A new exactly integrable hypergeometric potential for the Schr\"odinger equation

We introduce a new exactly integrable potential for the Schr\"odinger equation for which the solution of the problem may be expressed in terms of the Gauss hypergeometric functions. This is a potential step with variable height and steepness. We present the general solution of the problem, discuss the transmission of a quantum particle above the barrier, and derive explicit expressions for the reflection and transmission coefficients.


Introduction
The list of the solutions of the one-dimensional stationary Schrödinger equation in terms of the Gauss ordinary hypergeometric functions for potentials for which all the involved parameters are varied independently includes just three names. These are the Eckart [1] and Pöschl-Teller [2] classical potentials known from the early days of quantum mechanics and the third hypergeometric potential introduced recently [3]. In this paper we introduce another potential belonging to this class. This is an asymmetric potential-step with variable height and steepness. Though independent, the potential has much in common with the third exactly integrable hypergeometric potential which is also an asymmetric step-barrier [3]. In particular, both potentials are four-parameter and belong to the same general Heun family [4,5]. Besides, in both cases, the general solution of the problem has been expressed in terms of fundamental solutions, each of which is represented by an irreducible linear combination of two ordinary hypergeometric functions. It has also been shown that this combination may be represented by a single generalized hypergeometric function [6,7].
In this paper we present the explicit solution and discuss the quantum mechanical reflection and transmission of a quantum particle above the barrier. The reflection and transmission coefficients are given by simple formulas.

The potential and reduction to the Heun equation
The potential we introduce is given as where one may replace x by 0 x x  with arbitrary 0 x . The shape of the potential is shown in Fig. 1. This is an asymmetric step-barrier with controllable height and steepness. The potential involves four independent parameters, 0 1 , , V V  and 0 x , which stand, respectively, for the energy origin, the height, the steepness and the space position of the step (obviously, without any loss of the generality, one may put 0 0 V  and 0 0 x  ).
To solve the one-dimensional stationary Schrödinger equation for the potential (1) ). The inset presents the coordinate transformation ( ) z x for 1    .
We note that any combination of plus and minus signs for 1 2 ,   is applicable. Hence, by choosing different combinations, one can derive different fundamental solutions.

The solution of the Heun equation
The next step is to construct the solution of the Heun equation (9) in terms of the hypergeometric functions. One first checks that the known direct Heun-to-hypergeometric reductions (see, e.g., [26][27][28][29][30]) do not help. Instead, one may apply the series solutions in terms of the Gauss hypergeometric functions [31][32][33][34][35]. The cases when such expansions terminate thus resulting in finite-sum closed-form solutions have been discussed in [36]. It was shown that the series terminates if a characteristic exponent of a singularity of the Heun equation is a positive integer and the accessory parameter q satisfies a polynomial equation.
In our case 1    so that a characteristic exponent of the singularity Hence, a solution of the Heun equation in the form of a finite-sum of the Gauss hypergeometric functions may exist. For this to be the case, the necessary condition is We present the detailed derivation of this equation in Appendix. It may be readily checked that the equation is satisfied by parameters (10) , ; 1; In the similar way, the second independent fundamental solution is shown to be   Thus, the general solution of the Schrödinger equation may be written as with arbitrary 1 c , 2 c . This solution is valid for arbitrary (real or complex) set of all involved parameters (with the proviso that none of  and  is zero, one, or a negative integer).
We conclude this section by noting that Letessier noticed [37] (see also [38]) that the fundamental solutions (14), (15) This is a useful form that in several cases provides simpler derivations.

Above-barrier transmission
To discuss the quantum-mechanical reflection on transmission of a particle above the potential barrier, that is, when 0 1 E V V   , we note that the coordinate transformation given by equation (2) maps the real axes ( , ) x    onto the interval (1, ) z   and that the asymptotes at infinity for 0 Let the exponents 1  and 2  take the plus sign in equations (12). Then, the pre-factor of the general solution (16) at x   behaves as Demanding now that the wave function at x   involves only one plane wave (that is, only the transmitted wave), we derive 1 0 c  and Expanding then the solution at x   , we arrive at the asymptote where  is the Euler gamma-function. The transmission coefficient is then given as where we have introduced the notation In the limit 0 which is the result for the abrupt-step potential [40]. As expected, the correction term is shown to be positive, hence, because of the smoothness, the transmission above the potential (1),(2) is always greater than that for the abrupt-step potential. In the idealized infinitelysmooth limit    the potential becomes transparent. The reflection coefficient Fig. 2.

Discussion
Thus, we have introduced one more quantum mechanical potential for which the onedimensional stationary Schrödinger equation is solved in terms of the Gauss hypergeometric functions [1][2][3]. The new potential is a step-barrier with controllable height and width. We have discussed the transmission of a quantum particle above this barrier.
The potential is a four-parameter sub-potential of the fifth eight-parameter general Heun family defined by the triad 1 2 3 ( , , ) (1,1, 1) m m m   [25]. This is a remarkable family in that it generalizes the Eckart potential and involves, as independent particular cases, the third exactly integrable hypergeometric potential and the potential we have introduced. In addition, the family contains a number of conditionally integrable sub-potentials, in particular, the Dutt-Khare-Varshni [12] and López-Ortega potentials [16], as well as a variety of quasiexactly solvable potentials [21] or fixed-energy solutions [41].
An observation worth of some attention is that the two new hypergeometric potentials (as well as other recently reported exactly integrable potentials of the Heun class [42][43][44]) are four-parameter, while the classical ordinary hypergeometric potentials by Eckart and Pöschl-Teller (as well as the three classical confluent hypergeometric potentials [45][46][47][48]) are fiveparameter. We wonder if there exists a more general exactly integrable hypergeometric reduction of the fifth general Heun family, which involves as particular cases the two recent hypergeometric potentials. We hope to explore this possibility in the near future. ( 1 ) ( ) y q d u du u y y y a dy y y y a dy where the parameters obey the Fuchsian condition We examine an expansion of the solution of equation (A1) of the form where (compare with (A2)) In order to match the last equation with the Fuchsian condition (A2) for all n , we put  n n c w a n n a n n a a n n w n w q from which we obtain a three-term recurrence relation for the coefficients of the expansion: