Energy and momentum eigenspectrum of the Hulthén-screened cosine Kratzer potential using proper quantization rule and SUSYQM method

Using the Qiang-Dong proper quantization rule (PQR) and the supersymmetric quantum mechanics approach, we obtain the eigenspectrum of the energy and momentum for time-independent and time-dependent Hulth’en-screened cosine Kratzer potentials. For the suggested time-independent Hulthén-screened cosine Kratzer potential (HSCKP), we solve the Schrödinger equation in D dimensions. The Feinberg-Horodecki equation for time-dependent Hulth’en-screened cosine Kratzer potential (tHSCKP) also solves. To address the inverse square term in the time-independent and time-dependent equations, we employed the Greene-Aldrich approximation approach. We were able to extract time-independent and time-dependent potentials, as well as their accompanying energy and momentum spectra. In three-dimensional space, we estimate the rotational vibrational (RV) energy spectrum for many homodimers (H2, I2, O2) and heterodimers (MnH, ScN, LiH, HCl). We also use the recently introduced formula approach to obtain the relevant eigen function. We also calculate momentum spectra for the dimers MnH and ScN. The method is compared to prior methodologies for accuracy and validity using numerical data for heterodimer LiH, HCl and homodimer I2, O2, H2. The calculated energy and momentum spectra are tabulated and analyzed.


Introduction
In the literature, there are exact or approximate solvable potential models. These hypothetical models have sparked a lot of curiosity since they provide us a glimpse into the quantum mechanical system. To solve the various potentials, a variety of strategies are available. The asymptotic iteration method (AIM) [1,2], the ansatz method [3], the Nikiforov-Uvarov (NU) technique [4,5], the factorization technique [6,7], the formula method [8], and the supersymmetric quantum mechanics (SUSYQM) [9,10].
Combination of two or more potentials in both relativistic and non-relativistic realm has been great interest. The essence of mixing more than one physical potential model is to have many applications [23]. Kratzer potential and various type mixed Kratzer potential have been studied by different/same researcher via same/various methods such as Kratzer potential [24,25] by NU method same potential studies by Bayrak et al. [1] within the framework of AIM. Modified Kratzer plus ring shape potential studies by IKhdair [6] via NU method. Hellmann-Kratzer potential model investigated by Edet et al. [26,27] via WKB formalism. Screened Kratzer potential analyzed by Ikot et al. [28] using NU method. Recently, we studied screened cosine Kratzer potential and obtained various molecular properties using the NU method [29]. Screened Coulomb potential (SCP) plus Kratzer potential analyzed by Edet et al. [30] using the NU method. Manning-Rosen plus Hellmann potential studies by Ita et al. [11] via WKB approximation scheme and same potential investigated by Louis et al. [31] via proper quantization rule method in July 2019. Solutions of the energy-dependent molecular Kratzer potential have been obtained by A. N Ikot et al. [32] via asymptotic iteration method. Hulthén plus Kratzer potential in D space investigated by Obu et al. [33] under relativistic and non-relativistic treatments using NU method. Bound state solutions of the Schrödinger equation for inversely quadric Yukawa plus Kratzer-Fues potential have been obtained using the WKB method [34]. Approximate analytical solutions of Klein-Gordon equation obtained for Kratzer potential using the ansartz method [3]. In March 2020, Onyenegecha et al. [35] have been studied the modified Mobius square plus Kratzer potential via NU method.
The time-independent Hulthén potential defined as [18,19]: whereas recently investigated SCKP written as [29] −2D e a r − b 2r 2 e −δαr cosh δλαr (2) Newly proposed HSCKP given by 2r 2 e −δαr cosh δλαr (3) where D e is the dissociation energy, a ≡ re, b ≡ re 2 , δ, λ and α are the screening parameters, r e is the equilibrium bond length, and the interatomic distance r. We obtain energy eigenvalues for the HSCKP in D dimensional space using non-relativistic Schrödinger equation via PQR method in this paper. Using formula method given by Falaye et al. [8], we obtain wave function because the proper quantization rule is not applicable to compute the eigen functions of the given system.
In this paper, we also solve the time-dependent HSCKP (tHSCKP) written as The Feinberg-Horodecki (FH) equation is a spacelike counterpart of the Schrödinger equation. From the relativistic Feinberg equation [45], FH equation was derived by Horodecki [46]. Feinberg-Horodecki equation has been demonstrated in the possibility of describing biological systems [47,48] in terms of the time-like supersymmetric quantum mechanics [49]. The space-like solutions of the FH equation can be applied to test its relevance in various branches of science including medicine, biology, and physics. Since the last decade, numerous researchers solved the FH equation with different time-dependent potentials.
For FH equation, the exact momentum state solutions with the rotating time-dependent Deng-Fan oscillator are relevant within the pattern of generalized parametric NU (pNU) method. Bera and Sil [50] found exact solutions of the Feinberg-Horodecki equation for the time-dependent Wei-Hua oscillator and Manning-Rosen potentials by the NU method.
Recently, FH equation studied by Altug and Sever [51] for time-dependent pöschl-Teller potential and found its space-like coherent state. Celia et al. [52] have studied time-dependent mass and frequency and a perturbative potential to construct coherent states for two systems. Recently in 2020, Farout et al. [53] obtained the quantized momentum solution of the FH equation with general potential model using the NU method. In July 2020, Farout and Ikhdair solved the FH equation with the time-dependent screened Kratzer-Hellmann potential model [54] and obtained the approximated eigensolutions of momentum states. In August 2020, Farout et al. [55] obtained the quantized momentum eigenvalues with spacelike coherent eigenstates using FH equation with the Kratzer potential plus screened coulomb potential. In 2015, Ojonubah and Onate [56] obtained exact solutions of Feinberg-Horodecki equation for time-dependent Tietz-Wei diatomic molecular potential. The quantized momentum eigenvalues and corresponding wave functions are found in the framework of supersymmetric quantum mechanics.
We also obtain momentum eigensolution of the tHSCKP via supersymmetric quantum mechanics.
We calculate numerically eigen spectrum for the MnH, ScN, H Cl, LiH , and H 2 , I 2 , O 2 dimer in three dimensions for various values of the n and quantum numbers. At the end, we compare the numerical results of the energy spectrum for the HSCKP, screened Kratzer potential (SKP), and KP of the LiH, H Cl, H 2 , I 2 , O 2 dimer. We obtain momentum spectra for selected dimer.
The work is divided into six sections: "Reviews of the methodology". Solution of the HSCKP and tHSCKP are presented with deduced potentials in "Solution of the HSCKP and tHSCKP". In "Results and discussions", numerical results are discussed. The brief concluding remarks are given in "Conclusions".

Reviews of the methodology
We employ Qiang-Dong proper quantization rule (PQR) method to solve the Schrödinger equation in D dimensions for time-independent HSCKP and SUSYQM method has been employed to solve the Feinberg-Horodecki equation for time-dependent HSCKP.

Proper quantization rule and formula method
One-dimensional Schrödinger equation is expressed as Equation 5 can be written as where μ and k(z) denote the reduced mass of the interacting particles and momentum respectively. The Schrödinger equation can be written in the form of the Ricaati equation as Here, χ(z) = 1 dz . Several researchers applied PQR to physical potential problem [25][26][27][28][29][30][31][32][33][34][35] both in relativistic and non-relativistic regimes in the three and higher dimensions. Ma and Xu [12,13] by carefully studying the one-dimensional Schrödinger equation generalized to the three-dimension radial Schrödinger equation with spherically symmetric potential simplify making the replacement z → r. The proper quantization rule written as Here, r 1 and r 2 are two turning points determined by E = V eff (r). n is number of nodes of χ(r) in the region E ≥ V (r) and it is larger than the number of nodes of the wave function (r) by 1. Accordingly, it is required to first calculate the integral on the LHS of Eq. 8 and then replace spectrum of energy E n by the lowest energy level E 0 in the result to obtain the RHS integral of Eq. 8. PQR method has been applied in numerous physical systems to obtain the exact energy spectrum of many exactly solvable quantum mechanical systems [12, 13, 38-40, 42, 58].
In the formula method [8], the eigen equations with any physical potential of interest are converted into the second-order differential equation as [59] We obtain energy spectrum and corresponding eigen function respectively using the coordinate transformation variable of g = g(r) as and ψ(g) = N n g a 4 (1 − a 3 g) a 5 2 F 1 − n, n + 2(a 4 + a 5 ) where N n is constant of normalization and 2 F 1 is the hypergeometric function.

SUSYQM method
It is possible to define two nilpotent operators and † for N = 2 in SUSUQM. They satisfy the following commutation and anti-commutation relations [9,10,60] as H is the supersymmetric Hamiltonian operator and and † are known as the supercharge operators defined as [60,61] where S − is a bosonic operator and S + is its adjoint. The Hamiltonian H in terms of S − and S + can be defined where the H ± are named as the Hamiltonian of supersymmetric-partner. If we have zero ground state energy for H (i.e., E 0 = 0), we can always represent the Hamiltonian as a product of linear differential operators pairs in a factorable form. Therefore, the ground state 0 (z) obeys the schrodinger equation as follows: This result makes it possible to globally reconstruct the above potential from the information of its ground state wave functions that contain zero nodes. Therefore, factorizing of H is quite easy by using the following ansatz: where Now, the Riccati equation for U(z), Here, U(z) can express it in terms of 0 (z) by We obtain this solution by noticing that when S − 0 (z) = 0 is satisfied, we have H − 0 (z) = S + S − 0 (z) = 0. We then introduce the operator H + = S − S + which is written by reversing the order of the H − components. After a bit of simplification, we find that H + is nothing but the Hamiltonian for new potential V + (z) as, Here, V + (z) and V − (z) are supersymmetric partner potentials. For example, when the ground state energy of H 1 is E 1 0 with eigenfunction 1 0 forms Eq. 22, we can always write [60,61], (29) and The SUSY partner Hamiltonian H 2 is defined by with V 1 (z) and V 2 (z) are potentials corresponding to Hamiltonian H 1 and H 2 respectively. From Eq. 31, the energy eigenvalues and eigen functions corresponding to H 1 and H 2 are obtained as [60,61], where E m n represents the energy eigenvalue, with n and m denoting the energy level and mth Hamiltonian H m , respectively. Hence, it is clear that if H 1 has k ≥ 1 bound state with corresponding eigenvalues E 1 n , as well as eigen function ψ 1 n defined in 0 < n < k. Then, we can always generate a hierarchy of (k − 1) Hamiltonians, i.e., H 2 , H 3 , ......., H k such that the (H m ) has the same spectrum of eigenvalue as H 1 , apart from the fact that the first (m − 1) eigenvalue of the Hamiltonian H is absent in H [60,61]. where We also have By knowing all the eigen functions and eigenvalues of H 1 , we also obtain the corresponding eigenfunction 1 n and energy eigenvalues E 1 n of the (k − 1) Hamiltonian

Solution of the HSCKP and tHSCKP
In this section, we obtain solution of the HSCKP and tHSCKP using PQR method and SUSYQM method respectively.

Solution of the HSCKP in D dimensional space via PQR method
For spherically symmetric potential, D dimensional Schrödinger equation expresses as [62] where μ, V (r), , L D , are reduced mass, potential, reduced Planck constant, and angular coordinate, respectively, whereas E n is energy eigenvalue. The total wave function n m (r, L m ) is written as where Y m (L D ) is hyperspherical harmonic, whereas R n (r) hyper radial wave function. For the operator 2 D (L D ), the eigen spectrum of hyperspherical harmonic functions in D dimensional space is given by Using Eqs. 42, 43 and 44, 41 reduces as where Now, choosing the radial wave function as From Eq. 49 for δ = 1/2 and λ = 1 we obtain Now, employ a Greene-Aldrich approximation for the centrifugal term [63] 1 where To obtain solution for the proposed potential via the proper quantization rule, we apply p as Using this transformation, Eq. 53 can be written as where β = α 2 2 2μ S (S + 1), Now, we can determine two turning points p 1 and p 2 as The momentum k(p) between the two turning points p 1 and p 2 is given as Using Sturm-Liouville theorem and taking χ(p) = A 1 p + A 2 in Eq. 63, we obtain Equating coefficient in the above equation and after simplifying, we obtain [31] Using Eq. 66 into Eq. 137 and solving Eq. 8 (see Appendix), we obtain [31,58] To obtain eigen function of HSCKP via formula method [8], we uses an appropriate coordinate transformation p = e −αr in Eq. 10 where Equating Eq. 10 with Eq. 70, we obtain where

Solution of the FH equation for tHSCKP via SUSYQM method
The space-like counterpart of the Schrdinger equation presented by 2 2μc 2 where μ is mass of the particle, t is the space-like parameter x, c is light velocity, and P is quantized momentum with quantum number n = 0, 1, 2, 3..... Equation 75 was derived by Horodecki [46] from the relativistic Feinberg equation [45]. For δ = 1/2 and λ = 1, we obtain the time-dependent effective HSCKP Now, employ a Greene-Aldrich approximation for the centrifugal term [63] 1 Now, Eq. 75 can be written as Here and P n = μ cP n (81) where μ = 2μc 2 2 . and In SUSY quantum mechanics, the ground state wave function F 0 (r) in terms of superpotential U(t) is defined as [9,10,60], From Eqs. 25 and 27, the partner potential is obtained as Now, the associated Riccati equation is of the form where V eff presents in Eq. 81. Now, we proposed superpotential U(t) as (89) Using Eqs. 85 into 83, we obtained V − (t) and V + (t) as [49,64,65] Above two partner potentials, Eqs. 90 and 91 satisfy the condition of the shape invariant potential Where, b 0 = T 1 and b j is function of b 0 ; therefore, So, we observe that using mapping of the form T 1 → T 1 − α holds the shape invariance. From Eq. 92, we can write [64,65], . . .

Time-independent and time-dependent recovered potentials
In this subsection, we recovered time-independent and timedependent various potentials and corresponding energy and momentum spectrum.

Yukawa potential (YP)
The HSCKP reduces to the YP for b = δ = V 0 = 0 in Eq. 3 Corresponding energy spectrum for YP from Eq. 68 Time-dependent YP and corresponding momentum eigenvalues from Eq. 99

Hulthén potential (HP)
For D e = 0 in Eq. 3, HSCKP coverts into the Hulthén potential as [18,19] In D dimensions, energy eigenvalues for Hulthén potential, from Eq. 68 Time-dependent HP and corresponding momentum eigenvalues from Eq. 99

Hellmann potential
For V 0 = b = 0 and λ = 1/2, δ = 1 in Eq. 3, HSCKP coverts into the Hellmann potential as [11] Tacking V 1 = B for the first part of the above equation and V 1 = −C in exponential part of potential. We can write Hellman potential as In D dimensions, energy eigenvalues for Hellmann potential from Eq. 68 Time-dependent Hellmann potential and corresponding momentum eigenvalues from Eq. 99

Results and discussions
The behavior of the various potentials in terms of tHSCKP as a function of time t and dissociation energy D e is graphically depicted. The Hulthén-screened cosine Kratzer potential is provided as a function of several potential parameters such as r, D e , a, b, and α. The HSCKP's RV energy behavior is plotted using various parameters α, D e , and quantum number n. We use Eqs. 68 and 99 to determine the RV energy and momentum eigenvalues for the MnH and ScH dimers of the HSCKP and tHSCKP, which are presented in Tables 3  and 4, respectively. We also use Eq. 103 to determine the RV energy eigenvalues for the LiH and H 2 dimers for SCKP. Table 5 shows that the numerical results obtained in ref. [29] and the computed numerical results for SCKP coincide. We calculate the RV energy eigenvalues for H Cl, I 2 , O 2 , and LiH, dimer for the KP using Eq. 108.   Tables 6 and 7 show the numerical results for these dimers, together with a comparison of the values reported in refs. [29,70]. In Tables 8 and 9, we calculate the energy and momentum spectra for the time-independent and timedependent Hellmann potentials, respectively. With varied quantum numbers n, , we employed spectroscopic parameters from Tables 1 and 2. Through calculating the numerical results in Tables 3, 4, 5, 6, 7, 8 and 9, we utilized c = 1973.29eVÅ and 1 amu= 931.494028 × 10 6 eV/c 2 . Figure 1 shows plots of the various potentials vs. time t. It indicates that the tHSCKP and SCKP behave the same way, whereas for the MnH dimer, YP increases quickly at first and then grows slowly after t = 0.2. The behavior of the various potentials with regard to the dissociation energy D e for the MnH dimer is shown in Fig. 2. V 1 Table 3   and V 2 progressively increase, and V 5 gradually drops. In Fig. 3, we show the behavior of various potentials vs. time t for the ScN dimer. The plots demonstrate that V 1 and V 2 fall rapidly at first and then steadily grow after t = 1.5, whereas V 5 increases exponentially. Figure

Conclusions
In this paper, we used the proper quantization rule and the SUSYQM approach to solve the non-relativistic Schrödinger equation and the F-H equation for the HSCKP and tHSCKP to obtain the energy and momentum spectrum respectively. We were able to recover different timeindependent and time-dependent potentials, as well as their related energy and momentum spectra. SCKP, tSCKP, SKP, tSKP, KP, tKP, GCYP, tGCYP, YP, tYP, IQYP, tIQYP, HP, tHP, Hellmann potential, and time-dependent Hellmann potential are the derived potentials by establishing the parameters. Tables 5, 6, 7 and 8 show the computed RV energy eigenvalues for the MnH, ScN, H 2 , I 2 , O 2 , LiH , andH Cl dimers, which corresponds to the HSCKP and other potentials. The HSCKP vs. a radius r, D e , parameters a, b, and the screening parameter α are all discussed. The RV energy's behavior is displayed against the screening parameter α, dissociation energy D e , and vibrational    Tables 4 and 9 provide the numerical results of the momentum spectra for various dimers that correlate to the tHSCKP and Hellmann potentials, respectively. The HSCKP have applications in condensed matter, atomic-molecular physics, particle and nuclear physics, and quantum chemistry, among other fields of physical and chemical science.

Appendix: Solution of the HSCKP in D dimensional space via PQR method
From Eq. 66