Schrödinger oscillators in a deformed point-like global monopole spacetime and a Wu-Yang magnetic monopole: position-dependent mass correspondence and isospectrality.

Various kinds of topological defects depend on the topology of the vacuum manifold and are formed by the phase transition in the early universe Re1 ; Re2 ; Re021 . Among such topological defects are the cosmic string Re021 ; Re3 ; Re4 ; Re5 ; Re6 , domain walls Re2 ; Re021 , and global monopole Re7 . Cosmic strings and global monopoles are known to be topological defects that do not introduce gravitational interactions but they rather modify the geometry of spacetime Re3 ; Re6 ; Re7 ; Re8 . Global monopoles are formed as a consequence of spontaneous global $O(3)$ symmetry breakdown to $U\left(1\right)$ and are similar to elementary particles (with their energy mostly concentrated near the monopole core) Re7 . They are spherically symmetric topological defects that admit the general static metric

$$ds^{2}=-B\left(r\right)\,dt^{2}+A\left(r\right)\,dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\varphi^{2}\right).$$

(1)

Barriola and Vilenkin Re7 have reported that

$$B\left(r\right)=A\left(r\right)^{-1}=1-8\pi G\eta^{2}-\frac{2GM}{r},$$

(2)

where $M$ is a constant of integration and in flat space $M\sim M_{core}$ ( $M_{core}$ is the mass of the monopole core).  By neglecting the mass term and rescaling the variables $r$ and $t$ Re7 , one may rewrite the global monopole metric as

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

(3)

where $0<\alpha^{2}=1-8\pi G\eta^{2}\leq 1$, $\alpha$ is a global monopole parameter that depends on the energy scale $\eta$, $G$ is the gravitational constant, and $\alpha=1$ corresponds to flat Minkowski spacetime Re7 ; Re8 ; Re9 ; Re10 . Barriola and Vilenkin Re7 have shown that the monopole, effectively, exerts no gravitational force. The space around and outside the monopole has a solid deficit angle that deflects all light. This has motivated several studies, amongst are, vacuum polarization effects in the presence of Wu-Yang Re101  magnetic monopole Re102 , gravitating magnetic monopole Re103 , Dirac and Klein-Gordon (KG) oscillators Re11 , Schrödinger oscillators Re9 , KG particles with a dyon, magnetic flux and scalar potential Re8 , bosons in Aharonov-Bohm flux field and a Coulomb potential Re131 , Schrödinger particles in a Kratzer potential Re132 , Schrödinger particles in a Hulthėn potential Re1321 , and scattering by a monopole Re133 . In general, the influence of topological defects in spacetime on the spectroscopy of quantum mechanical systems (be it through the introduction of gravitational field interactions or merely a modification of spacetimes) has been a subject of research attention over the years. In relativistic quantum mechanics, for example, the harmonic oscillator is studied in the context of Dirac and Klein-Gordon (KG) Re11 ; Re12 ; Re121 ; Re13 ; Re14 ; Re15 ; Re16 ; Re17 ; Re18 ; Re19 ; Re20 ; Re21 ; Re211 ; Re22 ; Re23 ; Re24 ; Re25 ; Re26 ; Re27 in different spacetime backgrounds.

On the other hand, the effective position-dependent mass (PDM) Schrödinger equation introduced by von Roos Re271 finds it applications in nuclear physics, nanophysics, semiconductors, etc Re271 ; Re272 ; Re273 ; Re274 . Where, the von Roos PDM kinetic energy operator Re271 (in $\hbar=2m=1$ units) is given by

$$\hat{T}=-\frac{1}{2}\left[m\left(x\right)^{j}\,\partial_{x}\,m\left(x\right)^{k}\,\partial_{x}\,m\left(x\right)^{l}+m\left(x\right)^{l}\,\partial_{x}\,m\left(x\right)^{k}\,\partial_{x}\,m\left(x\right)^{j}\right],$$

(4)

with an effective PDM $m\left(x\right)=mf\left(x\right)$, and $m$ is the mass of Schrödinger particle. However, the continuity conditions at the abrupt heterojunction suggest that $j=l$ Re272 ; Re273 ; Re274 , where $j,k,l$ are called the ordering ambiguity parameters that satisfy the von Roos constraint $j+k+l=-1$ Re271 . Recently, it has been shown that under some coordinate deformation/transformation Re28 the PDM kinetic energy operator collapses into

$$\hat{T}=-m\left(x\right)^{-1/4}\,\partial_{x}\,m\left(x\right)^{-1/2}\,\partial_{x}\,m\left(x\right)^{-1/4},$$

(5)

where $j=l=-1/4$ and $k=-1/2$ (known in the literature as Mustafa-Mazharimousavi’s ordering Re28 ; Re281 ; Re282 . Inspired by Khlevniuk and Tymchyshyn Re29 observation that a point mass moving within the curved coordinates/space transforms into position-dependent mass in Euclidean coordinates/space, we, hereby, introduce a deformation/transformation of the global monopole spacetime metric (3) and show that the corresponding Schrödinger equation transforms into a one-dimensional von Roos Re271 PDM-Schrödinger equation. We proceed, under such settings, and discuss the corresponding effects on the spectroscopic structure of the PDM Schrödinger oscillators., including the Wu-Yang magnetic monopole and a hard-wall effects.

The organization of our manuscript is in order. In section 2, we show that a deformation/transformation in the global monopole spacetime metric (3) would yield an effective position-dependent mass (PDM) Schrödinger equation. We start with Schrödinger particles in the background of a deformed/transformed global monopole spacetime. We then connect our findings with the von Roos Re271  PDM Schrödinger equation. We discuss PDM Schrödinger oscillators in a global monopole spacetime, in section 3. We consider, in section 4, the PDM Schrödinger oscillators in a global monopole spacetime in the presence of a Wu-Yang magnetic monopole  Re101 . Within our deformed/transformed global monopole spacetime recipe, we show that all our PDM Schrödinger oscillators admit isospectrality and invariance with the Schrödinger oscillators in the regular global monopole spacetime in the presence of a Wu-Yang magnetic monopole. Nevertheless, the exclusive dependence of the thermodynamical partition function on the energy eigenvalues manifestly suggests that the Schrödinger oscillators and the PDM Schrödinger oscillators have the same thermodynamical properties as mandated by their isospectrality. We, therefore, report their thermodynamical properties (e.g., Re9 ; Re30 ; Re31 ; Re32 ; Re33 ; Re34 ; Re35 ), in section 5. Such properties are, in fact, shared by both Schrödinger oscillators and PDM Schrödinger oscillators in a global monopole spacetime without and with a Wu-Yang magnetic monopole. In section 6, we discuss the hard-wall effect on the energy levels PDM Schrödinger oscillators in a global monopole spacetime without and with a Wu-Yang magnetic monopole. The hard-wall confinement is studied by Bakke for a Landau-Aharonov-Casher system Re351 and for Dirac neutral particles Re352 , by Castro Re353 for scalar bosons. and by Vitória and Bakke Re354 for the rotating effects on the scalar field in spacetime with linear topological defects, to mention a few. Our concluding remarks are given in section 7. To the best of our knowledge, such a study has not been carried out elsewhere.

Let us consider Schrödinger particles interacting with a point-like global monopole (PGM) with a spacetime metric given by (3) and subjected to a point canonical transformation (PCT) in the form of

$$r=\int\sqrt{f\left(\rho\right)}d\rho=\sqrt{q\left(\rho\right)}\rho\Leftrightarrow\sqrt{f\left(\rho\right)}=\sqrt{q\left(\rho\right)}\left[1+\frac{q^{\prime}\left(\rho\right)}{2q\left(\rho\right)}\rho\right].$$

(6)

Then the PGM metric (3) transforms into

$$ds^{2}=-dt^{2}+\frac{f\left(\rho\right)}{\alpha^{2}}d\rho^{2}+q\left(\rho\right)\,\rho^{2}\left[d\theta^{2}+\sin^{2}\theta\,d\varphi^{2}\right],$$

(7)

where $q\left(\rho\right)$ and $f\left(\rho\right)$ are positive-valued scalar multiplier and $q\left(\rho\right)=1\Rightarrow f\left(\rho\right)=1$ (i.e., constant mass settings) recovers the PGM metric (3). Consequently, the corresponding deformed/transformed metric tensor is

$$g_{ij}=\left(\begin{tabular}[]{ccc}$\frac{f\left(\rho\right)}{\alpha^{2}}$&$0$&$0\vskip 6.0pt plus 2.0pt minus 2.0pt$\\ $0$&$q\left(\rho\right)\,\rho^{2}$&$0\vskip 6.0pt plus 2.0pt minus 2.0pt$\\ $0$&$0$&$q\left(\rho\right)\,\rho^{2}\sin^{2}\theta$\end{tabular}\right);\;i,j=\rho,\theta,\varphi,$$

(8)

to imply

$$\det\left(g_{ij}\right)=g=\frac{f\left(\rho\right)}{\alpha^{2}}q\left(\rho\right)^{2}\,\rho^{4}\sin^{2}\theta,$$

and

$$g^{ij}=\left(\begin{tabular}[]{ccc}$\frac{\alpha^{2}}{f\left(\rho\right)}$&$0$&$0\vskip 6.0pt plus 2.0pt minus 2.0pt$\\ $0$&$\frac{1}{q\left(\rho\right)\,\rho^{2}}$&$0\vskip 6.0pt plus 2.0pt minus 2.0pt$\\ $0$&$0$&$\frac{1\vskip 6.0pt plus 2.0pt minus 2.0pt}{q\left(\rho\right)\,\rho^{2}\sin^{2}\theta}$\end{tabular}\right).$$

(9)

Then, Schrödinger equation

$$\left\{\left(-\frac{\hbar^{2}}{2m_{\circ}}\frac{1}{\sqrt{g}}\partial_{i}\sqrt{g}g^{ij}\partial_{j}\right)+V\left(\mathbf{\rho},t\right)\right\}\Psi\left(\mathbf{\rho},t\right)=i\hbar\frac{\partial}{\partial t}\Psi\left(\mathbf{\rho},t\right),$$

(10)

would, with $V\left(\mathbf{\rho},t\right)=V\left(\rho\left(r\right)\right)$ and $\Psi\left(\mathbf{\rho},t\right)=e^{-iEt/\hbar}\psi\left(\rho\right)Y_{\ell m}\left(\theta,\varphi\right)$, yield

$$\left\{\frac{\hbar^{2}}{2m_{\circ}}\left(-\frac{1}{q\left(\rho\right)\,\sqrt{f\left(\rho\right)}\,\rho^{2}}\,\partial_{\rho}\left(\frac{q\left(\rho\right)\,\,\rho^{2}}{\sqrt{f\left(\rho\right)}}\,\partial_{\rho}\right)+\frac{\ell\left(\ell+1\right)}{\alpha^{2}q\left(\rho\right)\,\,\rho^{2}}\right)+\frac{1}{\alpha^{2}}V\left(\rho\left(r\right)\right)\right\}\psi\left(\rho\right)=\frac{1}{\alpha^{2}}E\psi\left(\rho\right),$$

(11)

where $Y_{\ell m}\left(\theta,\varphi\right)$ are the spherical harmonics, $\ell$ is the angular momentum quantum number, and $m$ is the magnetic quantum number. In a straightforward manner, equation (11) along with our PCT in (6), is transformed into

$$\left\{\frac{\hbar^{2}}{2m_{\circ}}\left(-\frac{1}{r^{2}}\,\partial_{r}\,r^{2}\partial_{r}+\frac{\tilde{\ell}\left(\tilde{\ell}+1\right)}{r^{2}}\right)+\frac{1}{\alpha^{2}}V\left(r\left(\rho\right)\right)\right\}\psi\left(r\left(\rho\right)\right)=\mathcal{E}\psi\left(r\left(\rho\right)\right)$$

(12)

to imply (with $\psi\left(r\right)=R\left(r\right)/r$)

$$\left[\frac{\hbar^{2}}{2m_{\circ}}\left(-\partial_{r}^{2}+\frac{\tilde{\ell}\left(\tilde{\ell}+1\right)}{r^{2}}\right)+\frac{1}{\alpha^{2}}V\left(r\left(\rho\right)\right)\right]R\left(r\right)=\mathcal{E}R\left(r\right),$$

(13)

where $\mathcal{E}=E/\alpha^{2}$, and

$$\tilde{\ell}\left(\tilde{\ell}+1\right)=\frac{\ell\left(\ell+1\right)}{\alpha^{2}}\Longrightarrow\tilde{\ell}=-\frac{1}{2}+\frac{\sqrt{\alpha^{2}+4\ell\left(\ell+1\right)}}{2\alpha}$$

(this would retrieve the regular angular momentum quantum number $\ell$ for a flat Minkowski spacetime at $\alpha=1$). Moreover, the two quantum mechanical systems in (11) and (12) are isospectral and invariant. That is, knowing the solution of one of them would immediately yield the solution of the other. Yet they both share the same energies.

Let us consider $V\left(r\left(\rho\right)\right)=\omega^{2}r^{2}$ in (13) to obtain

$$\left[-\partial_{r}^{2}+\frac{\tilde{\ell}\left(\tilde{\ell}+1\right)}{r^{2}}+\tilde{\omega}^{2}r^{2}\right]R\left(r\right)=\mathcal{E}R\left(r\right),$$

(22)

where $\tilde{\omega}=\omega/\alpha$ and $\mathcal{E}=E/\alpha^{2}$. This is the radial spherically symmetric Schrödinger oscillator equation that admits exact textbook solution in the form of

$$R\left(r\right)\sim r^{\tilde{\ell}+1}\exp\left(-\frac{\tilde{\omega}r^{2}}{2}\right)\;_{1}F_{1}\left(\frac{\tilde{\ell}}{2}+\frac{3}{4}-\frac{\mathcal{E}}{4\tilde{\omega}},\tilde{\ell}+\frac{3}{2},\tilde{\omega}r^{2}\right),$$

(23)

for the radial part, which is to be finite and square integrable through the condition that the confluent hypergeometric series is truncated into a polynomial of order $n_{r}=0,1,2,\cdots$. In this case,

$$\frac{\tilde{\ell}}{2}+\frac{3}{4}-\frac{\mathcal{E}}{4\tilde{\omega}}=-n_{r}\Rightarrow\mathcal{E}=2\tilde{\omega}\left(2n_{r}+\tilde{\ell}+\frac{3}{2}\right)\Rightarrow E=2\alpha\omega\left(2n_{r}+\frac{\sqrt{\alpha^{2}+4\ell\left(\ell+1\right)}}{2\alpha}+1\right)$$

(24)

for the energies (which is in exact accord with the result reported by Vitória and Belich in Eq. (11) of Re9 , $\omega=\omega_{VB}/2$)., and

$$R\left(r\right)\sim r^{\tilde{\ell}+1}\exp\left(-\frac{\tilde{\omega}r^{2}}{2}\right)\,L_{n_{r}}^{\tilde{\ell}+1/2}\left(\frac{\tilde{\omega}r^{2}}{2}\right)\Rightarrow\Psi\left(r,\theta,\varphi\right)=\mathcal{N}_{n_{r},\ell}\,r^{\tilde{\ell}}\exp\left(-\frac{\tilde{\omega}r^{2}}{2}\right)\,L_{n_{r}}^{\tilde{\ell}+1/2}\left(\frac{\tilde{\omega}r^{2}}{2}\right)Y_{\ell m}\left(\theta,\varphi\right).$$

(25)

for the radial part of the wave functions, where $\,L_{n_{r}}^{\tilde{\ell}+1/2}\left(\tilde{\omega}r^{2}/2\right)$ are the generalized Laguerre polynomials. Hereby, it should be noted that this quantum mechanical system is isospectral and invariant with the PDM one

$$\left\{-f\left(\rho\right)^{-1/4}\partial_{\rho}f\left(\rho\right)^{-1/2}\partial_{\rho}\,f\left(\rho\right)^{-1/4}+\frac{\tilde{\ell}\left(\tilde{\ell}+1\right)}{q\left(\rho\right)\,\rho^{2}}\,+\tilde{\omega}^{2}q\left(\rho\right)\,\rho^{2}\right\}\phi\left(\rho\right)=\mathcal{E}\,\phi\left(\rho\right),$$

(26)

where $f\left(\rho\right)$ and $q\left(\rho\right)$ are correlated through (6), provided that $R\left(r\left(\rho\right)\right)$ is given by (14). For example, for a power-low like dimensionless radial deformation $q\left(\rho\right)=A\rho^{\sigma}$, we obtain $f\left(\rho\right)=A\left(1+\sigma/2\right)^{2}\rho^{\sigma}$; $\sigma\neq 0,-2$, and the corresponding PDM Schrödinger oscillators system reads

$$\left\{-\left(\tilde{A}\rho^{\sigma}\right)^{-1/4}\partial_{\rho}^{-1/2}\left(\tilde{A}\rho^{\sigma}\right)\,\partial_{\rho}\,\left(\tilde{A}\rho^{\sigma}\right)^{-1/4}+\frac{\tilde{\ell}\left(\tilde{\ell}+1\right)}{A\rho^{\sigma+2}}\,+\tilde{\omega}^{2}A\rho^{\sigma+2}\right\}\phi\left(\rho\right)=\mathcal{E}\,\phi\left(\rho\right),$$

(27)

where $\tilde{A}=A\left(1+\sigma/2\right)^{2}$. Such a system represents just one of so many examples on PDM Schrödinger oscillators interacting with a PGM and share the same eigenvalues, (24) with that in (22).

In this section, we discuss PDM Schrödinger particles a PGM spacetime and a Wu-Yang magnetic monopole. Wu and Yang Re101 have proposed a magnetic monopole that is free of strings of singularities around it Re8 ; Re101 ; Re102 . They have defined the vector potential $A_{\mu}$ in two regions, $R_{A}$ and $R_{B}$, covering the whole space, outside the magnet monopole, and overlap in $R_{AB}$ so that

$R_{A}:0\leq\theta<\frac{\pi}{2}+\delta\vskip 6.0pt plus 2.0pt minus 2.0pt,\;$ $\;\;r>0,\;\;$ $0\leq\varphi<2\pi,$ $R_{B}:\frac{\pi}{2}-\delta<\theta\leq\pi,\;$ $\;\;r>0,\;\;$ $0\leq\varphi<2\pi,$ $R_{AB}:\frac{\pi}{2}-\delta<\theta<\frac{\pi}{2}+\delta,$ $\;\;r>0,\;$ $0\leq\varphi<2\pi,$

(28)

where $0<\delta\leq\pi/2$. Moreover, the vector potential has a non-vanishing component in each region given by

$$\begin{tabular}[]{ll}$A_{\varphi,A}=g\left(1-\cos\theta\right),\;\;$&$A_{\varphi,B}=-g\left(1+\cos\theta\right)$\end{tabular},$$

(29)

where $g$ is the Wu-Yang monopole strength and $A_{\varphi,A}$ and $A_{\varphi,B}$ are correlated by the gauge transformation Re8 ; Re102

$$A_{\varphi,A}=A_{\varphi,B}+\frac{i}{e}S\,\partial_{\varphi}\,S^{-1}\,;\;S=e^{2iq\varphi},\;q=eg.$$

(30)

We shall, for the sake of simplicity and economy of notation, use the form $A_{\varphi}=sg-g\cos\theta$, with $s=1$ for $A_{\varphi,A}$ and $s=-1$ for $A_{\varphi,B}$.

Under such settings, equation (10) would now read

$$\left\{\left(-\frac{1}{\sqrt{g}}\left(\partial_{i}-ieA_{i}\right)\sqrt{g}g^{ij}\left(\partial_{j}-ieA_{j}\right)\right)+V\left(\mathbf{\rho},t\right)\right\}\Psi\left(\mathbf{\rho},t\right)=i\frac{\partial}{\partial t}\Psi\left(\mathbf{\rho},t\right),$$

(31)

to imply

$$\left\{-\frac{\alpha^{2}}{r^{2}}\,\partial_{r}\left(r^{2}\,\partial_{r}\right)-\frac{1}{r^{2}}\left(\frac{1}{\sin\theta}\partial_{\theta}\sin\theta\;\partial\theta+\frac{1}{\sin^{2}\theta}\left[\partial_{\varphi}-ieA_{\varphi}\right]^{2}\right)+V\left(\mathbf{\rho}\left(r\right),t\right)\right\}\Psi\left(\mathbf{\rho},t\right)=i\frac{\partial}{\partial t}\Psi\left(\mathbf{\rho},t\right),$$

(32)

where $r$ is given by (6). We may now seek separation of variables for (32) and use the substitution $\Psi\left(\mathbf{\rho},t\right)=e^{-iEt}\;\psi\left(\rho\right)\;Y_{\tilde{q}\ell m}\left(\theta,\varphi\right)$, where $\tilde{q}=sq$ and $Y_{\tilde{q}\ell m}\left(\theta,\varphi\right)$ are the Wu-Yang monopole harmonics so that

$$\left(\frac{1}{\sin\theta}\partial_{\theta}\sin\theta\;\partial\theta+\frac{1}{\sin^{2}\theta}\left[\partial_{\varphi}-ieA_{\varphi}\right]^{2}\right)Y_{\tilde{q}\ell m}\left(\theta,\varphi\right)=-\lambda Y_{\tilde{q}\ell m}\left(\theta,\varphi\right).$$

(33)

Consequently (32) reduces to

$$\left\{-\frac{\alpha^{2}}{r^{2}}\,\partial_{r}\left(r^{2}\,\partial_{r}\right)+\frac{\lambda}{r^{2}}+V\left(\mathbf{\rho}\left(r\right)\right)\right\}\psi\left(\rho\right)=E\psi\left(\rho\right).$$

(34)

At this point, one should first solve for the eigenvalues $\lambda$ of  (33) using the substitution

$$Y_{\tilde{q}\ell m}\left(\theta,\varphi\right)=\exp\left(i\left(m+\tilde{q}\right)\varphi\right)\Theta_{\tilde{q}\ell m}\left(\theta\right);\;\tilde{q}=sq=seg,$$

(35)

to obtain

$$\left(\frac{1}{\sin\theta}\partial_{\theta}\sin\theta\;\partial\theta-\frac{1}{\sin^{2}\theta}\left[m+q\cos\theta\right]^{2}\right)\Theta_{\tilde{q}\ell m}\left(\theta\right)=-\lambda\Theta_{\tilde{q}\ell m}\left(\theta\right).$$

(36)

Notably, this equation does not depend on the value of $s$ in $\tilde{q}$ of (35) (i.e., $\Theta_{\tilde{q}\ell m}\left(\theta\right)=\left[\Theta_{q\ell m}\left(\theta\right)\right]_{A}=\left[\Theta_{q\ell m}\left(\theta\right)\right]_{B}=\Theta_{q\ell m}\left(\theta\right)$ as observed by Wu-Yang Re101 ) and consequently, with $x=\cos\theta$, would read

$$\left\{\left(1-x^{2}\right)\,\partial_{x}^{2}-2x\,\partial_{x}-\frac{\left(m+q\,x\right)^{2}}{1-x^{2}}\right\}\Theta_{q\ell m}\left(x\right)-\lambda\Theta_{q\ell m}\left(x\right).$$

(37)

Let us define

$$\Theta_{q\ell m}\left(x\right)=\left(1-x\right)^{\sigma/2}\left(1+x\right)^{\nu/2}\,P_{q\ell m}\left(x\right),$$

(38)

to obtain, with $\sigma=(|m|+q)$ and $\nu=(|m|-q)$ (this choice is motivated by the fact that the space around a monopole is without singularities and so is the wave function around the monopole Re101 ),

$$\left(x^{2}-1\right)\,P_{q\ell m}^{{}^{\prime\prime}}\left(x\right)+\left[2q+2\left(m+1\right)x\right]\,P_{q\ell m}^{{}^{\prime}}\left(x\right)+\left(m^{2}+m-q^{2}-\lambda\right)\,P_{q\ell m}\left(x\right)=0.$$

(39)

The exact solution of which admits the form of hypergeometric functions

$$P_{q\ell m}\left(x\right)=C\,_{1}F_{1}\left(|m|+\frac{1}{2}\pm\frac{1}{2}\sqrt{4q^{2}+4\lambda+1},|m|+1-q,\frac{1}{2}\left(1+x\right)\right).$$

(40)

However, to secure finiteness and square integrability of the quantum mechanical wave functions, we truncate the confluent hypergeometric series into a polynomial of order $n=0,1,2,\cdots$. In this case, we take

$$-n=|m|+\frac{1}{2}\pm\frac{1}{2}\sqrt{4q^{2}+4\lambda+1}\Longrightarrow\lambda=\left(n+|m|\right)\left(n+|m|+1\right)-q^{2}\Longrightarrow\lambda=\upsilon\left(\upsilon+1\right)-q^{2},$$

(41)

where $\upsilon=n+|m|=\ell=0,1,2,\cdots$ is, without loss of generality, the angular momentum quantum number. That is, when $q=0$ (i.e., the Wu-Yang monopole strength $g$ is zero) one should naturally retrieve the eigenvalue of the regular spherical harmonics as $\lambda=\ell\left(\ell+1\right)$. Obviously, this result is in exact accord with that reported by Wu and Yang Re8 ; Re101 who have named $Y_{\tilde{q}\ell m}\left(\theta,\varphi\right)$ as the monopole harmonics. At this point, one should observe that

$$Y_{\tilde{q}\ell m}\left(\theta,\varphi\right)=\left\{\begin{tabular}[]{ll}$e^{i\left(m+q\right)\varphi}\,\left(1-x\right)^{\sigma/2}\left(1+x\right)^{\nu/2}\,P_{q\ell m}\left(x\right);\;$&in region $R_{A}$\\ $e^{i\left(m-q\right)\varphi}\,\left(1-x\right)^{\sigma/2}\left(1+x\right)^{\nu/2}\,P_{q\ell m}\left(x\right);$&in region $R_{B}$\end{tabular}\right..$$

(42)

We may now rewrite the radial equation (34), with $V\left(\mathbf{\rho}\left(r\right)\right)=\omega^{2}r^{2}$, as

$$\left\{-\frac{1}{r^{2}}\,\partial_{r}\left(r^{2}\,\partial_{r}\right)+\frac{L\left(L+1\right)}{r^{2}}+\tilde{\omega}^{2}r^{2}\right\}\psi\left(\rho\left(r\right)\right)=\mathcal{E}\psi\left(\rho\left(r\right)\right),$$

(43)

where $\mathcal{E}=E/\alpha^{2}$,$\;\tilde{\omega}=\omega/\alpha$ and

$$L\left(L+1\right)=\frac{\ell\left(\ell+1\right)-q^{2}}{\alpha^{2}}\Longrightarrow L=-\frac{1}{2}+\sqrt{\frac{1}{4}+\frac{\ell\left(\ell+1\right)-q^{2}}{\alpha^{2}}}.$$

(44)

One should notice that the square root signature is chosen so that for $\alpha=1$ and $q=0$ one would retrieve $L=\ell$ the regular angular momentum quantum number. Moreover, the solution to (43) with $\psi\left(\rho\right)=R\left(\rho\right)/\rho$ would read

$$R\left(r\right)=R\left(r\left(\rho\right)\right)\sim r^{L+1}\exp\left(-\frac{\tilde{\omega}r^{2}}{2}\right)\;_{1}F_{1}\left(\frac{L}{2}+\frac{3}{4}-\frac{\mathcal{E}}{4\tilde{\omega}},L+\frac{3}{2},\tilde{\omega}r^{2}\right);\;r=\sqrt{q\left(\rho\right)}\rho.$$

(45)

However, finiteness and square integrability would again enforce the condition that the confluent hypergeometric series is truncated into a polynomial of order $n_{r}=0,1,2,\cdots$ so that $\frac{L}{2}+\frac{3}{4}-\frac{\mathcal{E}}{4\tilde{\omega}}=-n_{r}$ to imply

$$\mathcal{E}=2\tilde{\omega}\left(2n_{r}+L+\frac{3}{2}\right)\Rightarrow E_{n_{r},\ell,q}=2\alpha\omega\left(2n_{r}+\sqrt{\frac{1}{4}+\frac{\ell\left(\ell+1\right)-q^{2}}{\alpha^{2}}}+1\right).$$

(46)

The Schrödinger oscillators described in (43) are isospectral and invariant with the corresponding PDM Schrödinger oscillators

$$\left\{--\frac{1}{q\left(\rho\right)\,\sqrt{f\left(\rho\right)}\,\rho^{2}}\,\partial_{\rho}\left(\frac{q\left(\rho\right)\,\,\rho^{2}}{\sqrt{f\left(\rho\right)}}\,\partial_{\rho}\right)+\frac{L\left(L+1\right)}{q\left(\rho\right)\,\rho^{2}}+\tilde{\omega}^{2}q\left(\rho\right)\,\rho^{2}\right\}\psi\left(\rho\right)=\mathcal{E}\psi\left(\rho\right),$$

(47)

At this point, we may report that the energy levels of (46) are plotted in Figures 1 and 2.

In Figures 1(a), 1(b), and 1(c), we show the energy levels of (46) against the global monopole parameter $\alpha$ for different Wu-Yang magnetic monopole parameter values $q=0$, $q=\alpha/4$, and $q=\alpha/16$, respectively. For $q=0$ (i.e., No Wu-Yang monopole), we observe in Figure 1(a) that while the energy levels linearly increase with increasing $\alpha$, the spacing between the energy levels (for the same $n_{r}$ and $\ell=0,1,2,3$, where $n_{r}=1$ is used throughout) remains constant at each $\alpha$ value. This is a common characteristic for the Schrödinger oscillator in a flat Minkowski spacetime (i.e., $\alpha=1$). However, in 1(b) and 1(c) ( for $q=\alpha/4$, and $q=\alpha/16$, respectively), we notice that the equal spacing between energy levels is no longer valid. The maximum value for $q$ used are chosen so that $\alpha_{\max}=1$. In Figures 2(a), 2(b), and 2(c), we show  the energy levels at $\alpha=0.5$, $\alpha=0.9$, and $\alpha=1$, respectively, for different Wu-Yang magnetic monopole strengths $q=eg$. Where the maximum values for $q$ are now chosen so that the square root in (46) remains a real-valued one. We observe that the Wu-Yang monopole yields non-equally spaced energy levels. Moreover, it is clear that the energies are shifted up as the PGM parameter $\alpha$ increases for each value of the Wu-Yang monopole parameter $q$ (including $q=0$ for no Wu-Yang monopole).

In this section we shall study the thermodynamical properties of PDM Schrödinger oscillators in a global monopole spacetime background without and with a Wu-Yang magnetic monopole. In a straightforward manner one obtains the partition function

$$Z\left(\beta\right)=\sum\limits_{n_{r}=0}^{\infty}\exp\left(-\beta\,E_{n_{r},\ell,q}\right)=\frac{\exp\left(-2\alpha\beta\omega\tau\right)}{1-\exp\left(-4\alpha\beta\omega\right)};\text{ }\beta=\frac{1}{K_{B}T},$$

(48)

where $K_{B}$ is the Boltzmann constant, $T$ is the temperature and

$$\tau=1+\frac{1}{2\alpha}\sqrt{\alpha^{2}+4\ell\left(\ell+1\right)-4q^{2}}.$$

(49)

At this point, one should notice that for $q=eg=0$ represents PDM Schrödinger oscillators in a global monopole spacetime background without the Wu-Yang magnetic monopole. Moreover, the global monopole parameter $\alpha$ and the Wu-Yang monopole strength (through $q=eg$) are correlated in such a way that the value under the square root remains real. In this case, $0\leq q\leq\sqrt{\ell\left(\ell+1\right)+\alpha^{2}/4}$, and consequently $q_{\max}=\ell+1/2$, where $\alpha_{\max}=1$ corresponds to flat Minkowski spacetime.

To observe the effects of the global monopole spacetime background and the Wu-Yang magnetic monopole on some thermodynamical properties associated with such systems, we find that the Helmholtz free energy $f\left(T\right)$ is given by

$$f\left(T\right)=-\frac{1}{\beta}\ln\left(Z\left(\beta\right)\right)=2\alpha\omega\tau+K_{B}T\,\ln\left(1-\exp\left(-\frac{4\alpha\omega}{K_{B}T}\right)\right),$$

(50)

the Entropy $S\left(T\right)$

$$S\left(T\right)=-\frac{df\left(T\right)}{dT}=-K_{B}\,\ln\left(1-\exp\left(-\frac{4\alpha\omega}{K_{B}T}\right)\right)+\frac{4\alpha\omega}{T}\left[\frac{\exp\left(-\frac{4\alpha\omega}{K_{B}T}\right)}{1-\exp\left(-\frac{4\alpha\omega}{K_{B}T}\right)}\right],$$

(51)

the Specific heat $c\left(T\right)$

$$c\left(T\right)=T\,\frac{dS\left(T\right)}{dT}=\frac{16\alpha^{2}\omega^{2}}{T^{2}K_{B}}\left[\frac{\exp\left(-\frac{2\alpha\omega}{K_{B}T}\right)}{1-\exp\left(-\frac{4\alpha\omega}{K_{B}T}\right)}\right]^{2},$$

(52)

and Mean energy $U\left(T\right)$

$$U\left(T\right)=-\frac{dZ\left(\beta\right)}{d\beta}=2\alpha\omega\tau-\frac{4\alpha\omega}{1-\exp\left(\frac{4\alpha\omega}{K_{B}T}\right)}.$$

(53)

We observe that while the Helmholtz free energy $f\left(T\right)$ in (50) and the Mean energy $U\left(T\right)$ in (53) are affected by the Wu-Yang magnetic monopole through the parameter $\tau$ in (49), the Entropy $S\left(T\right)$ in (51) and the Specific heat $c\left(T\right)$ in (52) are not. However, all mentioned thermodynamical properties are affected by the global monopole through the parameter $\alpha$.

In Figures 3(a), 3(b), and 3(c), we show (for $\ell=1$ states) the effect of the Wu-Yang Re101 magnetic monopole on the Helmholtz free energies $f(T)$, Eq.(50), of the Schrödinger-oscillator in a point-like global monopole for $q=0$, $q=1$, and $q=1.4$, respectively. It is obvious that as $q=eg$ increases the Helmholtz free energy converges more rapidly to the zero value as the temperature $T$ grows up from just above zero. In Figures 4(a), 4(b), and 4(c), we show (for $\ell=1$ states) the effect of the Wu-Yang monopole on the mean energy $U\left(T\right)$, Eq. (53), for $q=0$, $q=1$, and $q=1.4$, respectively. We observe that as the Wu-Yang monopole strength increases (through $q=eg$) the mean energy decreases for each value of $T$. We also notice that the mean energy $U\left(T\right)$, for all allowed $\alpha$ values used, tend to cluster at very high temperatures for $q=0$ (i.e., no Wu-yang monopole). However, it is clear that as $q$ increases from zero, such clustering is slowed down. In Figure 5(a), we show (for $\ell=1$ states) the entropy $S\left(T\right)$, Eq. (51), as the temperature grows up from just above zero for the Schrödinger-oscillator in a point-like global monopole. Figure 5(b) shows the specific heat $c\left(T\right)$, Eq. (52), against the temperature for the Schrödinger-oscillator in a point-like global monopole. It is obvious that the ratio $c\left(T\right)/K_{B}\rightarrow 1\Rightarrow c\left(T\right)\rightarrow K_{B}$ as $T>>1$ for all allowed values of the point-like global monopole parameter $\alpha$. Notably, the Wu-Yang magnetic monopole has no effect on the entropy $S\left(T\right)$ or the specific heat $c\left(T\right)$ as the results in (51) and (52), respectively, suggest.

The same thermodynamical properties hold true for the PDM Schrödinger-oscillators in a PGM spacetime and a Wu-Yang magnetic monopole.

In this section, we consider that the system of PDM Schrödinger oscillators in a PGM background and a Wu-Yang magnetic monopole is now subjected to an impenetrable hard-wall potential at some radial distance $r_{\circ}=\sqrt{q\left(\rho_{\circ}\right)}\rho_{\circ}$. This would in turn restrict the motion of the PDM Schrödinger oscillators mentioned above to be confined within a spherical-box of radius $r_{\circ}$ with an impenetrable hard-wall. This would suggest that the the confluent hypergeometric polynomials $\;{}_{1}F_{1}\left(\frac{L}{2}+\frac{3}{4}-\frac{\mathcal{E}}{4\tilde{\omega}},L+\frac{3}{2},\tilde{\omega}r^{2}\right)$ in (45) vanish at $r=r_{\circ}$ to consequently yield that $R\left(r_{\circ}\right)=0$. One would then appeal to subsection 13.5 on the asymptotic expansions and limiting forms of Abramowitz and Stegum Re36 and recollect formula (13.5.14)

$$\lim\limits_{a\rightarrow-\infty}\,{}_{1}F_{1}\left(a,b,x\right)=\Gamma\left(b\right)\,e^{x/2}\,\pi^{-1/2}\left(\frac{bx}{2}-ax\right)^{1/4-b/2}\,\cos\left(\sqrt{\left(2b-4a\right)x}-\frac{b}{2}\pi+\frac{\pi}{4}\right)\left[1+O\left(|\frac{b}{2}-a|^{-1/2}\right)\right],$$

(54)

for a real $x$ and a bounded $b$. This formula immediately suggests that $a=\frac{L}{2}+\frac{3}{4}-\frac{\mathcal{E}}{4\tilde{\omega}}$, $b=L+\frac{3}{2}$, and $x=\tilde{\omega}r^{2}\Rightarrow x_{\circ}=\tilde{\omega}r_{\circ}^{2}$. Consequently, only at very high energies of PDM Schrödinger oscillators and/or very small values of the PGM parameter $\alpha$ (i.e., $\mathcal{E}=E/\alpha^{2}\mathcal{\longrightarrow\infty}$) one would have a vanishing radial function at some $r=r_{\circ}$, i.e., $R\left(r_{\circ}\right)=0$. Under such conditions,

$$\cos\left(\sqrt{\left(2b-4a\right)x_{\circ}}-\frac{b}{2}\pi+\frac{\pi}{4}\right)=0\Rightarrow\sqrt{\left(2b-4a\right)x}-\frac{b}{2}\pi+\frac{\pi}{4}=\left(n_{r}+\frac{1}{2}\right)\pi$$

(55)

one would, in a straightforward manner, obtain

$$E_{n_{r},\ell,q}=\frac{\pi^{2}\alpha^{2}}{4r_{\circ}^{2}}\left[2n_{r}+\sqrt{\frac{1}{4}+\frac{\ell\left(\ell+1\right)-q^{2}}{\alpha^{2}}}+\frac{3}{2}\right]^{2}$$

(56)

Comparing this result with that of (46) we observe that the hard-wall spherical box has indeed changed the corresponding energies for the PDM Schrödinger oscillators in a PGM background and a Wu-Yang magnetic monopole. In Figures 6(a),6(b), and 6(c), we show the hard-wall, at $r=r_{\circ}=1$, effects on the energy levels, Eq. (56) for $n_{r}=1$, and $\ell=0,1,2,3$.. In 6(a) and 6(b) the energy levels are plotted against the PGM parameter $\alpha$ at $q=0$ (i.e., no Wu-yang monopole) and $q=\alpha/4$, respectively. In 6(c) the energy levels are plotted against the Wu-Yang monopole parameter $q$ at $\alpha=0.5$.