A new shape invariance form of the trigonometric Scarf potential: Two-parameter cross-additivity shape invariance

The supersymmetry (SUSY) has permeated almost all fields of Physics: atomic and molecular physics, nuclear physics, statistical physics, and condensed matter physics [1–3]. During the process of solving the Schrödinger equation with supersymmetric quantum mechanical methods (SUSYQM), the most critical task is to find the superpotential of the solvable potential and to derive the shape invariance of the partner potentials based on the superpotential [4–15]. These references have shown that the solvable potentials can be solved through SUSYQM, the superpotential and shape invariant relations have been given. However, after a careful study of the shape invariance relationship of the partner potential, we find that the shape invariance of the corresponding superpotential is almost determined by the additivity of one parameter, while the values of other parameters are fixed (without additivity), such as the trigonometric Scarf potential, the hyperbolic Scarf potential, the hyperbolic Pöschl-Teller potential, the trigonometric Pöschl-Teller potential, and so on [4–15]. This is not rigorous both in the physical and mathematical sense. Since other parameters can also exhibit additivity characteristics simultaneously, they can also make the partner potentials have other forms of shape invariance. With this purpose in mind, we studied the trigonometric Scarf potential and obtained the expected results. Of course, the relevant theory of shape invariance of two parameters is certainly more complex than the shape invariance theory of one parameter. This paper will focus on the shape invariance of the superpotential with two parameters based on SUSYQM.

Another important content in SUSYQM is the potential algebra [10]. Through the potential algebra method, we can further normalize the relevant parameters in the solvable potential. Moreover, we can also use the Lie algebra theory to conduct a deeper study [11]. Similarly, at present, the SUSYQM studied by potential algebra theory is still dominated by one parameter, and there is nearly no case involving two parameters. Therefore, it is also of great necessity to study the algebra theory of the potential under two parameters in SUSYQM. In this paper, we compare the shape invariance of the partner potential and research the potential algebra of two parameters based on one-parameter potential algebra. The relevant results we obtained are also consistent with each other.

Therefore, this paper mainly involves the following aspects: In the next two sections, introductions to SUSYQM and two-parameter potential algebra are given. In the fourth section, we take the trigonometric Scarf potential with two parameters as an example for further study. In the fifth section we find that there are four cases of shape invariance. And this includes a new additive characteristic called two-parameter cross-additivity shape invariance. In the sixth and seventh sections, based on this cross-additivity, we discuss in detail the eigenvalues and the eigenfunctions of the Hamiltonian with the trigonometric Scarf potential. In the eighth section, we use the two-parameter potential algebra theory to calculate the potential and also obtain the energy spectrum and eigenfunctions. Tha last section is devoted to conclusions and discussions.

Through analysis, we find that there are many potentials with two parameters that can be solved accurately at present, such as the trigonometric (hyperbolic) Scarf potential, the hyperbolic (trigonometric) Pöschl-Teller potential, and so on. At present, almost all of the references treat them as the one-parameter cases [12]. There are only one-parameter corresponding results in the energy spectrum and the eigenwave function. In this article, we give further analyses through taking the trigonometric Scarf potential as an example which has two parameters.

Two parameters in the superpotential mean that there are two independent parameters a and b in addition to the independent variable x, that is, $W=W(x, a, b)$. And the increasing and decreasing operators are

$$\begin{equation} A^{\pm}=A^{\pm}(x, a, b)=\pm \frac{\mathrm{d}}{\mathrm{d} x}+W(x, a, b). \end{equation} \tag{ 1 }$$

The partner Hamiltonians are

$$\begin{equation} H^{\pm}=H^{\pm}(x, a, b)=A^{\mp}(x, a, b) A^{\pm}(x, a, b) . \end{equation} \tag{ 2 }$$

The partner potentials are

$$\begin{equation} V_{+}\left(x, a_{0}, b_{0}\right)=V_{-}\left(x, a_{1}, b_{1}\right)+R\left(a_{0}, b_{0}\right)\!, \end{equation} \tag{ 3 }$$

where the parameters a₁ and b₁ are two functions of the parameters a₀ and b₀, i.e., $a_{1}=f\left(a_{0}, b_{0}\right)$ and $b_{1}=h\left(a_{0}, b_{0}\right) \cdot R\left(a_{0}, b_{0}\right)$ is a function of the parameters a₀ and $b_{0}. V_{\pm}(x, a, b)$ have the same dependence on the independent variable x. The differences between $V_{\pm}(x, a, b)$ are the different values of the parameters a and b. There is also a corresponding relationship for $H^{\pm}(x, a, b)$:

$$\begin{equation} H^{+}\left(x, a_{0}, b_{0}\right)+g\left(a_{0}, b_{0}\right)=H\left(x, a_{1}, b_{1}\right)+g\left(a_{1}, b_{1}\right)\!, \end{equation} \tag{ 4 }$$

where $g\left(a_{0}, b_{0}\right)$ is a function of the parameters a₀ and b₀, and $g\left(a_{1}, b_{1}\right)$ is a function of the parameters a₁, b₁. We can also get $R\left(a_{0}, b_{0}\right)=g\left(a_{1}, b_{1}\right)-g\left(a_{0}, b_{0}\right)$. According to eq. (4), because there is a constant difference between the two Hamiltonian functions, their eigenvalues have also the same constant difference, and both have the same set of eigenfunctions, that is, for all values of n, there are

$$\begin{equation} E_{n}^{+}\left(a_{0}, b_{0}\right)=E_{n}^{-}\left(a_{1}, b_{1}\right)+g\left(a_{1}, b_{1}\right)-g\left(a_{0}, b_{0}\right) \end{equation} \tag{ 5 }$$

and

$$\begin{equation} \psi_{n}^{+}\left(x, a_{0}, b_{0}\right)=\psi_{n}^{-}\left(x, a_{1}, b_{1}\right) \end{equation} \tag{ 6 }$$

When the SUSY is not broken spontaneously, the eigenvalue of the ground-state is zero, i.e., $E_{0}^{-}\left(a_{i}, b_{i}\right)=0(i=0,1,2,3, \ldots, n)$. Through the iterative rule, the eigenvalue of $H^{-}\left(x, a_{0}, b_{0}\right)$ is

$$\begin{equation} E_{n}^{+}\left(a_{0}, b_{0}\right)=g\left(a_{n}, b_{n}\right)-g\left(a_{0}, b_{0}\right)\!. \end{equation} \tag{ 7 }$$

The eigenfunction of $H^{-}\left(x, a_{0}, b_{0}\right)$ is

$$\begin{align} \psi_{n}^{-}\left(x, a_{0}, b_{0}\right)\,\sim\,&A^{+}\left(x, a_{0}, b_{0}\right) A^{+}\left(x, a_{1}, b_{1}\right)\cdots\nonumber\\ &A^{+}\left(x, a_{n-1}, b_{n-1}\right) \psi_{0}^{-}\left(x, a_{n}, b_{n}\right)\!, \end{align} \tag{ 8 }$$

where the ground-state eigenfunction $\psi_{0}^{-}\left(x, a_{n}, b_{n}\right)$ is

$$\begin{equation} \psi_{0}^{-}\left(x, a_{n}, b_{n}\right)=N e^{-\int_{x_{0}}^{x} W\left(x, a_{n}, b_{n}\right) \mathrm{d} x}\!. \end{equation} \tag{ 9 }$$

The above is the principle of shape invariance with two parameters.

Based on the shape invariance of eq. (3), we can also adopt the relevant basis of the Lie algebra to derive the potential algebra shape invariance under the condition of the superpotential with two parameters. For this purpose, the Casimir operator J₃ and the operators $J_{\pm}$ are introduced. Similar to the one-parameter form [13], here we argue that these operators also contain two parameters. These operators are defined as

$$\begin{align} J_{+}&=e^{i\left(s_{A} \phi_{A}+s_{B} \phi_{B}\right)} \mathcal{A}^{+},\quad J_{-}=\mathcal{A}^{-} e^{-i\left(s_{A} \phi_{A}+s_{B} \phi_{B}\right)}\!, \end{align} \tag{ 10 }$$

$$\begin{align} J_{3}&=J_{3}^{A}+J_{3}^{B}=\left(k_{A}-i \frac{\partial_{\phi_{A}}}{s_{A}}\right)+\left(k_{B}-i \frac{\partial_{\phi_{B}}}{s_{B}}\right)\!, \end{align} \tag{ 11 }$$

$$\begin{eqnarray} \begin{array}{rcl} \hspace{-0.8pc}F\left(J_{3}\right)&=&F\left(J_{3}^{A}, J_{3}^{B}\right)\\ &=&f\left(\chi\left(s_{A} k_{A}-s_{A} J_{3}^{A},s_{B} k_{B}-s_{B} J_{3}^{B}\right)\right)\!, \end{array} \end{eqnarray} \tag{ 12 }$$

where s_A and s_B are constants which reflect the additive step length of parameters a and b, k is an arbitrary constant, the function χ must satisfy the compatibility equation $\chi\left(i \partial_{\phi_{A}}+s_{A}, i \partial_{\phi_{B}}+s_{B}\right)=\eta\left(\chi\left(i \partial_{\phi_{A}}, i \partial_{\phi_{B}}\right)\right)$ in which $\eta\left(\chi\left(i \partial_{\phi_{A}}, i \partial_{\phi_{B}}\right)\right)$ is a function of $\chi\left(i \partial_{\phi_{A}}, i \partial_{\phi_{B}}\right), \phi_{A}$ and $\phi_{B}$ is an auxiliary variable, the operator $\mathcal{A}^{-}=\mathcal{A}^{-}\left(z, \chi\left(i \partial_{\phi_{A}}, i \partial_{\phi_{B}}\right)\right)$ is obtained from $A^{-}\left(x, a_{0}, b_{0}\right)$ by introducing an auxiliary variable $\phi_{A}$, and regarding $\phi_{B}$ independent of z and replacing the parameter $\left(a_{0}, b_{0}\right)$ with an operator $\chi\left(i \partial_{\phi A}, i \partial_{\phi B}\right)$,

$$\begin{equation} \left\{\begin{array}{l} x \rightarrow z, \\ \left(a_{0}, b_{0}\right) \rightarrow \chi\left(i \partial_{\phi_{A}}, i \partial_{\phi_{B}}\right)\!, \\ \left(a_{1}, b_{1}\right) \rightarrow \chi\left(i \partial_{\phi_{A}}+s_{A}, i \partial_{\phi_{B}}+s_{B}\right)\!, \\ A\left(x, a_{0}, b_{0}\right) \rightarrow \mathcal{A}\left(z, \chi\left(i \partial_{\phi_{A}}, i \partial_{\phi_{B}}\right)\right)\!, \end{array}\right. \end{equation} \tag{ 13 }$$

and $J_{\pm}$ have the characteristics of raising and lowering operators. And J₃ satisfies the following properties:

$$\begin{equation} e^{\pm i s \phi} J_{3} e^{\mp i s \phi}=J_{3} \pm s, \quad e^{\pm i s \varphi} J_{3}^{2} e^{\mp i s \varphi}=\left(J_{3} \pm s\right)^{2}\!. \end{equation} \tag{ 14 }$$

Since we consider that the parameters a and b, $\phi_{A}$ and $\phi_{B}$, s_A and s_B are independent of each other. For example, $e^{\pm i\left(s_{A} \phi_{A}+s_{B} \phi_{B}\right)} J_{3}^{A} e^{\mp i\left(s_{A} \phi_{A}+s_{B} \phi_{B}\right)}\,=J_{3}^{A} \pm S_{A}$. $=J_{3}^{B} \pm s_{B}$. And for further discussion, see refs. [16,17]. The commutations of $J_{+}, J_{-}$ and J₃ are satisfied with

$$\begin{equation} \left[J_{3}^{A}, J_{\pm}^{A}\right]=\pm J_{\pm}^{A},\left[J_{+}^{A}, J_{-}^{A}\right]=F\left(J_{3}^{A}\right)\!, \end{equation} \tag{ 15 }$$

$$\begin{equation} \left[J_{3}^{B}, J_{\pm}^{B}\right]=\pm J_{\pm}^{B},\left[J_{+}^{B}, J_{-}^{B}\right]=F\left(J_{3}^{B}\right)\!. \end{equation} \tag{ 16 }$$

The remaining commutation relations are equal to zero. These operators are explicitly checked,

$$\begin{align} &J_{+} J_{-}-J_{-} J_{+}=F\left(J_{3}^{A}, J_{3}^{B}\right) = \nonumber\\ &G\left(J_{3}^{A}, J_{3}^{B}\right)-G\left(J_{3}^{A}-s_{A}, J_{3}^{B}-s_{A}\right)\!, \end{align} \tag{ 17 }$$

where $G\left(J_{3}^{A}, J_{3}^{B}\right)$ is a function of $J_{3}^{A}$ and $J_{3}^{B}$. Suppose $\mid\!\!h\rangle$ as an arbitrary eigenstate of J₃, and $J_{\pm}$ play the role of raising and lowering operators, and then there are

$$\begin{eqnarray} J_{3} \mid\!\!h \rangle &= h\left(h_{A}, h_{B}\right) \mid\!\!h \rangle, \end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray} J_{-} \mid\!\!h \rangle &= a\left(h_{A}, h_{B}\right) \mid\!\!h \rangle, \end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray} J_{+} \mid\!\!h \rangle &= a\left(h_{A}+s_{A}, h_{B}+s_{B}\right) \mid\!\!h+s \rangle, \end{eqnarray} \tag{ 20 }$$

where $h\left(h_{A}, h_{B}\right)$ and $a\left(h_{A}, h_{B}\right)$ are functions of eigenvalues h_A and h_B . According to $\left[J_{+}, J_{-}\right] \mid\!\!h \rangle =F\left(J_{3}\right) \mid\!\!h \rangle$, we obtain

$$\begin{align} &a^{2}\left(h_{A}, h_{B}\right)-a^{2}\left(h_{A}+s_{A}, h_{B}+s_{B}\right)= \nonumber\\ &G\left(h_{A \min }-s_{A}, h_{B \min }-s_{B}\right)-G\left(h_{A \min }, h_{B \min }\right)\!. \end{align} \tag{ 21 }$$

If $h=h_{\min }$, then $J_{-} \mid h_{\min} \rangle =0$ and $a\left(h_{\min }\right)=0$. By repeating the iterative calculations, we can get

$$\begin{align} &a^{2}\left(h_{A \min }+n s_{A}, h_{B \min }+n s_{B}\right)= \nonumber\\ &G\left(h_{A \min }-s_{A}, h_{B \min }-s_{B}\right) \nonumber\\ &-G\left(h_{A \min }+n s_{A}-s_{A}, h_{B \min }+n s_{B}-s_{B}\right)\!, \end{align} \tag{ 22 }$$

where n is a positive integer. If $h_{A \min }+n s_{A}=h_{A}, h_{B \min }+n s_{B}=h_{B}$ and $\mathcal{H}$ is allowed to act on the state $\psi_{n}(x)$, the following relation can be obtained:

$$\begin{align} &\mathcal{H}_{-} \psi_{n}(x)=J_{+} J_{-} \psi_{n}(x)=E_{n}^{-} \psi_{n}(x)= \nonumber\\ &{\left[G\left(h_{A}-n s_{A}-s_{A}, h_{B}-n s_{B}-s_{B}\right)\right.} \nonumber\\ &\left.-G\left(h_{A}-s_{A}, h_{B}-s_{B}\right)\right] \psi_{n}(x). \end{align} \tag{ 23 }$$

We can calculate the energy spectrum through eq. (23)

The above is the principle of SUSYQM under the condition of superpotential with two parameters.

The trigonometric Scarf potential is an important potential in the field of quantum mechanics and solid-state physics [18–20]. So, it is of great significance to study and discuss this potential (including its inverse potential) [21]. Studies also involve Scarf potential plus other potentials, and even extend to D-dimensional conditions [22,23]. The solution of the Schrödinger equation with the trigonometric Scarf potential through the SUSYQM method has also been discussed in many references [24–28]. The following is a brief introduction to the solution of SUSYQM of the trigonometric Scarf potential. The superpotential of the trigonometric Scarf potential is

$$\begin{equation} W(x, A, B)=A \tan \alpha x-B \sec \alpha x, \end{equation} \tag{ 24 }$$

where $x \in(-\frac{\pi}{2 \alpha}, \frac{\pi}{2 \alpha}),B<A$. The partner potentials are

$$\begin{align} V_{\pm}(x, A, B)=&\left(A^{2}+B^{2} \pm \alpha A\right) \sec^{2} \alpha x \nonumber\\ &-B(2 A \pm \alpha) \alpha x \operatorname{coth} \alpha x-A^{2}\!. \end{align} \tag{ 25 }$$

The figures of the superpotential and partner potentials are shown in fig. 1.

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As fig. 1(a) shows that only when B < A, W(x) can have an intersection point with the x-axis. When B  > A, W(x) has no intersection point with the axis, we call the SUSY broken at this time [6], this article does not discuss such a situation.

Select $a_{0}=A$ and $a_{1}=A+\alpha$, we can obtain the shape invariance of $V_{\pm}(x, A, B)$, and $g\left(a_{0}\right)=a_{0}^{2}$. Furtherly, we have

$$\begin{align} E_{n}^{-}&=(A+n \alpha)^{2}-A^{2}\!, \end{align} \tag{ 26 }$$

$$\begin{align} \psi_{0}^{(-)}&=N e^{-\int^{(x)} A \tan \alpha x-B \sec \alpha x \mathrm{d} x}\!. \end{align} \tag{ 27 }$$

However, the superpotential of the trigonometric Scarf potential contains two parameters: A and B. It is incomplete to choose only A and its consistent additivity to obtain the shape invariance and establish the energy spectrum and the eigenfunction. We define the case with only one parameter to discuss as one-parameter shape invariance.

To make a comprehensive discussion of the shape invariance of the trigonometric Scarf potential, we need to consider the case of two parameters. The specific expression for the partner potentials in eq. (25) is as follows:

$$\begin{align} V_{+}\left(x, A_{0}, B_{0}\right)&=\left(A_{0}^{2}+B_{0}^{2}+A_{0} \alpha\right) \sec^{2} \alpha x \nonumber\\ &-B_{0}\left(2 A_{0}+\alpha\right) \tan \alpha x \sec \alpha x-A_{0}^{2}, \end{align} \tag{ 28 }$$

$$\begin{eqnarray} \begin{array}{rcl} V_{-}\left(x, A_{1}, B_{1}\right)&=&\left(A_{1}^{2}+B_{1}^{2}-A_{1} \alpha\right) \sec ^{2} \alpha x \\ &-&B_{1}\left(2 A_{1}-\alpha\right) \tan \alpha x \sec \alpha x-A_{1}^{2}. \end{array} \end{eqnarray} \tag{ 29 }$$

To satisfy the shape invariance in eq. (3), the coefficients of the variable x in eqs. (28) and (29) have to cancel each other. So the following needs to be satisfied:

$$\begin{equation} \left\{\begin{array}{l} A_{0}^{2}+B_{0}^{2}+A_{0} \alpha=A_{1}^{2}+B_{1}^{2}-A_{1} \alpha, \\ B_{0}\left(2 A_{0}+\alpha\right)=B_{1}\left(2 A_{1}-\alpha\right)\!. \end{array}\right. \end{equation} \tag{ 30 }$$

Through rigorous calculation, we can obtain four solutions that satisfy the shape invariance of eq. (3):

Case i): $A_{1}=-A_{0}, B_{1}=-B_{0}$.

Case ii): $A_{1}=A_{0}+\alpha, B_{1}=B_{0}$.

Case iii): $A_{1}=-B_{0}+\frac{1}{2} \alpha, B_{1}=-A_{0}-\frac{1}{2} \alpha$.

Case iv): $A_{1}=B_{0}+\frac{1}{2} \alpha, B_{1}=A_{0}+\frac{1}{2} \alpha$.

For Case i), the solution does not exhibit additivity properties, we do not discuss it here. In Case ii), only A changes additively, while the other parameter B remains unchanged, so the situation is essentially a single-parameter additive shape invariance. This is not what this paper will focus on, as this single-parameter additive shape invariance has been discussed in too many articles before. The discussion of it can be found in ref. [12]. In Case iii) and Case iv), both parameters A and B are changing. However, parameter A is increasing in Case iii), while parameter B is decreasing, that is, the additivity is more complex. We will study this case in our later research.

In Case iv), the parameters A and B not only exhibit additive changes, but also show staggered characteristics. Therefore, the study of this situation will be more complicated than the existing two-parameter pure additive change situation. This computationally rigorously derived a new shape invariance form of the trigonometric Scarf potential, which has never been discovered by supersymmetry researchers before. Therefore, our research team calls it as two-parameter cross-additivity shape invariance. Let us now turn to what exciting and different results this new shape invariance form can bring us from the previous one-parameter additive shape invariance.

According to the shape invariance requirements of eq. (3), and the additivity characteristic of the parameters $A_{1}=B_{0}+\frac{1}{2} \alpha$ and $B_{1}=A_{0}+\frac{1}{2} \alpha$. Although we say that they show pure additivity characteristics, it is not only simply additivity, but also crossover additivity. The increase of the latter A is always based on the former B. Similarly, the increasing of the latter B is always based on the former A. Using the iteration, we can derive recursive relations between the parameters that satisfy the shape invariant requirement which are shown in table 1.

Table 1:. The parameters expressions of the partner potentials $(k=0,1,2,3,\ldots).$

$\left(A_{n}, B_{n}\right)$	expressions
$\left(A_{0}, B_{0}\right)$	$\left(A_{0}, B_{0}\right)$
$\left(A_{1}, B_{1}\right)$	$\left(B_{0} + \dfrac{1}{2} \alpha, A_{0} + \dfrac{1}{2} \alpha\right)$
$\left(A_{2}, B_{2}\right)$	$\left(A_{0} + \alpha, B_{0} + \alpha\right)$
...	...
$\left(A_{2k}, B_{2k}\right)$	$\left(A_{0}+ \dfrac{n}{2} \alpha, B_{0} + \dfrac{n}{2} \alpha\right)$
$\left(A_{2k+1}, B_{2k+1}\right)$	$\left(B_{0} + \dfrac{n}{2} \alpha, A_{0} + \dfrac{n}{2} \alpha\right)$

From table 1, we find $A_{n}=B_{n-1}+\frac{1}{2} \alpha \operatorname{and} B_{n}=A_{n-1}+\frac{1}{2} \alpha$. As can be seen from the relationships listed above, we need to discuss the index partition parity.

1) When n is an even number, we set $n=2 k$,

$$\begin{align} &V_{-}\left(x, A_{2 k}, B_{2 k}\right)=\left(A_{2 k}^{2}+B_{2 k}^{2}-\alpha A_{2 k}\right) \operatorname{csch}^{2} \alpha x \nonumber\\ &-B_{2 k}\left(2 A_{2 k}-\alpha\right) \tan \alpha x \sec \alpha x-A_{2 k}^{2}, \end{align} \tag{ 31 }$$

$$\begin{align} &V_{+}\left(x, A_{2 k-1}, B_{2 k-1}\right)=-B_{2 k-1}\left(2 A_{2 k-1}+\alpha\right) \tan \alpha x \nonumber\\ &\times\sec \alpha x +\left(A_{2 k-1}^{2}+B_{2 k-1}^{2}+\alpha A_{2 k-1}\right) \sec^{2} \alpha x-A_{2 k-1}^{2}. \end{align} \tag{ 32 }$$

2) When n is an odd number, we set $n=2 k+1$,

$$\begin{align} &V_{-}\left(x, A_{2 k+1}, B_{2 k+1}\right)=\bigg[\left(B_{2 k}+\frac{\alpha}{2}\right)^{2}+\left(A_{2 k}+\frac{\alpha}{2}\right)^{2} \nonumber\\ &-\alpha\left(B_{2 k}+\frac{\alpha}{2}\right)\bigg] \sec^{2} \alpha x-\left(A_{2 k}+\frac{\alpha}{2}\right)\bigg[2 \left(B_{2 k}\right. \nonumber\\ &\left.+\frac{\alpha}{2}\right)-\alpha\bigg] \tan \alpha x \sec \alpha x-\left(B_{2 k}+\frac{\alpha}{2}\right)^{2}\!, \end{align} \tag{ 33 }$$

$$\begin{align} &V_{+}\left(x, A_{2 k}, B_{2 k}\right)=\left(A_{2 k}^{2}+B_{2 k}^{2}+\alpha A_{2 k}\right) \sec^{2} \alpha x \nonumber\\ &-B_{2 k}\left(2 A_{2 k}+\alpha\right) \tan \alpha x \sec \alpha x-A_{2 k}^{2} \end{align} \tag{ 34 }$$

The partner potentials are shown in fig. 2.

Note that the abscissa represents x.

As can be seen in fig. 2,

$$\begin{equation} \begin{cases} V_{+}\left(x, A_{2 k}, B_{2 k}\right)<V_{-}\left(x, A_{2 k+1}, B_{2 k+1}\right)\!,\\ V_{+}\left(x, A_{2 k+1}, B_{2 k+1}\right)>V_{-}\left(x, A_{2 k+2}, B_{2 k+2}\right), \end{cases} \end{equation} \tag{ 35 }$$

this means that $R\left(A_{2 k}, B_{2 k}\right)<0$, which is not allowed.While $R\left(A_{2 k+1}, B_{2 k+1}\right)$ is greater than 0, the correct energy level can be obtained. The partner potentials shown in fig. 2 inspire us to also discuss the parity of the energy level $E_{n}^{-}$.

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From eq. (7), we have

$$\begin{equation} E_{n}^{-}\left(A_{0}, B_{0}\right)=\sum_{k=1}^{n}\left[g\left(A_{k}, B_{k}\right)-g\left(A_{k-1}, B_{k-1}\right)\right]. \end{equation} \tag{ 36 }$$

Further, we can obtain

$$\begin{equation} E_{n}^{-}\left(A_{0}, B_{0}\right)= \begin{cases}\left(B_{0}+\dfrac{n}{2} \alpha\right)^{2}-A_{0}^{2} & (\mathit{n}=2 \mathit{k}+1), \\ \left(A_{0}+\dfrac{n}{2} \alpha\right)^{2}-A_{0}^{2} & (\mathit{n}=2 \mathit{k}). \end{cases} \end{equation} \tag{ 37 }$$

By analogy, we can conclude that when k is an even number

$$\begin{equation} E_{n}^{-}\left(A_{k}, B_{k}\right)= \begin{cases}\left(B_{k}+\dfrac{n}{2} \alpha\right)^{2}-A_{k}^{2} & (\mathit{n}=2 \mathit{k}+1), \\ \left(A_{k}+\dfrac{n}{2} \alpha\right)^{2}-A_{k}^{2} & (\mathit{n}=2 \mathit{k}),\end{cases} \end{equation} \tag{ 38 }$$

when k is an odd number,

$$\begin{equation} E_{n}^{-}\left(A_{k}, B_{k}\right)= \begin{cases}\left(B_{k}+\dfrac{n}{2} \alpha\right)^{2}-B_{k}^{2} & (\mathit{n}=2 \mathit{k}+1), \\ \left(A_{k}+\dfrac{n}{2} \alpha\right)^{2}-B_{k}^{2} & (\mathit{n}=2 \mathit{k}).\end{cases} \end{equation} \tag{ 39 }$$

The energy levels are shown in fig. 3.

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Based on fig. 3 and eqs. (37)–(39), our discussion can be divided into the following four cases.

i) When k is an even number and n is an even number, too, $E_{n}^{-}\left(A_{k}, B_{k}\right) \geq 0$.

ii) When k is an even number, n is an odd number and small enough, that is, to ensure that $g\left(A_{n+k}, B_{n+k}\right)<g\left(A_{k}, B_{k}\right), E_{n}^{-}\left(A_{k}, B_{k}\right)<0$. The wave function does not exist, namely, the wave function diverges. It is worth noting that, as can be seen from fig. 3, when k is an odd number, as k increases, the situation $g\left(A_{n+k}, B_{n+k}\right) \geq g\left(A_{k}, B_{k}\right)$ will definitely occur, which means $E_{n}^{-}\left(A_{k}, B_{k}\right) \geq 0$. The minimum value of n that can satisfy the above situation is $n=\left[A_{0}-B_{0}\right]$.

iii) When k is an odd number and n is an even number, $E_{n}^{-}\left(A_{k}, B_{k}\right) \geq 0$.

iv) When k is an odd number and n is an odd number, too, $E_{n}^{-}\left(A_{k}, B_{k}\right) \geq 0$. The summary of the energy levels are shown in table 2.

Table 2:. A summary of the energy levels. Note $k=0,1,2,3, \ldots,\, j=0,1,2,3, \ldots$ .

Exisiting	Non-exisiting
energy levels	energy levels
${E}_{2j}^{(-)}\left(x, A_{k}, B_{k}\right)$	None
${E}_{2j+1}^{(-)}\left(x, A_{2 k+1}, B_{2 k+1}\right)$	${E}_{2j+1}^{(-)}\left(x, A_{2 k}, B_{2 k}\right)$
$E_{2j+1}^{(-)}\left(x, A_{2 k}, B_{2 k}\right)$	$\left(n<\left[A_{0}-B_{0}\right]\right)$
$\left(n \geq\left[A_{0}-B_{0}\right]\right)$

For $E_{n}^{-}\left(A_{k}, B_{k}\right) \geq 0$, whether the eigenwave function exists will be further discussed below. And we also can deduce the ground-state wave functions

$$\begin{equation} \psi_{0}^{(-)}\left(A_{n}, B_{n}\right)=N_{0}^{\prime}(\cos \alpha x)^{\frac{A_{n}}{\alpha}}(\sec \alpha x+\tan \alpha x)^{\frac{B_{n}}{\alpha}}\!, \end{equation} \tag{ 40 }$$

where $N_{0}^{\prime}$ is the normalized constant. Because we have given A  > B in advance, i.e., A₀ > B₀, when $n=2 j\, (j=1,2, \cdots)$, the zero-energy ground-state wave functions $\psi_{0}^{(-)}\left(x, A_{n}, B_{n}\right)$ are normalizable. But, when $n=2 j-1\, (j=1,2, \cdots)$, the zero-energy ground-state wave functions $\psi_{0}^{(-)}\left(x, A_{n}, B_{n}\right)$ are unnormalizable. Based on eq. (8), the excited states of the trigonometric scarf potential can be obtained by using the rising operator $A^{+}(x, A, B)$. We will discuss this in detail.

According to table 1, the alternating additivity of the coefficients of the partner potentials can be summarized as

$$\begin{equation} \left(A_{n}, B_{n}\right)=\left\{\begin{array}{l} \left(B_{0}+\dfrac{n}{2} \alpha,A_{0}+\dfrac{n}{2} \alpha\right) \quad(\mathit{n}=2\mathit{k}+1),\\ \left(A_{0}+\dfrac{n}{2} \alpha,B_{0}+\dfrac{n}{2} \alpha\right)\quad(\mathit{n}=2\mathit{k}). \end{array}\right. \end{equation} \tag{ 41 }$$

There are two classes of the ground-state wave functions $\psi_{0}^{(-)}\left(A_{n}, B_{n}\right)$.

When $n=2 j\, (j=0,1,2, \ldots)$, if A₀ > B₀, i.e., A_n  > B_n , the ground-state wave functions $\psi_{0}^{(-)}\left(A_{n}, B_{n}\right)$ are normalizable. The first five zero-energy ground-state wave functions are shown in fig. 4(a).

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When $n=2 j-1\, (j=1,2, \ldots)$, if A₀ > B₀, i.e., A_n  < B_n , the ground-state wave functions $\psi_{0}^{(-)}\left(A_{n}, B_{n}\right)$ are unnormalizable.

The first excited state wave functions can be written as

$$\begin{align} & \psi_{1}^{-}\left(x, A_{n-1}, B_{n-1}\right) =\nonumber\\ & A^{+}\left(x, A_{n-1}, B_{n-1}\right) \psi_{0}^{(-)}\left(x, A_{n}, B_{n}\right)\!. \end{align} \tag{ 42 }$$

There are two classes of the first excited state wave functions $\psi_{1}^{(-)}\left(x, A_{n-1}, B_{n-1}\right)$.

Although the zero-energy ground-state wave functions $\psi_{0}^{(-)}\left(x, A_{n}, B_{n}\right)$ (where $n=2 j\,(j=1,2, \ldots))$ is normalized when A_n  > B_n is satisfied, $\psi_{1}^{(-)}\left(x, A_{n-1}, B_{n-11}\right)$ is not normalized. The reason is that $A^{+}\left(x, A_{n-1}, B_{n-1}\right)$ acts on $\psi_{0}^{(-)}\left(x, A_{n}, B_{n}\right)$, and $A_{n-1}$ and $B_{n-1}$ in $A^{+}\left(x, A_{n-1}, B_{n-1}\right)$ satisfy $A_{n-1}<B_{n-1}$,$\psi_{1}^{(-)}\left(x, A_{n-1}, B_{n-1}\right)$ is spontaneously broken.

Conversely, although the zero-energy ground-state wave functions $\psi_{0}^{(-)}\left(x, A_{n}, B_{n}\right)$ (where $n=2 j-1\,(j=1,2, \ldots))$ is not normalized when A_n  < B_n is satisfied, $\psi_{1}^{(-)}\left(x, A_{n-1}, B_{n-11}\right)$ is normalized. The reason is that $A^{+}\left(x, A_{n-1}, B_{n-1}\right)$ acts on $\psi_{0}^{(-)}\left(x, A_{n}, B_{n}\right)$, and $A_{n-1}$ and $B_{n-1}$ in $A^{+}\left(x, A_{n-1}, B_{n-1}\right)$ satisfy $A_{n-1}>B_{n-1}$. The first five normalizable wave functions in the first excited state are shown in fig. 4(b).

What about the second excited state? Whether its characteristics are similar to the ground state or the first excited state, let us explore further. The second excited state wave functions are

$$\begin{align} & \psi_{2}^{-}\left(x, A_{n-1}, B_{n-1}\right) =\nonumber\\ & A^{+}\left(x, A_{n-1}, B_{n-1}\right) \psi_{1}^{(-)}\left(x, A_{n}, B_{n}\right)\!. \end{align} \tag{ 43 }$$

There are two classes of the second excited state wave functions $\psi_{2}^{(-)}\left(A_{n-1}, B_{n-1}\right)$.

i) When A_n  < B_n is satisfied, the first excited state wave functions $\psi_{1}^{(-)}\left(x, A_{n}, B_{n}\right)$ (where $n=2 j-1\, (j=1,2, \ldots))$ are unnormalizable, functions $\psi_{2}^{(-)}\left(x, A_{n-1}, B_{n-11}\right)$ are normalizable.

ii) Conversely, although the first excited state wave functions $\psi_{1}^{(-)}\left(x, A_{n}, B_{n}\right)$ (where $n=2 j\, (j=1,2, \ldots))$ is normalizable when A_n  > B_n is satisfied, $\psi_{1}^{(-)}\left(x, A_{n-1}, B_{n-11}\right)$ is unnormalizable.

At this point, this indicates that the cross-additive change of two parameters will lead to a part of the wave function breaking, and which part of the wave function breaking should also be discussed in terms of odd and even energy levels. The m-th excited state wave functions

$$\begin{align} & \psi_{m}^{-}\left(x, A_{n-1}, B_{n-1}\right)= \nonumber\\ & A^{+}\left(x, A_{n-1}, B_{n-1}\right) \psi_{m-1}^{(-)}\left(x, A_{n}, B_{n}\right)\!. \end{align} \tag{ 44 }$$

There are two classes, too.

i) When m is an even number, if the parameters satisfy A_n  < B_n , although the $(\left.m-1\right){\text{-th}}$ excited state wave functions $\psi_{m-1}^{(-)}\left(x, A_{n}, B_{n}\right)$, where $n=2 j-1\, (j=1,2, \ldots)$ are unnormalizable $\psi_{m}^{(-)}\left(x, A_{n-1}, B_{n-11}\right)$ are normalizable.

ii) When m is an odd number, if the parameters satisfy A_n  < B_n , although the $(m-1){\text{-th}}$ excited state wave functions $\psi_{m-1}^{(-)}\left(x, A_{n}, B_{n}\right)$ (where $n=2 j-1\, (j=1,2, \ldots))$ are normalizable, $\psi_{m}^{(-)}\left(x, A_{n-1}, B_{n-1}\right)$ are unnormalizable. A summary of the ground-state and the excited state is shown in table 3.

Table 3:. A summary of the ground-state and the excited state. Note$k=0,1,2,3, \ldots, j=0,1,2,3, \ldots$.

Normalizable	Non-mmormalizable
state	state
$\psi_{2 j}^{(-)}\left(x, A_{2 k}, B_{2 k}\right)$	$\psi_{2 j}^{(-)}\left(x, A_{2 k+1}, B_{2 k+1}\right)$
$\psi_{2j+1}^{(-)}\left(x, A_{2 k+1}, B_{2 k+1}\right)$	$\psi_{2j+1}^{(-)}\left(x, A_{2 k}, B_{2 k}\right)$

For $A_{1}=B_{0}+\frac{1}{2} \alpha, B_{1}=A_{0}+\frac{1}{2} \alpha$, we can define operators $J_{3}, J_{+}$ and $J_{-}(J_{3}$ is a Casimir operator):

$$\begin{equation} \begin{aligned} &J_{+}=e^{i\left(s_{A} \phi_{A}+s_{B} \phi_{B}\right)}\left(-\frac{\mathrm{d}}{\mathrm{d} z}+W\left(z, \chi\left(i \partial_{\phi_{A}}, i \partial_{\phi_{B}}\right)\right)\right) \\ &\rightarrow e^{i\left(s_{A} \phi_{A}+s_{B} \phi_{B}\right)}\left(-\frac{\mathrm{d}}{\mathrm{d} z}+i \partial_{\phi_{A}} \tan \alpha z-i \partial_{\phi_{B}} \sec \alpha z\right)\!, \end{aligned} \end{equation} \tag{ 45 }$$

$$\begin{equation} \begin{aligned} &J_{-}=\left(\frac{\mathrm{d}}{\mathrm{d} z}+W\left(z, \chi\left(i \partial_{\phi_{A}}, i \partial_{\phi_{B}}\right)\right)\right) e^{-i\left(s_{A} \phi_{A}+s_{B} \phi_{B}\right)} \\ &\rightarrow\left(\frac{\mathrm{d}}{\mathrm{d} z}+i \partial_{\phi_{A}} \tan \alpha z-i \partial_{\phi_{B}} \sec \alpha z\right) e^{-i\left(s_{A} \phi_{A}+s_{B} \phi_{B}\right)}\!. \end{aligned} \end{equation} \tag{ 46 }$$

Since $J_{+} J_{-}$ embodies the additive characteristic of parameter interchanging, operators must also reflect the additive characteristic of such interchanging, i.e., $i \partial_{\phi_{A}}=s_{B}\left(k_{B}-J_{3}^{B}\right)$ and $i \partial_{\phi_{B}}=s_{A}\left(k_{A}-J_{3}^{A}\right)$. Therefore, for the calculation of the first excited state energy spectrum with the potential algebra method, we need to interchange the coefficients in $J_{+} J_{-}$ to reflect additivity, while we need not interchange the coefficients in $J_{-} J_{+}$. According to eqs. (45) and (46), we have

$$\begin{equation} s_{A} \alpha=2 s_{A}^{3}, s_{B} \alpha=2 s_{B}^{3} \text { and } s_{B}^{2} \alpha=s_{A}^{4}+s_{B}^{4}. \end{equation} \tag{ 47 }$$

We can obtain $s_{A}^{2}=s_{B}^{2}=\frac{\alpha}{2}$. Further, we have

$$\begin{equation} J_{+} J_{-}-J_{-} J_{+}=s_{B}^{2}\left(k_{B}-J_{3}^{B}+s_{B}\right)^{2}-s_{A}^{2}\left(k_{A}-J_{3}^{A}\right)^{2}\!. \end{equation} \tag{ 48 }$$

When calculating the energy spectrum of the second excited state with the potential algebra method, the coefficients in $J_{+} J_{-}$ need to interchange again based on the first excited state to reflect the corresponding additivity characteristics. However, the coefficients in $J_{-} J_{+}$ remain the same as those in the first excited state and need not interchange to correspond to each other, so we still have

$$\begin{equation} s_{A} \alpha=2 s_{A}^{3}, s_{B} \alpha=2 s_{B}^{3} \text { and } s_{B}^{2} \alpha=s_{A}^{4}+s_{B}^{4}. \end{equation} \tag{ 49 }$$

From eq. (23), we know

$$\begin{align} E_{n}^{-}=\,&G\left(h_{A}-n s_{A}-s_{A}, h_{B}-n s_{B}-s_{B}\right) \nonumber\\ &-G\left(h_{A}-s_{A}, h_{B}-s_{B}\right)\!. \end{align} \tag{ 50 }$$

If we want to determine the energy spectrum, it is obvious that we also need to consider the characteristics of cross-correspondence between two coefficients. Therefore, we have

$$\begin{align} &G\left(h_{A}-n s_{A}-s_{A}, h_{B}-n s_{B}-s_{B}\right)= \nonumber\\ &\begin{array}{rcl} s_{A}^{2}\left(k_{A}-\left(h_{A}-n s_{A}-s_{A}\right)\right)^{2} \quad (n \text { is even}),\\ s_{B}^{2}\left(k_{B}-\left(h_{B}-n s_{B}-s_{B}\right)\right)^{2} \quad (n \text { is odd}), \end{array} \end{align} \tag{ 51 }$$

$$\begin{equation} G\left(h_{A}-s_{A}, h_{B}-s_{B}\right)=s_{A}^{2}\left(k_{A}-\left(h_{A}-s_{A}\right)\right)^{2}\!. \end{equation} \tag{ 52 }$$

Set

$$\begin{equation*}A_{k}=s_{A}\left[k_{A}-\left(h_{A}-s_{A}\right)\right], \quad B_{k}=s_{B}\left[k_{B}-\left(h_{B}-s_{B}\right)\right],\end{equation*}$$

and we get

$$\begin{align} &G\left(h_{A}-n-1, h_{B}-n-1\right)= \nonumber\\ &\begin{cases}\left(A_{k}+n s_{A}^{2}\right)^{2}=\left(A_{k}+\dfrac{\alpha}{2} n\right)^{2} & (n \text { is even}),\\ \left(B_{k}+n s_{B}^{2}\right)^{2}=\left(B_{k}+\dfrac{\alpha}{2} n\right)^{2} & (n \text { is odd}).\end{cases} \end{align} \tag{ 53 }$$

This result is the same as eqs. (37)–(39). This shows that the shape invariance of the partner potential derived in this paper is consistent with the shape invariance of the potential algebra, and the theory is self-consistent.

The shape invariant potential obtained by changing the two parameters at the same time brings new physical features to the wave function and energy spectrum, which is by no means a repeated study of the original super potential. In this paper, we take the Scarf potential as an example, deriving the shape invariance of the partner potentials of the two-parameter superpotential. The problems related to shape invariance are deeply studied. Our results exceed the theoretical expectation of two-parameter shape invariance: the parameters show cross-additive characteristics. For the shape invariance of the partner potentials of the trigonometric Scarf potential, we focus on the shape invariance with two parameters, and derive the energy eigenvalues and the eigenwave functions. It is worth noting that in the shape invariance of partner potentials, the two-parameter's additivity is alternating. These additive characteristics also bring new features to the eigenwave functions. These have been discussed in detail in the paper. Further, we study the potential algebraic shape invariance of the partner potentials of the trigonometric Scarf potential, which also focuses on the algebraic shape invariance of the potential with two parameters, and derive the same energy eigenvalues as for the partner potentials. The analysis of the example also shows that many of the solvable two-parameter superpotentials involved may not be comprehensive enough, and the shape invariance of the solvable potential in many references [4,12,16,17] needs further investigation. It also shows that the Schrödinger equation with two-parameter superpotential requires further investigation. According to the recent results of our study group, a further study of the shape invariances of the two parameters is well expected.

Data availiability statement: All data that support the findings of this study are included within the article (and any supplementary files).