KG-oscillators in Som-Raychaudhuri rotating cosmic string spacetime in a mixed magnetic field

We investigate Klein-Gordon (KG) oscillators in a G\"{o}% del-type Som-Raychaudhuri spacetime in a mixed magnetic field (given by the vector potential $A_{\mu }=\left( 0,0,A_{\varphi },0\right) $, with $% A_{\varphi }=B_{1}r^{2}/2+B_{2}r$). The resulting KG equation takes a Schr% \"{o}dinger-like form (with an oscillator plus a linear plus a Coulomb-like interactions potential) that admits a solution in the form of biconfluent Heun functions/series $H_{B}\left( \alpha ,\beta ,\gamma ,\delta ,z\right) $% . The usual power series expansion of which is truncated to a polynomial of \ order $n_{r}+1=n\geq 1$ through the usual condition $\gamma =2\left( n_{r}+1\right) +\alpha $. However, we use the very recent recipe suggested by Mustafa \cite{1.29} as an alternative parametric condition/correlation. i.e., $\delta =-\beta \left( 2n_{r}+\alpha +3\right) $, to facilitate conditional exact solvability of the problem. We discuss and report the effects of the mixed magnetic field as well as the effects of the G\"{o}del-type SR-spacetime background on the KG-oscillators' spectroscopic structure.


I. INTRODUCTION
Modern superstring theories predict cosmic strings as one-dimensional stable configurations of matter that are formed, along with other topological defects in spacetime, during cosmological phase transitions in the early universe [1][2][3][4][5].Cosmic strings are of particular interest since their gravitational fields play a crucial role in galaxy formation [4,5].Their gravitational fields introduce intriguing effects like, to mention a few, self-interacting particles [6,7], gravitational lensing [8], and high energy particles [9][10][11].Nevertheless, their gravitational lensing [8] is believed to be the most effective way for their detection.Cosmic strings (static or rotating) are characterized, in = c = 1 units, by the wedge parameter α = 1 − 4μG, which is a measure of angle deficit produced by the string, where G is the gravitational Newton constant and μ is the linear mass density of the string.A spinning/rotating cosmic string has, however, an additional characteristic represented by a rotational/spinning parameter.The spacetime metric that describes the structure generated by a rotating cosmic string is given by where A, B, and C are functions of the radial coordinate r only [12][13][14].Which would, with A = 1, B = αΩr 2 , and C = αr, yield a Gödel-type [15,16] Som-Raychaudhuri (SR) [17,18] spacetime metric where 0 < α < 1 in general relativity, α = 1 corresponds to Minkowski spacetime, and α > 1 is used in the geometric theory of topological defects in condense matter physics.Moreover, further other solutions on rotating cosmic strings are investigated in [19][20][21][22].
In the current methodical proposal, however, we shall be interested in Klein-Gordon (KG) oscillators in Som-Raychaudhuri rotating cosmic string spacetime (2) in a mixed magnetic field.The corresponding covariant and contravariant metric tensors of which are given by Yet, the metric element g ϕϕ > 0 would suggest an upper limit for the radial coordinate so that 0 ≤ r < 1/|Ω|.
Moreover, we shall consider a 4-vector potential A µ = (0, 0, A ϕ , 0), with Here, B 1 denotes the strength of a uniform magnetic field and B 2 denotes the strength of a non-uniform magnetic field.KG-particles in the Gödel-type Som-Raychaudhuri cosmic string spacetime background have been intensively studied with a uniform or a non-uniform magnetic fields [18,, but never with the mixed magnetic field above.
Owing to the fact that Schrödinger, Dirac, and KG oscillators are quantum mechanically of fundamental pedagogical interest, the study of their spectroscopic structure under the effects of the gravitational fields, introduced by different spacetime fabrics, should be of fundamental pedagogical interest in quantum gravity (c.f., e.g., sample of references [45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60], and related references cited therein).We have very recently studied KG-particles in a cosmic string rainbow gravity spacetime in the mixed magnetic field [23].Therein, we have observed that the competition between the two magnetic field strengths B 1 and B 2 has generated energy levels crossing which, consequently, turned the spectra upside down.That was the only mixed magnetic field scenario in the literature, to the best of our knowledge.It would be, therefore, interesting to study the gravitational effects of the Gödel-type Som-Raychaudhuri cosmic string spacetime on the KG-particles in the mixed magnetic field.Our motivation to carry out the current study is clear, therefore.
The organization of this study is in order.In section 2, we start with KG-particles in Gödel-type SR-spacetime in a mixed magnetic field and including the KG-oscillators.The resulting differential equation is a Schrödinger-like (with an effective potential that includes an oscillator plus a linear plus a Coulomb like interactions) and admits a solution in the form of biconfluent Heun functions/series [61,62].A three terms recursion relation is manifestly introduced and a conditionally exact solution is reported.Such a conditionally exact solution reproduces the usual truncation of the biconfluent Heun function/series H B ( α, β, γ, δ, z) to a polynomial of order n r + 1 = n ≥ 1 (through the condition γ = 2 (n r + 1) + α) and provides a parametric correlation that allows us to find conditionally exact solutions.We report the effects of the mixed magnetic field as well as that of the Gödel-type SR-spacetime background.Our concluding remarks are given in section 3.

II. KG-PARTICLES IN G ÖDEL-TYPE SR-SPACETIME IN A MIXED MAGNETIC FIELD
The KG-particles are described by the KG-equation where Dµ = ∂ µ − ieA µ + F µ with F µ ∈ R. One should notice that F µ = (0, F r , 0, 0) is in a non-minimal coupling form, whereas the 4-vector potential A µ = (0, 0, A ϕ , 0) is in the usual minimal coupling form, and m • denotes the rest mass energy (i.e., m • ≡ m • c 2 , with = c = 1 units to be used throughout).With would read where m is the the magnetic quantum number m = m ± = ±|m| = 0, ±1, ±2, • • • , and We now use A ϕ = B 1 r 2 /2 + B 2 r and incorporate the KG-oscillators through the substitution F r = ηr [45,46] to obtain with We may now substitute in ( 8) to obtain This equation is known to have a solution in the form of biconfluent Heun functions so that where Obviously, we have to take N 2 = 0 to secure finiteness of the solution at r = 0. Consequently, one may cast the solution as We now need to truncate the biconfluent Heun function to a polynomial of order n r + 1.This is done when the condition γ = 2 (n r + 1) + α is sought [61,62] to obtain Moreover, the biconfluent Heun series could admit further truncation through the assumption that the n r +1 coefficient in the series expansion is a polynomial of degree n r in δ, provided that δ is a root of this polynomial which consequently cancels , the n r +1 subsequent coefficients and the series truncates to degree n r for H B ( α, β, γ, δ, z) [61].In the current methodical proposal, however, we shall follow a new recipe that manifestly introduces a clear correlation between the physical parameters involved and truncates the series to a polynomial of order n r + 1 = n ≥ 1.This is done in the sequel.
We follow the power series expansion of the biconfluent Heun function as usual so that to obtain This necessarily implies that where ν = +| m| is the value to be adopted (otherwise, the wave function is divergent at r = 0), and Consequently, we obtain a three terms recursion relation in the form of Which in turn gives for j = 0 and so on we find the coefficients C ′ j s for the power series (17).However, finiteness and square integrability of the wave function requires that the power series should be truncated into a polynomial.To do so, we follow the recipe we used in [43,60].That is, ∀j = n r we take C nr +2 = 0, C nr +1 = 0, and C nr = 0.One should notice that the condition C nr +2 = 0 would allow us to obtain a polynomial of order n r + 1 ≥ 1.However, we further require that the coefficients of C nr+1 = 0 and C nr = 0 to vanish identically to allow conditional exact solvability of the problem at hand.Under such conditional exact solvability, we obtain and where ñ = n r + | m| + 1.At this point, one should notice that whilst condition (23) offers conditional exact solvability through a parametric correlation, the second condition ( 24) is in exact accord with that ( 16) and provides the KGoscillators energies as with One would observe that this quadratic energy equation is unlikely to be analytically solvable.However, to observe the effects of the uniform B 1 and the non-uniform B 2 magnetic fields, and vorticity Ω, we plot the KG-particles' and antiparticles' energies in (25) in figures 1 and 2. For some fixed values of (α, η, k, m • ) = (0.5, 1, 1, 1) we plot in Fig. 1 the energies E against vorticity Ω so that 1(a) for B 1 = 0, B 2 = 1, 1(b) for B 1 = 1, B 2 = 0, and 1(c) for Where as, in Fig. 2 we plot the energies E against B 1 , in 2(a) and 2(b), and against B 2 , in 2(c) and 2(d), for vorticity Ω = ±1.A common characteristic of all such figures is that the symmetry of the energies about E = 0 is broken.We observe that, while the minima of |E ± | are located at Ω = 0 value when the uniform magnetic field is switched off (B 1 = 0), they shift to (E + , Ω + ) and (E − , Ω − ) quarters of 1(b) and 1(c) for B 1 = 0.This is due to the fact that we effectively have Where it is obvious that the term ΩE = ± |ΩE| is competing with B1 /2 ≥ 0. That is, the first term under the square root in (27) takes the values FIG. 1: The energy levels against the vorticity Ω for (nr, m) states with nr = 1 and m = ±1, ±2 given by Eq. ( 25), at α = 0.5, and m• = 1 = η = k, so that Fig. 1(a) for B1 = 0 and B2 = 1, 1(b) for B1 = 1 and B2 = 0, and 1(c) for B1 = 1 and B2 = 1.
and consequently suggest that Obviously, therefore, the competition between an effective energy dependent vorticity (i.e., Ὼ (E) = ΩE = ± |Ω| |E|) and the uniform magnetic field strength B1 in ( 27) plays a crucial role in shaping the spectroscopic structure of the KG-oscillators in Som-Raychaudhuri rotating cosmic string spacetime in a mixed magnetic field.This would explain the similar behaviours of the curves in the first and third quarters for (E + , Ω + ) and (E − , Ω − ), respectively, as well as the similar behaviours of the curves in the second and fourth quarters for (E − , Ω + ) and (E + , Ω − ), respectively. of figures reported, we clearly observe that the symmetry of the energies about E = 0 value is broken mainly because of the effect of the uniform magnetic field B 1 (which is obvious in (29)).This is to be shown in the sequel.
A. Switching off the uniform magnetic field, B1 = 0 When B 1 is switched off, the problem reduces into that for an effective potential one obtains we have and for |ΩE| = −Ω ∓ E ± we have Under such settings, we obtain for |ΩE| = Ω ± E ± , and for |ΩE| = −Ω ∓ E ± .That is, E ± in (34) should exactly read and Similarly, we obtain E + and E − for (35) as and One may clearly observe, without doubt, the symmetry of the energies about E = 0 value for the set |ΩE| = Ω ± E ± in ( 36) and (37), and the set |ΩE| = −Ω ∓ E ± in ( 38) and ( 39), respectively.

III. CONCLUDING REMARKS
In this work, we have studied the effects Som-Raychaudhuri rotating cosmic string spacetime on the KG-oscillators in a mixed magnetic field.We have observed that the corresponding KG-equation takes a Schrödinger-like form with an interaction potential in the form of (i.e., it includes an oscillator, a linear, and a Coulomb-like interactions).Such a Schrödinger-like equation is shown to admit a solution in the form of biconfluent Heun functions/series H B (α, β, γ, δ, z), where the usual power series expansion is truncated to a polynomial of order n r + 1 = n ≥ 1 through the usual condition γ = 2 (n r + 1) + α.
We have also used the very recently developed parametric correlation δ = −β (2n r + α + 3) used by Mustafa [43] and Mustafa et al. [60] to obtain a conditionally exact solution to the problem at hand.Such a parametric correlation identifies an alternative condition, than that suggested by Ronveaux [61].Consequently, we were able to discuss and report the effects the mixed magnetic fields and the vorticity of the Som-Raychaudhuri rotating cosmic string spacetime on the KG-oscillators spectroscopic structure (documented in as such a competition no longer exists.
To the best of our knowledge, the current study has never been discussed elsewhere in the literature.Yet, the current methodical proposal provide a gateway to the study the thermodynamic properties of several quantum systems

Fig.s 1
and 2).Interestingly, we have observed that the competition between an effective energy dependent vorticity (i.e., Ὼ (E) = ΩE = ± |Ω| |E|) and the uniform magnetic field strength B1 in(27) plays a crucial role in shaping the spectroscopic structure of the KG-oscillators and breaks the symmetry of the corresponding energies about E = 0 value.However, when the uniform magnetic field is switched off (i.e., B 1 = 0) such symmetry is retrieved (clearly documented in(35))