RELATIVISTIC SYMMETRIES OF BOSONIC PARTICLES AND ANTIPARTICLES IN THE BACKGROUND OF THE POSITION-DEPENDENT MASS FOR THE IMPROVED DEFORMED HULTHÉN PLUS DEFORMED TYPE-HYPERBOLIC POTENTIAL IN 3D-EQM SYMMETRIES

The bound state solutions of the deformed Klien-Gordon equation have been determined in the three-dimensional extended relativistic quantum mechanics 3D-ERQM symmetries using position-dependent mass (PDM) with unequal scalar and vector potential for the improved Hulthén plus improved deformed type-hyperbolic potential (PDM-SVID(H-TP)) models. PDM with unequal scalar and vector potential for the Hulthén plus deformed type-hyperbolic potential (PDM-(SVH-DTP)) models, as well as a combination of radial terms, which are coupled with the coupling L  , which explains the interaction of the physical features of the system with the topological deformations of space-space. The new relativistic energy eigenvalues have been derived using the parametric Bopp shift method and standard perturbation theory which is sensitive to the atomic quantum numbers ( , , , j l s m ), mixed potential depths ( 0 0 1 1 , , , V S V S ), the rest, and perturbed mass   0, 1 m m , the screening parameter's inverse  , and noncommutativity parameters   , ,  . Within the framework of 3D-ERQM symmetries, we have treated certain significant particular instances that we hope will be valuable to the specialized researcher. We have also treated the nonrelativistic limit and applied our obtained results to generate the mass spectra of heavy-light mesons (HLM) such as cc and bb under PDM-SE with improved deformed Hulthén plus improved hyperbolic potential (PDM-ID(H-TP)) models. When the three simultaneous limits   , ,  were applied, we recovered the normal results of relativistic in the literature ( 0,0,0 ) for the PDM-ID(H-TP)) models.


INTRODUCTION
One of the significant issues in quantum mechanics (QM) and noncommutative quantum mechanics (NCQM) or extended quantum mechanics (EQM) is the investigation of solutions to the nonrelativistic Schrödinger equation (SE) or relativistic Klien-Gordon (KG), Dirac and Duffin-Kemmer-Petiau equations for a particle with spin 0, 1/2 or (1,2...) under the real physical potentials. The hyperbolic and Hulthén potentials are considered to be one of the most important interactions that have received great attention. It has been the subject of an in-depth study by many researchers within the framework of fundamental equations [1][2][3][4][5] whether it is a single treatment or a combination of both. In their study of the bound and scattering states of the KGE with deformed Hulthén plus deformed hyperbolical potential for arbitrary states, Ikot et al. used supersymmetry quantum mechanics and factorization techniques [6]. The variable mass formalism provides relevant and practical theoretical predictions of a variety of experimental properties for many-body quantum systems for this goal [7,8]. The effective mass notion has been applied to numerous important issues in the literature, including nuclei, metallic clusters, 3 He clusters, quantum liquids, and nuclei [9][10][11][12][13]. In the present work, we aim to investigate the solution of KG and SE with deformed Hulthén plus deformed-type hyperbolic potential in 3D-ERQM and 3D-ENRQM symmetries to develop the physical concepts in ref. [6]. We aspire through this work to reveal more new applications within the framework of extended postulates that include more comprehensive axioms than we know about relativistic quantum mechanics (see below). These new postulates were connected to the deformation space-space and phase-phase. The divergence problem of the standard model, gravity quantization, the problem of unifying it with the rest of the fundamental interactions, and other significant physical problems have emerged despite the brilliant successes of quantum mechanics in treating physical and chemical systems in various research fields [14][15][16][17][18][19][20][21]. It should be mentioned that before the renormalization approach was created and gained popularity, Heisenberg proposed the idea of extended noncommutativity to the coordinates as a possible treatment for eliminating the limitless number of field theories in 1930. Snyder published the first work on QFT's history in 1947 [22], and Connes introduced its geometric analysis in 1991 and 1994 [23,24] to standardize QFT. I believe that this research will contribute to further subatomic scale investigations and scientific knowledge of elementary particles. The position-dependent mass with unequal scalar and vector potential for the improved deformed Hulthén plus improved type-hyperbolic potential (PDM-SVID(H-TP)) models in the 3D-ERQM symmetries was motivated by the fact that it had not been reported in the literature for bosonic particles and antiparticles. † Cite as: A. Maireche The corresponding generalizing momentums ( The outline of the paper is as follows: Sect. 2 presents an overview of the 3D-KGE under the PDM-SVID(H-TP) model. Sect. 3 is devoted to investigating the 3D-DKGE using the well-known Bopp's shift method to obtain the effective potential of the PDM-SVID(H-TP) model. Furthermore, using standard perturbation theory, we find the expectation values of some radial terms to calculate the corrected relativistic energy generated by the effect of the perturbed effective potential   ht pert W r , and we derive the global corrected energies for bosonic particles and bosonic antiparticles whose spin quantum number has an integer value ( 0,1, 2... ). Sect. 4 is reserved for the study of important relativistic particular cases in 3D-ERQM symmetries. The next section is reserved for the nonrelativistic limits for PDM-SVID(H-TP) models in 3D-ENRQM symmetries and we apply these results to generate mass spectra of HLM systems. Finally, we present our conclusion in Sec. 7.

AN OVERVIEW OF KGE UNDER THE PDM-SVD(H-TP) MODEL IN RQM SYMMETRY
where z equal   where  

SOLUTIONS OF PDM-SVID(H-TP) MODELS IN 3D-ERQM SYMMETRIES
By applying the new principles which we have seen in the introduction, Eqs. (3) and (4), summarized in new relationships MASCCCRs and the notion of the Weyl-Moyal star product. These data allow us to rewrite the usual radial KG equations in Eq. (5) in 3D-ERQM symmetries as follows: There are two approaches to including non-commutativity in the quantum field theory: either through the Moyal product on the space of ordinary functions or by redefining the field theory on a coordinate operator space that is inherently noncommutative [35][36][37]. It is known to specialists that the star product can be translated into the ordinary product known in the literature using what is called Bopp's shift method. F. Bopp was the first to consider pseudo-differential operators obtained from a symbol by the quantization rules   , respectively. This procedure is known as Bopp's shifts (BS) method, and this quantization procedure is known as Bopp quantization [38][39][40][41][42][43][44][45]. It is worth motioning that the BS method permutes us to reduce Eq. (10) in the simplest form: The Taylor expansion of   nc W r can be expressed as in the 3D-ERQM symmetries, as [42][43][44][45][46][47][48][49][50][51][52]: Substituting Eq. (12) into Eq. (11), we obtain the following, as in the Schrödinger equation: By comparing Eqs. (5) and (11), we observe an additive potential   pert ht W r dependent on new radial terms, which are coupled with the coupling L that explains the interaction of the physical features of the system with the topological deformations of space-space: Eq. (13) cannot be solved analytically for any state l 0  because of the centrifugal term and the studied potential itself. The effective perturbative potential   pert ht W r in Eq. (15) has a strong singularity 0 r  , we need to use the suitable approximation of the centrifugal term proposed by Kurniawan et al. [46] and applied by Ikot et al. [47]. The radial part of the 3D-DKGE with the PDM-SVID(H-TP) models contains the centrifugal term 2 ( 1)/ l l r  and 4 ( 1)/ l l r  since we assume l 0  . However, the PDM-SVID(H-TP) model is a kind of potential that cannot be solved exactly when the centrifugal term is taken into account unless 0 l  is assumed. The conventional approximation used in this paper is as follows: This gives the perturbative effective potential as follows: The PDM-SVID(H-TP) model is extended by including new radial terms   proportional to the infinitesimal coupling L , this is logical from a physical point of view because it explains the interaction between the physical properties of the studied potential L and the topological properties resulting from the deformation of space-space  . This allows us to consider the additive effective potential as a perturbation potential compared with the main potential    [48] and applied by Zhang [49], to obtain the general excited state directly. We calculate the integrals in Eqs. (20) with help of the special integral formula: here  

The corrected energy for the PDM-SVID(H-TP) models
The crucial goal of this sub-section is to identify the contribution under the PDM-SVID(H-TP) models, in 3D-ERQM symmetries, arising from deformation space-space using the method we have successfully applied in the past and are always working to develop. We can confirm that the PDM-SVD(H-TP) models are in place, which we provided through a summary of the bosonic particles and bosonic antiparticles in Eq. Additionally, we use the following transformation which is well known in QM symmetries: The global expectation values   ht nlm K for the bosonic particles and bosonic antiparticles, which were created from the effect of the PDM-SVID(H-TP) models, are determined from the following expression: The second principal physical contribution for the perturbed potential We chose a rotational velocity  parallel to the ( Oz ) axis ( z  e  ) to simplify the calculations. The perturbed generated spin-orbit coupling is then transformed into new physical phenomena as follows: All of these data allow for the discovery of the new corrected square improved energy It is worth noting that the authors of ref. [48] were studied rotating isotropic and anisotropic harmonically confined ultra-cold Fermi gases in two and 3D space at zero temperature, but in this case, the rotational term was manually added to the Hamiltonian operator, whereas, in our study, the rotation operator Here nl E  are usual relativistic energies under the PDM-SVID(H-TP) model obtained from equations of energy in Eq. (9). It should be noted that the positive and negative sign denotes the improved energy of the bosonic particles which corresponds to the positive and negative energy of the bosonic antiparticles which corresponds to the negative energy. We can now generalize our obtained energies by using the unit step function (also known as a viside step function  

STUDY OF IMPORTANT RELATIVISTIC PARTICULAR CASES IN 3D-ERQM SYMMETRIES
We will look at some specific examples involving the new bound state energy eigenvalues in Eq. (27) in this section. By adjusting relevant parameters of the PDM-SVID(H-TP) models in the 3D-ERQM, we could derive some specific potentials useful for other physical systems for much concern the specialist reach.
(1). If we choose, 1 0 V  , 0 Here   h V r presents the GHP in 3D-RQM symmetries [51], while is determined from the limits: The first two parts hp nl E  describe the relativistic energies of bosonic particles and bosonic antiparticles. In 3D-RQM symmetries, the rest of the terms present the topological effect of the deformation space-space (TDSS) on the thesis's main energies hp nl E  .
(2). If we choose, 1 R where 0 V ,  and R are the potential depth, the width of the potential, and the surface thickness whose values correspond to the ionization energies, respectively,

SE WITH PDM-ID(H-TP) MODES IN 3D-ENRQM SYMMETRIES
To realize a study of the nonrelativistic limit, in 3D extended nonrelativistic QM (3D-ENRQM) symmetries, for the PDM-ID(H-TP) models, two steps must be applied. The first corresponds to the NR limit, in 3D-NRQM symmetries. This is done by applying the following simultaneous replacements, (

Spin-averaged mass spectra of HLM under PDM-ID(H-TP) modes
The quark-antiquark interaction potentials, are spherically symmetrical and provide a good description of HLM such as cc and bb under PDM-ID(H-TP) modes. This would give us a strong incentive to dedicate this section to the purpose to determine the modified spin-averaged mass spectra of HLM under the PDM-ID(H-TP) modes interaction by using the following formula: The LHS of Eq. (34) describes spin-averaged mass spectra of HLM in usual QM symmetries [53][54][55][56][57], while the RHS is our self-generalization to this formula in 3D-ENRQM symmetries, q m and q m are the quark mass and the antiquark By substituting Eqs. (36) and (35) with ht nc np