It is well known that the exact solutions of quantum systems play an important role since the early foundation of the quantum mechanics. Generally speaking, two typical examples such as the hydrogen atom and harmonic oscillator have been taken in classical quantum mechanics textbooks.^[1–2] Until now, some main methods have been used to solve the quantum soluble systems. First, it is the functional analysis method. That is to say, one solves the second-order differential equation and obtains their solutions^[3] given by some well-known special functions. Second, it is called the algebraic method. This method can be realized by studying the Hamiltonian of quantum system and is also related with supersymmetric quantum mechanics (SUSYQM),^[4] further closely with the factorization method.^[5] Third, it is so-called the exact quantization rule method^[6] and proper quantization rule.^[7] The latter shows more beauty and symmetry than exact quantization rule. It should be recognized that almost all soluble potentials mentioned above belong to single-well potentials. For example, the classical single-well potential Rosen-Morse potential^[8] has been used to treat the vibrations of diatomic molecules such as Na₂ and SiC radical^[9–10] and to predict successfully the molar enthalpy values, molar entropy values and Gibbs free energies for molecular NO and P₂ in a wide temperature range.^[11–13] Scattering States of l-wave Schrödinger equation with modified Rosen-Morse potential was also studied.^[14] This potential can also be used to construct a double-well potential to treat the vibrations of polyatomic molecules such as the ammonia molecule.^[8] The double well potentials^[15–23] have been studied for a long time due to their complications and they could be used in the quantum theory of molecules to describe the motion of the particle in the presence of two centers of force, the heterostructures, Bose-Einstein condensates, and superconducting circuits, etc.

Thirty years ago, Konwent proposed an interesting potential^[24]
where V₀, A, and a are positive constants. As we will see below, the parameter A can also be taken negative. In Fig. 1, we plot it as the function of the variables x with various A and a, in which we take V₀ = 5 and a = 1. We see that the potential has a flat bottom in a single well for A ≥ 1, while it has the form of a double well for A ∈ (0, 1) as shown in Fig. 1(a). The parameter a is related with the width of the potential well, which is inversely proportional to the a as seen in Fig. 1(b). The double-well case is illustrated in Fig. 1(c) for small a and A = 1.

Fig. 1

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	Fig. 1 (Color online) A plot of potential as function of the variables y and A.

Through the series expansion around the origin, we have
which shows that V(x) is symmetric to variable x, i.e., V(−x) = V(x). We find that the minimum value of the potential V_min(x) = 0 at minimum values x = ± arccosh (1/A)/a for A∈(0, 1) and x = 0 for A ≥ 1. Konwent presented the so-called exact solutions by using the “polynomial method" following the Razavy’s approach.^[17] After studying Konwent’s work carefully, we find that the solutions cannot be given exactly due to the complicated three-term recurrence relation. The method presented there^[24] is more like the Bethe Ansatz method as summarized in our recent book.^[25]

That is, the solutions cannot be expressed as one of special functions because of three-term recurrence relations. In order to obtain some so-called exact solutions, the authors have to take some constraints on the coefficients in the recurrence relations as shown in Refs. [17, 24]. As shown in recent study of the hyperbolic type potential well,^[26–34] we have found that their solutions can be exactly expressed by the confluent Heun function.^[29] In this work we attempt to study the solutions of the Konwent potential. We shall find that the solutions can be written as the confluent Heun function but their energy levels have to be calculated numerically since the energy level term is involved in the parameter η of the confluent Heun function H_c (α, β, γ, δ, η z). This constraints us to use the traditional Bathe ansatz method to get the energy levels. It should be pointed out that the Heun function has been studied well, but its main topics are focused in the mathematical area. Only recent connections with the physical problems have been discovered, in particular the quantum systems for those hyperbolic type potential have been studied.^[26–34] For this reason, it is not surprising why the authors did not find their solutions related to Heun function.^[17,24] The terminology “semi-exact" solutions used in Ref. [27] arise from the fact that the wave functions can be obtained analytically, but the eigenvalues cannot be written out explicitly. The eigenvalues can be calculated by taking the series expansion and then by studying the behaviors of the wave functions at the infinity or by other numerical method.

This paper is organized as follows. In Sec. 2, we present the solutions of the Schrödinger equation with the Konwent potential. It should be recognized that the Konwent potential is single or double well depends on the potential parameter A. In Sec. 3 some fundamental properties of the wave functions are studied. The energy levels for different A are calculated numerically. It is interesting to see that the energy spectra increase with the potential parameter |A| ≥ 1 and A ∈ (−1, 0), while decrease with |A| for the parameter interval A ∈ (0, 1). Some concluding remarks are given in Sec. 4.

Let us consider the one-dimensional Schrödinger equation,
Substituting potential (1) into (3), we have
where
How to solve Eq. (4) becomes our main task. Take the wave functions of the form
Substituting this into Eq. (4) allows us to obtain
Take a new variable z = cosh²(y/2). The above equation becomes
which can be re-arranged as
where ϵ₁ = ϵ − (A²+ 1 ) m². When comparing this with the confluent Heun differential equation in the simplest uniform form^[29]
we find the solution to (9) is given by an acceptable confluent Heun function H_c (α, β, γ, δ, η; z) with
from which we are able to calculate the parameters δ and η involved in H_c (α, β, γ, δ, η; z) as

It is found that the parameter η related to energy levels is involved in the confluent Heun function. The wave functions given by this function seem to be analytical, but the key issue is how we first get the energy levels. Otherwise, the solutions become incomplete. Generally, the confluent Heun function can be expressed as a series expansion around the origin
The coefficients v_n are given by a three-term recurrence relation A_n v_n − B_n v_n−1 − C_n v_{n − 2} = 0, v₋₁ = 0, v₀ = 1, where A_n, B_n, and C_n depend on the parameters of the Heun functions.^[29] For the present case, however, we have z ≥ 1. Thus, the series expansion method around the origin becomes invalid. Since this case is very similar to our previous discussions,^[26–27] we have to study the eigenvalues numerically as done in Refs. [26–27].

In this section we are going to study some basic properties of the wave functions as shown in Figs. 2–5. We first consider the positive A. Since the energy spectrum cannot be given explicitly we have to solve the second order differential equation (3) numerically. Originally, we are going to calculate the energy levels numerically by using the MAPLE, which includes the confluent Heun function which is unavailable in MATHEMATICA. However, the variable z ≥ 1 makes the calculation difficult or impossible. We finally used the command “ParametricNDSolveValue” in MATHEMATICA to solve this differential equation (3). We denote the energy levels as ϵ_i(i ∈ [1, 7]) and list them in Table 1.

Fig. 2

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	Fig. 2 (Color online) The characteristics of the wave function as a function of the position y. We take A = 0.5, 1, 2, 4, and V0 = 5, a = 1.

Fig. 3

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	Fig. 3 (Color online) The characteristics of the wave function as a function of the position y. We take A = −1, −2, −3, −4, and V0 = 5, a = 1.

Fig. 4

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	Fig. 4 (Color online) Same as above case but A = 0.01, 0.03, 0.05, 0.08, and V0 = 5, a = 1.

Fig. 5

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	Fig. 5 (Color online) Same as above case but A = −0.01, −0.08, and V0 = 5, a = 1.

Table 1

Table 1

Table 1 Energy levels of the Schrödinger equation with potential (1). . A ϵ1 ϵ2 ϵ3 ϵ4 ϵ5 ϵ6 ϵ7 A = −4 135.209 156.03 177.639 200.002 223.089 246.873 271.333 A = −3 87.9451 104.217 121.234 138.957 157.357 176.413 196.124 A = −2 50.6605 62.3303 74.6793 87.6871 101.386 115.907 131.51 A = −1 23.3202 30.2943 46.9152 57.3065 69.5639 83.8798 100.339 A = −0.8 19.0409 25.0949 32.0881 40.454 50.5962 62.7581 77.0484 A = −0.5 13.375 18.1533 24.2696 32.1829 42.1418 54.2419 68.5156 A = −0.2 8.659 31 12.4182 18.042 25.7665 35.6567 47.7300 61.9921 A = 0.2 4.050 09 7.064 05 12.506 20.1651 30.0263 42.0844 56.3376 A = 0.5 2.020 63 4.987 46 10.4737 18.1513 28.0230 40.0875 54.3448 A = 0.8 1.241 69 4.587 76 10.238 17.9999 27.9157 40.0049 54.2771 A = 1.0 1.367 13 5.185 92 11.0603 18.936 28.9071 41.0245 55.312 A = 2.0 8.540 19 16.253 25.1475 35.1689 46.4578 59.2694 73.8672 A = 4.0 53.045 69.6912 87.3988 106.079 125.664 146.102 167.353

Table 1 Energy levels of the Schrödinger equation with potential (1). .

We find that the energy levels ϵ_i increase with the increasing |A| ≥ 1. However, it is seen that the ϵ_i increase with the parameter |A| for the parameter interval A ∈ (−1, 0), while they decrease with |A| for the parameter interval A ∈ (0, 1). As we know, the wave functions are given by ψ(z) = e^{m A(2z − 1)} H_c(α, β, γ, δ, η; z). Generally speaking, the wave functions require ψ(z) → 0 when z → ∞, i.e., x → ∞. Unfortunately, the present study is unlike our previous study,^[26–27] in which z → 1 when x goes to infinity. The energy spectra can be calculated by taking the limit z → 1 after the series expansion. In addition, on the contrary to the case discussed by Konwent,^[24] in which he supposed the A is taken positive, we also notice that the A can also be taken negative. In this case, the minimum value of single well will be lifted relatively. The wave functions relevant for the positive and negative A are plotted in Figs. 2 and 3.

We find that the wave functions are shrunk towards the origin with the increasing |A|. This makes the amplitude of the wave functions be increasing. For the small positive and negative A, however, the change of the wave functions is very sensitive to the parameter A, the wave functions are also shrunk towards the origin with the increasing |A| as shown in Figs. 4 and 5. In particular, the amplitude of wave function of the second excited state around the origin moves towards the origin when the parameter A decreases (see Fig. 4).

In this work we have studied the quantum system with the symmetric Konwent potential and shown how its exact solutions are found by transforming the original differential equation into a confluent type Heun differential equation. It is found that the solutions can be expressed by the confluent Heun function Hc (α, β, γ, δ, η; z), in which the energy levels are involved inside the parameter η. This makes us calculate the eigenvalues numerically in another way. The properties of the wave functions depending on the parameter A are illustrated graphically for given potential parameters V₀ and a. We have also noticed that the energy levels ϵ_i increase with the increasing potential parameter |A| ≥ 1 corresponding to a single well potential. Finally, we see that the ϵ_i increase with the increasing |A| for the parameter interval 0 > A > − 1, while they decrease with the increasing |A| for the parameter interval 0 < A < 1.