Relativistic Motions of Spin-Zero Quantum Oscillator Field in a Global
Monopole Space-Time with External Potential and AB-effect

Faizuddin Ahmed
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Department of Physics, University of Science & Technology Meghalaya, Ri-Bhoi,
Meghalaya-793101, India

###### Abstract

: In this paper, we analyze a spin-zero relativistic quantum oscillator in the
presence of the Aharonov-Bohm magnetic flux in a space-time background
produced by a point-like global monopole (PGM). Afterwards, we introduce a
static Coulomb-type scalar potential and subsequently with the same type of
vector potential in the quantum system. We solve the generalized Klein–Gordon
oscillator analytically for different functions (e. g. Coulomb- & Cornell-type
functions) and obtain the bound-states solutions in each case. We discuss the
effects of topological defects associated with the scalar curvature of the
space-time and the Coulomb-type external potentials on the energy profiles and
the wave function of these oscillator fields. Furthermore, we show that the
obtained energy eigenvalues depend on the magnetic quantum flux which gives
rise to the gravitational analogue of the Aharonov-Bohm (AB) effect.

Keywords: Topological defects, Relativistic wave equations, solutions of wave
equations: bound states

PACS Number(s): 03.65.Pm, 03.65.Ge, 14.80.Hv

##  1 Introduction

The relativistic quantum motions of scalar and spin-half particles under
gravitational effects produced by different curved space-time geometries, for
instance, Gödel and Gödel-type space-times AA1 ; AA2 , topologically trivial
AA3 , and nontrivial LCNS  space-times, topological defects (which will
discuss in the next paragraphs) have been of interest among researchers. The
spin-$0$ scalar particles (bosons) are described by the Klein-Gordon equation
while the Dirac equation for the spin-half particles (fermion). These wave
equations have been solved using interaction potentials (scalar and vector) of
different kinds in quantum systems by different techniques (e. g., power
series method, supersymmetric approach, factorization method, the Nikiforov-
Uvarov method). Moreover, a harmonic oscillator acts as a prototype model in
different areas of physics, such as condensed matter physics, quantum
statistical mechanics, and quantum field theory. This quantum field theory in
curved space-time is considered the first approximation to quantum gravity. In
the gravitational background, it is necessary to analyze a single-particle
state to make a consistent quantum field theory. A well-known version for the
relativistic harmonic oscillator or additional form of interaction was
proposed in Ref. MM  for a spin-half particle which is called the Dirac
oscillator. This new alternative form of interaction furnishes the Schrödinger
equation with a harmonic oscillator potential in the non-relativistic scheme
SP . Inspired by this Dirac oscillator model, a similar model for spin-zero
particles (bosons) is studied by the substitution of the radial momentum
operator $\vec{p}\to(\vec{p}-i\,M\,\omega\,\vec{r})$ in the Klein-Gordon
equation, and this new model is called the Klein-Gordon oscillator. Several
researchers have been investigated this Klein-Gordon oscillator, for instance,
in noncommutative space bb , with Coulomb-type scalar bb1  and vector
potentials bb2 , under the influence of Coulomb-type and linear confining
potential bb3 , with Cornell-type potential in fifth dimensional Minkowski
space-time using the Kaluza-Klein theory EHR .

Some Grand Unified Theories suggested that topological defects may have been
produced during the phase transition in the early universe through a
spontaneous symmetry breaking mechanism TWBK ; AVV . Various topological
defects includes cosmic strings WAH ; SZ2 , domain walls AVV ; AVV2 , and
global monopoles MBAV ; DPB ; TRPC ; ERBM . In condensed matter physics AVV3 ;
TV ; MOK ; HK ; TWBK2 ; CF22 ; CF33 , these linear defects are related to
screw dislocations (or torsion) and disclinations (curvature) MOK ; RAP . A
global monopole is a heavy object characterized by spherically symmetry and
divergent mass. The gravitational field of a static global monopole was found
by Barriola et al. MBAV  and are expected to be stable against spherical as
well as polar perturbation. The effects of global monopole in quantum
mechanical systems have been studied, for instance, in the non-relativistic
limit, the harmonic oscillators CF44 , quantum scattering of charged or
massive particles ALCO2 ; ERBM33 ; ERBM44 , solutions of the Klein-Gordon
equation in the presence of a dyon, magnetic flux and scalar potential ALCO .
In the relativistic limit, studies on hydrogen and pionic atom ERFM2 , exact
solutions of scalar bosons in the presence of Coulomb potential ABHA , the
Dirac and Klein–Gordon oscillators EAFB , the generalized Klein-Gordon
oscillator SZ , and the Klein–Gordon oscillator with rainbow gravity SZ3 . In
addition, global monopole has been studied, for instance, in scalar self-
energy for a charged particle AAS ; AAS2 , induced self-energy on a static
scalar charged particle AAS3 , vacuum polarization for mass-less scalar fields
AAS4 , vacuum polarization for mass-less spin-$1/2$ fields AAS5 , vacuum
polarization effects in the presence of the Wu–Yang magnetic monopole AAS6 .
Moreover, gravitational deflection of light by a rotating global monopole
space-time have been investigated in Ref. AAS7  and more recently in Ref. AAS9
, and a charged global monopole in Ref. AAS8 . Other topological defects, such
as cosmic strings which are one-dimensional linear defects have been studied
in quantum system, for example, the Klein-Gordon oscillator in fifth-
dimensional cosmic string space-time using the Kaluza-Klein theory JC , the
Klein-Gordon oscillator in a cosmic string space-time with an external
magnetic field AB , the Dirac field and oscillator in a spinning cosmic string
MH5 , the Dirac oscillator under the influence of non-inertial effects in
cosmic string space-time KB10 , the relativistic quantum dynamics of
Klein–Gordon scalar fields subject to Cornell-type potential in a spinning
cosmic-string space-time MH , spin-zero bosons in an elastic medium with a
screw dislocation SZ4 , the generalized DKP oscillator in a spinning cosmic
string SZ5 , the generalized Klein-Gordon oscillator under a uniform magnetic
field in a spinning cosmic string space-time cc1 , the modified Klein–Gordon
oscillator under a scalar and electromagnetic potentials in rotating cosmic
string space-time IJGMMP , the spin-$0$ DKP equation and oscillator with a
Cornell interaction in a cosmic-string space-time cc13 , the interaction of a
Cornell-type non-minimal coupling with the scalar field under the topological
defects cc14 , the influence of topological defects space–time with a spiral
dislocation on spin-$0$ bosons field (via the DKP equation formalism) cc12 ,
and the generalized Klein-Gordon oscillator in fifth-dimensional cosmic string
space-time without or with external potential in context of the Kaluza-Klein
theory AHEP2 ; EPJC ; SR , relativistic vector bosons with non-minimal
coupling in the spinning cosmic string space-time cc15 .

Our interest in magnetic monopole is due to the new reach made by the MoEDAL
detector, an experiment for the search of magnetic monopole and other highly
ionizing and long-lived particles at the CERN’s Large Hadron Collider. MoEDAL
is a largely passive detector that consists of three detection systems,
namely, nuclear track detector, trapping detector array and time pix array BA
. In recent times, an analysis of 13-TeV pp collisions with the trapping
detector during the 2015-2017 period provided the mass limits in the 1500–3750
GeV range of magnetic charges up to 5gD for monopole of spin-$0$,
$\frac{1}{2}$ and $1$ in Ref. BA2 . MoEDAL also performed the first search for
dyons, and based on a Drell-Yan production model, excluded dyons with a
magnetic charge up to 5gD and electric charge up to 200e for the mass limits
in the range 870-3120 GeV, and monopole with magnetic charge up to 5gD with
the mass limits in the range 870–2040 GeV in Ref. BA3 . In this paper, we will
study the relativistic quantum oscillator via the generalized Klein-Gordon
oscillator formalism in the presence of the Aharonov-Bohm magnetic flux
subject to Coulomb-type interaction potentials in a point-like global monopole
space-time. We discuss the effects of topological defects, the magnetic flux,
and interaction potentials on the energy profile and the wave function of
these oscillator fields. Potential applications of this work include condensed
matter systems, linear defects in elastic solids, impurities and vacancies in
elastic continuous solids. This paper contains a theoretical analysis of
relativistic quantum motions of spin-zero oscillator field by solving the
generalized KG-oscillator in the point-like global monopole space–time subject
to interaction potential.

This contribution is organized as follows: in section 2, we discuss in details
the generalized Klein-Gordon oscillator in the background space-time produced
by a point-like global monopole, and then determine its solutions by choosing
respectively, Coulomb-type function $f(r)=\frac{b}{r}$ without external
potential (sub-section 2.1), Cornell-type function
$f(r)=\Big{(}a\,r+\frac{b}{r}\Big{)}$ without external potential (sub-section
2.2), Coulomb-type function $f(r)=\frac{b}{r}$ with Coulomb-types scalar
$S(r)(\propto\frac{1}{r})$ and vector $A_{0}(\propto\frac{1}{r})$ potentials
(sub-section 2.3), Cornell-type function $f(r)=\Big{(}a\,r+\frac{b}{r}\Big{)}$
with Coulomb-types scalar $S(r)(\propto\frac{1}{r})$ and vector
$A_{0}(\propto\frac{1}{r})$ potentials (sub-section 2.4); and finally
conclusions in section 3. Here, we have used the natural units $c=1=\hbar$.

##  2 Generalized KG-Oscillator in a Global Monopole Space-time with
Interaction Potential

We study the quantum dynamics of oscillator fields in a space-time background
produced by a point-like global monopole (PGM) without or with external
potential. By solving the generalized Klein-Gordon oscillator analytically, we
discuss the effects of the topological defects characterise by the parameter
$\alpha$ of the space-time on the energy profile of these oscillator fields.
Thereby, we begin this section by introducing the line element of a point-like
global monopole which is a static and spherically symmetric metric in
coordinates $(t,r,\phi,\theta)$ described by CF44 ; ERBM44 ; ALCO ; ERFM2 ;
ABHA ; EAFB ; SZ ; MBAV

$ds^{2}=-dt^{2}+\frac{dr^{2}}{\alpha^{2}}+r^{2}\,(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}),$
(1)

where $\alpha^{2}=\Big{(}1-8\,\pi\,\eta_{0}^{2}\Big{)}<1$ depends on the
energy scale $\eta_{0}$. The parameter $\eta_{0}$ represents the dimensionless
volumetric mass density of the point-like global monopole (PGM) defect. Here,
the different coordinates are in the ranges $-\infty<t<+\infty$, $0\leq
r<\infty$, $0\leq\theta\leq\frac{\pi}{2}$, and $0\leq\phi<2\,\pi$. This point-
like global monopole space-time have some interesting features: $(i)$ it is
not globally flat, and possesses a naked curvature singularity on the axis
given by the Ricci scalar,
$R=R^{\,\mu}_{\mu}=\frac{2\,(1-\alpha^{2})}{r^{2}}$; $(ii)$ the area of a
sphere of radius $r$ in this manifold is not $4\,\pi\,r^{2}$ but rather it is
equal to $4\,\pi\,\alpha^{2}\,r^{2}$; $(iii)$ the surface
$\theta=\frac{\pi}{2}$ presents the geometry of a cone with the deficit angle
$\nabla\,\phi=8\,\pi^{2}\,\eta_{0}^{2}$; and $(iv)$ there is no Newtonian-like
gravitational potential: $g_{tt}=-1$. Furthermore, in this topological defect
space-time geometry the solid angle of a sphere of radius $r$ is
$4\,\pi^{2}\,r^{2}\,\alpha^{2}$ which is smaller than $4\,\pi^{2}\,r^{2}$, and
hence, there is a solid angle deficit
$\nabla\,\Omega=32\,\pi^{2}\,\eta_{0}^{2}$. In condensed matter systems, this
point-like global monopole space-time describes an effective metric produced
in super-fluid ${}^{3}He$-by a monopole with the angle deficit $\alpha$. In
that case, the topological defect has a negative mass GEV . As this point-like
global monopole geometry has no gravitational fields, some global effects of
this geometry has been measured, for example, scattering cross section for
mass-less bosonic POM , and fermionic particles propagating on it HR . Note
that in the limit $\alpha\to 1$, one can obtain the Minkowski flat space line
element in spherically symmetric.

The relativistic quantum dynamics of spin-zero scalar particles with an
external potential $S(r)$ is described by the following wave equation ALCO ;
JC ; AHEP2 ; EPJC ; SR

$\displaystyle\Bigg{[}-\frac{1}{\sqrt{-g}}\,D_{\mu}\,\Big{\\{}\sqrt{-g}\,g^{\mu\nu}\,D_{\nu}\Big{\\}}+\xi\,R+\Big{(}M+S(r)\Big{)}^{2}\Bigg{]}\,\Psi=0,$
(2)

where $D_{\mu}\equiv\Big{(}\partial_{\mu}-i\,e\,A_{\mu}\Big{)}$, $e$ is the
electric charges, $A_{\mu}=(-A_{0},\vec{A})$ is the electromagnetic four-
vector potential, $g$ is the determinant of the metric tensor with
$g^{\mu\nu}$ its inverse, $\xi$ is an arbitrary coupling constant with the
background curvature, $R$ is the Ricci scalar or the scalar curvature, and $M$
is the rest mass of the scalar particle. Here we followed Refs. HGD ; WG ; WG2
; AHEP2 ; EPJC ; SR , where it was suggested that a non-electromagnetic or
static scalar potential $S(r)$ can introduce by modifying the rest mass of the
scalar particle via transformation $M^{2}\to\Big{(}M+S(r)\Big{)}^{2}$ in the
wave equation. This new formalism has been first used in Ref. WG ; WG2  to
analyse the Dirac particle in the presence of a Coulomb and static scalar
potentials proportional to the inverse of the radial distance, i. e.,
$S(r)\propto\frac{1}{r}$. Later on, this new formalism has been studied in
various space-times background in quantum systems by several researchers
(Refs. AHEP2 ; EPJC ; SR  and related references there in).

In this contribution, we examine a spin-zero relativistic quantum oscillator
(via the generalized Klein-Gordon oscillator) in the point-like global
monopole (PGM) space-time background. We performed the replacement in the
radial momentum vector $p_{\mu}\to\Big{(}p_{\mu}-i\,M\,\omega\,X_{\mu}\Big{)}$
and $p^{\,\dagger}_{\mu}\to\Big{(}p_{\mu}+i\,M\,\omega\,X_{\mu}\Big{)}$ AB ;
MH ; JC ; EHR ; AHEP2 ; EPJC ; SR  where, $\omega$ is the oscillation
frequency and $X_{\mu}=f(r)\,\delta^{r}_{\mu}$ is a four-vector with $f(r)\neq
r$ an arbitrary function. Note that if one choose $f(r)=r$ in the above
replacement, e. g., $p_{\mu}\to(p_{\mu}-i\,M\,\omega\,r\,\delta^{r}_{\mu})$ or
$\vec{p}\to(\vec{p}-i\,M\,\omega\,\vec{r})$, then the quantum system is called
the Klein-Gordon oscillator AB ; MH ; JC ; EHR . As we are focusing on the
generalized Klein-Gordon oscillator, so we have replaced $r\to f(r)$, and, we
can do replacement
$\vec{p}^{\,\,2}\to\Big{(}\vec{p}-i\,M\,\omega\,f(r)\,\hat{r}\Big{)}\bullet\Big{(}\vec{p}+i\,M\,\omega\,f(r)\,\hat{r}\Big{)}$
in the wave equation. It is worth mentioning that this type of transformation
where $r\to f(r)$ was first used in Ref. KBCF  to study the Dirac oscillator.
So with the choice $X_{\mu}=(0,f(r),0,0)$, the invariant of the Lorentz
transformation of particle (or the active invariant) is broken. Indeed, this
coupling preserves the invariant of the Lorentz transformation of observers as
the behaviour of a genuine background field.

Therefore, the generalized Klein-Gordon oscillator from Eq. (2) becomes

$\displaystyle\Bigg{[}-\frac{1}{\sqrt{-g}}\,\Big{(}D_{\mu}+M\,\omega\,X_{\mu}\Big{)}\,\Big{\\{}\sqrt{-g}\,g^{\mu\nu}\,\Big{(}D_{\nu}-M\,\omega\,X_{\nu}\Big{)}\Big{\\}}+\xi\,R+\Big{(}M+S(r)\Big{)}^{2}\Bigg{]}\,\Psi=0,$
(3)

By the method of separation of variables, one can always write the total wave
function $\Psi(t,r,\theta,\phi)$ in terms of different variables. Suppose, we
choose a possible wave function in terms of a radial wave function $\psi(r)$
as :

$\Psi(t,r,\theta,\phi)=e^{-i\,E\,t}\,Y_{l,m}(\theta,\phi)\,\psi(r),$ (4)

where $E$ is the energy of the scalar particles, $Y_{l,m}(\theta,\phi)$ is the
spherical harmonics, and $l,m$ are respectively the angular momentum and
magnetic moment quantum numbers.

Also, we have chosen the three-vector electromagnetic potential $\vec{A}$
given by Refs. ALCO ; ABHA

$A_{r}=0=A_{\theta},\quad
A_{\phi}=\frac{\Phi_{B}}{2\,\pi\,r\,\sin\theta},\quad\Phi_{B}=\Phi\,\Phi_{0},\quad\Phi_{0}=2\,\pi\,e^{-1},$
(5)

where $\Phi_{B}=const$ is the Aharonov-Bohm magnetic flux, $\Phi_{0}$ is the
quantum of magnetic flux, and $\Phi$ is the amount of magnetic flux which is a
positive integer. Note that the Aharonov–Bohm effect YA ; MP  is a quantum
mechanical phenomena and has been investigated in different branches of
physics including bound-states of massive fermions VRK2 , and in the context
of Kaluza–Klein theory JC ; EHR ; AHEP2 ; EPJC ; SR  etc..

Thereby, in the space-time background (1) and using Eqs. (4)–(5) into the Eq.
(3), we obtain the following differential equation:

$\displaystyle\frac{1}{\psi}\left[\alpha^{2}\,\left\\{\left(\frac{\partial}{\partial
r}+M\,\omega\,f\right)\left(r^{2}\,\frac{\partial\psi}{\partial
r}-M\,\omega\,r^{2}\,f\,\psi\right)\right\\}\right]+\frac{1}{\psi}\left[\left\\{-\frac{2\,\xi\,(1-\alpha^{2})}{r^{2}}+\Big{(}E-e\,A_{0}\Big{)}^{2}-\Big{(}M+S(r)\Big{)}^{2}\right\\}\,\psi\right]$
(6) $\displaystyle+$
$\displaystyle\frac{1}{r^{2}\,Y_{l,m}}\,\left[\frac{1}{\sin\theta}\,\frac{\partial}{\partial\theta}\,\left(\sin\theta\,\frac{\partial
Y_{l,m}}{\partial\theta}\right)+\frac{1}{\sin^{2}\theta}\,\left(\frac{\partial}{\partial\phi}-i\,\Phi\right)^{2}\,Y_{l,m}\right]=0.$

The effective total momentum and the $z$-component of the angular momentum
operators are follows:

$\displaystyle
L^{2}_{eff}=\left[-\frac{1}{\sin\theta}\,\frac{\partial}{\partial\theta}\,\left(\sin\theta\,\frac{\partial}{\partial\theta}\right)-\frac{1}{\sin^{2}\theta}\,\left(\frac{\partial}{\partial\phi}-i\,\Phi\right)^{2}\right],\quad
L^{eff}_{z}=-i\,\left(\frac{\partial}{\partial\phi}-i\,\Phi\right).$ (7)

The eigenvalues of the various operator terms involve in Eq. (7) are as
follows ALCO ; ABHA

$\displaystyle L_{z}\,Y_{l,m}(\theta,\phi)=m\,Y_{l,m}(\theta,\phi)\quad,\quad
L^{eff}_{z}\,Y_{l,m}(\theta,\phi)=(m-\Phi)\,Y_{l,m}(\theta,\phi),$
$\displaystyle
L^{2}\,Y_{l,m}(\theta,\phi)=\lambda\,Y_{l,m}(\theta,\phi)\quad,\quad
L^{2}_{eff}\,Y_{l,m}(\theta,\phi)=\lambda^{\prime}\,Y_{l,m}(\theta,\phi)$
$\displaystyle\lambda=l\,(l+1)\quad,\quad\lambda^{\prime}=l^{\prime}\,(l^{\prime}+1)\quad,\quad
l^{\prime}=l-\Phi.$ (8)

Thereby, using the above operators in Eq. (6), the radial wave equation for
the generalized Klein-Gordon oscillator becomes:

$\displaystyle\psi^{\prime\prime}(r)+\frac{2}{r}\,\psi^{\prime}(r)-\Big{[}2\,M\,\omega\,\Big{(}\frac{f}{r}+\frac{f^{\prime}}{2}\Big{)}+M^{2}\,\omega^{2}\,f^{2}(r)\Big{]}\,\psi(r)$
(9) $\displaystyle+$
$\displaystyle\frac{1}{\alpha^{2}}\,\Bigg{[}-\frac{2\,\xi\,(1-\alpha^{2})+l^{\prime}\,(l^{\prime}+1)}{r^{2}}+\Big{(}E-e\,A_{0}\Big{)}^{2}-\Big{(}M+S(r)\Big{)}^{2}\Bigg{]}\,\psi(r)=0.\quad\quad$

To study the generalized Klein-Gordon oscillator in a point-like global
monopole space-time, we have chosen two types of function $f(r)$, namely, a
Coulomb-type function ($f(r)\propto\frac{1}{r}$), and a Cornell-type potential
form function (linear plus Coulomb-type function) and obtain the bound-states
energy eigenvalues and the wave function of these oscillator fields.

###  2.1 Coulomb-Type Potential Form Function $f(r)=\frac{b}{r}$ Without
External Potential $A_{0}=0=S$.

In this section, we study the generalized Klein-Gordon oscillator by choosing
a function $f(r)$ proportional to the inverse of the radial distance, i. e.,
$f(r)\propto\frac{1}{r}\Rightarrow f(r)=\frac{b}{r}$ where, $b>0$ is a
constant in the point-like global monopole (PGM) space-time subject to zero
scalar and vector potential, $A_{0}=0=S$. Several authors have been used this
type of function to study the generalized KG- and/or the Dirac oscillator in
quantum systems Refs. AHEP2 ; EPJC ; SZ2 ; SZ ; IJGMMP . We discuss the
effects of the topological defects of the space-time as well as the above
function $f(r)$ on the energy profiles of these oscillator fields.

Thereby, substituting the function $f(r)=\frac{b}{r}$ and considering
$A_{0}=0=S$ into the Eq. (9), the radial wave equation becomes

$\psi^{\prime\prime}(r)+\frac{2}{r}\,\psi^{\prime}(r)+\Big{(}\zeta^{2}-\frac{\sigma^{2}}{r^{2}}\Big{)}\,\psi(r)=0,$
(10)

where

$\zeta=\frac{\sqrt{E^{2}-M^{2}}}{\alpha},\quad\sigma=\sqrt{\frac{2\,\xi\,(1-\alpha^{2})+l^{\prime}\,(l^{\prime}+1)}{\alpha^{2}}+M\,\omega\,b\,+M^{2}\,\omega^{2}\,b^{2}}.$
(11)

Now, we perform a transformation via $\psi(r)=\frac{U(r)}{\sqrt{r}}$ into the
Eq. (10), we have

$r^{2}\,U^{\prime\prime}(r)+r\,U^{\prime}(r)+\Big{(}\zeta^{2}\,r^{2}-\tau^{2}\Big{)}\,U(r)=0,$
(12)

where $\tau=\sqrt{\sigma^{2}+\frac{1}{4}}$.

Equation (12) is the well-known Bessel’s differential equation GBA . Since
$\tau$ is always positive, the general solution to the Bessel equation (12) is
in the form: $U(r)=C_{1}\,J_{|\tau|}(\zeta\,r)+C_{2}\,Y_{|\tau|}(\zeta\,r)$,
where $J_{|\tau|}(\zeta\,r)$ and $Y_{|\tau|}(\zeta\,r)$ are the Bessel
function of first kind and second kind GBA , respectively. The Bessel function
of second kind $Y_{|\tau|}(\zeta\,r)$ diverges at the origin; so, we must take
$C_{2}=0$ in the general solution, since we are mainly interested in the well-
behaved solution. Thus, the regular solution to the Eq. (12) at the origin is
given by

$U(r)=C_{1}\,J_{|\tau|}(\zeta\,r).$ (13)

Let us restrict the motions of scalar fields to the region where a hard-wall
potential is present. This kind of confinement is described by the boundary
condition: $U(r_{0})=0$, which means that the wave function $\psi(r)$ vanishes
at a fixed radius $r=r_{0}$; that is, this boundary condition corresponds to
the scalar field subject to a hard-wall confining potential. This type
confining potential has been studied in quantum system, for instance, on
scalar particles under non-inertial effects mm6 ; mm3  (and relatd references
there in). Let us consider a particular case where $\zeta\,r_{0}>>1$. In this
particular case, we can write (13) in the following form

$J_{|\tau|}(\zeta\,r_{0})\propto\cos\,\Big{(}\zeta\,r_{0}-\frac{\tau\,\pi}{2}-\frac{\pi}{4}\Big{)}.$
(14)

Thereby, substituting Eq. (14) into the Eq. (13) and using the boundary
condition $U(r_{0})=0$, one can find the relativistic energy profile of these
oscillator fields as follows:

$E_{n,l}=\pm\,\sqrt{M^{2}+\Bigg{\\{}n+\frac{1}{2}\,\Bigg{(}\sqrt{\frac{1}{4}+\frac{2\xi\,(1-\alpha^{2})+l^{\prime}\,(l^{\prime}+1)}{\alpha^{2}}+M\,\omega\,b+M^{2}\,\omega^{2}\,b^{2}}+\frac{3}{2}\Bigg{)}\Bigg{\\}}^{2}\,\frac{\alpha^{2}\,\pi^{2}}{r^{2}_{0}}},$
(15)

where $l^{\prime}=(l-\Phi)$ and $n=0,1,2,3,...$.

(a) $l=1=\omega,\Phi=1/4$

(b) $l=1$, $\Phi=1/2$

(c) $l=1$

(d) $l=1$, $\alpha=1/2$

Figure 1: Eigenvalues $E_{n,l}$ keeping fixed $M=1=r_{0}=b$, $\xi=1/2$ [unit
of different parameters are chosen in natural units $(c=1=\hbar)$].

Thus we can see that the background space-time produced by a point-like global
monopole modified the relativistic energy eigenvalues of these oscillator
fields through the presence of the topological defect parameter $\alpha^{2}$
subject to a hard-wall confining potential. In addition, we see that the
energy eigenvalues $E_{n,l}$ depends on the geometric quantum phase
$\Phi_{B}$, and this dependence of the eigenvalue on the quantum phase gives
us the gravitational analogue to the Aharonov-Bohm effect YA ; MP . We have
plotted few graphs showing the effects of different parameters on the energy
spectrum of these oscillator fields (fig. 1).

###  2.2 Cornell-Type Potential Form Function
$f(r)=\Big{(}a\,r+\frac{b}{r}\Big{)}$ Without External Potential $A_{0}=0=S$.

In this section, we study the generalized Klein-Gordon oscillator in the
point-like global monopole space-time by choosing the Cornell-type potential
form function $f(r)=\Big{(}a\,r+\frac{b}{r}\Big{)}$ KBCF  with zero scalar and
vector potential, $A_{0}=0=S$. This type of function has been used for the
studies of the generalized Klein-Gordon oscillator in Refs. AHEP2 ; EPJC ; SR
; SZ2 ; SZ ; cc1 ; IJGMMP , and the generalized Dirac oscillator in Refs. LFD
in quantum systems. We solve the generalized KG-oscillator analytically and
discuss the effects of topological defects as well as this type of function on
the energy profile of these oscillator fields.

Thereby, substituting the Cornell-type function
$f(r)=\left(a\,r+\frac{b}{r}\right)$ and considering $A_{0}=0=S$ into the Eq.
(9), we have obtained the following radial wave equation:

$\psi^{\prime\prime}(r)+\frac{2}{r}\,\psi^{\prime}(r)+\Big{(}\Lambda-\frac{j^{2}}{r^{2}}-M^{2}\,\omega^{2}\,a^{2}\,r^{2}\Big{)}\,\psi(r)=0,$
(16)

where we have defined different parameters

$\displaystyle\Lambda=\frac{E^{2}-M^{2}-3\,M\,\omega\,a\,\alpha^{2}-2\,a\,b\,M^{2}\,\omega^{2}\,\alpha^{2}}{\alpha^{2}},\quad
j=\sqrt{\frac{2\,\xi\,(1-\alpha^{2})+l^{\prime}\,(l^{\prime}+1)}{\alpha^{2}}+M\,\omega\,b+M^{2}\,\omega^{2}\,b^{2}}.$
(17)

Transforming the above equation (17) via $\psi(r)=\frac{U(r)}{r^{3/2}}$, we
have

$U^{\prime\prime}(r)-\frac{1}{r}\,U^{\prime}(r)+\Bigg{[}\Lambda-\frac{(j^{2}-\frac{3}{4})}{r^{2}}-M^{2}\,\omega^{2}\,a^{2}\,r^{2}\Bigg{]}\,U(r)=0.$
(18)

Introducing a new variables via $s=M\,\omega\,a\,r^{2}$ into the above Eq.
(18), we have obtained the following second order differential equation:

$U^{\prime\prime}(s)+\left(\frac{1-4\,\mu^{2}}{4\,s^{2}}\right)\,U(s)+\frac{\nu}{s}\,U(s)-\frac{1}{4}\,U(s)=0,$
(19)

where different parameters are defined as

$2\,\mu=\sqrt{j^{2}+\frac{1}{4}}\quad,\quad\nu=\frac{\Lambda}{4\,M\,\omega\,a}.$
(20)

Equation (19) is the Whittaker differential equation KDM  and $U(s)$ is the
Whittaker function which can be written in terms of the confluent
hypergeometric function ${}_{1}F_{1}(s)$ as

$U(s)=s^{\frac{1}{2}+\mu}\,e^{-\frac{s}{2}}\,{}_{1}F_{1}\left(\mu-\nu+\frac{1}{2},2\,\mu+1;s\right).$
(21)

In order to obtain the bound-states solutions, it is necessary that the
confluent hypergeometric function
${}_{1}F_{1}\left(\mu-\nu+\frac{1}{2},2\,\mu+1;s\right)$ must be a power
series expansion of $s$ with degree $n$, and the quantity
$\Big{(}\mu-\nu+\frac{1}{2}\Big{)}$ should be a negative integer, that is,
$\Big{(}\mu-\nu+\frac{1}{2}\Big{)}=-n$, where $n=0,1,2,..$. After
simplification of this condition, we have obtained the following eigenvalues
expression for the quantum system

$\displaystyle E_{n,l}=\pm\,\sqrt{\begin{aligned}
&M^{2}\,\Big{(}1+2\,a\,b\,\omega^{2}\,\alpha^{2}\Big{)}+2\,M\,\omega\,\alpha^{2}\\\
&\times\left(2\,n+\sqrt{\frac{1}{4}+\frac{2\,\xi\,(1-\alpha^{2})+l^{\prime}\,(l^{\prime}+1)}{\alpha^{2}}+M\,\omega\,b+M^{2}\,\omega^{2}\,b^{2}}+\frac{5}{2}\right)\end{aligned}}.\quad\quad$
(22)

(a) $l=1=\omega,\Phi=1/4$

(b) $l=1,\Phi=1/4$

(c) $l=1=\omega$

(d) $l=1,\alpha=1/4,\Phi=1/2$

(e) $l=1,\alpha=1/4$

(f) $l=1$

Figure 2: Eigenvalues $E_{n,1}$ keeping fixed $M=1=b=a$, $\xi=1/2$ [unit of
different parameters are chosen in natural units $(c=1=\hbar)$].

The normalized radial wave functions are given by

$\psi_{n,l}(s)=D_{n,l}\,(M\,\omega\,a)^{3/4}\,s^{\mu-\frac{1}{4}}\,e^{-\frac{s}{2}}\,{}_{1}F_{1}\left(\mu-\nu+\frac{1}{2},2\,\mu+1;s\right),$
(23)

(a) $l=1=\omega,\Phi=1/4,\alpha=1/2$

(b) $l=1=\omega,\Phi=1/4$

(c) $l=1=\omega$

(d) $l=1,\Phi=1/4$

(e) $n=1=l$

(f) $n=1=l=\omega$

Figure 3: Radial wave function $\psi_{n,l}$ keeping fixed $M=1=b=a$, $\xi=1/2$
[unit of different parameters are chosen in natural units $(c=1=\hbar)$].

where $D_{n,l}$ is a constant which can be determined by the normalization
condition for the radial wave function

$\frac{1}{\alpha}\,\int^{\infty}_{0}\,r^{2}\,dr\,|\psi(r)|^{2}=1.$ (24)

To solve the integrals of the radial wave function, we can write the confluent
hypergeometric function in terms of the associated Laguerre polynomials by the
relation GBA

${}_{1}F_{1}\left(-n,2\,\mu+1;x\right)=\frac{n!\,(2\,\mu)!}{(n+2\,\mu)!}\,L^{(2\,\mu)}_{n}(x).$
(25)

Then, taking into account $s=M\,\omega\,r^{2}$, and with the help of AP  to
solve the integrals, the normalization constant is given by

$D_{n,l}=\frac{1}{(2\,\mu)!}\,\sqrt{\frac{2\,\alpha\,(n+2\,\mu)!}{n!}}=\frac{1}{\left(\sqrt{j^{2}+\frac{1}{4}}\right)!}\,\sqrt{\frac{2\,\alpha\,\left(n+\sqrt{j^{2}+\frac{1}{4}}\right)!}{n!}}.$
(26)

Equation (22) is the relativistic energy spectrum that stems from the
interaction of the scalar field with the generalized Klein-Gordon oscillator
in a point-like global monopole space-time background. We see that the
topological defects associated with the scalar curvature of the space-time,
and the Cornell-type function $f(r)=\Big{(}a\,r+\frac{b}{r}\Big{)}$ modified
the energy profiles of these oscillator fields. Furthermore, the energy
eigenvalues $E_{n,l}$ depends on the geometric quantum phase $\Phi_{B}$ which
gives us the gravitational analogue to the Aharonov-Bohm effect YA ; MP . We
have plotted graphs showing the effects of different parameters on the energy
spectrum (fig. 2), and the wave function (fig. 3) of these oscillator fields.

In the below analysis, we insert an external potential through a minimal
substitution to study the generalized Klein-Gordon oscillator in the point-
like global monopole space-time with the function $f(r)$ considered in the
previous two analysis. We have chosen the first component of the
electromagnetic four-vector potential $A_{0}$ and a static scalar potential
proportional to the inverse of the radial distance bb1 ; bb2 ; bb3 , i. e.,

$A_{0}\propto\frac{1}{r}\Rightarrow
A_{0}=\frac{\kappa}{r}=\pm\,\frac{|\kappa|}{r}\quad,\quad
S\propto\frac{1}{r}\Rightarrow S=\frac{\eta}{r},$ (27)

where $\kappa,\eta$ characterize the potential parameters. This type of
potential have widely been used in different branches of physics, for
instance, $1$-dimensional systems aa4 ; aa5 , topological defects in solids
aa20 , quark-antiquark interaction aa30 , propagation of gravitational waves
aa21 , quark models aa22 , and in the relativistic quantum systems mm3 ;
IJGMMP . Here we followed Refs. HGD ; WG ; WG2  to introduce the scalar
potential in the wave equation by modifying the mass term via transformation
$M\to M+S(t,r)$, where $S(t,r)$ is the scalar potential.

###  2.3 Coulomb-Type Function $f(r)=\frac{b}{r}$ With Coulomb-Type Vector
$A_{0}=\pm\,\frac{|\kappa|}{r}$ and Scalar $S=\frac{\eta}{r}$ Potentials.

In this section, we study the generalized Klein-Gordon oscillator subject to a
vector and scalar potentials of Coulomb-types (27) with a Coulomb-type
function $f(r)=\frac{b}{r}$ in the presence of an Aharonov-Bohm magnetic flux
field in the space-time background produced by a point-like global monopole.

Thereby, substituting the function $f(r)=\frac{b}{r}$ and Coulomb-types scalar
and vector potential into the Eq. (9), we have obtained the following radial
wave equation:

$\psi^{\prime\prime}(r)+\frac{2}{r}\,\psi^{\prime}(r)+\left(-\chi^{2}-\frac{\beta^{2}}{r^{2}}-\frac{2\,\gamma}{r}\right)\,\psi(r)=0,$
(28)

where we have defined different parameters

$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2\,\xi\,(1-\alpha^{2})+l^{\prime}\,(l^{\prime}+1)+\eta^{2}-e^{2}\,\kappa^{2}}{\alpha^{2}}+M\,\omega\,b+M^{2}\,\omega^{2}\,b^{2}},\quad\gamma=\frac{(M\,\eta-e\,E\,\kappa)}{\alpha^{2}},$
$\displaystyle\chi$ $\displaystyle=$
$\displaystyle\frac{\sqrt{M^{2}-E^{2}}}{\alpha}.$ (29)

Following the radial wave transformation via $\psi(r)=\frac{U(r)}{\sqrt{r}}$
into the Eq. (28), we have

$U^{\prime\prime}(r)+\frac{1}{r}\,U^{\prime}(r)+\left(-\chi^{2}-\frac{\beta^{2}}{r^{2}}-\frac{2\,\gamma}{r}\right)\,U(r)=0.$
(30)

We do another transformation on the radial coordinate via $\rho=2\,\chi\,r$
into the Eq. (30), we have

$U^{\prime\prime}(\rho)+\frac{1}{\rho}\,U^{\prime}(\rho)+\left(-\frac{\beta^{2}}{\rho^{2}}-\frac{\gamma}{\chi\,\rho}-\frac{1}{4}\right)\,U(\rho)=0.$
(31)

Now, we impose requirement of the wave function that the radial wave function
$R(\rho)$ must be well-behaved at the origin, since it is a singular point of
the Eq. (31). In this case, for $lim_{\rho\to 0}U(\rho)=0$, the solution is
$U(\rho)\sim\rho^{\beta}$. Furthermore, for $lim_{\rho\to\infty}U(\rho)=0$,
the solution is $U(\rho)\sim e^{-\frac{\rho}{2}}$. Thus, a possible solution
to the Eq. (31) is given by

$U(\rho)=\rho^{\beta}\,e^{-\frac{\rho}{2}}\,F(\rho),$ (32)

where $F(\rho)$ is an unknown function. Substituting this solution (32) into
the Eq. (31), we have

$\rho\,F^{\prime\prime}(\rho)+(2\,\beta+1-\rho)\,F^{\prime}(\rho)+\left(-\beta-\frac{\gamma}{\chi}-\frac{1}{2}\right)\,F(\rho)=0.$
(33)

Equation (33) is the confluent hypergeometric second order differential
equation and the function
$F(\rho)={}_{1}F_{1}(\beta+\frac{\gamma}{\chi}+\frac{1}{2},2\,\beta+1;\rho)$
is called the confluent hypergeometric function. In order to find the the
bound-states solutions of the quantum system, this confluent hypergeometric
function must be a finite degree polynomial $n$, and the quantity
$\left(\beta+\frac{\gamma}{\chi}+\frac{1}{2}\right)$ should be a negative
integer. This condition implies that
$\left(\beta+\frac{\gamma}{\chi}+\frac{1}{2}\right)=-n$, where $n=0,1,2,...$.
After simplifying this condition, one will find the following expression of
the energy spectrum

$E_{n,l}=\frac{1}{\alpha^{2}\,\Delta^{2}+e^{2}\,\kappa^{2}}\,\left[M\,e\,\kappa\,\eta\pm\,M\Delta\,\alpha^{2}\,\sqrt{\Delta^{2}+\frac{e^{2}\,\kappa^{2}-\eta^{2}}{\alpha^{2}}}\right],$
(34)

where

$\Delta=\left(n+\sqrt{\frac{2\,\xi\,(1-\alpha^{2})+l^{\prime}\,(l^{\prime}+1)+\eta^{2}-e^{2}\,\kappa^{2}}{\alpha^{2}}+M\,\omega\,b+M^{2}\,\omega^{2}\,b^{2}}+\frac{1}{2}\right).$
(35)

(a) $l=1,\Phi=1/4$

(b) $l=1,\Phi=1/4$

(c) $l=1,\omega=1$

(d) $l=1,\omega=1$

Figure 4: Energy $E_{n,l}$ with parameters keeping fixed $M=1=b=e$, $\xi=1/2$
[units of different parameters are chosen in natural units $(c=1=\hbar)$].

The radial wave function is given by

$\displaystyle U_{n,l}(\rho)$ $\displaystyle=$
$\displaystyle\rho^{\beta}\,e^{-\frac{\rho}{2}}\,{}_{1}F_{1}\Big{(}\beta+\frac{\gamma}{\chi}+\frac{1}{2},2\,\beta+1;\rho\Big{)},$
$\displaystyle\psi_{n,l}(\rho)$ $\displaystyle=$
$\displaystyle\Big{(}2\,\chi\Big{)}^{1/2}\,\rho^{(\beta-\frac{1}{2})}\,e^{-\frac{\rho}{2}}\,{}_{1}F_{1}\Big{(}\beta+\frac{\gamma}{\chi}+\frac{1}{2},2\,\beta+1;\rho\Big{)}.$
(36)

We can see that the energy spectrum (34) and the radial wave function (36) of
oscillator field are influenced by the topological defects parameter
$\alpha^{2}$ which is associated with the curvature of the space-time, the
function $f(r)=\frac{b}{r}$ as well as the Coulomb-types scalar and vector
potentials present in the system. Furthermore, the energy eigenvalues
$E_{n,l}$ depends on the Aharonov-Bohm magnetic flux $\Phi_{B}$ which gives us
the gravitational analog of the Aharonov-Bohm effect YA ; MP . We have plotted
graphs showing the effects of different parameters on the energy spectrum
(fig. 4), and the wave function (fig. 5) of these oscillators field.

(a) $\Phi=1/2,\omega=1=l,\alpha=1/2$

(b) $n=1=l,\Phi=1/2=\alpha$

(c) $n=1=l,\eta=1/2,\kappa=1/4$

(d) $n=1=l$

Figure 5: Wave function $U_{n,l}$ keeping fixed $M=1=b=e$, $\xi=1/2$ [unit of
different parameters are chosen in natural units $(c=1=\hbar)$].

###  2.4 Cornell-Type Function $f(r)=\left(a\,r+\frac{b}{r}\right)$ With
Coulomb-Type Vector $A_{0}=\pm\,\frac{|\kappa|}{r}$ and Scalar
$S=\frac{\eta}{r}$ Potentials.

In this section, we study the generalized Klein-Gordon oscillator subject to a
vector and scalar potentials of Coulomb-types (27) with a Cornell-type
function, $f(r)=\left(a\,r+\frac{b}{r}\right)$ in the presence of an Aharonov-
Bohm flux in a point-like global monopole space-time. We discuss the
influences on the energy profiles of these oscillators.

Thereby, substituting the Cornell-type function
$f(r)=\left(a\,r+\frac{b}{r}\right)$ and Coulomb-type scalar and vector
potential into the Eq. (9), we have obtained the following radial wave
equation:

$\psi^{\prime\prime}(r)+\frac{2}{r}\,\psi^{\prime}(r)+\left(-\delta^{2}-\frac{\beta^{2}}{r^{2}}-\frac{2\,\gamma}{r}\right)\,\psi(r)=0,$
(37)

where we have defined

$\delta^{2}=3\,M\,\omega\,a+2\,a\,b\,M^{2}\,\omega^{2}+\frac{1}{\alpha^{2}}\,(M^{2}-E^{2}).$
(38)

Now, we perform a transformation via $\psi(r)=\frac{U(r)}{\sqrt{r}}$ into the
Eq. (37), we have

$U^{\prime\prime}(r)+\frac{1}{r}\,U^{\prime}(r)+\left(-\delta^{2}-\frac{\beta^{2}}{r^{2}}-\frac{2\,\gamma}{r}\right)\,U(r)=0.$
(39)

Considering a transformation of the radial coordinate via $\rho=2\,\delta\,r$
into the Eq. (39), we have

$U^{\prime\prime}(\rho)+\frac{1}{\rho}\,U^{\prime}(\rho)+\left(-\frac{\beta^{2}}{\rho^{2}}-\frac{\gamma}{\delta\,\rho}-\frac{1}{4}\right)\,U(\rho)=0.$
(40)

As stated earlier, the radial wave-function $U(\rho)\to 0$ for $\rho\to 0$ and
$\rho\to\infty$. Suppose, a possible solution to the Eq. (40) given by

$U(\rho)=\rho^{\beta}\,e^{-\frac{\rho}{2}}\,F(\rho),$ (41)

where $F(\rho)$ is an unknown function. Substituting this solution into the
Eq. (40), we have

$\rho\,F^{\prime\prime}(\rho)+(2\,\beta+1-\rho)\,F^{\prime}(\rho)+\left(-\beta-\frac{\gamma}{\delta}-\frac{1}{2}\right)\,F(\rho)=0.$
(42)

Equation (42) is the well-known confluent hypergeometric equation which is a
second order linear homogeneous differential equation. The solution of this
Eq. (42) that is regular for $\rho\to 0$ is given in terms of the confluent
hypergeometric function as

$F(\rho)={}_{1}F_{1}\Big{(}\beta+\frac{\gamma}{\delta}+\frac{1}{2},2\,\beta+1;\rho\Big{)}.$
(43)

As state earlier, in order to have a bound-states solutions of this equation,
the confluent hypergeometric function
${}_{1}F_{1}(\beta+\frac{\gamma}{\delta}+\frac{1}{2},2\,\beta+1;\rho\to\infty)$
must be a finite degree polynomial of degree $n$, and the quantity
$\left(\beta+\frac{\gamma}{\delta}+\frac{1}{2}\right)$ is a negative integer,
that is, $\left(\beta+\frac{\gamma}{\delta}+\frac{1}{2}\right)=-n$, where
$n=0,1,2,..$. After simplifying this condition, we have the following energy
eigenvalues of the system

$E_{n,l}=\frac{1}{\alpha^{2}\,\Delta^{2}+e^{2}\,\kappa^{2}}\,\left[M\,e\,\kappa\,\eta\pm\,M\,\Delta\,\alpha^{2}\,\sqrt{\Delta^{2}+\frac{e^{2}\,\kappa^{2}-\eta^{2}}{\alpha^{2}}+\Sigma^{2}}\right],$
(44)

where $\Delta$ is given earlier and

$\Sigma=\sqrt{\omega\,a\,\left(\frac{3}{M}+2\,\omega\,b\right)\,(\alpha^{2}\,\Delta^{2}+e^{2}\,\kappa^{2})}.$
(45)

The radial wave-function is given by

$\displaystyle U_{n,l}(\rho)$ $\displaystyle=$
$\displaystyle\rho^{\beta}\,e^{-\frac{\rho}{2}}\,{}_{1}F_{1}\Big{(}\beta+\frac{\gamma}{\delta}+\frac{1}{2},2\,\beta+1;\rho\Big{)},$
$\displaystyle\psi_{n,l}(\rho)$ $\displaystyle=$
$\displaystyle\Big{(}2\,\delta\Big{)}^{1/2}\,\rho^{(\beta-\frac{1}{2})}\,e^{-\frac{\rho}{2}}\,{}_{1}F_{1}\Big{(}\beta+\frac{\gamma}{\delta}+\frac{1}{2},2\,\beta+1;\rho\Big{)}.$
(46)

We can see that the energy spectrum (44)–(45) of oscillator field are
influenced by the topological defects parameter $\alpha^{2}$ of the space-
time, the modified function $f(r)=\left(a\,r+\frac{b}{r}\right)$ as well as
Coulomb-types scalar and vector potentials. Furthermore, the energy
eigenvalues $E_{n,l}$ depends on the Aharonov-Bohm magnetic flux $\Phi_{B}$
which gives us the gravitational analog of the Aharonov-Bohm effect YA ; MP .

##  3 Conclusions

We have studied a spin-zero relativistic quantum oscillator (via the
generalized Klein-Gordon oscillator) interacting with the gravitational field
produced by the topological defects in a point-like global monopole (PGM)
space-time. Later on, we insert scalar and vector potentials of Coulomb-types
in the quantum system. We analyze the behaviour of the oscillator field under
the topological defect that is associated with the scalar curvature of the
space-time and see that the energy eigenvalues and the wave function are
modified in comparison to the standard Landau levels. The present analysis can
be used for simulation of a series of physical systems, for instance, the
vibrational spectrum of diatomic molecules SMI , binding of heavy quarks CQ ;
MC , quark–antiquark interaction EE  etc.. The modified energy spectrum may be
suitable to demonstrate the existence of these kinds of topological defects.
However, from the observational point of view, it is clear that to have an
observable modification in the energy spectrum of a physical system, we need a
huge amount of particles unless the magnitude of this deviation from the
original spectrum may not be strong enough to observe.

In this work, we have analyzed the effects of topological defects on a quantum
oscillator field in the presence of external potentials. The radial wave
equation of the generalized Klein-Gordon oscillator is derived. Then in sub-
section 2.1, we have chosen a Coulomb-type function $f(r)=\frac{b}{r}$ and
considering zero vector and scalar potentials, $A_{0}=0=S$ into the radial
equation, a second-order differential equation of the Bessel form is achieved.
We solved this Bessel equation using the boundary condition of the hard-wall
confining potential condition and obtained the relativistic energy eigenvalue
by Eq. (15). We have seen that the energy spectrum gets modified by the
topology of the point-like global monopole space-time. In sub-section 2.2, we
have considered a Cornell-type function $f(r)=\Big{(}a\,r+\frac{b}{r}\Big{)}$
and considering zero vector and scalar potentials into the radial wave
equation, the Whittaker differential equation KDM  form which is a second-
order is achieved. After solving this Whittaker equation, the bound-states
eigenvalues by the Eq. (22) and the radial wave function by the Eq. (23) of
these oscillator fields are obtained. We have seen that the topological
defects of the space-time geometry, as well as, the new function $f(r)\neq r$
modified the spectrum of energy and the wave function. In sub-section 2.3, we
have considered the Coulomb-type function $f(r)=\frac{b}{r}$ and Coulomb-type
scalar and vector potentials (27) into the derived radial wave equation. We
have achieved the confluent hypergeometric differential equation form and
imposed the boundary condition for the bound-states solutions, one can find
the bound-states energy eigenvalues by the Eq. (34) and the radial wave
function by the Eq. (36) of these oscillator fields. One can see that the
obtained energy eigenvalues, as well as the wave function, are modified by the
topological defects and the Coulomb-type scalar and vector potentials
considered in the quantum system. In sub-section 2.4, we have considered
Cornell-type function $f(r)=\Big{(}a\,r+\frac{b}{r}\Big{)}$ with Coulomb-types
(27) vector and scalar potentials into the derived radial wave equation. Again
we have achieved the confluent hypergeometric differential equation form, and
following the previous procedure, we have obtained the bound-states energy
eigenvalues by the Eq. (44) and the radial wave function Eq. (46) of these
oscillator fields. Here also, we have seen that the energy spectrum and the
wave unction are modified by the topological defects, the Coulomb-type scalar
and vector potentials, and the considered function $f(r)\neq r$ on these
oscillator fields.

An interesting feature of the presented results is that the energy profile of
these oscillator fields in addition to the topological defect parameter
$\alpha^{2}$ associated with the scalar curvature of the space-time and
Coulomb-type potential depends on the Aharonov-Bohm magnetic flux $\Phi_{B}$.
This is because the angular momentum quantum number $l$ is shifted, that is,
$l\to l_{eff}=\left(l-\frac{e\,\Phi_{B}}{2\pi}\right)$, an effective angular
momentum quantum number. Thus, the relativistic energy eigenvalue of the
oscillator field is a periodic function of the geometric quantum phase
$\Phi_{B}$, and we have that
$E_{n,l}(\Phi_{B}\pm\,\Phi_{0}\,\tau)=E_{n,l\mp\tau}(\Phi_{B})$, where
$\tau=0,1,2,...$. This dependence of the energy eigenvalue on the geometric
quantum phase gives us the gravitational analogue of the Aharonov-Bohm (AB)
effect YA ; MP . Several authors have investigated this AB-effect in quantum
mechanical systems Refs. EHR ; AHEP2 ; EPJC ; SR ; IJGMMP ; cc2 ; cc3 ; cc4 .
It is well-known in condensed matter physics that the dependence of the energy
eigenvalues on the geometric quantum phase gives rise to persistent currents
that will discuss in future work.

Therefore, in this paper, we have shown some results for quantum systems where
general relativistic effects are taken into account with the Aharonov-Bohm
magnetic flux, which in addition with the previous results Refs. SZ ; SZ3
present several interesting effects. This is a fundamental subject in physics,
and the connections between these theories are not well understood.

## Conflict of Interest

There is no conflict of interests regarding publication of this paper.

## Contribution

F. Ahmed has done solely the whole work.

## Funding Statement

There is no funding agency associated with this manuscript.

## Data Availability Statement

All data generated or analysed during this study are included in this article
[and its supplementary information files].

## Acknowledgement

Author would like to thank the anonymous kind referee(s) for their valuable
comments and suggestion which have improved this paper a lot.

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# Lower bounds for the scalar curvatures of Ricci flow singularity models

Pak-Yeung Chan∗,††Department of Mathematics, University of California San
Diego, La Jolla, CA 92093 Bennett Chow∗, Zilu Ma∗∗,††∗ Department of
Mathematics, Rutgers University, Piscataway, NJ 08854 Yongjia Zhang∗∗∗††∗∗
School of Mathematical Sciences, Shanghai Jiao Tong University, Minhang
District, Shanghai, China 200240

###### Abstract

In a series of papers, Bamler [Bam20a, Bam20b, Bam20c] further developed the
high-dimensional theory of Hamilton’s Ricci flow to include new monotonicity
formulas, a completely general compactness theorem, and a long-sought partial
regularity theory analogous to Cheeger–Colding theory. In this paper we give
an application of his theory to lower bounds for the scalar curvatures of
singularity models for Ricci flow. In the case of $4$-dimensional non-Ricci-
flat steady soliton singularity models, we obtain as a consequence a quadratic
decay lower bound for the scalar curvature.

##  1 Introduction and background

To help formulate Ricci flow with surgery, one would like to better understand
singularity models of the Ricci flow. This has been highly successful in
dimension $3$ by the Hamilton–Perelman theory leading to the proof of the
Poincaré and Thurston geometrization conjectures by Hamilton’s Ricci flow
[Per02, Per03a, Per03b]. A deeper understanding of Ricci flow and singularity
formation in dimension 3 has led to solutions to a pair of fundamental
conjectures of Perelman; see Brendle [Bre13, Bre20] and Bamler [Bam18] and the
references therein. Bamler and Kleiner [BK22, BK21a] proved the generalized
Smale conjecture using $3$-dimensional Ricci flow with surgery. Furthermore,
the understanding of $3$-dimensional ancient solutions essentially complete.
See Brendle, Daskalapoulos, and Sesum [BDS21], Angenent, Brendle,
Daskalapoulos, and Sesum [ABDS22], Bamler and Kleiner [BK21b], and Lai [Lai20,
Lai22].

Among the higher dimensions, dimension 4 is the most hopeful. In this case,
Bamler’s theory [Bam20a, Bam20b, Bam20c] yields the strongest results
regarding Ricci flow singularity formation and compactness. In general,
regarding singularity models, one is interested in Ricci solitons since they
are prototypical [Ham88, Ham93] and [Per02, Per03a]. Among a vast literature
on Ricci solitons, important progress has been made by Munteanu and Wang (see
[MW19] and the references therein), especially on the geometric understanding
of shrinkers.

A general question is: what curvature estimates hold for singularity models,
and in particular, Ricci solitons? In this paper we apply Bamler’s theory to
make some further progress on lower bounds for the scalar curvature.

If a singularity model is Ricci flat, then by Perelman’s no local collapsing
theorem [Per02], it has Euclidean volume growth. If, in addition, the
dimension is $4$, then by a theorem of Cheeger and Naber [CN15, Corollary
8.86], it is an asymptotically locally Euclidean (ALE) space. The
Eguchi–Hanson metric, which is such a space, occurs as a singularity model of
$4$-dimensional Ricci flow by Appleton [App19]. In this paper we consider the
complementary case of non-Ricci-flat singularity models.

Let $(M^{n},g_{t})_{t\leq 0}$ be an ancient Ricci flow, i.e.,
$\partial_{t}g_{t}=-2\operatorname{Ric}_{g_{t}}$, and $t\leq 0$ means that
$t\in(-\infty,0]$, whose time slices are complete. The conjugate heat operator
is defined as $\Box^{*}=-\partial_{t}-\Delta_{g_{t}}+R_{g_{t}}$. We write the
conjugate heat kernel based at the space-time point $(o,0)$ as

$K(o,0\,|\,x,t)=(4\pi|t|)^{-\frac{n}{2}}e^{-f_{t}(x)}.$

So, $\Box^{*}_{x,t}K=0$ and $\lim_{t\to 0_{-}}K(o,0\,|\,\cdot,t)=\delta_{o}$.
The pointed Nash entropy based at $(o,0)$ is

$\mathcal{N}_{o,0}(\tau):=\int_{M}f_{t}(x)\,K(o,0\,|\,x,t)\,dg_{t}(x)-\frac{n}{2},$
(1.1)

where $\tau=|t|$ (we will henceforth use this notation).

Henceforth, we assume that $(M^{n},g_{t})_{t\leq 0}$ either is a finite-time
singularity model or has bounded curvature over compact time intervals with a
pointed _Nash entropy lower bound_ , namely,

$\inf_{\tau>0}\mathcal{N}_{o,0}(\tau)=:\mu_{\infty}>-\infty,$ (NELB)

for some point $o\in M$ (and thus for any point since $\inf\mathcal{N}$ is
independent of the base point by [MZ21, Proposition 4.6]). By a _finite-time
singularity model_ we mean a complete and non-flat ancient solution
$(M^{n},g_{t})_{t\leq 0}$ which arises as a blow-up limit of a Ricci flow
$(\overline{M}^{n},\overline{g}(t))_{t\in[0,T)}$ on a closed manifold
$\overline{M}$, with a finite-time singularity at $T<\infty$. More precisely,
one can find sequences of $(x_{i},t_{i})\in\overline{M}\times[0,T)$ and
$\lambda_{i}\to\infty$ such that the rescaled solutions
$\big{(}\overline{M},\lambda_{i}\overline{g}(t_{i}+\tfrac{t}{\lambda_{i}})\big{)},t\in(-\lambda_{i}t_{i},0]$,
$\mathbb{F}$-converge on compact time intervals to $(M,g(t)),t\in(-\infty,0]$,
in the sense of Bamler [Bam20b] (see also [CFSZ20, BCDMZ21]).

For each $t<0,$ let $(z_{t},t)$ be an $H_{n}$-center of $(o,0)$, as defined in
[Bam20a]. That is,
$\operatorname{Var}_{t}\big{(}\delta_{z_{t}},K(o,0\,|\,\cdot,t)\mskip
1.5mudg_{t}\big{)}\leq H_{n}|t|$, where $H_{n}:=\frac{(n-1)\mskip
1.5mu\pi^{2}}{2}+4$. Such space-time points effectively represent the past of
a point $(o,0)$ of a Ricci flow. We shall consider the following
$H_{n}$-center _scalar curvature lower bound_ assumption

$\inf\Big{\\{}(1+\tau)\,R(x,t)\,\Big{|}\,x\in
B_{t}(z_{t},D\sqrt{1+\tau}),t<0\Big{\\}}:=a_{0}>0,$ (SCLB)

for some constant $D<\infty.$ As we will see below, this assumption is
suitable for applications to noncollapsed steady solitons.

##  2 Statements of the results

The main results of this paper stem from the following global scalar curvature
decay estimate for the ancient solutions in which we are interested.

######  Theorem 2.1.

Assume that a complete ancient solution $(M^{n},g_{t})$ is a finite-time
singularity model or that $(M,g_{t})$ has bounded curvature over compact time
intervals with (NELB). Further assume (SCLB). If the constant $D$ in (SCLB)
satisfies $D\geq\underline{D}(n,\mu_{\infty})$, then

$(1+\tau)\,R(x,t)\geq\frac{c(n,\mu_{\infty},a_{0})}{f_{t}(x)+C(n)-\mu_{\infty}},$

for $(x,t)\in M\times(-\infty,0)$, where $a_{0}$ is the infimum in (SCLB) and
$\mu_{\infty}$ is as in (NELB).

The following says that we can replace the assumption of an $H_{n}$-center
scalar curvature lower bound by the smoothness of the tangent flow at
infinity.

######  Corollary 2.2.

Assume that $(M,g_{t})$ is a finite-time singularity model or that $(M,g_{t})$
has bounded curvature over compact time intervals with (NELB). Suppose further
that each tangent flow at infinity of $(M,g_{t})$ (as defined in [Bam20b]) is
smooth. Then

$(1+\tau)R(x,t)\geq\frac{a}{f_{t}(x)+C(n)-\mu_{\infty}},$

where $a>0$ is a constant depending on the geometry of $(M,g_{t})_{t\leq 0}$.

Alternative versions of this result include that it holds under the type-I
assumption or under another more technical assumption stated below. In
particular, we have the following.

######  Corollary 2.3.

Let $(M^{n},g_{t})_{t\leq 0}$ be a complete and non-flat ancient flow with
(NELB). Suppose that either:

  1. 1.

$(M,g_{t})$ is of type-I, i.e., $\tau|{\operatorname{Rm}}|\leq C$ uniformly
over $M\times(-\infty,0]$ for some constant $C<\infty$; or

  2. 2.

$(M,g_{t})$ satisfies Hamilton’s trace Harnack estimate, i.e., for any smooth
vector field $X$ on $M$,

$\frac{\partial R}{\partial t}-2\langle X,\nabla
R\rangle+2\operatorname{Ric}(X,X)\geq 0,$ (2.1)

and $|{\operatorname{Rm}}|\leq CR$ for some constant $C<\infty.$

Then

$(1+\tau)R(x,t)\geq\frac{a}{f_{t}(x)+C(n)-\mu_{\infty}},$

where $a>0$ is a constant depending on the geometry of $(M,g_{t})_{t\leq 0},$
and $\mu_{\infty}$ is as in (NELB).

We then apply the results above to steady gradient Ricci solitons. Recall that
a triple $(M^{n},g,f)$ is called a steady gradient Ricci soliton, if
$(M^{n},g)$ is a Riemannian manifold and $f$ is a smooth function on $M$
satisfying

$\operatorname{Ric}=\nabla^{2}f.$

If we denote by $(\Phi_{t})_{t\in\mathbb{R}}$ the $1$-parameter group of
diffeomorphisms generated by $-\nabla f$ with $\Phi_{0}={\rm id}$, then
$g_{t}:=\Phi_{t}^{*}g$ solves the Ricci flow, and we call
$(M^{n},g_{t})_{t\in\mathbb{R}}$ the canonical form of $(M^{n},g,f).$ It was
proved by Hamilton [Ham93] that on any steady gradient Ricci soliton,
$\nabla\left(|\nabla f|^{2}+R\right)\equiv 0$ holds, and hence

$|\nabla f|^{2}+R=C_{1},$ (2.2)

for some nonnegative constant $C_{1}$. Let $|xy|:=d(x,y)$ denote the distance
between $x$ and $y$; if we add a subscript $t$, then this means that the
distance is with respect to $g_{t}$.

######  Definition 2.4.

A steady gradient Ricci soliton $(M^{n},g,f)$ is called _noncollapsed_ if its
canonical form $(M^{n},g_{t})_{t\in\mathbb{R}}$ has pointed Nash entropy
bounded from below, i.e., (NELB) holds for some $o\in M$.

It is to be remarked that our definition of noncollapsing is different from
that of Perelman [Per02]. Bamler [Bam20a, Theorem 6.1] and [MZ21, Proposition
4.6] show that Definition 2.4 implies Perelman’s $\kappa$-noncollapsing on all
scales. Nevertheless, Definition 2.4 is natural for applications, since due to
the entropy monotonicity, any finite-time singularity model is necessarily
noncollapsed in the sense of Definition 2.4.

In the case of noncollapsed steady solitons with nonnegative Ricci curvature,
we can use the tangent flows at infinity to obtain curvature bounds near the
$H_{n}$-centers, and then our estimate provides a global scalar curvature
lower bound. Thus, we have the following.

######  Corollary 2.5.

Let $(M^{n},g,f)$ be a complete and noncollapsed steady gradient Ricci
soliton. Suppose that $\operatorname{Ric}\geq 0$ and
$|{\operatorname{Rm}}|\leq CR$ for some constant $C<\infty$. If $(M^{n},g)$ is
not Ricci flat, then for any $x\in M,\tau\geq\underline{\tau}(n),$ we have
that

$R(x)\geq\frac{a}{(C-\mu_{\infty})\tau+|xx_{\tau}|^{2}},$

where $x_{\tau}\in\partial B_{\tau}(o)$ satisfies
$\lambda(x_{\tau},\tau_{0})\leq C,$ for some $\tau_{0}\in[\tau/C,C\tau].$ (See
the definition of $\lambda$ in Section 3.3 or [BCMZ21, Section 2.1].) Here,
$C=C(n),$ and $a>0$ depends on the geometry of $g.$

We remark that this lower bound is not sharp for the hypothesized class of
steady solitons, e.g., Lai’s noncollapsed examples in dimension $4$ [Lai20].
However, this estimate is sharp for $3$-cylindrical steady solitons. In
particular, if $f$ grows linearly and the level sets
$\Sigma_{\tau}=\\{f=\tau\\}$ have diameters bounded by $C\sqrt{\tau}$ (this is
the case when $R(x)\leq\frac{C}{1+|xo|}$; see Deng and Zhu [DZ20b, Proposition
3.3]), then Corollary 2.5 gives the improved estimate

$R(x)\geq c/\tau\geq c^{\prime}/|xo|,$

for any $x\in\Sigma_{\tau}$, if $\tau\geq\underline{\tau}(n).$ This can be
viewed as an alternative proof of a result by Deng and Zhu [DZ20a, Theorem
2.5] in this case. Note that we do not need to assume the existence of
critical points of $f.$

In fact, the arguments in [DZ20a, Theorem 2.5] or [DZ18, Proposition 4.3]
prove the following.

######  Theorem 2.6 (Deng and Zhu).

Let $(M^{n},g,f)$ be a complete noncollapsed steady gradient soliton. Suppose
that $\operatorname{Ric}>0$ and $|{\operatorname{Rm}}|\leq CR$ for some
constant $C<\infty.$ If $f$ admits a critical point, then

$R(x)\geq c/|xo|,$

for any $x\notin B_{r_{0}}(o)$, for some constant $c>0$ and $r_{0}<\infty.$

###### Sketch of the proof.

[DZ20a, Theorem 2.5] uses the same arguments as [DZ18, Proposition 4.3], where
they assumed nonnegative curvature operator or bisectional curvature only to
obtain Hamilton’s trace Harnack estimate (2.1) which implies a Harnack
estimate for the scalar curvature [DZ18, (3.3)]. As observed in [MZ21, Lemma
5.1], nonnegative Ricci curvature on steady solitons implies Hamilton’s trace
Harnack estimate. The rest of the arguments are the same as in the proof of
[DZ18, Proposition 4.3]. ∎

The estimate in the Corollary 2.5 implies that

$\limsup_{x\to\infty}\,R(x)|xo|>0,$

which is a statement stronger than infinite ASCR. Indeed, by taking
$x=x_{\tau}\in\partial B_{\tau}(o)$, we see that for all $\tau\gg 1$,

$R(x_{\tau})\geq\frac{a}{(C-\mu_{\infty})\tau}=\frac{a}{(C-\mu_{\infty})|x_{\tau}o|}.$

Hence, $\limsup_{x\to\infty}\,R(x)|xo|>0$.

A main consequence of Theorem 2.1, applicable to $4$-dimensional steady
soliton singularity models, is:

######  Corollary 2.7.

Let $(M^{4},g,f)$ be a $4$-dimensional noncollapsed and non-Ricci-flat steady
gradient Ricci soliton with bounded curvature. Let $o\in M$ be a fixed point.
Then there exists a positive constant $c$ depending on the soliton and $o$
such that

$R(x)\geq\frac{c}{1+|xo|^{2}}.$

Hence, any $4$-dimensional steady soliton singularity model either is
$\mathbb{R}^{4}/\Gamma$, or is a Ricci flat ALE, or has scalar curvature
decaying at most quadratically.

By Munteanu, Sung, and Wang [MSW19] (generalizing [CLY11]), the general
exponential lower bound for steady solitons $(M^{n},g,f)$ with potential
functions bounded from below is:

$R\geq ce^{-f}.$

Another asymptotic curvature estimate for steady solitons satisfying the
conditions in Corollary 2.5 is due to Ma and Zhang [MZ21, Corollary 5.2]:

${\rm ASCR}(g):=\limsup_{x\to\infty}R(x)|xo|^{2}=\infty.$

We mention a related result by Han in [Han20], where he proves the following
asymptotic curvature estimates: Let $(M^{n},g,f)$ be a complete steady
gradient Ricci soliton with $\sec\geq 0,$ $\operatorname{Ric}>0$, and where
the scalar curvature decays uniformly. Then

$\liminf_{x\to\infty}R(x)|xo|^{\alpha}=0,$

for any $\alpha\in(0,1).$

We remark that one may conjecture that $R$ has an inverse linear lower bound
in distance for general steady gradient Ricci solitons without nonnegative
curvature conditions.

##  3 Proofs of the results

###  3.1 Proof of Theorem 2.1

###### Proof of Theorem 2.1.

By [Bam20a, Theorem 7.2], for any $x\in M,$ $t<0,$ we have the Gaussian upper
bound for the conjugate heat kernel

$|t|^{n/2}K(o,0\,|\,x,t)\leq
C\exp\left(-\mu_{\infty}-\frac{|xz_{t}|_{t}^{2}}{9|t|}\right),$

where $(z_{t},t)$ is an $H_{n}$-center of $(o,0)$ and $\mu_{\infty}$ is the
constant in (NELB). Thus,

$f_{t}(x)\geq-C_{0}+\mu_{\infty}+1+\frac{|xz_{t}|_{t}^{2}}{9\tau},$

for some $C_{0}=C_{0}(n)>0.$ In the following, we write

$\rho_{t}(x):=\sqrt{1+\tau}\left(f_{t}(x)+C_{0}-\mu_{\infty}\right)\geq\sqrt{1+\tau}\left(1+\frac{|xz_{t}|_{t}^{2}}{9\tau}\right).$
(3.1)

Since $\partial_{t}f=-\Delta f+|\nabla f|^{2}-R+\frac{n}{2\tau}$, we have that

$\tau\Box f=f-\tfrac{n}{2}-w,$

where $\Box=\partial_{t}-\Delta_{g_{t}}$ is the heat operator. By Perelman’s
differential Harnack estimate [Per02, §9], we have

$w:=\tau(2\Delta f-|\nabla f|^{2}+R)+f-n\leq 0.$

So

$\displaystyle\tau\Box\rho$
$\displaystyle\geq-\frac{\tau}{2(1+\tau)}\rho+\sqrt{1+\tau}\,\big{(}f-\tfrac{n}{2}\big{)}\geq\tfrac{1}{2}\rho-A\sqrt{1+\tau},$

where

$A=A(n,\mu_{\infty}):=C_{0}-\mu_{\infty}+\tfrac{n}{2}.$

Then

$\displaystyle\tau\Box\rho^{-1}$
$\displaystyle=-\tau\rho^{-2}\,\Box\rho-2\tau\rho^{-3}\,|\nabla f|^{2}\vskip
6.0pt plus 2.0pt minus 2.0pt$
$\displaystyle\leq-\tfrac{1}{2}\rho^{-1}+A\sqrt{1+\tau}\,\rho^{-2}.$

Similarly,

$\displaystyle\tau\Box\rho^{-2}$
$\displaystyle\leq-\rho^{-2}+2A\sqrt{1+\tau}\,\rho^{-3},$

and

$\tau\Box(\sqrt{1+\tau}\,\rho^{-2})\leq-\sqrt{1+\tau}\,\rho^{-2}+2A(1+\tau)\rho^{-3}.$

So

$\tau\Box\big{(}\rho^{-1}+4A\sqrt{1+\tau}\,\rho^{-2}\big{)}\leq-\tfrac{1}{2}\left(\rho^{-1}+4A\sqrt{1+\tau}\,\rho^{-2}\right)-A\sqrt{1+\tau}\,\rho^{-3}\left(\rho-8A\sqrt{1+\tau}\,\right).$

Recall that

$\Box(\sqrt{1+\tau}\,R)=-\frac{R}{2\sqrt{1+\tau}}+2\sqrt{1+\tau}\,|{\operatorname{Ric}}|^{2}\geq-\frac{R}{2\sqrt{1+\tau}}.$

Let us define

$F:=\sqrt{1+\tau}\,R-c\rho^{-1}-4cA\sqrt{1+\tau}\,\rho^{-2},$ (3.2)

where $c=c(n,\mu_{\infty},a_{0})$ will be determined in the course of the
proof, and where $\mu_{\infty}$ and $a_{0}$ are the constants in (NELB) and
(SCLB), respectively. Then

$\tau\Box
F\geq-\tfrac{1}{2}F+cA\sqrt{1+\tau}\,\rho^{-3}\left(\rho-8A\sqrt{1+\tau}\,\right).$
(3.3)

Next, we aim to show that $\inf_{M\times(-\infty,0)}F\geq 0.$ To apply the
maximum principle to (3.3), we need to verify the boundary conditions. Note
that since the operator $\tau\Box$ and the function $\rho$ become singular as
$t\to 0-$, we also check the boundary condition on $M\times\\{0\\}$ to avoid
invoking advanced maximum principles.

Claim. _By taking
$c\leq\overline{c}(n,\mu_{\infty},a_{0}):=\frac{a_{0}}{1+4A}$, we have_

$\liminf_{x\to\infty}F(x,t)\geq 0,$ (3.4)

_uniformly for $t\in I$, where $I\subset(-\infty,0)$ is any compact interval,
and_

$\liminf_{t\to 0-}\left(\inf_{x\in M}F(x,t)\right)\geq 0,$ (3.5)
$\liminf_{t\to-\infty}\left(\inf_{x\in M}F(x,t)\right)\geq 0.$ (3.6)

###### Proof of the claim..

Inequality (3.4) is a direct consequence of Chen’s result that $R\geq 0$ (see
[Che09, Corollary 2.5]), the quadratic growth of $\rho$ from (3.1), the
equivalence of the metrics on a compact time interval, and the fact that the
$H_{n}$-centers of a fixed point do not drift with infinite velocity (a
consequence of the estimate of Perelman’s $\ell$-distance).

To prove (3.5), we fix an arbitrary time $t\in(-\infty,0)$. Let $x\in M$ be an
arbitrary point. If $x\in B_{t}(z_{t},D\sqrt{1+\tau}\,)$, then by (SCLB) and
(3.1), we have

$\displaystyle
F(x,t)\geq\frac{a_{0}}{\sqrt{1+\tau}}-\frac{c(1+4A)}{\sqrt{1+\tau}}\geq 0,$

provided we take $c\leq\frac{a_{0}}{1+4A}$. If $x\not\in
B_{t}(z_{t},D\sqrt{1+\tau})$, then we have

$\rho(x,t)\geq\sqrt{1+\tau}\left(1+\frac{D^{2}(1+\tau)}{\tau}\right)\geq\frac{D^{2}}{\tau},$

and thus

$\displaystyle
F(x,t)\geq-\frac{c\tau}{D^{2}}-\frac{4cA\tau^{2}\sqrt{1+\tau}}{D^{4}}.$

Therefore, we have

$\inf_{x\in M}F(x,t)\geq-C(A,D)\tau\to 0\quad\text{ as }\quad t\to 0-,$

and (3.5) follows immediately.

Finally, for (3.6), recall that $R\geq 0$ and $\rho\geq\sqrt{1+\tau}$. We have

$\inf_{x\in M}F(x,t)\geq-\frac{c(1+4A)}{\sqrt{1+\tau}}\quad\text{ for all
}\quad t<0,$

and (3.6) is proved. ∎

Assume, for a contradiction, that $\inf_{M\times(-\infty,0)}F<0$. By (3.4),
(3.5), and (3.6), we have that the space-time infimum of $F$ must be attained
at some point $(x_{0},t_{0})\in M\times(-\infty,0)$. Applying (3.3) at
$(x_{0},t_{0})$, we have

$0\geq\tau\Box
F\geq-\tfrac{1}{2}F+cA\sqrt{1+\tau}\,\rho^{-3}\left(\rho-8A\sqrt{1+\tau}\,\right)\quad\text{at}\quad(x_{0},t_{0}).$

Since $F(x_{0},t_{0})<0$, we have

$8A\sqrt{1+|t_{0}|}\geq\rho(x_{0},t_{0})\geq\sqrt{1+|t_{0}|}\left(1+\frac{|x_{0}z_{t_{0}}|_{t_{0}}^{2}}{9|t_{0}|}\right),$

where we have applied (3.1). Thus, we have

$|x_{0}z_{t_{0}}|_{t_{0}}\leq\sqrt{72A(1+|t_{0}|)}.$ (3.7)

If we require the constant $D$ in (SCLB) to satisfy $D\geq\sqrt{100A}$, then
by (SCLB), we have

$R(x_{0},t_{0})\geq\frac{a_{0}}{1+|t_{0}|}.$

By the definitions of $F$ in (3.2) and of $\rho$ in (3.1), we have

$\displaystyle
F(x_{0},t_{0})\geq\frac{a_{0}}{\sqrt{1+|t_{0}|}}-\frac{c(1+4A)}{\sqrt{1+|t_{0}|}}\geq
0,$

provided we take $c\leq\frac{a_{0}}{1+4A}$; this is a contradiction, and the
proof of Theorem 2.1 is complete. ∎

###  3.2 Proofs of Corollaries 2.2 and 2.3

###### Proof of Corollary 2.2.

First of all, the ancient solution in question cannot be Ricci-flat. Since the
tangent flow at infinity of a Ricci-flat static Ricci flow is its own blow-
down limit, which necessarily contains a singularity unless the manifold is
flat.

We show by contradiction that (SCLB) holds for any $D>0$. Suppose that (SCLB)
is false for some $D>0$. Since non-Ricci-flat ancient Ricci flows have
positive scalar curvature [Che09], there must be a sequence $(x_{i},t_{i})$
with $t_{i}\to-\infty$ and $x_{i}\in B_{t_{i}}(z_{t_{i}},D\sqrt{1+|t_{i}|})$,
such that $(1+|t_{i}|)R(x_{i},t_{i})\to 0$. Take $|t_{i}|$ as the scaling
factors when obtaining the tangent flow and let $z_{t_{i}}\to z_{\infty}\in
M_{\infty}$. Then there is a point $x_{\infty}\in B(z_{\infty},D)$ such that
$R_{\infty}(x_{\infty})=0$. By the strong maximum principle, we immediately
have that $(M_{\infty},g_{\infty},f_{\infty})$ is the Euclidean space, and
$\mu_{\infty}=0$. The ancient solution in question must also be Euclidean
space by Perelman’s monotonicity formula; this is a contradiction. ∎

###### Proof of Corollary 2.3.

By [MZ21, Proposition 2.4], under the assumptions of Hamilton’s trace Harnack
estimate (2.1) and the curvature pinching condition $|{\operatorname{Rm}}|\leq
CR,$ Perelman’s asymptotic shrinkers exist and are smooth. Thus, by [CMZ21a],
each tangent flow at infinity is smooth. We may now apply Corollary 2.2.

For the type-I case, Cao and Zhang proved the existence of Perelman’s
asymptotic shrinkers in [CZ11, Theorem 4.1] and thus we can apply Corollary
2.2. ∎

###  3.3 Proofs of Corollaries 2.5 and 2.7

Corollary 2.5 bounds the scalar curvature from below according to the distance
to certain anchor points on the level sets which play the same role as
$H_{n}$-centers or $\ell$-centers (points where the $\ell$-distance is bounded
by a fixed constant).

For steady solitons, it is often more convenient to work with the static
metric as compared to the induced Ricci flow. Let us recall some notions from
[BCMZ21]. Let $(M^{n},g,f)$ be a steady gradient Ricci soliton and let
$\Phi_{t}$ be the $1$-parameter group of diffeomorphisms generated by $-\nabla
f.$ Fix a point $o\in M.$ We define

$\Lambda(x,\tau):=\inf\int_{0}^{\tau}\sqrt{s}\,\big{(}R_{g}+|\dot{\gamma}-\nabla
f|_{g}^{2}\big{)}(\gamma(s))\,ds,$

where the infimum is taken over all $\gamma:[0,\tau]\to M$ with $\gamma(0)=o$
and $\gamma(\tau)=x$. Accordingly, define

$\lambda(x,\tau):=\ell(\Phi_{\tau}(x),\tau)=:\frac{1}{2\sqrt{\tau}}\Lambda(x,\tau),$

where $\ell$ is Perelman’s $\ell$-function. Arguing as Perelman in [Per02,
Section 7.1], we have that, for any $\tau>0,$ there is a point $p_{\tau}\in M$
such that

$\lambda(p_{\tau},\tau)=\ell(\Phi_{\tau}(p_{\tau}),\tau)\leq\frac{n}{2}.$

Any such point $p_{\tau}$ is called a $\lambda$-center at time $-\tau$. See
[BCMZ21, Section 2.1] for more details.

###### Proof of Corollary 2.5.

Recall that by Perelman’s space-time comparison geometry, the trace Harnack
estimate (2.1) implies that

$\tau|\nabla\ell|^{2}\leq C\ell.$ (3.8)

By [BCMZ21, Lemma 2.4], for $\tau\geq\bar{\tau}(n),$ we can find a point
$x_{\tau}\in\partial B_{\tau}(o)$ satisfying

$\lambda(x_{\tau},\tau_{0})\leq C$

for some $\tau_{0}\in[\tau/C,C\tau]$, where $C=C(n).$ We write
$\bar{x}_{\tau}=\Phi_{\tau_{0}}(x_{\tau}).$ Then

$\ell(\bar{x}_{\tau},\tau_{0})=\lambda(x_{\tau},\tau_{0})\leq C.$

By [Per02, Corollary 9.5] and (3.8), for any $\bar{x}=\Phi_{\tau_{0}}(x)\in
M,$

$f_{-\tau_{0}}(\bar{x})\leq\ell(\bar{x},\tau_{0})\leq
C+C\frac{|\bar{x}\bar{x}_{\tau}|_{-\tau_{0}}^{2}}{\tau_{0}}\leq
C+C\frac{|xx_{\tau}|^{2}}{\tau},$

where $(4\pi|t|)^{-\frac{n}{2}}e^{-f_{t}(x)}=K(o,0\,|\,x,t).$

By [MZ21, Theorem 1.13], Perelman’s asymptotic shrinkers exist and are smooth.
We may now apply Corollary 2.2 to conclude that, for any
$\bar{x}=\Phi_{\tau_{0}}(x)\in M,$ $\tau\geq\underline{\tau}(n),$

$\displaystyle R(x)$
$\displaystyle=R(\Phi_{-\tau_{0}}(\bar{x}))=R(\bar{x},-\tau_{0})$
$\displaystyle\geq\frac{a}{\tau_{0}(f_{-\tau_{0}}(\bar{x})+C(n)-\mu_{\infty})}$
$\displaystyle\geq\frac{a^{\prime}}{(C(n)-\mu_{\infty})\tau+|xx_{\tau}|^{2}}.$

∎

###### Proof of Corollary 2.7.

Let $(M,g_{t})_{t\in(-\infty,\infty)}$ be the canonical form of the steady
soliton in question. Since the curvature and the pointed Nash entropy are both
uniformly bounded, Bamler’s theorems [Bam20a, Bam20b, Bam20c] can be applied,
due to Bamler’s explanation in [Bam21]. Thus, we may apply [BCDMZ21,
Proposition 3.1] to the ancient solution $(M,g_{t})_{t\in(-\infty,0]}$. By the
classification of tangent flows at infinity therein and the non-Ricci-flat
assumption, we have that the tangent flow at infinity is unique and smooth.
Thus, we may apply Corollary 2.2. Finally, fixing, say, $\tau=1$, and applying
a standard Gaussian lower bound to the conjugate heat kernel (see e.g.
[CCGGIIKLLN10, Theorem 26.31]), we obtain the quadratic decay lower bound for
the scalar curvature. More precisely, by the Bishop–Gromov volume comparison
theorem, the volumes of unit balls are uniformly bounded from above: for any
$x\in M$,

$|B_{1}(x)|_{g}\leq C,$

where $|B_{1}(x)|_{g}$ is the volume of the unit geodesic ball taken with
respect to the soliton metric $g$ and $C$ depends on the curvature bound of
$g$. It can be seen from [CCGGIIKLLN10, Theorem 26.31] that

$(4\pi)^{-\frac{n}{2}}e^{-f_{-1}(x)}=K(o,0\,|\,x,-1)\geq\frac{c_{1}}{\sqrt{|B_{1}(o)|_{g}}\sqrt{|B_{1}(x)|_{g}}}e^{-\frac{|ox|^{2}}{c_{2}}}\geq\frac{c_{1}}{C}e^{-\frac{|ox|^{2}}{c_{2}}},$

where $c_{1}$ and $c_{2}$ are positive constants depending only on the
curvature bound of the soliton. Hence, by $|\nabla f|\leq\sqrt{C_{1}}$ (2.2),
for any $x\in M$,

$f_{-1}(\Phi_{1}(x))\leq C(|o\Phi_{1}(x)|^{2}+1)\leq
2C(|x\Phi_{1}(x)|^{2}+|ox|^{2}+1)\leq 2C(|ox|^{2}+C_{1}+1).$

The quadratic decay bound of the scalar curvature now follows by applying
Corollary 2.2 to the canonical form $(M,g_{t})_{t\in(-\infty,0]}$. ∎

###  3.4 Recovering the shrinker estimate for singularity models

In this section we show that we can consider Theorem 2.1 as a generalization
of the lower scalar curvature estimate for shrinkers of [CLY11], at least in
the case of singularity models.

Let $(M^{n},g,\bar{f})$ be a complete shrinking gradient Ricci soliton
satisfying

$\operatorname{Ric}+\nabla^{2}\bar{f}=\tfrac{1}{2}g,\quad
R+|\nabla\bar{f}|^{2}=\bar{f}.$

Let $(\Phi_{t})_{t\in\mathbb{R}}$ be the $1$-parameter group of
diffeomorphisms generated by $\nabla\bar{f}.$ Then

$g_{t}:=|t|\Phi^{*}_{-\ln|t|}g$

solves the Ricci flow and is called the canonical form generated by the
shrinking soliton. Recall that if we set

$\bar{f}_{t}:=\Phi_{-\ln|t|}^{*}\bar{f},$

then

$\operatorname{Ric}_{g_{t}}+\nabla^{2}_{g_{t}}\bar{f}_{t}=\tfrac{1}{2\tau}g_{t},\quad\tau(R_{g_{t}}+|\nabla\bar{f}_{t}|^{2}_{g_{t}})=\bar{f}_{t}.$

We have the following simple lemma.

######  Lemma 3.1.

Let $(M^{n},g_{t})_{t<0}$ be the canonical form induced by a complete
shrinking gradient Ricci soliton $(M^{n},g,\bar{f})$. Then for any $x\in M,$
$t<-1,$

$\int_{M}|t|\bar{f}_{t}\,d\nu_{x,-1|t}=\bar{f}(x)+\tfrac{n}{2}(-1-t).$

###### Proof.

For any $t<-1,$ writing $\nu_{t}=\nu_{x,-1|t},$

$\displaystyle\partial_{t}\int_{M}\bar{f}_{t}\,d\nu_{t}$
$\displaystyle=\int_{M}\Box\bar{f}_{t}\,d\nu_{t}=\frac{1}{\tau}\int_{M}\bar{f}_{t}\,d\nu_{t}-\frac{n}{2\tau}.$

See, e.g., [LW20] for the justification of integration by parts at infinity.
Integrating the equation above from $t$ to $-1$, we have

$\bar{f}(x)-\tau\int_{M}\bar{f}_{t}\,d\nu_{t}=-\tfrac{n}{2}(-1-t),$

and the conclusion follows. ∎

As a corollary, the $H_{n}$-centers of any minimum point $o$ of the potential
function $\bar{f}$ on shrinkers are not far away from $o$.

######  Proposition 3.2.

Let $(M^{n},g_{t})_{t<0}$ be the canonical form induced by a complete
shrinking gradient Ricci soliton $(M^{n},g,\bar{f})$. Let $o$ be a minimum
point of $\bar{f}$. Then for any $t<-1,$ if $(z,t)$ is an $H_{n}$-center of
$(o,-1),$ then

$|oz|_{t}\leq C(n)\sqrt{\tau}.$

###### Proof.

By the fundamental growth estimate of the potential function $\bar{f}$ by Cao
and Zhou [CZ10], for any $y\in M,t<-1,$

$\tau\bar{f}_{t}(y)\geq\tfrac{1}{4}\left(|oy|_{t}-C\sqrt{\tau}\right)_{+}^{2},$

where $C=C(n).$ Write $\nu_{t}=\nu_{o,-1|t}.$ Then for any $t<-1,$

$\displaystyle\int_{M}|oy|_{t}^{2}\,d\nu_{t}(y)$
$\displaystyle\leq\int_{M}\left(|oy|_{t}-C\sqrt{\tau}\right)_{+}^{2}\,d\nu_{t}(y)+C\tau$
$\displaystyle\leq C\tau+4\tau\int_{M}\bar{f}_{t}\,d\nu_{t}$
$\displaystyle\leq C\tau,$

where we have used the previous lemma and the fact that $\bar{f}(o)\leq n/2$
in the last inequality. Thus, by [Bam20a, Lemma 3.2],

$\displaystyle|oz|_{t}$ $\displaystyle\leq{\rm
dist}_{W_{1}}^{g_{t}}(\delta_{o},\nu_{t})+{\rm
dist}_{W_{1}}^{g_{t}}(\delta_{z},\nu_{t})\leq C\sqrt{\tau}.$

∎

######  Corollary 3.3.

Let $(M^{n},g_{t})_{t<0}$ be the canonical form induced by a complete non-flat
shrinking gradient Ricci soliton $(M^{n},g,\bar{f})$. Suppose also that it is
a singularity model or $(M^{n},g)$ has bounded curvature. Then for any
$D<\infty,$ the condition (SCLB) holds for some $a_{0}>0$, after shifting the
time by $-1$. As a consequence of Theorem 2.1, we can recover the main theorem
of [CLY11] in this case.

###### Proof.

Let $o$ be a minimum point of $\bar{f}.$ By Proposition 3.2, for any $t<-1$,
and any $H_{n}$-center $(z_{t},t)$ of $(o,-1),$

$|oz_{t}|_{t}\leq C\sqrt{\tau}.$

Since $(M^{n},g)$ is not Ricci-flat, we can choose $a_{0}>0$ such that

$\inf_{B(o,2(D+C))}R\geq a_{0}.$

Note that

$B_{t}\big{(}z_{t},D\sqrt{1+\tau}\,\big{)}\subset
B_{t}\big{(}o,(D+C)\sqrt{1+\tau}\,\big{)}=\Phi_{\ln|t|}\left(B\left(o,(D+C)\sqrt{(1+\tau)/\tau}\right)\right).$

Recall that $R(x,t)=R(\Phi_{-\ln|t|}(x))/\tau.$ Thus, (SCLB) holds. ∎

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