The third five-parametric hypergeometric quantum-mechanical potential

We introduce the third five-parametric ordinary hypergeometric energy-independent quantum-mechanical potential, after the Eckart and P\"oschl-Teller potentials, which is proportional to an arbitrary variable parameter and has a shape that is independent of that parameter. Depending on an involved parameter, the potential presents either a short-range singular well (which behaves as inverse square root at the origin and vanishes exponentially at infinity) or a smooth asymmetric step-barrier (with variable height and steepness). The general solution of the Schr\"odinger equation for this potential, which is a member of a general Heun family of potentials, is written through fundamental solutions each of which presents an irreducible linear combination of two Gauss ordinary hypergeometric functions.


Introduction
The solutions of the Schrödinger equation in terms of special mathematical functions for energy-independent potentials which are proportional to an arbitrary variable parameter and have a shape independent of that parameter are very rare [1][2][3][4][5][6][7][8][9][10] (see the discussion in [11]). It is a common convention to refer to such potentials as exactly solvable in order to distinguish them from the conditionally integrable ones for which a condition is imposed on the potential parameters such that the shape of the potential is not independent of the potential strength (e.g., a parameter is fixed to a constant or different term-strengths are not varied independently). While there is a relatively large set of potentials of the latter type (see, e.g., [12][13][14][15][16][17][18][19][20] for some examples discussed in the past, and [21][22][23][24][25] for some recent examples), the list of the known exactly integrable potentials is rather limited even for the potentials of the most flexible hypergeometric class. The list of the exactly solvable hypergeometric potentials currently involves only ten items [1][2][3][4][5][6][7][8][9][10]. Six of these potentials are solved in terms of the confluent hypergeometric functions [1][2][3][4][5][6]. These are the classical Coulomb [1], harmonic oscillator [2] and Morse [3] potentials and the three recently derived potentials, which are the inverse square root [4], the Lambert-W step [5] and Lambert-W singular [6] V z x z .
However, in several cases the coordinate transformation ( ) x z is inverted thus producing explicitly written potentials given as All these cases are achieved by fixing the parameter a to a particular value, hence, all these particular potentials are four-parametric. The mentioned two recently presented fourparametric ordinary hypergeometric potentials [9,10] are just such cases.
The potential we present is either a singular well (which behaves as the inverse square root in the vicinity of the origin and exponentially vanishes at infinity) or a smooth asymmetric step-barrier (with variable height, steepness, and asymmetry). The general solution of the Schrödinger equation for this potential is written through fundamental solutions each of which presents an irreducible linear combination of two ordinary hypergeometric functions 2 1 F . The singular version of the potential describes a short-range interaction and for this reason supports only a finite number of bound states. We derive the exact equation for energy spectrum and estimate the number of bound states.

The potential
The potential is given parametrically as it is seen that for real rational a the transformation is rewritten as a polynomial equation for z , hence, in several cases it can be inverted. Since 0,1 a  , the possible simplest case is when the polynomial equation is quadratic. This is achieved for 1,1 / 2, 2 a   . It is checked, however, that these three cases lead to four-parametric sub-potentials which are equivalent in the sense that each is derived from another by specifications of the involved parameters. For 1 a   the potential reads [9]: The next are the cubic polynomial reductions of equation (3) which are achieved in six cases: 2, 1/ 2, 1/ 3, 2 / 3, 3 / 2, 3 a    . It is again checked, however, that these choices produce only one independent potential. This is the four-parametric potential presented in [10]: where one should replace x by 0 x x  . Similar potentials in terms of elementary functions through quartic and quintic reductions of equation (3) are rather cumbersome; we omit those.
For arbitrary real 0,1 a  , assuming (0,1) z  and shifting 0 0 the potential (1),(2) presents a singular well. In the vicinity of the origin it behaves as 1/2 x  : and exponentially approaches a constant, 0 1 V V  , at infinity: The inset presents the coordinate transformation ( ) (0,1) The potential and the two asymptotes are shown in Fig. 1 we get an asymmetric step-barrier the height of which depends on 0 V and 1 V , while the asymmetry and steepness are controlled by the parameters a and  . The shape of the potential is shown in Fig. 2 for 2 a   and 1.25 a  . We note that in the limit 0   the potential turns into the abrupt-step potential and that the sub-family of barriers generated by variation of  at constant 0 V and 1 V has a  -independent fixed point located at

Reduction to the general Heun equation
The solution of the one-dimensional Schrödinger equation for potential (1),(2): is constructed via reduction to the general Heun equation [27][28][29] The details of the technique are presented in [11] and [25]. It has been shown that the energyindependent general-Heun potentials, which are proportional to an arbitrary variable parameter and have shapes which are independent of that parameter, are constructed by the coordinate transformation ( ) z z x  of the Manning form [30] given as where 1,2,3 m are integers or half-integers and  is an arbitrary scaling constant. As it is seen, the coordinate transformation is solely defined by the singularities 1,2,3 a of the general Heun equation. The canonical form of the Heun equation assumes two of the three finite singularities at 0 and 1, and the third one at a point a , so that 1,2,3 (0,1, ) a a  [27][28][29].
However, it may be convenient for practical purposes to apply a different specification of the singularities, so for the moment we keep the parameters 1,2,3 a unspecified.
The coordinate transformation is followed by the change of the dependent variable and application of the ansatz The form of this ansatz and the permissible sets of the parameters 1,2,3 m are revealed through the analysis of the behavior of the solution in the vicinity of the finite singularities of the general Heun equation [11]. This is a crucial point which warrants that all the parameters involved in the resultant potentials can be varied independently.
It has been shown that there exist in total thirty-five permissible choices for the coordinate transformation each being defined by a triad  [25]. The potential (1), (2) belongs to the fifth independent family with 1,2,3 (1,1, 1) m   for which from equation (14) we have with arbitrary 0,1,2,3,4 const V  , and, from equation (12), It is now convenient to have a potential which does not explicitly involve the singularities.
Hence, we put 3 0 a  and apply the specification 1,2,3 ( ,1,0) a a  to get the potential The solution of the Schrödinger equation (10) for this potential is written in terms of the general Heun function G H as where the involved parameters , , , ,      and q are given through the parameters 0,1,2,3,4 V of the potential (17) and the exponents 1,2,3  of the pre-factor by the equations [25]     the exponents 1,2,3  of the pre-factor being defined by the equations

The solution of the Schrödinger equation in terms of the Gauss functions
Having determined the parameters of the Heun equation, the next step is to examine the cases when the general Heun function G H is written in terms of the Gauss hypergeometric functions 2 1 F . An observation here is that the direct one-term Heun-tohypergeometric reductions discussed by many authors (see, e.g., [27][28][31][32][33][34] Pöschl-Teller potentials [25]. For the first nontrivial case 1    the termination condition for singularities 1,2,3 ( ,1,0) a a  takes a particularly simple form: The solution of the Heun equation for a root of this equation is written as [39]   This solution has an alternative representation through Clausen's generalized hypergeometric function 3 This equation generally defines a conditionally integrable potential in that the potential  (17) is reduced to that given by equation (1). Furthermore, since  is arbitrary, in order for equation (18) to exactly reproduce the coordinate transformation (2), we replace With this, the solution of the Schrödinger equation (10) for potential (1) is written as ; This solution applies for any real or complex set of the involved parameters. Furthermore, we note that any combination for the signs of 1

Bound states
Consider the bound states supported by the singular version of potential (1),(2), achieved by shifting 0 (2). Since the potential vanishes at infinity exponentially, it is understood that this is a short-range potential. The integral of the function ( ) xV x over the semi-axis (0, ) x   is finite, hence, according to the general criterion [43][44][45][46][47], the potential supports only a finite number of bound states. These states are derived by demanding the wave function to vanish both at infinity and in the origin (see the discussion in [48]). We recall that for this potential the coordinate transformation maps the with some constants 1,2 A . Since for positive 2  the first term diverges, we conclude 1 0 The condition ( 0) 0 z    then gives the following exact equation for the spectrum: The graphical representation of this equation is shown in Fig. 3.

SE
According to the general theory, the number of bound states is equal to the number of zeros (not counting 0 x  ) of the zero-energy solution, which vanishes at the origin [43][44][45][46][47].
We note that for 0 E  the lower parameter of the second hypergeometric function in Eq.  For practical purposes, it is useful to have an estimate for the number of bound states.
The absolute upper limit for this number is given by the integral [43,44]     where 2 Li is Jonquière's polylogarithm function of order 2 [49,50]. Though of general importance, however, in many cases this is a rather overestimating limit. Indeed, for the parameters applied in Fig. 3 it gives 24 B n I   . More stringent are the estimates by Calogero [45] and Chadan [46] which are specialized for everywhere monotonically non-decreasing attractive central potentials.
Calogero's estimate reads C n I  with [45]   We note that 2 The result by Chadan further tunes the upper limit for the number of bound states to the half of that by Calogero, that is / 2 C n I  [46]. For the parameters applied in Fig. 3

Discussion
Thus, we have presented the third five-parametric quantum-mechanical potential for which the solution of the Schrödinger equation is written in terms of the Gauss ordinary hypergeometric functions. The potential involves five (generally complex) parameters which are varied independently. Depending on the particular specifications of these parameters, the potential suggests two different appearances. In one version we have a smooth step-barrier  The potential belongs to the general Heun family 1,2,3 (1, 1, 1) m   . This family allows several conditionally integrable reductions too [25]. A peculiarity of the exactly integrable potential that we have presented here is that the location of a finite singularity of the general Heun equation is not fixed to a particular point of the complex z -plane but serves as a variable potential-parameter. In the step-barrier version of the potential, this parameter stands for the asymmetry of the potential. functions [57][58][59][60][61] when the solutions eventually reduce to quasi-polynomials, e.g., discussed in the context of quasi-exactly solvability [59][60][61]. We note that owing to the contiguous functions relations [42], the two-term structure of the solution is a general property of all finite-sum hypergeometric reductions of the general Heun functions achieved via termination of series solutions. It is checked that in our case the linear combination of the involved Gauss functions is expressed through a single generalized hypergeometric function 3 2 F [40,41].
We have presented the explicit solution of the problem and discussed the bound states supported by the singular version of the potential. We have derived the exact equation for the energy spectrum and estimated the number of bound states. The exact number of bound states is given by the number of zeros of the zero-energy solution which we have also presented.