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Bound-states for generalized trigonometric and hyperbolic P\"oschl-Teller potentials

We use the"tridiagonal representation approach"to solve the time-independent Schr\"odinger equation for the bound states of generalized versions of the trigonometric and hyperbolic P\"oschl-Teller potentials. These new solvable potentials do not belong to the conventional class of exactly solvable problems. The solutions are finite series of square integrable functions written in terms of the Jacobi polynomial.


Introduction
In quantum mechanics, potential models are used to describe the behavior of physical systems as they interact with their environment. The success of a model depends on how well it represents the system and predicts its behavior in a given scenario. In principle, there is an infinite number of such potential models. However, the most relevant ones are those that could be solved exactly since then one can explore all physical scenarios that may arise, especially at critical couplings to the environment where the system makes a dramatic transition from one phase to another. Moreover, exact solutions can easily give access to the most significant properties of the system such as the energy spectrum of bound states, phase shift of the scattering states, thermal properties, density of states, etc. On the other hand, it is worth mentioning that the class of exactly solvable potentials is very small. The most well-known members of this class include the Coulomb, harmonic oscillator, Morse, Pöschl-Teller, Eckart, and Scarf potentials [1,2]. Many researchers over the years (present company included) devised different methods in an attempt to find exact solutions of the wave equation (e.g., Schrödinger and Dirac) for new potential models. If and when successful, only very few discovered potentials join the class of exactly solvable potentials. The most popular methods of solution of the wave equation include factorization, supersymmetry, path integral, point canonical transformation, shape invariance, Darboux transformation, asymptotic iteration, and the Nikiforov-Uvarov method. Our choice is the Tridiagonal Representation Approach (TRA). It is an algebraic method for solving linear ordinary differential equations of the second order [3][4][5].
For a detailed description of the method and how it is used to solve quantum mechanical problems, one may consult [6,7] and references therein.
In this work, we employ the TRA to obtain exact bound-state solution of the time-independent one-dimensional Schrödinger equation for the following four-parameter potential models Without the most singular 0 V -term, they become the hyperbolic and trigonometric Pöschl-Teller potentials, respectively [2,8]. All potential parameters are real with 0 0 V  and 0 D  . Under the transformation I , these potentials get reflected on the 0 x  axis but remain physically the same (isospectral). Moreover, the transformation interchanges the trigonometric functions as cos sin  and results in a horizontal displacement of the potential along the x-axis by a  but also remains isospectral. Consequently, the potential II is isospectral to II ( ) V x . Moreover, it is easy to see that the complex transformation i x x  in the Schrödinger equation with this potential changes it to that for . This stands to show that complex extension of quantum mechanics [9] and doing quantum mechanics on complex coordinates provide a unified treatment for potentials that seemingly have different physical characteristics. Now, both potentials (1a) and (1b) do not belong to the conventional class of exactly solvable quantum mechanical problems. Potential I ( ) V x is inverse-quartic and inverse-square singular at the origin with singularity strength of 0 V and A, respectively. Moreover, it is short-range with  being the range measure, which is a positive scale parameter of inverse length dimension. On the other hand, II ( ) V x is a potential well of width a with an inverse-square singularity at the left wall of strength D. However, on the right wall, it is inverse-quartic and inverse-square singular with singularity strength of 0 V and C, respectively. It is obvious that the energy spectrum for II ( ) V x is discrete, infinite, and bounded from below. Whereas, with a proper choice of values of the parameters, I ( ) V x could support a finite number of bound states in addition to a resonance structure. These proper parameter values are required to produce one or two positive real roots for the cubic equation that results from the condition   0 I 0 x dV dx  . A necessary, but not sufficient, condition for the existence of bound states is that I 0 ( ) 0 V x  . The existence of two different real positive roots of the cubic equation implies the possibility of resonance. This cubic equation, which is associated with I ( ) V x , reads as follows 3 where 2 sinh ( ) s x   . Since the constant term in this equation is positive, then Descartes' rule of signs [10] dictates that the existence of one positive real root requires that the coefficient of the 3 s term be negative. That is, a necessary (may not be sufficient) condition for bound states Figure 1 shows several plots of the two potential functions (in units of 0 V ) for several values of the associated parameters. Figure 2 is the spectral phase diagram (SPD) for I ( ) V x showing the distribution of the energy spectrum (scattering states, bound states, and resonances) as a function of the potential parameters. The SPD for potential systems was introduced in Ref. [11] and is explained here briefly in section B.3 of Appendix B.
In sections 2 and 3, we formulate the problem within the TRA and solve it for the two potential models, respectively. We conclude in section 4 with some remarks and a discussion of our main findings.

TRA solution of the hyperbolic potential
We start by writing the solution of the Schrödinger equation, where W(y) is a node-less entire function and   , , where we wrote Now, to solve for the continuous spectrum or for an infinite discrete spectrum of a given system, completeness of the set   n  implies that it is an infinite set. However, for systems with a finite number of bound states (e.g., the system associated with the potential I V ), a finite basis set   0 N n n   could produce a faithful representation of the physical system provided that the number of bound states is less than or equal to the size of the basis 1 N  . Additionally, for quasi-exact solution where one looks for a finite portion of the infinite discrete spectrum (e.g., the system associated with the potential II V ), such finite basis set could lead to a very good approximation for that portion of the spectrum with an accuracy that increases with N.
is the finite Jacobi polynomial defined on the semi-infinite real line ( ) 1 y x  as shown in Appendix A. The real basis parameters   , , ,     are to be determined below in terms of the physical parameters by the TRA constraints. Moreover, 0,1,.., To solve this equation, we substitute the series ( ) ( ) ( 1) . Now, the TRA constraint (3) and the recursion relation of the Jacobi polynomials (A3) dictate that terms inside the curly brackets in (8) must be linear in y. Thus, we should choose the Jacobi polynomial parameters as Reality dictates that 0 E  and 2 Using the three-term recursion relation for the Jacobi polynomials (A3) in this equation and comparing the result to the TRA constraint (3), we obtain  (4) will form a set of orthogonal polynomials with 2 0 ( ) f z as their positive definite weight function [7,14,15] and where z is some proper parameter that depends on the potential parameters 0 V and A. If we define the polynomial Then the recursion relation (4) The condition cos 1 The condition cosh 1   dictates that 0 A  , which is the same condition that guarantees reality of z. Finally, the m th bound state wavefunction with energy    [16][17][18]. On the other hand, for a small number of terms in the wavefunction series (14), we can easily compute this polynomial explicitly for all these low degrees using its recursion relation and initial values. First, however, we need to obtain the energy spectrum. To do that, we use two independent numerical techniques as explained in Appendix B with results shown in Table 1 for two different sets of physical parameters. Once the bound state energies   m E are known, the wavefunction is now fully determined by (14). Figure 3 shows the unnormalized wavefunctions corresponding to the energy spectrum given by Table 1.

TRA solution of the trigonometric potential
We repeat the same treatment in section 2 above for the potential model II ( ) V x . However, we choose the following coordinate transformation and basis parameters where again we have used the differential equation of the Jacobi polynomial (A2). The TRA constraint (3) and the recursion relation of the Jacobi polynomials (A3) dictate that we assign the following values to the Jacobi polynomial parameters Using the three-term recursion relation for the Jacobi polynomials (A3) in this equation and comparing the result to the TRA constraint (3), we obtain Consequently, the recursion relation (4) written in terms of   n P defined in (12) The condition cos 1   dictates that 0 0 V C    , which is the same condition that guarantees reality of z. On the other hand, comparing the recursion relation (4) written for   n P to Eq.
(A8), we conclude that ( , ) The condition cosh 1   . Here again and due to the lack of knowledge of the complete analytic properties of these TRA polynomials, we resort to numerical means to determine the energy spectrum as explained in Appendix B. Table 2 gives the lowest energy spectrum for two sets of potential parameters using two independent numerical routines. With the energy eigenvalues being found, the complete wavefunction is now fully determined as given by (21). Figure 4 is a plot of the un-normalized wavefunctions corresponding to the lowest energy eigenvalues in Table 2.

Conclusion and Discussion
This work is an additional testimony to the power and viability of the tridiagonal representation approach in solving novel quantum mechanical problems. The approach is endowed with robust mathematical underpinnings for solving ordinary second order differential equations of which the Schrödinger equation is but an example. The advantage of the approach in being algebraic is reinforced by the analytic power of orthogonal polynomials and special functions. On the computational side, it is favored as being reliant on powerful numerical techniques that deal with tridiagonal matrices and Gauss quadrature. Here, we presented bound state solutions for two generalizations of the trigonometric and hyperbolic Pöschl-Teller potentials, which are known to have a wide range of applications in atomic, molecular and nuclear physics. The generalized models are believed to have similar characteristics and fruitful applications.
It should be obvious from the structure of the hyperbolic potential I ( ) V x , which is also visually evident in Fig. 1a, that it can be used to give an improved modeling of molecular binding that could be adjusted using four parameters   0 , , , V A B  instead of the usual two or three potential parameters. Additionally, with a proper choice of these parameters, the potential may have local minimum and maximum that could place the system in the green phase of the SPD with a mix of bound states and resonances, which enriches the model depending on the value of the energy barrier measured by the difference between the two potential extrema V  . On the other hand, the trigonometric potential II ( ) V x , could be used in nuclear and particle physics as an alternative confining potential model (with a 4-parameter flexibility control) to the frequently used square potential box or the 1D Woods-Saxon potential well.
Nonetheless, a shortcoming of the TRA in the current problem is the absence of knowledge of the full analytic properties of the polynomials ( , ) ( ; , ) These analytic properties contain all physical features of the system such as the energy spectrum. On the other hand, due to the typically small number of terms in the solution series (14) and (21), these polynomials could be evaluated explicitly using their three-term recursion relation and initial values. However, we had to obtain the energy spectrum by numerical means.
Due to the finiteness of the energy spectrum of the potential I ( ) V x , the finite series (14) should give a faithful representation of the solution. However, since the energy spectrum of the potential II ( ) V x is infinite, the finite series (21) represents a quasi-exact solution for the lowest part of the spectrum whose accuracy could be improved by increasing the number of terms in the series (i.e., increasing the size of the basis).

Appendix A: Jacobi polynomial on the semi-infinite line
These polynomials are defined over the semi-infinite interval 1 y  whereas the conventional Jacobi polynomials are defined on the finite interval 1 1 y     . To distinguish it from the conventional Jacobi polynomials ( , ) ( ) n P y   we use the notation ( , ) ( ) It also satisfies the following three-term recursion relation The associated orthogonality relation reads as follows   Table 2 where we compare them against those obtained via the DVR. Again, a good agreement between the two methods is evident.

B.3 Spectral phase diagram (SPD)
Assi et al [11] introduced the concept of a spectral phase diagram (SPD) for non-confining potentials. The diagram shows the different phases that the system can exist in for various choice of parameters. The SPD is generally divided into four regions: Bound states, mix of bound and resonance states, resonances only, and scattering states only. Those regions are based on the necessary (may not be sufficient) conditions for the potential to support any of these states.
The SPD is useful for choosing values of the potential parameters to place the system in a certain spectral phase. Clearly, the SPD for the potential well II ( ) V x is trivial where it has only one region as it only supports bound states. Thus, we have generated the SPD only for I ( ) V x as given in Figure 2. In the figure, "B" stands for bound states (blue color), "R" stands for resonance states (red color), "B&R" stands for regions where a mix of bound states and resonances exist (green color), and "S" where only scattering states can exist (grey color). Note, of course, that scattering states occur in all regions of the SPD. In the figure we show the TRA solution space as the rectangular area indicated by the TRA constraints 2 It is also clear from Fig. 2 that a necessary condition for pure bound states is that ( ) 0 A B   , which is consistent with the observation made below Eq. (2) above. Table 1. The complete energy spectrum (in atomic units) for the hyperbolic potential I ( ) V x obtained using DVR and HOFD techniques with two sets of potential parameters:     Table 1. The top row corresponds to the parameter set 1 S and the bottom row corresponds to the parameter set 2 S . The horizontal axis is x  .  Table 2. The top row corresponds to the parameter set 3 S and the bottom row corresponds to the parameter set 4 S . The horizontal axis is x a .