2\tilde{\beta}}{\sqrt{\tilde{\omega}}},$
(48)

with the condition $\gamma^{\prime}=2\left(2n_{r}+1\right)+\alpha^{\prime}$ to
consequently yield

$E=\,\,\Omega\left(2n_{r}+\left|\tilde{\ell}\right|+\,\tilde{\ell}\,+1\right)\pm\sqrt{\Omega^{2}\left(2n_{r}+\left|\tilde{\ell}\right|+\,\tilde{\ell}\,+1\right)^{2}+m^{2}+k_{z}^{2}+\beta^{2}-4\Omega\beta_{\circ}\tilde{B}}.$
(49)

In each of the three PDM-KG Coulombic particles above, we notice that the
first term of the energy eigenvalues (i.e., equations (35),(42), and (49))
lifts the degeneracies associated with the magnetic quantum number
$\ell=\pm\left|\ell\right|$.

###  IV.4 A quasi-free PDM KG-oscillator in Gödel SR-type cosmic string
spacetime background

We now consider a free PDM KG-particle with not only position-dependent but
also energy-dependent scalar multiplier deformation function in the form of

$f\left(r\right)=\exp\left(2\xi Er^{2}\right),$ (50)

which implies that

$M\left(r\right)=E^{2}\xi^{2}r^{2}+2E\xi.$ (51)

Let us assume that this PDM KG-particle is free from the Lorentz potentials
(i.e., $V\left(r\right)=S\left(r\right)=0$ and no magnetic field effect
$eA_{\varphi}=0$, hence quasi-free). In this case, equation (26) reads

$\left\\{\partial_{r}^{2}+\frac{\left(\tilde{\ell}\,^{2}-1/4\right)}{r^{2}}-\tilde{\Omega}^{2}r^{2}+{\Large\tilde{\varepsilon}}\right\\}R\left(r\right)=0,$
(52)

which resembles the 2D radial Schrödinger harmonic oscillator problem with

${\Large\tilde{\varepsilon}=}E^{2}-2\tilde{\ell}\Omega
E-2E\xi-\left(m^{2}+k_{z}^{2}\right),\;\tilde{\Omega}^{2}=E^{2}\left(\Omega^{2}+\xi^{2}\right).$
(53)

Under such settings, the textbook solution is therefore given by

${\Large\tilde{\varepsilon}}=2\tilde{\Omega}\left(2n_{r}+\left|\tilde{\ell}\right|+1\right)$
(54)

and

$R\left(r\right)\sim
r^{\left|\tilde{\ell}\right|+1/2}\exp\left(-\frac{\tilde{\Omega}r^{2}}{2}\right)L_{n_{r}}^{\left|\tilde{\ell}\right|}\left(\tilde{\Omega}r^{2}\right),$
(55)

where $L_{n_{r}}^{\left|\tilde{\ell}\right|}\left(\tilde{\Omega}r^{2}\right)$
are the associated Laguerre polynomials. One should notice that the choice of
the deformation in (50) is just to facilitate the calculation to be able to
get

$E=\tilde{\gamma}\pm\sqrt{\tilde{\gamma}^{2}+m^{2}+k_{z}^{2}};\;\tilde{\gamma}=\tilde{\ell}\Omega+\xi+\left(\Omega^{2}+\xi^{2}\right)\left(2n_{r}+\left|\tilde{\ell}\right|+1\right).$
(56)

This particular example shows that PDM-KG particles (PDM-particles in general)
may very well create their own byproducted effective interactions that may
lead to bound states. Hence the notion of quasi-free particles is unavoidable
in the process. Moreover, we observe that quasi-Landau type energy levels
manifestly introduced by the vorticity parameter $\Omega$ in
$\;\tilde{\gamma}$. Hence, the vorticity parameter have similar effect as that
of the magnetic field..

##  V Concluding remarks

The harmonic oscillator and Coulomb problems form the most fundamental systems
in non-relativistic and relativistic quantum mechanics in different spacetime
backgrounds. Their exact solvability and impressive/superb
pedagogical/research implementation makes them systems of great physical
relevance. In the current methodical proposal, we considered KG-particles
(harmonic oscillator and Coulombic types) in Gödel SR-type cosmic string
spacetime backgrounds and under the influence of different 4-vector and/or
scalar Lorentz potentials settings Quigg 1979 . We started with the KG-
oscillators in Gödel SR-type cosmic string spacetime backgrounds and reported
their exact solutions. Hereby, we have emphasized that the mathematical
collapse of the KG-equation into the two dimensional radially symmetric
Schrödinger-oscillator (hence the metaphoric notion of _KG-oscillator_ emerged
in the process) does not mean that the parametric characterizations are copied
from one to the other. The parameter $\tilde{\omega}$ of (8) is not related to
the angular frequencies of the harmonic oscillator (e.g., Vitoria 2018 ;
Francisco 2020 ; Ahmed 2020 ; Mirza 2004 ) but rather admits values given by
$\tilde{\omega}=\pm\left|\Omega E\right|$ (e.g., Fernandez 2021 ). The usage
of $\tilde{\omega}$ as the angular frequency of the harmonic oscillator had
the consequences of losing at least half of the spectra (only positive
energies were reported and anti-particle solutions were dismissed from the
relativistic theory). The recycling of the KG-oscillators in Gödel SR-type
cosmic string spacetime backgrounds in section 2 is unavoidable, therefore. We
have, in the same section, pinpointed that the results reported by Carvalho et
al Carvalho 2014  on the quantum influence of topological defects in Gödel
spacetime, by Vitória et al Vitoria 2018  on the linear confinement of a
scalar particle in Gödel-type spacetime (including the related comment by Neto
et al Francisco 2020 ), and those reported by Ahmed Ahmed 2020  on KG-
Coulombic type particles should be redirected to reflect the results of the
current proposal, therefore. Not only they have lost all negative energies but
also they have lost all nodeless states form the spectra (excluding Carvalho
et al Carvalho 2014 , where nodeless states, with $n_{r}=0$, are reported
therein). Here, we have just mentioned few of so many references that are not
cited herein but lie far beyond the scope of the current study. Subsequently,
we have reported (in section 3) the KG-oscillators in pseudo-Gödel SR-type
cosmic string spacetime that admit invariance and isospectrality with the KG-
oscillator.

On the PDM settings side of the current methodical proposal, we have
introduced a general recipe for PDM-KG particles in a 4-vector and scalar
Lorentz potentials and a magnetic field in Gödel SR-type cosmic string
spacetime background (in section 4). Therein, we have considered four
illustrative examples of fundamental nature: (i) a PDM-KG Coulomb particle in
4-vector and scalar Coulombic potentials and a magnetic field in Gödel SR-type
cosmic string spacetime background, (ii) a PDM-KG Coulombic particle in
4-vector and scalar Coulombic potentials and magnetic field in Gödel SR-type
spacetime at zero vorticity, (iii) a PDM-KG Coulombic particle in equally
mixed 4-vector and scalar Lorentz potentials and a magnetic field in Gödel SR-
type spacetime, and (iv) a quasi-free PDM KG-oscillator in Gödel SR-type
cosmic string spacetime background.

In the light of our experience on the PDM KG-particles (this would also
include constant mass settings with $f\left(r\right)=1$, in general) in Gödel
SR-type cosmic string spacetime background, the following notes are inevitably
unavoidable:

  1. 1.

For the PDM-KG Coulomb particle in 4-vector and scalar Coulombic potentials
and a magnetic field, the corresponding one-dimensional Schrödinger equation
(30), indulges within a harmonic oscillator-like in a Cornel type potentials
Quigg 1979 , that admits a biconfluent Heun polynomial type solution (33)
which consequently yields some conditionally exact energies reported in (35).
However, it is obvious that neither the biconfluent Heun polynomials in (33)
nor the eigenvalues in (35) collapse into the corresponding known Schrödinger-
like KG-Coulombic solution, when the linear plus oscillator like potential
terms are switched off..

  2. 2.

We have deliberately considered a PDM-KG Coulombic particle in Gödel SR-type
spacetime at zero vorticity to provide a brut-force evidence on the drawback
of the use of the biconfluent Heun polynomials approach. We have observed that
the more general and conditionally exact solution for system (30) does not
collapse into that for the pure Coulombic one of (37).

  3. 3.

The equally mixed 4-vector and scalar Lorentz potentials is a common practice
in quarkonium spectroscopy Quigg 1979 ; Mustafa 1998 . We have therefore
introduced such feasible mixture/model of interaction potentials. Again, this
system suffers from the drawback of the conditional exact solvability of the
biconfluent Heun polynomial approach, as in point 1 above.

  4. 4.

PDM-KG particles (PDM-particles in general) may generate their own byproducted
effective interactions that support bound states. They should, therefore, be
labelled as quasi-free PDM-particles. Yet, quasi-Landau type energy levels
could be introduced by the vorticity $\Omega.$of spacetime that has similar
effect of the magnetic field.

The drawback in the biconfluent Heun polynomial approach (a power series
manifested solution) inspires the search for a more sophisticated/reliable
approach. We foresee that an alternative could be sought in the well known
shifted $1/N$ or shifted-$\ell$ expansion quasi-perturbation method (e.g.,
Imbo 1983 ; Imbo 1984 ; Mustafa 1993 ; Mustafa 1994 ; Mustafa 1991 ; Mustafa
1998 ; Mustafa 2000 ; Mustafa 2002 ). Such quasi-perturbation methods result
the exact energies for the harmonic oscillator like and Coulomb like
interactions (especial cases of (33)) from the zeroth (leading) order
correction, where all higher order correction identically vanish. This method
is also reported to provide highly accurate results for the Coulomb plus
oscillator Mustafa 1993 ; Mustafa 1994 , a close form of potential to the one
used in section 4-A and 4-C. Work in this direction is in progress.

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