Constructions of the soluble potentials for the non-relativistic quantum system by means of the Heun functions

The Schr\"{o}dinger equation $\psi"(x)+\kappa^2 \psi(x)=0$ where $\kappa^2=k^2-V(x)$ is rewritten as a more popular form of a second order differential equation through taking a similarity transformation $\psi(z)=\phi(z)u(z)$ with $z=z(x)$. The Schr\"{o}dinger invariant $I_{S}(x)$ can be calculated directly by the Schwarzian derivative $\{z, x\}$ and the invariant $I(z)$ of the differential equation $u_{zz}+f(z)u_{z}+g(z)u=0$. We find an important relation for moving particle as $\nabla^2=-I_{S}(x)$ and thus explain the reason why the Schr\"{o}dinger invariant $I_{S}(x)$ keeps constant. As an illustration, we take the typical Heun differential equation as an object to construct a class of soluble potentials and generalize the previous results through choosing different $\rho=z'(x)$ as before. We get a more general solution $z(x)$ through integrating $(z')^2=\alpha_{1}z^2+\beta_{1}z+\gamma_{1}$ directly and it includes all possibilities for those parameters. Some particular cases are discussed in detail.


Introduction
The exact solution of the Schrödinger equation with physical potentials has played an important role in quantum mechanics. Generally speaking, for a given external field, one of our main tasks is to show how to solve the differential equation through choosing suitable variables and then find its solutions can be expressed by some special functions. Here we focus on how to construct a class of the solvable potentials within the framework of the non-relativistic Schrödinger equation. Similar works have been carried out [1,2,3,4,5], but it is worth pointing out that Ishkhanyan and his co-authors took ρ 2 ∝ (z − a 1 ) m 1 (z − a 2 ) m 2 (z − a 3 ) m 3 , where the parameters a 1,2,3 are three singularity points, to construct the soluble potentials with some constraints on the parameters −1 ≤ m 1,2,3 ≤ 1 and 1 ≤ m 1 + m 2 + m 3 ≤ 3 [4,5]. By choosing different values of these parameters which satisfy these constraints, some interesting results have been obtained. However, the approaches which were taken by Natanzon [2], who constructed a class of the soluble potentials related to the hypergeometric functions and Bose [1], who discussed the Riemann and Whittaker differential equations are different from Ishkhanyan et al. Nevertheless, in Bose's classical work [1] he only studied a few special cases for the differential equation (z ′ ) 2 = α 1 z 2 + β 1 z + γ 1 . Its general solutions were not presented at that time due to the limit on the possible computation condition. In this work our aim is to construct the soluble potentials within the framework of the Schrödinger invariant I S (x) through solving z ′ (x) differential equation directly and then obtaining its more general solutions but not only considering several special cases for the parameters α 1 , β 1 and γ 1 .
The rest of this work is organized as follows. In Section 2 we present the Schwarzian derivative {z, x} and the invariant I(z) of the differential equation u zz + f (z)u z + g(z)u = 0 through acting the similarity transformation ψ(z) = φ(z)u(z) on the Schrödinger equation. In Section 3 as an illustration we take the Heun differential equation as a typical example but with different approach taken by Ishkhanyan et al. The all soluble potentials are obtained completely in Section 4. Some concluding remarks are given in Section 5.
2 Similarity transformations to the Schrödinger equation As we know, the Schrödinger equation has the form where we call k 2 an energy term and V (x) an external potential.
Through choosing a similarity transformation ψ(z) = φ(z)u(z) where z = z(x), we are able to obtain the following differential equation which can be rewritten as which implies that Integrating the first differential equation allows us to obtain Substitution of this into the second differential equation of Eqs. (4) yields from which we define the expression [6] as the invariant 1 of Eq. (3). Using Schwarzian derivative where we have used the relation z ′′ (x) = ρ d dx = ρρ z and considering equation (7), then equation (6) can be rewritten as where I S (x) is defined as the Schrödinger invariant [1,2]. Thus, the problem of the construction of the soluble potentials for the original Schrödinger equation (1) is solvable on the basis of the functions corresponding to a given I(z) (7) becomes a problem of deciding transformations z(x) such that the relation ρ 2 I(z) is thus characterized by two elements, i. e. I(z) and the Schwarzian derivative {z, x}, which is directly related to the function z(x).

Application to Heun differential equation
The Heun differential equation is given by [7,8,9,10,11] where the parameters satisfy the Fuchsian relation For this equation, the Heun Invariant I h can be calculated as where the parameters A, B, C, D, F depend on the parameters a and α, β, γ, δ, ǫ, i. e. suggests us to take the class of functions defined by 2 where α 1 , β 1 , γ 1 are arbitrary constants. Such a choice is to make the ρ 2 I h (z) (9) generate a constant to cancel the energy level term k 2 . Otherwise, the expansion terms for ρ 2 I h (z) without including a constant will make the k 2 = 0, which means the particle moving in a free field. In terms of this equation (14), one has where we have used the relation {z, x} = ρρ zz − 1 2 ρ 2 z and ρ = z ′ (x). To solve equation (14), we first obtain its general solutions for arbitrary parameters. In this 2 It should be pointed out that present choice is different from previous one [5], in which the ρ 2 ∝ (z − a 1 ) m1 (z − a 2 ) m2 (z − a 3 ) m3 is chosen in order to adapt the mathematical character of the Heun invariant (12).

case, one has
where c 1 is an integral constant and we take c 1 = 0 for simplicity.
On the other hand, it was recalled that [1,3] {z t , x} = {z, x}, z t ≡ where A 1 , B 1 , C 1 , D 1 are constants but A 1 D 1 − B 1 C = 0. From Eq. (22), we have If differentiating z t given in Eq. (22) with respect to x and eliminating the variable z, one has where z ′ (x) is given by Eq. (14). It is not difficult to see that the solutions of Eq. (24) are also possible transformations since it is a generalization of Eq. (14). Up to now, we have found a class of functions for transforming I h to I S . It is worth noting that this class of functions can be characterized differently. We are going to give a useful remark on the z ′ t (x) given in Eq. (24). If we use this to calculate the Schrödinger invariant I S (x) (9), then we will find that the soluble potentials would become rather complicated and do not consider this for simplicity, but it should be recognized that the variable z t is just z(x) as given in Eq. (14).
We are now in the position to construct the simple Schrödinger invariants corresponding to the general Heun differential equation Invariant with the aid of the transformation (15) we obtained above. First, let us consider the simpler transform (14). Substituting equations (12) and (15) into Eq. (9) allows us to obtain the following useful Schrödinger invariant i) ii) iii) iv) v) Here, we have used the symbol (H n (±,a,b) ) to denote the above invariants, H referring to I h and n (±,a,b) to z (±,a,b) n . Let us analyze these potentials through expanding them as follows: For the i) case, we have where For the ii) case, we have where For the iii) case, one has where For the special case iv), we have where For the v) case, one has where Obviously, the potential given in case i) is more complicated than the usual Eckart potential.
The potentials discussed in cases ii) and iii) are more complicated than the first and second Now, let us study the wave function. In terms of Eqs. (5) and the function given in (10), one has the following form where ρ given in Eq. (14) depends on the solutions (16), while those particular cases given in Eqs. (17) to (21). The partial wave function u(x) involved in the whole wave function ψ(z) = φ(z)u(z) is given by the Heun functions H l (a, q, α, β, γ, δ, ǫ; z).

Concluding remarks
The Schrödinger equation is rewritten as a more popular form of a second order differential equation through taking a similarity transformation. We find that this classical equation is closely related to the Schwarzian derivative and the invariant identity of the differential equation u zz + f (z)u z + g(z)u = 0. As a typical differential equation, the corresponding mathematical properties of the Heun differential equation are studied. Before ending this work, we give a useful remark on the Schrödinger invariant I S (x). First, let us consider the Schrödinger equation (1) and equation (9). We find that the Schrödinger equation can also be rewritten as ∇ 2 ψ(x) = −I S (x)ψ(x). Since ∇ 2 represents the kinetic term T of the moving particle, it should keep invariant for the same particle. This is also reflection of the conservation of energy Competing Interests