Nonrelativistic Atomic Spectrum for Companied Harmonic Oscillator Potential and its Inverse in Both NC-2D: RSP

. A novel study for the exact solvability of nonrelativistic quantum spectrum systems for companied Harmonic oscillator potential and its inverse (the isotropic harmonic oscillator plus inverse quadratic potential) is discussed used both Boopp’s shift method and standard perturbation theory in both noncommutativity two dimensional real space and phase (NC-2D: RSP), furthermore the exact corrections for the spectrum of studied potential was depended on two infinitesimals parameters  and  which plays an opposite rolls, this permits us to introduce a new fixing gauge condition and we have also found the corresponding noncommutative anisotropic Hamiltonian.


INTRODUCTION
One of the important problems in relativistic and nonrelativistic quantum mechanics is to find exact solutions to the Dirac, Klein-Gordon and Schrödinger equations for spherical and non spherical potentials that are used in different fields of physics and materials sciences, in commutative and noncommutative spaces-phases at two and three dimensional spaces . In particular, the fermionic particle with spin ( 2 / 1 ) interacted with proton and in other hand interacted with external field produced by Harmonic oscillator and its inverse under investigation [7]. The application of noncommutativity properties on physics fields satisfied by imposing one or two new commutators   1    c [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]: It's important to notice that, the above two fundamental commutators are satisfied in particulars' cases from the general star product between two arbitrary functions   x f and   x g in the first order of two parameters  and ij  as follow [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]: The two parts are represent the effects of the noncommutativity of space and phase, respectively, the parameters ij  and ij  are antisymmetric real matrixes. At (NC-N D: RSP), a Boopp's shift method can be used, instead of solving any quantum systems by using directly star product procedure: In present work, the star product replaced by usual product together with a Boopp's shift in (NC-2D: RSP) [26][27][28][29][30][31][32][33][34][35]:  , its worth to mention, that the above two uncertainties, can be deduced from the generalized incertitude relations to the special coordinates and impulsions in noncommutative N dimensional space and phase, respectively: The main aim of this paper is to present and study a companied Harmonic potential and its inverse (h.p.i) in (NC-2D RSP) to discover the new symmetries and a possibility to obtain another application to this potential in different fields. The rest of present search is organized as follows: In next section, we briefly review the Schrödinger equation with companied Harmonic potential and it's inverse. The Section 3, reserved to derive the deformed Hamiltonians of the Schrödinger equation with companied (h.p.i) and by applying both Boopp's shift method and standard perturbation theory we find the quantum spectrum of the lowest excitations in (NC-2D RSP) for studied potential. Finally, the important found results and the conclusions are discussed in the four and last section.

REVIEW OF COMPAINED HARMONIC POTANTIAL AND ITS INVERSE IN ORDINARY TWO DIMENSIONAL SPACES
Let's present a brief review of time independent Schrödinger equation for a fermionic particle like electron of rest mass   (15) And nl C is the normalization constant, determined from the condition [7]: The complete orthonormalization eignenfunctions and the energy eigenvalues respectively in two dimensional spaces [7]:

TWO DIMENSIONAL NONCOMMUTATIVE REAL SPACE AND PHASE FOR COMPAINED HARMONIC POTANTIAL AND ITS INVERSE
In this section, we present some fundamental principles of Schrödinger equation in (NC-2D: RSP); applying the important 4-steps on the ordinary quantum Schrödinger equation [27][28][29][30][31][32][33][34][35][36]: Now, we apply the Boopp's shift method on the above equation to obtain, the reduced Schrödinger equation (without star products): . As a direct result of the eq. (4), the two operators 2 r and 2 p in (NC-2D RSP) can be written as follows [24][25][26][27][28][29][30] It's important to notice that, the two obtained results in eq. (21) conserved the symmetry between Which allow us to obtaining the global potential operator It's clearly, that the two first terms are given the ordinary companied (h.p.i.) in 2D space, while the rest terms are proportional's with two infinitesimals parameters ( and ) and then gives the terms of perturbations This can be writing to the equivalent form: We orient the spin to the (Oz) which appear parallel with z L , which allow us to write, the perturbative term as follows: We have replaced   are the modifications to the energy levels, associate with spin up and spin down, respectively, at first order of two parameters and  obtained by applying the perturbation theory, as follows:   Applying the following special integration [38]: Then, we can prove that first integral in equation (33) is: , the eq. (35) can be simplified to the form: And the second integral in equation (36) can be written as follow: winches represent the roll of (NC-2D) space as: Now, we apply the integral [7,38]: Which allow us to writing (NC-2D) phase contribution p(h.p.i.) T as: The modification to the energy levels, associate with spin up and spin down, at first order of  and  for companied Harmonic potential and its inverse in both (NC-2D: RSP) are given by: The first two terms in above two equations are clearly determine the physics contributions of (NC-2D) space, while the second parts are gives physics contributions of (NC-2D) phase, we conclude, from Eqs. (47) and (32)

H
, respectively for companied Harmonic potential and its inverse in (NC-2D: RSP) as: The above operator represents two fundamentals interactions: the first one between spin and external uniform magnetic field (ordinary Zeeman Effect) while the other is a new coupling between the momentum of electron and external uniform magnetic field. As it's mentioned previously, the two parameters  and  was played an opposite rolls which allow us to introduce the following gauge fixing condition of Maireche:  is a positive constant, our obtained spectrum, will be proportioned only to one infinitesimal parameter  and positive constant.

RESULTS AND CONCLUSIONS
In present work, we solved the Schrödinger equation with the companied Harmonic oscillator potential and its inverse in both noncommutativity two dimensional of real space and phase, we used both Boopp's shift method and standard perturbation theory to obtain the following resumed structure of the energy levels in a new symmetric, in two tables: