Confined Klein-Gordon oscillator from a (2+1)-dimensional G\"{u}rses to a G\"{u}rses or a pseudo-G\"{u}rses space-time backgrounds: Invariance and isospectrality

We study the Klein-Gordon (KG) oscillator with a Cornell-type scalar confinement in (2+1)-dimensional G\"{u}rses space-time backgrounds and report their exact solutions. The effect of the vorticity parameter $\Omega$ on the energy levels is found to yield some interesting features like; energy levels-crossings, partial clustering of positive and negative energy levels, and shifting the energy gap upwards or downwards. Such confined KG-oscillators are also studied in a general deformed G\"{u}rses space-time background. Moreover, we consider the confined-deformed KG-oscillator from a (2+1)-dimensional G\"{u}rses to G\"{u}rses and pseudo-G\"{u}rses space-time backgrounds. The resulting confined-deformed KG-oscillators are found to admit invariance and isospectrality with each other

On the other hand, the introduction of Mathews-Lakshmanan oscillator [22] has activated intensive research studies on "effective" position-dependent mass (PDM in short), both in classical and quantum mechanics [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39].PDM is a metaphoric manifestation of the coordinate deformation/transformation [26][27][28].Nevertheless, Khlevniuk [31] has argued that a point mass in the curved space may effectively be transformed into a PDM in Euclidean space.Such coordinate transformation/deformation affects, in turn, the form of the canonical momentum in classical and the momentum operator in quantum mechanics (e.g., [26,27,30,34] and related references therein).In classical mechanics, it has been shown that negative the gradient of the potential force field is no longer the time derivative of the canonical momentum p = m (x) ẋ, but it is rather related to the time derivative of the pseudo-momentum (also called Noether momentum) π (x) = m (x) ẋ [27].In quantum mechanics, however, the PDM momentum operator is constructed [30] and used to find the PDM creation and annihilation operators for the Schrödinger oscillator [26].
It would be interesting, therefore, to investigate the effects of such PDM recipe [26,27,30,34] on the KG-oscillator Figures 1-4.Such figures exhibit some interesting features like, energy levels crossings, partial clustering of positive and negative energy levels, and shifting the energy gap upwards or downwards.In section 3, we consider the confined KG-oscillator in a generalized deformed (2+1)-dimensional Gürses space-time background.We show that the resulting confined-deformed KG-oscillator is, in fact, invariant and isospectral with that in the (2+1)-dimensional Gürses spacetime background of section 1.We discuss, in section 4, a confined-deformed KG-oscillator from a (2+1)-dimensional Gürses to yet another Gürses space-time backgrounds.That is, the deformation in the (2+1)-dimensional Gürses space-time metric ds 2 is chosen so that it belongs to a another Gürses space-time metric but with different Gürsestype parametric settings.Moreover, we consider (in section 5) a confined-deformed KG-oscillator from Gürses to a pseudo-Gürses space-time backgrounds.The notion of pseudo-Gürses space-time metric is manifested by the fact that its parametric settings do not belong to the set of parameters of Gürses space-time metric, but it can be transformed into Gürses space-time metric within a transformation (24) below.The resulting confined-deformed KG-oscillators are found to admit invariance and isospectrality with the confined KG-oscillator in a (2+1)-dimensional Gürses space-time background discussed in section 2. Our concluding remarks are given in section 6.

BACKGROUND
In this section, we recollect the basic formulation of the KG-oscillator in a (2+1)-dimensional Gürses space-time background.Hence, we consider the (2+1)-dimensional Gürses space-time metric [15] with a 0 = b 0 = e 0 = 1, b 1 = c 0 = λ 0 = 0, and the vorticity Ω = −µ/3, in the Gürses metric (i.e., as in Eq.( 5) of [15]), where The covariant and contravarian metric tensors in this case, respectively, read Moreover, we may now couple the KG-oscillator using the recipes in [20,21] and allow the momentum operator to indulge the oscillator through with η denoting the frequency of the oscillator and χ µ = (0, r, 0).This would, in effect, transform KG-equation ( 5) Which consequently yields A textbook substitution in the form of would result in a one-dimensional Schrödinger-like KG-oscillator with a confinement S(r) so that where Obviously, equation ( 10) represents, with S(r) = 0, the 2-dimensional radial harmonic oscillator with an effective oscillation frequency ω and consequently inherits its textbook eigenvalues and radial eigenfunctions At this point we may move further and use a Cornell-type scalar potential so that equation ( 10) now reads where Equation ( 15) admits a finite/bounded solution in the form of biconfluent Heun functions where for Then the Kummer relation [41] in this case suggests that where n r = 0, 1, 2, • • • is the radial quantum number (number of nodes in the wave function).This would in turn establish the relation between α ′ and γ ′ and allow us to write where 2n r ≥ 0 =⇒ n r ≥ 0 satisfies the condition of a finite polynomial nature of the biconfluent Heun functions.
Therefore, for our 17) to imply a quantum mechanically viable and acceptable solution, we adopt the parametric relation (19).This would consequently yield that In this case, we get the relation for the energy eigenvalues as One should notice that this result collapses into that in (12) when the parameters A and B of the Cornell potential ( 14) are set zeros.This is indeed the natural tendency of a viable solution of a more general case (i.e., with the Cornell confinement).Moreover, it is obvious that this equation is hard to solve analytically, not impossible though.
Yet, it is clear that the second term (−2 Ω ℓ E) on the L.H.S. plays a critical role in shaping the fate of energy levels' E nr ,ℓ structure when plotted against the vorticity parameter Ω for some different parametric values, and against the oscillator frequency ω.The effect of the term (−2 Ω ℓ E ± ) is identified (± stands for positive and negative energy regions, respectively ) as follows: (i) the positive energy E + , with Ω and ℓ are both positive or both negative, will be boosted upward for ℓ = 0, whereas (ii) the negative energy E − , with positive/negative Ω and negative/positive ℓ, the energy levels are boosted downward for ℓ = 0.This is documented in Figures 1-2.In Figure 3, we observe that the effect of the second term is clearly exemplified through shifting the energy gap upwards (when Ω and ℓ are both positive or both negative) and downwards (when Ω is negative/positive and ℓ is positive/negative, respectively).As a result, we observe that the energy levels crossings (documented in Fig. 1), the energy levels partial clustering (documented in Fig. 2(b)-2(f)), and the energy gap shifting (documented in Fig. 3(a) and 3(b)) are unavoidable manifestations in the process.Furthermore, when ℓ = 0 the second term effect dies out and the spectrum retains its regular ordering format, but remains infected with the effect of the vorticity parameter though.

BACKGROUND
In this section, we consider that the (2+1)-dimensional Gürses space-time metric (1) is deformed in such a way that Then the corresponding confined KG-oscillator is given, with χµ = (0, r, 0), by We may now use the transformation to connect the deformed Gürses space-time (22) with the formal one in (1).Moreover, the relation between m (r) and Q (r) is given by In this case, the deformed Gürses metric (22) reads and consequently the corresponding space-time metric tensor is with its determinant det (g) = −m (r) Q (r) r 2 and its inverse The deformed (2+1)-dimensional space-time metric (26) may very well be called a pseudo-Gürses metric for it may return to the Gürses one through reversing the transformation recipe.Under such settings, one may, in a straightforward manner, obtain which immediately inherits the form of ( 17) so that Now multiplying from the left by m (r) 1/4 we get where that admits isospectrality with the one dimensional Schrödinger-like confined KG-oscillator in (15) with its eigenvalues given in (21) and its eigen functions given by (30).Hence the radial wave function is eventually given by Obviously, the confined KG-oscillators of (32) and that of ( 15) are isospectral and share the same energy eigenvalues of (21), therefore.Yet we may safely conclude that the two confined KG-oscillators in a the (2+1)-dimensional Gürses to deformed-Gürses space-time backgrounds, ( 7) and ( 23), respectively, are invariant and isospectral.Moreover, one should notice that equation ( 32) resembles an effective position-dependent mass (PDM) particles in the onedimensional von Roos PDM Hamiltonian [23] with Mustafa and Mazharimusavi parametric settings [26,29,30]).

IV. CONFINED-DEFORMED KG-OSCILLATOR FROM A (2+1)-DIMENSIONAL G ÜRSES TO A G ÜRSES SPACE-TIME BACKGROUNDS
Let us rewrite the deformed (2+1)-dimensional Gürses space-time metric (26), using the transformation recipe in (24), as and compare it with (2) of [15] to imply, that a 0 = 1, Consequently, our Q(r) reads to imply, through (25), that Obviously, the resulted parametric structures agree with those of Gürses [15] in (2), where Then the corresponding metric tensor is given by with det(g) = −e 2 0 r 2 , and its inverse Moreover, we may now report the corresponding confined-deformed KG-oscillator in the deformed Gürses space-time (35) background as where This confined-deformed KG-oscillator is isospectral and invariant with that of (15) and, therefore, shares the eigen energies given in (21).The corresponding eigenfunctions are readily given in (30) with Q (r) and m (r) are as defined in (37) and (38), respectively.The confined KG-oscillators, (42) and ( 15), in Gürses space-time backgrounds are invariant and isospectral, therefore.

V. CONFINED-DEFORMED KG-OSCILLATOR FROM A PSEUDO-G ÜRSES TO A G ÜRSES SPACE-TIME BACKGROUNDS
In this section, we consider a deformation in the radial coordinate through which would, through the correlation in (25), imply that Consequently, equation ( 32) yields where V (r) is now given by ( 33) as Such confined-deformed KG-oscillators, ( 46) and ( 15), in pseudo-Gürses to Gürses space-time backgrounds are invariant and isospectral, therefore.Moreover, this system corresponds to a (2+1)-dimensional pseudo-Gürses deformed space-time metric (26) of the form The notion of pseudo-Gürses space-time metric is manifested by the fact that this metric may yield a Gürses spacetime like metric (1) (discussed in section 1) if the transformation is reversed.Having said that, it is obvious that the only feasible Gürses to Gürses space-time metric backgrounds case is the one discussed section 4, where Q (r) and m (r) are, respectively, given by ( 37) and (38).Any other structure for Q (r) and m (r) in (26) should be classified as pseudo-Gürses to Gürses space-time metric backgrounds.

VI. CONCLUDING REMARKS
In this study, we have considered (in section 2) the KG-oscillator confined in a Cornell-type scalar potential S (r) of ( 14) in some (2+1)-dimensional Gürses space-time backgrounds.We started with a confined KG-oscillator in a Gürses space-time background (i.e.Gürses space-time metric ds 2 (1) at specific parametric settings) and reported the corresponding exact solution in (17) and (21).Hereby, we argue that the natural tendency of a physically viable solution of a more general case ( confined KG-oscillator in our case) should necessarily collapse into that of a less complicated KG-oscillator ones in ( 12) and ( 13), when the confinement parameters are switched off.So is the tendency of our reported exact solution in (17) and (21).
To observe the effect of the vorticity parameter Ω = −µ/3 (in our case) on the energy levels E nr ,ℓ , we have reported E nr ,ℓ (21) versus Ω in Figures 1-2, and E nr ,ℓ vs η (the KG-oscillator frequency) in Figure 3.The mathematical structure as well as the reported figures document the critical role of the second term, (−2 Ω ℓ E) on the L.H.S. of (21), in shaping the energies of the confined KG-oscillator.This effect is summarized as follows: (i) the positive energy E + , with Ω and ℓ are both positive or both negative, is boosted upward for ℓ = 0, (ii) the negative energy E − , with positive/negative Ω and negative/positive ℓ, respectively, is boosted downward for ℓ = 0, and (iii) for a fixed Ω the energy gap is shifted upwards (for positive Ω and ℓ) and downwards (for negative Ω and ℓ), whilst the gap increases as the KG-oscillator frequency η increases.As a result of such effects, the energy levels crossings (documented in Fig. 1), the energy levels clustering (documented in Fig. 2  E 0,0 , E 0,±1 , E 0,±2 form bottom to top for the positive energies and from top to bottom for negative energies.As we move far from Ω = 0, we see that E 0,+1 and E 0,−1 switch their ordering in moving from Ω > 0 to Ω < 0 regions (similar behaviour occurs for all E nr>0,+|ℓ| and E nr >0,−|ℓ| states).To make sure that energy levels clustering do not indulge levels crossings we plot Next, we have considered (in section 3) the confined KG-oscillator in a general deformation of the (2+1)-dimensional Gürses space-time background (26).Hereby, we have shown that the resulting confined and deformed KG-oscillator is invariant and isospectral with of the confined KG-oscillator (15) in the (2+1)-dimensional Gürses space-time background (1).We have further considered (in section 4) a confined and deformed KG-oscillator in a deformed (2+1)dimensional Gürses (26) to a Gürses space-time backgrounds (2).That is, the deformation in the (2+1)-dimensional Gürses space-time metric ds 2 (26) is chosen so that it belongs to a another Gürses space-time metric (but with different Gürses-type parametric setting) in (2).We have shown that such a confined and deformed KG-oscillator is invariant and isospectral with of the confined KG-oscillator (15) in the (2+1)-dimensional Gürses space-time background (1).
Finally, we have considered (in section 5) a confined and deformed KG-oscillator in pseudo-Gürses to Gürses spacetime backgrounds.The pseudo-Gürses space-time metric is manifested by the fact that its parametric settings do
(b)-2(f)), and the energy gap shifting (documented in Fig.3(a) and 3(b)) are unavoidable natural manifestations in the process.Moreover, to figure out how the energy levels behave near Ω = 0 (i.e., flat space-time background) we have also reported Figure 4.