Fredholm Determinant and Wronskian Representations of the Solutions to the Schrödinger Equation with a KdV-Potential

Abstract

From the finite gap solutions of the KdV equation expressed in terms of abelian functions we construct solutions to the Schrödinger equation with a KdV potential in terms of fourfold Fredholm determinants. For this we establish a connection between Riemann theta functions and Fredholm determinants and we obtain multi-parametric solutions to this equation. As a consequence, a double Wronskian representation of the solutions to this equation is constructed. We also give quasi-rational solutions to this Schrödinger equation with rational KdV potentials.

1. Introduction

We consider the KdV equation

$$u_t=6u{u}_x-u_{xxx},$$

(1)

where, as usual, the subscripts x and t denote partial derivatives.

This was introduced by Korteweg and de Vries [1] for the first time in 1895. The equation describes in particular the propagation of waves with weak dispersion in various nonlinear media.

A lot of works have been devoted to this equation, starting from the celebrated paper by Gardner et al. [2] who proposed a method for solving the Equation (1) in 1967. Four years later, Zakharov and Faddeev in [3] proved that this equation is a completely integrable system, while Hirota constructed solutions by using the bilinear method [4]. In 1975, Its and Matveev expressed these solutions in terms of Riemann theta functions [5] in 1975, and in the same year, Lax gave the expressions of periodic and almost periodic solutions [6]. We can also mention, for example, Airault et al. in 1977 [7], Adler and Moser in 1978 [8], Ablowitz and Cornille in 1979 [9], Freeman and Nimmo in 1984 [10], Matveev in 1992 [11], Ma in 2004 [12], and Kovalyov in 2005 [13].

In this paper, we consider the algebro-geometric approach given by Its and Matveev in 1975. We degenerate the solutions to this KdV equation given in terms of Riemann theta functions as in [14] to obtain solutions in terms of Fredholm determinants. We consider solutions to the Schrödinger equation defined by

$$\begin{array}{c}\hfill -\psi {(x,t)}^{″}+u(x,t)\psi (x,t)=z^2\psi (x,t)\end{array}$$

(2)

with $u(x,t)$ as the solution to the KdV Equation (1). We also degenerate the solutions of this Equation (2) given by Riemann theta functions to obtain solutions expressed in terms of Fredholm determinants. Some examples are given for the first orders.

We deduce solutions of the Schrödinger equation with a KdV potential in terms of Wronskians. We constuct some examples of these solutions.

By performing a limit when one parameter tends to 0, we construct quasi-rational solutions to the Schrödinger Equation (2). Some examples are proposed for the first orders.

2. The Schrödinger Equation and Its Solutions in Terms of Fredholm Determinants with a KdV Potential

2.1. Solutions to the KdV Equation

Let $\mathsf{\Gamma }$ be the Riemann surface of the algebraic curve defined by

$${\omega }^2=\prod _{j=1}^{2g+1}(z-E_j),$$

where $E_j\ne E_k$, for $j\ne k$. Let D be some divisor $D={\sum }_{j=1}^g{P}_j$, $P_j\in \mathsf{\Gamma }$. The finite gap solution of the KdV equation

$$u_t=6u{u}_x-u_{xxx}$$

(3)

can be given in the form [5]

$$u(x,t)=-2{\partial }_x^2\left[ln\theta (xg+tv+l)\right]+C.$$

(4)

In (4), $\theta$ is the Riemann function defined by

$$\theta \left(z\right)=\sum _{k\in {\mathbf{Z}}^g}exp\{\pi i\left(\mathcal{B}k\right|k)+2\pi i\left(k\right|z)\}$$

(5)

and constructed from the matrix $\mathcal{B}$ of the B-periods of the surface $\mathsf{\Gamma }$.

The vectors g, v, and l are defined by

$$g_j=2i{c}_{j1},$$

(6)

$$v_j=8i({\displaystyle \frac{c_{j1}}{2}}\sum _{k=1}^{2g+1}{E}_k+c_{j2}),$$

(7)

$$l_j=-\sum _{k=1}^g{\int }_{\infty }^{P_k}d{U}_j+{\displaystyle \frac{j}{2}}-{\displaystyle \frac{1}{2}}\sum _{k=1}^g{\mathcal{B}}_{kj},$$

(8)

$$C=\sum _{k=1}^{2g+1}{E}_k-2\sum _{k=1}^g{\int }_{a_k}zd{U}_k.$$

(9)

The coefficients $c_{jk}$ are related with abelian differential $d{U}_j$ by

$$d{U}_j={\displaystyle \frac{{\sum }_{k=1}^g{c}_{jk}{z}^{g-k}}{\sqrt{{\prod }_{k=1}^{2g+1}(z-E_k})}}dz,$$

(10)

and can be obtained by solving the system of linear equations

$${\int }_{a_k}d{U}_j={\delta }_{jk},1\le j\le g,1\le k\le g.$$

In [14], we have made the degeneracy of these solutions following the ideas first presented by Its and Matveev, given for example.

For real $E_j$, such that $E_m<E_j$ if $m<j$, we have evaluated the limits of all objects in Formula (4); when $E_{2m}$, $E_{2m+1}$ tends to $-{\kappa }_m^2$, with ${\kappa }_m>0$, for $1\le m\le g$, and $E_1$ tends to 0.

In [14], we have given different representations in terms of Fredholm determinants; in particular, we obtained the following representation of the solutions to the KdV equation.

Theorem 1.

The function u defined by

$$\begin{array}{c}\hfill u(x,t)=-2{\partial }_x^{2}ln(det(I+B)),\end{array}$$

(11)

with B the matrix defined by $B={\left(b_{jk}\right)}_{1\le j,k\le m}$

$$\begin{array}{c}\hfill b_{jk}={(-1)}^{j}exp\left[2({\kappa }_{j}x-4{\kappa }_j^{3}t+{\kappa }_j{k}_j)\right]\prod _{l\ne j}\left|{\displaystyle \frac{{\kappa }_l+{\kappa }_k}{{\kappa }_l-{\kappa }_j}}\right|,\end{array}$$

(12)

and the ${\kappa }_j$, $k_j$ arbitrary real parameters, offer a solution to the KdV Equation (1).

Proof. 

We recall the proof given in [14].

We evaluate all the limits of the expressions in the formula when $E_{2m}$; $E_{2m+1}$ tends to $-{\alpha }_m$, $-{\alpha }_m=-{\kappa }_m^2$, ${\kappa }_m>0$, for $1\le m\le g$.

• Limit of $P\left(z\right)={\prod }_{j=1}^{2g+1}(z-E_j)$

The limit of $P\left(z\right)={\prod }_{j=1}^{2g+1}(z-E_j)$ is equal to $\tilde{P}\left(z\right)=z{\prod }_{j=1}^g{(z+{\alpha }_j)}^2$

• Limit of $d{U}_m={\displaystyle \frac{{\sum }_{k=1}^g{c}_{mk}{z}^{g-k}}{\sqrt{{\prod }_{k=1}^{2g+1}(z-E_k})}}dz$

The limit of $d{U}_m$ is equal to $\widetilde{d{U}_m}={\displaystyle \frac{{\phi }_m\left(z\right)}{\sqrt{z}{\prod }_{j=1}^g(z+{\alpha }_j)}}dz$, where ${\phi }_m\left(z\right)={\sum }_{k=1}^g{\tilde{c}}_{mk}{z}^{g-k}$. In the limit, the normalisation condition can be written as

$${\int }_{a_k}d{U}_j\to {\displaystyle \frac{2\pi i{\phi }_j(-{\alpha }_k)}{-i{\kappa }_k{\prod }_{m\ne k}(-{\alpha }_k+{\alpha }_m)}}={\delta }_{kj},$$

(13)

which proves that $-{\alpha }_m$, $m\ne k$ are the zeros of the polynomials ${\phi }_k\left(z\right)$, and so ${\phi }_k\left(z\right)$ can be expressed as ${\phi }_k\left(z\right)={\tilde{c}}_{k1}{\prod }_{m\ne k}(z+{\alpha }_m)$. Through (13), in the limit, we obtain

$${\tilde{c}}_{k1}=-{\displaystyle \frac{{\kappa }_k}{2\pi i}}.$$

So

$$d{\tilde{U}}_k=-{\displaystyle \frac{{\kappa }_k}{2\pi i\sqrt{z}(z+{\alpha }_k)}}dz$$

• Limit of $v_k$ and $g_k$

We identify the powers of $z^{g-2}$ in (14) to obtain

$$\widetilde{{\phi }_k}\left(z\right)=\widetilde{c_{k1}}\prod _{l\ne k}(z+{\alpha }_l)=\sum _{j=1}^g\widetilde{c_{kj}}{z}^{g-j},$$

(14)

In the limit, we obtain

$${\tilde{c}}_{k1}\sum _{l=1}^g{\alpha }_l={\tilde{c}}_{k2}.$$

The limit values of $v_k$ and $g_k$ can be expressed as

$${\tilde{v}}_k={\displaystyle \frac{4i{\kappa }_k^3}{\pi }}$$

and

$${\tilde{g}}_k={\displaystyle \frac{-i{\kappa }_k}{\pi }}.$$

• Limit of $U_k\left(P\right)$ and ${\mathcal{B}}_{mk}$

For ${\lambda }_0=-{\alpha }_m=-{\kappa }_m^2$, $I={\int }_{{\lambda }_0}^{0}d{U}_k\to {\displaystyle \frac{1}{2}}{\tilde{\mathcal{B}}}_{mk}$. We evaluate the integral I along the real axis on the upper sheet of surface $\mathsf{\Gamma }$ and we obtain

$$I\to {\displaystyle \frac{i}{2\pi }}ln\left|{\displaystyle \frac{{\kappa }_m+{\kappa }_k}{{\kappa }_m-{\kappa }_k}}\right|.$$

We can deduce the limit values of the matrix $\mathcal{B}$:

$${\tilde{\mathcal{B}}}_{mk}={\displaystyle \frac{i}{\pi }}ln\left|{\displaystyle \frac{{\kappa }_m+{\kappa }_k}{{\kappa }_m-{\kappa }_k}}\right|.$$

So $i{\mathcal{B}}_{kk}$ tends to $-\infty$ and we have

$$\begin{array}{c}\hfill {\int }_{\infty }^{P}d{U}_j\to -{\displaystyle \frac{i}{2\pi }}ln\left|{\displaystyle \frac{{\kappa }_j-\sqrt{z_P}}{{\kappa }_j+\sqrt{z_P}}}\right|.\end{array}$$

(15)

• • Limit of argument of exponential in $\theta \left(p\right)$

We denote $A_0$ the argument of exponential in $\theta \left(p\right)={\sum }_{k\in {\mathbf{Z}}^g}exp\{\pi i\left(\mathcal{B}k\right|k)+2\pi i\left(k\right|p)\}$. $A_0$ can be expressed as

$$\begin{array}{c}\hfill A_0=\pi i\sum _{j=1}^g{\mathcal{B}}_{jj}{k}_j(k_j-1)+2\pi i\sum _{j>m}{\mathcal{B}}_{mj}{k}_m{k}_j+\sum _{j=1}^g\pi i(2p_j+{\mathcal{B}}_{jj})k_j.\end{array}$$

(16)

Using the relation $k_j(k_j-1)\ge 0$ for all $k\in {\mathbf{Z}}^g$ and that $i{\mathcal{B}}_{kk}$ tends to $-\infty$, we can express the limit $\tilde{\theta }$ of $\theta \left(p\right)$ to a finite sum only over vectors $k\in {\mathbf{Z}}^g$ such that each $k_j$ must be equal to 0 or 1.

Denoting A the argument of $\theta (xg+tv+l)$, this can be expressed as

$$A=\pi i\sum _{j=1}^g{\mathcal{B}}_{jj}{k}_j(k_j-1)+2\pi i\sum _{j>m}{\mathcal{B}}_{mj}{k}_m{k}_j+\sum _{j=1}^g{k}_j[2\pi i(g_{j}x+v_{j}t)$$

$$-\pi i(-j+2\sum _{k=1}^g{\int }_{\infty }^{P_k}d{U}_j+\sum _{m\ne j}{\mathcal{B}}_{mj})].$$

It can be rewritten as

$$A=\pi i\sum _{j=1}^g{\mathcal{B}}_{jj}{k}_j(k_j-1)+2\pi i\sum _{j>m}{\mathcal{B}}_{mj}{k}_m{k}_j+\sum _{j=1}^g{k}_j{Q}_j,$$

with

$$Q_j=2\pi i(g_{j}x+v_{j}t)+{\beta }_j$$

and

$${\beta }_j=-\pi i(-j+2\sum _{k=1}^g{\int }_{\infty }^{P_k}d{U}_j+\sum _{m\ne j}{\mathcal{B}}_{mj}).$$

The quantity ${\beta }_j$ has a finite limit value ${\tilde{\beta }}_j$ which is independent from x and t.

• Limit of $\theta (xg+vt+l)$

From the relation $k_j(k_j-1)\ge 0$ for all $k\in {\mathbf{Z}}^{\mathbf{g}}$ and the fact that $i{\mathcal{B}}_{kk}$ tends to $-\infty$, we can deduce that the limit $\tilde{\theta }$ of $\theta (xg+tv+l)$ reduces to a finite sum only over vectors $k\in {\mathbf{Z}}^{\mathbf{g}}$ because each $k_j$ must be equal to 0 or 1.

$$\tilde{\theta }=\sum _{k\in {\mathbf{Z}}^g,k_j=0or1}exp\{\sum _{m>j}2ln\left|{\displaystyle \frac{{\kappa }_m-{\kappa }_j}{{\kappa }_m+{\kappa }_j}}\right|k_m{k}_j$$

$$+(\sum _{j=1}^{g}2{\kappa }_{j}x-8{\kappa }_j^{3}t+2{\kappa }_j{x}_j+\pi ji+\sum _{m\ne j}ln\left|{\displaystyle \frac{{\kappa }_m+{\kappa }_j}{{\kappa }_m-{\kappa }_j}}\right|)k_j\},$$

with

$$x_j={\displaystyle \frac{1}{2{\kappa }_j}}\sum _{k=1}^{g}ln\left|{\displaystyle \frac{\sqrt{z_k}-i{\kappa }_j}{\sqrt{z_k}+i{\kappa }_j}}\right|.$$

This can be expressed as

$$\begin{array}{c}\hfill \tilde{\theta }=\sum _{J\subset \{1,\dots ,g\}}\prod _{j\in J}{(-1)}^j\prod _{j\in Jk\notin J}\left|{\displaystyle \frac{{\kappa }_j+{\kappa }_k}{{\kappa }_j-{\kappa }_k}}\right|exp\sum _{j\in J}2({\kappa }_{j}x-4{\kappa }_j^{3}t+{\kappa }_j{x}_j).\end{array}$$

(17)

• Limit of the coefficient C

C is defined in (9) by

$$C=\sum _{k=1}^{2g+1}{E}_k-2\sum _{k=1}^g{\int }_{a_k}zd{U}_k=C_1+C_2,$$

It can be computed in the following way.

$$C_2=-2\sum _{k=1}^g{\int }_{a_k}zd{U}_k=-2\sum _{k=1}^g{\int }_{a_k}{\displaystyle \frac{-{\kappa }_{k}zdz}{2\pi \sqrt{z}(z+{\alpha }_k)}}=\sum _{k=1}^g{\displaystyle \frac{{\kappa }_k}{\pi }}{\int }_{a_k}{\displaystyle \frac{\sqrt{z}dz}{(z+{\alpha }_k)}}$$

$$=\sum _{k=1}^g{\displaystyle \frac{{\kappa }_k}{\pi }}2i\pi (-i{\kappa }_k)=\sum _{k=1}^{g}2{\kappa }_k^2=2\sum _{k=1}^g{\alpha }_k.$$

So when the gaps tend to points, the coefficient C tends to

$$\tilde{C}=2\sum _{k=1}^g-{\alpha }_k+2\sum _{k=1}^g{\alpha }_k=0.$$

• Degenerate solution to the KdV equation

So we obtain the degenerate solution to the KdV equation. The function u defined by

$$\begin{array}{c}\hfill u(x,t)=-2{\partial }_x^{2}ln\left(\sum _{J\subset \{1,\dots ,g\}}\prod _{j\in J}{(-1)}^j\prod _{j\in Jk\notin J}\left|{\displaystyle \frac{{\kappa }_j+{\kappa }_k}{{\kappa }_j-{\kappa }_k}}\right|exp\left(\sum _{j\in J}2({\kappa }_{j}x-4{\kappa }_j^{3}t+{\kappa }_j{x}_j)\right)\right),\end{array}$$

(18)

with ${\kappa }_j$ and $x_j$ as arbitrary real parameters, which is a solution to the KdV Equation (1).

• We evaluate $det(I+B)$ with $B={\left(b_{jk}\right)}_{1\le j,k\le m}$ in the following matrix:

  $$b_{jk}={(-1)}^{j}exp\left[2({\kappa }_{j}x-4{\kappa }_j^{3}t+{\kappa }_j{k}_j)\right]\prod _{l\ne j}\left|{\displaystyle \frac{{\kappa }_l+{\kappa }_k}{{\kappa }_l-{\kappa }_j}}\right|$$

  with $x_j$ being an arbitrary parameter.

$det(I+B)$ can be expressd as

$$\begin{array}{c}\hfill det(I+B)=\sum _{J\subset \{1,\dots ,N\}}exp\left(\sum _{j\in J}2({\kappa }_{j}x-4{\kappa }_j^{3}t+{\kappa }_j{x}_j)\right)\prod _{j\in J}{(-1)}^j\prod _{j\in Jk\notin J}\left|{\displaystyle \frac{{\kappa }_j+{\kappa }_k}{{\kappa }_j-{\kappa }_k}}\right|,\end{array}$$

(19)

On the other hand

$$\begin{array}{c}\hfill \tilde{\theta }=\sum _{J\subset \{1,\dots ,g\}}\prod _{j\in J}{(-1)}^j\prod _{j\in Jk\notin J}\left|{\displaystyle \frac{{\kappa }_j+{\kappa }_k}{{\kappa }_j-{\kappa }_k}}\right|exp\left(\sum _{j\in J}(2({\kappa }_{j}x-4{\kappa }_j^{3}t+{\kappa }_j{x}_j)\right).\end{array}$$

(20)

If we compare (19) to (20), we obtain the equality with $g=N$

$$\begin{array}{c}\hfill \tilde{\theta }=det(I+B).\end{array}$$

(21)

□

2.2. Solutions to the Schrödinger Equation with a KdV Potential

It is well known that if $u(x,t)$ is a solution to the KdV Equation (1) given by $u(x,t)=-2{\partial }_x^{2}ln\theta (xg+tv+l)+C$, then the function defined by

$$\begin{array}{c}\hfill \psi (x,t,P)={\displaystyle \frac{\theta \left(U\right(P)+xg+tv+l)\theta (tv+l)}{\theta \left(U\right(P)+tv+l)\theta (xg+tv+l)}}{e}^{i\omega \left(P\right)x},\end{array}$$

(22)

is a solution to the Schrödinger Equation (2), with $\omega \left(P\right)=\sqrt{z}$, and $U\left(P\right)={\int }_{\infty }^{P}d{U}_k$.

We obtain the following result.

Theorem 2.

Let A, B, C, D be the matrices defined by

$$\begin{array}{c}\hfill \begin{array}{c}{a}_{jk}={(-1)}^j\left({\displaystyle \frac{z+i{\kappa }_j}{z-i{\kappa }_j}}\right)exp\left[2({\kappa }_{j}x-4{\kappa }_j^{3}t+{\kappa }_j{k}_j)\right]{\prod }_{l\ne j}\left|{\displaystyle \frac{{\kappa }_l+{\kappa }_k}{{\kappa }_l-{\kappa }_j}}\right|,\hfill \\ b_{jk}={(-1)}^{j}exp\left[2({\kappa }_{j}x-4{\kappa }_j^{3}t+{\kappa }_j{k}_j)\right]{\prod }_{l\ne j}\left|{\displaystyle \frac{{\kappa }_l+{\kappa }_k}{{\kappa }_l-{\kappa }_j}}\right|,\hfill \\ c_{jk}={(-1)}^j\left({\displaystyle \frac{z+i{\kappa }_j}{z-i{\kappa }_j}}\right)exp\left[2(-4{\kappa }_j^{3}t+{\kappa }_j{k}_j)\right]{\prod }_{l\ne j}\left|{\displaystyle \frac{{\kappa }_l+{\kappa }_k}{{\kappa }_l-{\kappa }_j}}\right|,\hfill \\ d_{jk}={(-1)}^{j}exp\left[2(-4{\kappa }_j^{3}t+{\kappa }_j{k}_j)\right]{\prod }_{l\ne j}\left|{\displaystyle \frac{{\kappa }_l+{\kappa }_k}{{\kappa }_l-{\kappa }_j}}\right|,\hfill \end{array}\end{array}$$

(23)

let u be the potential defined by

$$\begin{array}{c}\hfill u(x,t)=-2{\partial }_x^{2}ln(det(I+B)),\end{array}$$

(24)

and let ${\kappa }_j$, $k_j$ be arbitrary real parameters; then, the function ψ defined by

$$\begin{array}{c}\hfill \psi (x,t)={\displaystyle \frac{det(I+A)det(I+D)}{det(I+B)det(I+C)}}{e}^{izx},\end{array}$$

(25)

which is a solution to the Schrödinger equation

$$\begin{array}{c}\hfill -\psi {(x,t)}^{″}+u(x,t)\psi (x,t)=z^2\psi (x,t)\end{array}$$

(26)

Proof. 

We have proven in [14] that the limit of $\theta (cg+tv+l)$ is equal to $\tilde{\theta }(xg+tv+l)=det(I+B)$.

All of the evaluated limits for the quantities appearing in (22) have been conducted.

We have to evaluate the limit of $U\left(P\right)={\int }_{\infty }^{P}d{U}_k$. As in [14], it is easy to obtain

$$\begin{array}{c}\hfill {\int }_{\infty }^{P}d{\tilde{U}}_k={\displaystyle \frac{1}{2i\pi }}ln{\displaystyle \frac{\sqrt{z}+i{\kappa }_k}{\sqrt{z}-i{\kappa }_k}}\end{array}$$

(27)

We make the change in variable $z\to z^2$.

Using the same arguments as shown in [14] to compute the limits of the terms in $\theta$, it is easy to prove the following equalities:

• $\tilde{\theta }(U\left(P\right)+xg+tv+l)=det(I+A)$,
• $\tilde{\theta }(U\left(P\right)+tv+l)=det(I+C)$,
• $\tilde{\theta }(tv+l)=det(I+D)$.
• So we obtain a result. □

2.3. Some Examples

Example 1.

Solution of order 1: the function ψ is defined by

$$\psi (x,t)={\displaystyle \frac{n(x,t)}{d(x,t)}}$$

• $n(x,t)=(-z+i{K}_1+{\mathrm{e}}^{-2K_1(4t{K_1}^2-x-k_1)}z+i{\mathrm{e}}^{-2K_1(4t{K_1}^2-x-k_1)}{K}_1)(-1+{\mathrm{e}}^{-2K_1(4t{K_1}^2-k_1)}){\mathrm{e}}^{izx}$
• $d(x,t)=(-1+{\mathrm{e}}^{-2K_1(4t{K_1}^2-x-k_1)})(-z+i{K}_1+{\mathrm{e}}^{-2K_1(4t{K_1}^2-k_1)}z+i{\mathrm{e}}^{-2K_1(4t{K_1}^2-k_1)}{K}_1)$
• is a solution to the Schrödinger equation

$$\begin{array}{c}\hfill -\psi {(x,t)}^{″}+u(x,t)\psi (x,t)=z^2\psi (x,t)\end{array}$$

with

$$u(x,t)={\displaystyle \frac{n_u(x,t)}{d_u(x,t)}}$$

• $n_u(x,t)=8{\mathrm{e}}^{-2K_1(4t{K_1}^2-x-k_1)}{K_1}^2$
• $d_u(x,t)={(-1+{\mathrm{e}}^{-2K_1(4t{K_1}^2-x-k_1)})}^2$

Example 2.

Solution of order 2: the function ψ is defined by

$$\psi (x,t)={\displaystyle \frac{n(x,t)}{d(x,t)}}$$

with 

• $n(x,t)=(-{K_1}^2{K}_2+{K_1}^2{\mathrm{e}}^{-2K_1(4t{K_1}^2-x-k_1)}{K}_2-{K_1}^2{\mathrm{e}}^{-2K_2(4t{K_2}^2-x-k_2)}{K}_2-$
• $i{K_1}^2{\mathrm{e}}^{2K_{1}x-8{K_1}^{3}t+2K_1{k}_1+2K_{2}x-8{K_2}^{3}t+2K_2{k}_2}z-iz{K_2}^2{\mathrm{e}}^{-2K_2(4t{K_2}^2-x-k_2)}$
• $-i{\mathrm{e}}^{-2K_1(4t{K_1}^2-x-k_1)}z{K_2}^2+{K_1}^2{\mathrm{e}}^{2K_{1}x-8{K_1}^{3}t+2K_1{k}_1+2K_{2}x-8{K_2}^{3}t+2K_2{k}_2}{K}_2+$
• $iz{K_2}^2-K_1{\mathrm{e}}^{2K_{1}x-8{K_1}^{3}t+2K_1{k}_1+2K_{2}x-8{K_2}^{3}t+2K_2{k}_2}{z}^2+K_1{K_2}^2+K_1{\mathrm{e}}^{-2K_1(4t{K_1}^2-x-k_1)}{z}^2+$
• $K_1{z}^2-K_1{z}^2{\mathrm{e}}^{-2K_2(4t{K_2}^2-x-k_2)}-K_1{\mathrm{e}}^{2K_{1}x-8{K_1}^{3}t+2K_1{k}_1+2K_{2}x-8{K_2}^{3}t+2K_2{k}_2}{K_2}^2+$
• ${K_2}^2{\mathrm{e}}^{-2K_1(4t{K_1}^2-x-k_1)}{K}_1-{K_2}^2{\mathrm{e}}^{-2K_2(4t{K_2}^2-x-k_2)}{K}_1-iz{K_1}^2+i{K_1}^2{\mathrm{e}}^{-2K_2(4t{K_2}^2-x-k_2)}z-$
• $z^2{K}_2+i{K_1}^2{\mathrm{e}}^{-2K_1(4t{K_1}^2-x-k_1)}z+i{\mathrm{e}}^{2K_{1}x-8{K_1}^{3}t+2K_1{k}_1+2K_{2}x-8{K_2}^{3}t+2K_2{k}_2}z{K_2}^2+$
• ${\mathrm{e}}^{2K_{1}x-8{K_1}^{3}t+2K_1{k}_1+2K_{2}x-8{K_2}^{3}t+2K_2{k}_2}{K}_2{z}^2+K_2{\mathrm{e}}^{-2K_1(4t{K_1}^2-x-k_1)}{z}^2$
• $-K_2{z}^2{\mathrm{e}}^{-2K_2(4t{K_2}^2-x-k_2)}\left)\right({\mathrm{e}}^{-2K_2(4t{K_2}^2-k_2)}{K}_1-K_1-{\mathrm{e}}^{-2K_1(4t{K_1}^2-k_1)}{K}_1+$
• $K_1{\mathrm{e}}^{-8{K_1}^{3}t+2K_1{k}_1-8{K_2}^{3}t+2K_2{k}_2}-{\mathrm{e}}^{-8{K_1}^{3}t+2K_1{k}_1-8{K_2}^{3}t+2K_2{k}_2}{K}_2+K_2+{\mathrm{e}}^{-2K_2(4t{K_2}^2-k_2)}{K}_2-$
• ${\mathrm{e}}^{-2K_1(4t{K_1}^2-k_1)}{K}_2){\mathrm{e}}^{izx}$
• and
• $d(x,t)=(i{K_1}^2{\mathrm{e}}^{-8{K_1}^{3}t+2K_1{k}_1-8{K_2}^{3}t+2K_2{k}_2}z+{K_1}^2{K}_2-{K_1}^2{\mathrm{e}}^{-8{K_1}^{3}t+2K_1{k}_1-8{K_2}^{3}t+2K_2{k}_2}{K}_2-$
• $i{K_1}^2{\mathrm{e}}^{-2K_1(4t{K_1}^2-k_1)}z+{K_1}^2{\mathrm{e}}^{-2K_2(4t{K_2}^2-k_2)}{K}_2+iz{K_2}^2{\mathrm{e}}^{-2K_2(4t{K_2}^2-k_2)}-$
• ${K_1}^2{\mathrm{e}}^{-2K_1(4t{K_1}^2-k_1)}{K}_2-iz{K_2}^2-K_1{K_2}^2+K_1{\mathrm{e}}^{-8{K_1}^{3}t+2K_1{k}_1-8{K_2}^{3}t+2K_2{k}_2}{z}^2-$
• $K_1{\mathrm{e}}^{-2K_1(4t{K_1}^2-k_1)}{z}^2+K_1{z}^2{\mathrm{e}}^{-2K_2(4t{K_2}^2-k_2)}-K_1{z}^2+K_1{\mathrm{e}}^{-8{K_1}^{3}t+2K_1{k}_1-8{K_2}^{3}t+2K_2{k}_2}{K_2}^2+$
• ${K_2}^2{\mathrm{e}}^{-2K_2(4t{K_2}^2-k_2)}{K}_1-{K_2}^2{\mathrm{e}}^{-2K_1(4t{K_1}^2-k_1)}{K}_1+iz{K_1}^2+z^2{K}_2-i{K_1}^2{\mathrm{e}}^{-2K_2(4t{K_2}^2-k_2)}z+$
• $i{\mathrm{e}}^{-2K_1(4t{K_1}^2-k_1)}z{K_2}^2-i{\mathrm{e}}^{-8{K_1}^{3}t+2K_1{k}_1-8{K_2}^{3}t+2K_2{k}_2}{K_2}^{2}z-{\mathrm{e}}^{-8{K_1}^{3}t+2K_1{k}_1-8{K_2}^{3}t+2K_2{k}_2}{K}_2{z}^2-$
• $K_2{\mathrm{e}}^{-2K_1(4t{K_1}^2-k_1)}{z}^2+K_2{z}^2{\mathrm{e}}^{-2K_2(4t{K_2}^2-k_2)}\left)\right(-{\mathrm{e}}^{-2K_2(4t{K_2}^2-x-k_2)}{K}_1+K_1+$
• ${\mathrm{e}}^{-2K_1(4t{K_1}^2-x-k_1)}{K}_1-K_1{\mathrm{e}}^{2K_{1}x-8{K_1}^{3}t+2K_1{k}_1+2K_{2}x-8{K_2}^{3}t+2K_2{k}_2}+$
• ${\mathrm{e}}^{2K_{1}x-8{K_1}^{3}t+2K_1{k}_1+2K_{2}x-8{K_2}^{3}t+2K_2{k}_2}{K}_2-K_2-{\mathrm{e}}^{-2K_2(4t{K_2}^2-x-k_2)}{K}_2+{\mathrm{e}}^{-2K_1(4t{K_1}^2-x-k_1)}{K}_2)$
• is a solution to the Schrödinger equation

$$\begin{array}{c}\hfill -\psi {(x,t)}^{″}+u(x,t)\psi (x,t)=z^2\psi (x,t)\end{array}$$

with

$$u(x,t)={\displaystyle \frac{n_u(x,t)}{d_u(x,t)}}$$

• $n_u(x,t)=16{\mathrm{e}}^{2K_{1}x-8{K_1}^{3}t+2K_1{k}_1+2K_{2}x-8{K_2}^{3}t+2K_2{k}_2}{K_1}^4$
• $-8{K_1}^4{\mathrm{e}}^{2K_{1}x-8{K_1}^{3}t+2K_1{k}_1+4K_{2}x-16{K_2}^{3}t+4K_2{k}_2}-8{\mathrm{e}}^{-2K_1(4t{K_1}^2-x-k_1)}{K_1}^4$
• $+8{K_1}^2{\mathrm{e}}^{-2K_2(4t{K_2}^2-x-k_2)}{K_2}^2-32{K_1}^2{\mathrm{e}}^{2K_{1}x-8{K_1}^{3}t+2K_1{k}_1+2K_{2}x-8{K_2}^{3}t+2K_2{k}_2}{K_2}^2+$
• $8{K_1}^2{K_2}^2{\mathrm{e}}^{-16{K_1}^{3}t+4K_{1}x+4K_1{k}_1+2K_{2}x-8{K_2}^{3}t+2K_2{k}_2}$
• $+8{K_1}^2{\mathrm{e}}^{2K_{1}x-8{K_1}^{3}t+2K_1{k}_1+4K_{2}x-16{K_2}^{3}t+4K_2{k}_2}{K_2}^2+8{K_2}^2{\mathrm{e}}^{-2K_1(4t{K_1}^2-x-k_1)}{K_1}^2+$
• $16{\mathrm{e}}^{2K_{1}x-8{K_1}^{3}t+2K_1{k}_1+2K_{2}x-8{K_2}^{3}t+2K_2{k}_2}{K_2}^4-8{\mathrm{e}}^{-2K_2(4t{K_2}^2-x-k_2)}{K_2}^4$
• $-8{K_2}^4{\mathrm{e}}^{-16{K_1}^{3}t+4K_{1}x+4K_1{k}_1+2K_{2}x-8{K_2}^{3}t+2K_2{k}_2}$
• and 
• $d_u(x,t)=({\mathrm{e}}^{-2K_2(4t{K_2}^2-x-k_2)}{K}_1-K_1-{\mathrm{e}}^{-2K_1(4t{K_1}^2-x-k_1)}{K}_1$
• $+K_1{\mathrm{e}}^{2K_{1}x-8{K_1}^{3}t+2K_1{k}_1+2K_{2}x-8{K_2}^{3}t+2K_2{k}_2}-{\mathrm{e}}^{2K_{1}x-8{K_1}^{3}t+2K_1{k}_1+2K_{2}x-8{K_2}^{3}t+2K_2{k}_2}{K}_2+K_2+$
• ${\mathrm{e}}^{-2K_2(4t{K_2}^2-x-k_2)}{K}_2-{\mathrm{e}}^{-2K_1(4t{K_1}^2-x-k_1)}{K_2)}^2$.

• We could go on and give more examples, but even in the simple case of order 3, the only expression of the solution $\psi$ of the Schrödinger equation takes more than 6 pages. For this reason we cannot give examples for greater orders.

3. The Schrödinger Equation and Its Solutions in Terms of Wronskians with a KdV Potential

3.1. Link between Fredholm Determinants and Wronskians

In [14], we have given a link between Fredholm determinants and Wronskians. We use here these results.

We define some notations. We consider the following functions

$$\begin{array}{c}\hfill \begin{array}{c}{\varphi }_j^a\left(x\right)=sinh({\kappa }_{j}x-4{\kappa }_j^{3}t+{\kappa }_j{k}_j+{\displaystyle \frac{1}{2}}ln\left({\displaystyle \frac{z+i{\kappa }_j}{z-i{\kappa }_j}}\right))=sinh\left({\theta }_j^a\right),\hfill \\ {\varphi }_j^b\left(x\right)=sinh({\kappa }_{j}x-4{\kappa }_j^{3}t+{\kappa }_j{k}_j)=sinh\left({\theta }_j^b\right),\hfill \end{array}\end{array}$$

(28)

with arbitrary $k_j$, $K_j$.

• $W=W({\varphi }_j,\dots ,{\varphi }_N)(x,t)$ is the classical Wronskian $W=det\left[{\left({\partial }_x^{j-1}{\varphi }_i\right)}_{i,j\in [1,\dots ,N]}\right]$.
• We consider the matrix $B={\left(b_{jk}\right)}_{j,k\in [1,\dots ,N]}$, defined by

  $$\begin{array}{c}\hfill b_{jk}={(-1)}^{j}exp\left[2({\kappa }_{j}x-4{\kappa }_j^{3}t+{\kappa }_j{k}_j)\right]\prod _{l\ne j}\left|{\displaystyle \frac{{\kappa }_l+{\kappa }_k}{{\kappa }_l-{\kappa }_j}}\right|.\end{array}$$

  Then, we recall the result [14]

Theorem 3.

$$\begin{array}{c}\hfill det(I+B)={\displaystyle \frac{2^N{(-1)}^{\frac{N(N+1)}{2}}exp\left({\sum }_{j=1}^N{\theta }_j^b\right)}{{\prod }_{j=2}^N{\prod }_{i=1}^{j-1}({\kappa }_j-{\kappa }_i)}}W({\varphi }_1^b,\dots ,{\varphi }_N^b)(x,t)\end{array}$$

(29)

Proof. 

We recall the proof given in [14].

For simplicity, in the proof, we replace ${\varphi }_j^b\left(x\right)$ by ${\varphi }_j\left(x\right)$ and ${\theta }_j^b$ by ${\theta }_j$.

We remove the factor $2^{-1}{e}^{{\theta }_j}$ in each row j in the Wronskian W for $1\le j\le N$.

• So

$$\begin{array}{c}\hfill W=\prod _{j=1}^N{e}^{{\theta }_j}{2}^{-N}\times W_1,\end{array}$$

(30)

with

$$W_1=\left|\begin{array}{cccc}(1-e^{-2{\theta }_1})& {\kappa }_1(1+e^{-2{\theta }_1})& \dots & {\left({\kappa }_1\right)}^{N-1}(1+{(-1)}^N{e}^{-i{\theta }_1})\\ (1-e^{-2{\theta }_2})& {\kappa }_2(1+e^{-2{\theta }_2})& \dots & {\left({\kappa }_2\right)}^{N-1}(1+{(-1)}^N{e}^{-2{\theta }_2})\\ ⋮& ⋮& ⋮& ⋮\\ (1-e^{-2{\theta }_N})& {\kappa }_N(1+e^{-2{\theta }_N})& \dots & {\left({\kappa }_N\right)}^{N-1}(1+{(-1)}^N{e}^{-2{\theta }_N})\end{array}\right|$$

$W_1$ can be expressed as

$$W_1=det({\alpha }_{jk}{e}_j+{\beta }_{jk}),$$

with ${\alpha }_{jk}={(-1)}^k{\left({\kappa }_j\right)}^{k-1}$, $e_j=e^{-2{\theta }_j}$, and ${\beta }_{jk}={\left({\kappa }_j\right)}^{k-1}$.

• Let us denote $U={\left({\alpha }_{ij}\right)}_{i,j\in [1,\dots ,N]}$, $V={\left({\beta }_{ij}\right)}_{i,j\in [1,\dots ,N]}$; then, the determinant of U can be expressed as

  $$\begin{array}{c}\hfill det\left(U\right)={(-1)}^{{\displaystyle \frac{N(N+1)}{2}}}\prod _{N\ge l>m\ge 1}({\kappa }_l-{\kappa }_m).\end{array}$$

  (31)

  □

We need the following lemma:

Lemma 1.

Let $B={\left(b_{ij}\right)}_{i,j\in [1,\dots ,N]}$, $U={\left(u_{ij}\right)}_{i,j\in [1,\dots ,N]}$,

• ${\left(H_{ij}\right)}_{i,j\in [1,\dots ,N]}$, the matrix formed by replacing the jth row of B by the ith row of U
• Then

  $$\begin{array}{c}\hfill det(b_{ij}{x}_i+u_{ij})=det\left(b_{ij}\right)\times det\left({\delta }_{ij}{x}_i+{\displaystyle \frac{det\left(H_{ij}\right)}{det\left(b_{ij}\right)}}\right)\end{array}$$

  (32)

Proof. 

Denoting $\tilde{B}={\left({\tilde{b}}_{ij}\right)}_{i,j\in [1,\dots ,N]}$ the matrix of cofactors of B, we use the relation $B{\times }^t\tilde{B}=detB\times I$.

• Therefore, we have $det\left(\tilde{B}\right)={(det\left(B\right))}^{N-1}$.

The general term of the product ${\left(c_{ij}\right)}_{i,j\in [1,\dots ,N]}={(b_{ij}{x}_i+u_{ij})}_{i,j\in [1,\dots ,N]}\times ({\tilde{b}}_{ij}{)}_{i,j\in [1,\dots ,N]}$ can be expressed as

• $c_{ij}={\sum }_{s=1}^N(b_{is}{x}_i+u_{is})\times {\tilde{b}}_{js}$
• $=x_i{\sum }_{s=1}^N{b}_{is}{\tilde{b}}_{js}+{\sum }_{s=1}^N{u}_{is}{\tilde{b}}_{js}$
• $={\delta }_{ij}det\left(B\right)x_i+det\left(H_{ij}\right)$.

So we obtain

• $det\left(c_{ij}\right)=det(b_{ij}{x}_i+u_{ij})\times {(det\left(B\right))}^{N-1}={(det\left(B\right))}^N\times det({\delta }_{ij}{x}_i+{\displaystyle \frac{det\left(H_{ij}\right)}{det\left(B\right)}})$.

Therefore, $det(b_{ij}{x}_i+u_{ij})=det\left(B\right)\times det({\delta }_{ij}{x}_i+{\displaystyle \frac{det\left(H_{ij}\right)}{det\left(B\right)}})$.

We use the lemma (32) to obtain

$$det({\alpha }_{ij}{e}_i+{\beta }_{ij})=det\left({\alpha }_{ij}\right)\times det({\delta }_{ij}{e}_i+{\displaystyle \frac{det\left(H_{ij}\right)}{det\left({\alpha }_{ij}\right)}}),$$

with ${\left(H_{ij}\right)}_{i,j\in [1,\dots ,N]}$, the matrix is defined by replacing the j-th row of B by the i-th row of U, as defined previously.

We calculate $det\left(H_{ij}\right)$ and we obtain

$$\begin{array}{c}\hfill det\left(H_{ij}\right)={(-1)}^{{\displaystyle \frac{N(N+1)}{2}}+1}\prod _{N\ge l>m\ge 1,l\ne j,m\ne j}({\kappa }_l-{\kappa }_m)\prod _{l<j}(-{\kappa }_k-{\kappa }_l)\prod _{l>j}({\kappa }_k+{\kappa }_l).\end{array}$$

(33)

The quotient $q={\displaystyle \frac{det\left(H_{ij}\right)}{det\left({\alpha }_{ij}\right)}}$ can be simplified

$$\begin{array}{c}\hfill q={\displaystyle \frac{{(-1)}^j{\prod }_{l\ne j}|{\kappa }_l+{\kappa }_k|}{{\prod }_{l\ne j}|{\kappa }_l-{\kappa }_j|}}.\end{array}$$

Thus, $det({\delta }_{jk}{e}_j+{\displaystyle \frac{det\left(H_{jk}\right)}{det\left({\alpha }_{jk}\right)}})$ can be written as

$$det({\delta }_{jk}{e}_j+{\displaystyle \frac{det\left(H_{jk}\right)}{det\left({\alpha }_{jk}\right)}})=\prod _{j=1}^N{e}^{-2{\theta }_j}det({\delta }_{jk}+{(-1)}^j\prod _{l\ne j}\left|{\displaystyle \frac{{\kappa }_l+{\kappa }_k}{{\kappa }_l-{\kappa }_j}}\right|e^{2{\theta }_j}).$$

and thus

$$det({\delta }_{jk}{e}_j+{\displaystyle \frac{det\left(H_{jk}\right)}{det\left({\alpha }_{jk}\right)}})=\prod _{j=1}^N{e}^{-2{\theta }_j}det(I+B).$$

The Wronskian can be expressed as

$$W({\varphi }_1,\dots ,{\varphi }_N)=\prod _{j=1}^N{e}^{{\theta }_j}{2}^{-N}{(-1)}^{{\displaystyle \frac{N(N+1)}{2}}}\prod _{j=2}^N\prod _{i=1}^{j-1}({\kappa }_j-{\kappa }_i)\prod _{j=1}^N{e}^{-2{\theta }_j}det(I+B)$$

then we have

$$\begin{array}{c}\hfill det(I+B)={\displaystyle \frac{e^{{\sum }_{j=1}^N{\theta }_j}{\left(2\right)}^N{(-1)}^{\frac{N(N+1)}{2}}}{{\prod }_{j=2}^N{\prod }_{i=1}^{j-1}({\kappa }_j-{\kappa }_i)}}W({\varphi }_1,\dots ,{\varphi }_N)\end{array}$$

(34)

□

3.2. Solutions to the Schrödinger Equation in Terms of Wronskians

Theorem 4.

Let ${\varphi }_j^a$, ${\varphi }_j^b$, ${\varphi }_j^c$, ${\varphi }_j^d$, be the functions defined in (28) by

$$\begin{array}{c}\hfill \begin{array}{c}{\varphi }_j^a\left(x\right)=sinh({\kappa }_{j}x-4{\kappa }_j^{3}t+{\kappa }_j{k}_j+{\displaystyle \frac{1}{2}}ln\left({\displaystyle \frac{z+i{\kappa }_j}{z-i{\kappa }_j}}\right))=sinh\left({\theta }_j^a\right),\hfill \\ {\varphi }_j^b\left(x\right)=sinh({\kappa }_{j}x-4{\kappa }_j^{3}t+{\kappa }_j{k}_j)=sinh\left({\theta }_j^b\right),\hfill \\ {\varphi }_j^c\left(x\right)=sinh(-4{\kappa }_j^{3}t+{\kappa }_j{k}_j+{\displaystyle \frac{1}{2}}ln\left({\displaystyle \frac{z+i{\kappa }_j}{z-i{\kappa }_j}}\right))=sinh\left({\theta }_j^c\right),\hfill \\ {\varphi }_j^d\left(x\right)=sinh(-4{\kappa }_j^{3}t+{\kappa }_j{k}_j)=sinh\left({\theta }_j^d\right),\hfill \end{array}\end{array}$$

(35)

with arbitrary $k_j$, $K_j$.

• u be the potential defined by

$$\begin{array}{c}\hfill u(x,t)=-2{\partial }_x^{2}ln(W({\varphi }_1^b,\cdots ,{\varphi }_N^b)(x,t),\end{array}$$

${\kappa }_j$, $x_j$ be arbitrary real parameters, then the function

$$\begin{array}{c}\hfill \psi (x,t)={\displaystyle \frac{W({\varphi }_1^a,\dots ,{\varphi }_N^a)(x,t)W({\varphi }_1^d,\dots ,{\varphi }_N^d)(x,t)}{W({\varphi }_1^b,\dots ,{\varphi }_N^b)(x,t)W({\varphi }_1^c,\dots ,{\varphi }_N^c)(x,t)}}{e}^{izx},\end{array}$$

(36)

is a solution to the Schrödinger Equation (26)

$$\begin{array}{c}\hfill -\psi {(x,t)}^{″}+u(x,t)\psi (x,t)=z^2\psi (x,t)\end{array}$$

Proof. 

We have proven in [14] that for the matrix B defined by

$$b_{jk}={(-1)}^{j}exp\left[2({\kappa }_{j}x-4{\kappa }_j^{3}t+{\kappa }_j{k}_j)\right]\prod _{l\ne j}\left|{\displaystyle \frac{{\kappa }_l+{\kappa }_k}{{\kappa }_l-{\kappa }_j}}\right|,$$

we have

$$det(I+B)={\displaystyle \frac{e^{{\sum }_{j=1}^N{\theta }_j^b}{\left(2\right)}^N{(-1)}^{\frac{N(N+1)}{2}}}{{\prod }_{j=2}^N{\prod }_{i=1}^{j-1}({\kappa }_j-{\kappa }_i)}}W({\varphi }_1^b,\dots ,{\varphi }_N^b).$$

Using the same arguments, it is easy to prove the folowing relations

$$det(I+A)={\displaystyle \frac{e^{{\sum }_{j=1}^N{\theta }_j^a}{\left(2\right)}^N{(-1)}^{\frac{N(N+1)}{2}}}{{\prod }_{j=2}^N{\prod }_{i=1}^{j-1}({\kappa }_j-{\kappa }_i)}}W({\varphi }_1^a,\dots ,{\varphi }_N^a),$$

$$det(I+C)={\displaystyle \frac{e^{{\sum }_{j=1}^N{\theta }_j^c}{\left(2\right)}^N{(-1)}^{\frac{N(N+1)}{2}}}{{\prod }_{j=2}^N{\prod }_{i=1}^{j-1}({\kappa }_j-{\kappa }_i)}}W({\varphi }_1^c,\dots ,{\varphi }_N^c),$$

$$det(I+D)={\displaystyle \frac{e^{{\sum }_{j=1}^N{\theta }_j^d}{\left(2\right)}^N{(-1)}^{\frac{N(N+1)}{2}}}{{\prod }_{j=2}^N{\prod }_{i=1}^{j-1}({\kappa }_j-{\kappa }_i)}}W({\varphi }_1^d,\dots ,{\varphi }_N^d).$$

Using the previous theorem and performing some simplifications, the solution to the Schrödinger Equation (26) can be written as

$$\psi (x,t)={\displaystyle \frac{W({\varphi }_1^a,\dots ,{\varphi }_N^a)(x,t)W({\varphi }_1^d,\dots ,{\varphi }_N^d)(x,t)}{W({\varphi }_1^b,\dots ,{\varphi }_N^b)(x,t)W({\varphi }_1^c,\dots ,{\varphi }_N^c)(x,t)}}{e}^{izx}.$$

which proves the result. □

4. Quasi-Rational Solutions to the Schrödinger Equation with a KdV Rational Potential

To obtain rational solutions to the Schrödinger equation, we choose $K\left(j\right)$ and $k_j$ as functions of e for each integer j and we perform a limit when the parameter e tends to 0.

4.1. Quasi-Rational Solutions as a Limit Case

We have the following statement:

Theorem 5.

Let $\tilde{A}$, $\tilde{B}$, $\tilde{C}$, $\tilde{D}$ be the matrices defined by

$$\begin{array}{c}\hfill \begin{array}{c}{\tilde{a}}_{jk}={(-1)}^j\left({\displaystyle \frac{z+i{\tilde{\kappa }}_j}{z-i{\tilde{\kappa }}_j}}\right)exp\left[2({\tilde{\kappa }}_{j}x-4{\tilde{\kappa }}_j^{3}t+{\tilde{\kappa }}_j{\tilde{k}}_j)\right]{\prod }_{l\ne j}\left|{\displaystyle \frac{{\tilde{\kappa }}_l+{\tilde{\kappa }}_k}{{\tilde{\kappa }}_l-{\tilde{\kappa }}_j}}\right|,\hfill \\ {\tilde{b}}_{jk}={(-1)}^{j}exp\left[2({\tilde{\kappa }}_{j}x-4{\tilde{\kappa }}_j^{3}t+{\tilde{\kappa }}_j{\tilde{k}}_j)\right]{\prod }_{l\ne j}\left|{\displaystyle \frac{{\tilde{\kappa }}_l+{\tilde{\kappa }}_k}{{\tilde{\kappa }}_l-{\tilde{\kappa }}_j}}\right|,\hfill \\ c_{jk}={(-1)}^j\left({\displaystyle \frac{z+i{\kappa }_j}{z-i{\kappa }_j}}\right)exp\left[2(-4{\kappa }_j^{3}t+{\kappa }_j{\tilde{k}}_j)\right]{\prod }_{l\ne j}\left|{\displaystyle \frac{{\tilde{\kappa }}_l+{\tilde{\kappa }}_k}{{\tilde{\kappa }}_l-{\tilde{\kappa }}_j}}\right|,\hfill \\ {\tilde{d}}_{jk}={(-1)}^{j}exp\left[2(-4{\tilde{\kappa }}_j^{3}t+{\tilde{\kappa }}_j{\tilde{k}}_j)\right]{\prod }_{l\ne j}\left|{\displaystyle \frac{{\tilde{\kappa }}_l+{\tilde{\kappa }}_k}{{\tilde{\kappa }}_l-{\tilde{\kappa }}_j}}\right|,\hfill \end{array}\end{array}$$

(37)

u the potential

$$\begin{array}{c}\hfill u(x,t)=-2\underset{e\to 0}{lim}{\partial }_x^{2}ln(det(I+\tilde{B})),\end{array}$$

(38)

then the function

$$\begin{array}{c}\hfill \psi (x,t)=\underset{e\to 0}{lim}{\displaystyle \frac{det(I+\tilde{A})det(I+\tilde{D})}{det(I+\tilde{B})det(I+\tilde{C})}}{e}^{izx},\end{array}$$

(39)

is a quasi-rational solution to the Schrödinger equation:

$$\begin{array}{c}\hfill -\psi {(x,t)}^{″}+u(x,t)\psi (x,t)=z^2\psi (x,t).\end{array}$$

(40)

Proof. 

When e tends to 0, we proceed to a passage to the limit, which leads to the result. □

4.2. First Order Quasi-Rational Solutions

Here, we replace $K\left(1\right)$ by $K\left(1\right)e$ and choose $k_1$ independent of e. For order $N=1$, we obtain the following:

Proposition 1.

The function

$$\begin{array}{c}\hfill \psi \left(x\right)=-2{\displaystyle \frac{K_1\left(3z^{3}x+3z^3{k}_1+3i{z}^2\right)k_1{\mathrm{e}}^{izx}}{\left(-2K_{1}x-2K_1{k}_1\right)\left(3z^3{k}_1+3i{z}^2\right)}},\end{array}$$

(41)

with the potential u

$$\begin{array}{c}\hfill u\left(x\right)={\displaystyle \frac{2}{x^2+2k_{1}x+{k_1}^2}}\end{array}$$

(42)

is a solution to the Schrödinger Equation (2).

Remark 1.

In this case, we note that ψ and u are independent of t; moreover, the parameter $K_1$ does not appear.

4.3. Second Order Quasi-Rational Solutions

Here, we replace $K_j$ by $K_{j}e$ and $k_j$ by $k_j{e}^2$. Then we obtain the following:

Proposition 2.

The function

$$\begin{array}{c}\hfill \psi \left(x\right)={\displaystyle \frac{n(x,t)}{d(x,t)}},\end{array}$$

where

• $n(x,t)={\mathrm{e}}^{izx}(12t{z}^2{K_1}^2-3z^2{k}_1+3z^2{k}_2+x^3{z}^2{K_1}^2-x^3{z}^2{K_2}^2+3i{x}^{2}z{K_1}^2-3i{x}^{2}z{K_2}^2-$
• $3{K_1}^{2}x+3{K_2}^{2}x-12t{z}^2{K_2}^2)$
• and
• $d(x,t)=z^2(12{K_1}^{2}t+{K_1}^2{x}^3-12{K_2}^{2}t-{K_2}^2{x}^3-3k_1+3k_2)$
• form a solution to the Schrödinger Equation (2),
• with the potential

  $$\begin{array}{c}\hfill u(x,t)={\displaystyle \frac{n_u(x,t)}{d_u(x,t)}}\end{array}$$

  (43)

  where
• $n_u(x,t)=-6x(24{K_1}^{4}t-6{K_1}^2{k}_1-48{K_2}^2{K_1}^{2}t+6{K_2}^2{k}_1+24{K_2}^{4}t-6{K_2}^2{k}_2+6{K_1}^2{k}_2-$
• ${K_1}^4{x}^3+2{K_2}^2{K_1}^2{x}^3-{K_2}^4{x}^3)$
• and
• $d_u(x,t)=72{K_1}^{2}t{k}_2-48{K_2}^2{K_1}^{2}t{x}^3-6{K_1}^2{k}_1{x}^3+72{K_2}^2{k}_{1}t-18k_1{k}_2+6{K_2}^2{k}_1{x}^3+$
• $6{K_1}^2{x}^3{k}_2-72{K_2}^{2}t{k}_2+24{K_2}^{4}t{x}^3-6{K_2}^2{k}_2{x}^3-72{K_1}^{2}t{k}_1+24{K_1}^{4}t{x}^3-288{K_2}^2{K_1}^2{t}^2+$
• $144{K_1}^4{t}^2+9{k_1}^2+{K_1}^4{x}^6-2{K_2}^2{K_1}^2{x}^6+144{K_2}^4{t}^2+9{k_2}^2+{K_2}^4{x}^6$.

4.4. Quasi-Rational Solutions of Order Three

We choose $K_j=K_{j}e$ and $l_j$ independent of e. Then we obtain the following solutions:

Proposition 3.

The function

$$\begin{array}{c}\hfill \psi \left(x\right)={\displaystyle \frac{n(x,t)}{d(x,t)}}\end{array}$$

(44)

with

• $n(x,t)=(xz{K_1}^2{k}_2-xz{K_1}^2{k}_3-xz{K_2}^2{k}_1+xz{K_2}^2{k}_3+xz{K_3}^2{k}_1-xz{K_3}^2{k}_2+z{K_1}^2{k}_1{k}_2-$
• $z{K_1}^2{k}_1{k}_3-z{K_2}^2{k}_1{k}_2+z{K_2}^2{k}_2{k}_3+z{K_3}^2{k}_1{k}_3-z{K_3}^2{k}_2{k}_3+i{K_1}^2{k}_2+i{K_2}^2{k}_3-i{K_2}^2{k}_1+$
• $i{K_3}^2{k}_1-i{K_3}^2{k}_2-i{K_1}^2{k}_3)({K_1}^2{k}_1{k}_2+{K_2}^2{k}_2{k}_3+{K_3}^2{k}_1{k}_3-{K_1}^2{k}_1{k}_3-{K_2}^2{k}_1{k}_2-{K_3}^2{k}_2{k}_3){\mathrm{e}}^{izx}$
• and
• $d(x,t)=(x{K_1}^2{k}_2-x{K_1}^2{k}_3-x{K_2}^2{k}_1+x{K_2}^2{k}_3+x{K_3}^2{k}_1-x{K_3}^2{k}_2+{K_1}^2{k}_1{k}_2+{K_2}^2{k}_2{k}_3+$
• ${K_3}^2{k}_1{k}_3-{K_1}^2{k}_1{k}_3-{K_2}^2{k}_1{k}_2-{K_3}^2{k}_2{k}_3)(z{K_1}^2{k}_1{k}_2-z{K_1}^2{k}_1{k}_3-z{K_2}^2{k}_1{k}_2+z{K_2}^2{k}_2{k}_3+$
• $z{K_3}^2{k}_1{k}_3-z{K_3}^2{k}_2{k}_3+i{K_1}^2{k}_2-i{K_1}^2{k}_3-i{K_2}^2{k}_1+i{K_2}^2{k}_3+i{K_3}^2{k}_1-i{K_3}^2{k}_2),$
• forms a solution to the Schrödinger Equation (2),
• with the potential u
• $u(x,t)={\displaystyle \frac{n_u(x,t)}{d_u(x,t)}},$
• where
• $n_u(x,t)=2{({K_1}^2{k}_2-{K_1}^2{k}_3-{K_2}^2{k}_1+{K_2}^2{k}_3+{K_3}^2{k}_1-{K_3}^2{k}_2)}^2$
• and
• $d_u(x,t)=(x{K_1}^2{k}_2-x{K_1}^2{k}_3-x{K_2}^2{k}_1+x{K_2}^2{k}_3+x{K_3}^2{k}_1-x{K_3}^2{k}_2+{K_1}^2{k}_1{k}_2+{K_2}^2{k}_2{k}_3+$
• ${K_3}^2{k}_1{k}_3-{K_1}^2{k}_1{k}_3-{K_2}^2{k}_1{k}_2-{K_3}^2{k}_2{k}_3{)}^2$.

4.5. Quasi-Rational Solutions of Order Four

We replace $K_j$ with $K_{j}e$ and $l_j$ with $l_j{e}^2$. We obtain the following solutions:

Proposition 4.

The function

$$\begin{array}{c}\hfill \psi \left(x\right)={\displaystyle \frac{n(x,t)}{d(x,t)}}\end{array}$$

(45)

with

• $n(x,t)=(3i{x}^{2}z{K_1}^4{K_4}^2{k}_2-3i{x}^{2}z{K_1}^4{K_4}^2{k}_3-3i{x}^{2}z{K_1}^2{K_2}^4{k}_3+3i{x}^{2}z{K_1}^2{K_2}^4{k}_4+$
• $3i{x}^{2}z{K_1}^2{K_3}^4{k}_2-3i{x}^{2}z{K_1}^2{K_3}^4{k}_4-3i{x}^{2}z{K_1}^2{K_4}^4{k}_2+3i{x}^{2}z{K_1}^2{K_4}^4{k}_3+3i{x}^{2}z{K_2}^4{K_3}^2{k}_1-$
• $3i{x}^{2}z{K_2}^4{K_3}^2{k}_4-3i{x}^{2}z{K_2}^4{K_4}^2{k}_1+3i{x}^{2}z{K_2}^4{K_4}^2{k}_3-3i{x}^{2}z{K_2}^2{K_3}^4{k}_1+3i{x}^{2}z{K_2}^2{K_3}^4{k}_4+$
• $3i{x}^{2}z{K_2}^2{K_4}^4{k}_1-3i{x}^{2}z{K_2}^2{K_4}^4{k}_3+3i{x}^{2}z{K_3}^4{K_4}^2{k}_1-3i{x}^{2}z{K_3}^4{K_4}^2{k}_2-3i{x}^{2}z{K_3}^2{K_4}^4{k}_1+$
• $3i{x}^{2}z{K_3}^2{K_4}^4{k}_2+3i{x}^{2}z{K_1}^4{K_2}^2{k}_3-3i{x}^{2}z{K_1}^4{K_2}^2{k}_4-3i{x}^{2}z{K_1}^4{K_3}^2{k}_2+3i{x}^{2}z{K_1}^4{K_3}^2{k}_4-$
• $3x{K_1}^4{K_2}^2{k}_3+3x{K_1}^4{K_2}^2{k}_4+3x{K_1}^4{K_3}^2{k}_2-3x{K_1}^4{K_3}^2{k}_4-3x{K_1}^4{K_4}^2{k}_2+3x{K_1}^4{K_4}^2{k}_3+$
• $3x{K_1}^2{K_2}^4{k}_3-3x{K_1}^2{K_2}^4{k}_4-3x{K_1}^2{K_3}^4{k}_2+3x{K_1}^2{K_3}^4{k}_4+3x{K_1}^2{K_4}^4{k}_2-3x{K_1}^2{K_4}^4{k}_3-$
• $3x{K_2}^4{K_3}^2{k}_1+3x{K_2}^4{K_3}^2{k}_4+3x{K_2}^4{K_4}^2{k}_1-3x{K_2}^4{K_4}^2{k}_3+3x{K_2}^2{K_3}^4{k}_1-3x{K_2}^2{K_3}^4{k}_4-$
• $3x{K_2}^2{K_4}^4{k}_1+3x{K_2}^2{K_4}^4{k}_3-3x{K_3}^4{K_4}^2{k}_1+3x{K_3}^4{K_4}^2{k}_2+3x{K_3}^2{K_4}^4{k}_1-3x{K_3}^2{K_4}^4{k}_2-$
• $3z^2{K_1}^2{K_4}^2{k}_1{k}_2+3z^2{K_1}^2{K_4}^2{k}_1{k}_3+3z^2{K_1}^2{K_4}^2{k}_2{k}_4-3z^2{K_1}^2{K_4}^2{k}_3{k}_4-3z^2{K_2}^2{K_3}^2{k}_1{k}_2+$
• $3z^2{K_2}^2{K_3}^2{k}_1{k}_3+3z^2{K_2}^2{K_3}^2{k}_2{k}_4-3z^2{K_2}^2{K_3}^2{k}_3{k}_4+3z^2{K_2}^2{K_4}^2{k}_1{k}_2-3z^2{K_2}^2{K_4}^2{k}_1{k}_4-$
• $3z^2{K_2}^2{K_4}^2{k}_2{k}_3+3z^2{K_2}^2{K_4}^2{k}_3{k}_4-3z^2{K_3}^2{K_4}^2{k}_1{k}_3+3z^2{K_3}^2{K_4}^2{k}_1{k}_4+3z^2{K_3}^2{K_4}^2{k}_2{k}_3-$
• $3z^2{K_3}^2{K_4}^2{k}_2{k}_4-x^3{z}^2{K_1}^2{K_4}^4{k}_2+x^3{z}^2{K_1}^2{K_4}^4{k}_3+x^3{z}^2{K_2}^4{K_3}^2{k}_1-x^3{z}^2{K_2}^4{K_3}^2{k}_4-$
• $x^3{z}^2{K_2}^4{K_4}^2{k}_1+x^3{z}^2{K_2}^4{K_4}^2{k}_3-x^3{z}^2{K_2}^2{K_3}^4{k}_1+x^3{z}^2{K_2}^2{K_3}^4{k}_4+x^3{z}^2{K_2}^2{K_4}^4{k}_1-$
• $x^3{z}^2{K_2}^2{K_4}^4{k}_3+x^3{z}^2{K_3}^4{K_4}^2{k}_1-x^3{z}^2{K_3}^4{K_4}^2{k}_2-x^3{z}^2{K_3}^2{K_4}^4{k}_1+x^3{z}^2{K_3}^2{K_4}^4{k}_2+$
• $12t{z}^2{K_1}^4{K_2}^2{k}_3-12t{z}^2{K_1}^4{K_2}^2{k}_4-12t{z}^2{K_1}^4{K_3}^2{k}_2+12t{z}^2{K_1}^4{K_3}^2{k}_4+12t{z}^2{K_1}^4{K_4}^2{k}_2-$
• $12t{z}^2{K_1}^4{K_4}^2{k}_3-12t{z}^2{K_1}^2{K_2}^4{k}_3+12t{z}^2{K_1}^2{K_2}^4{k}_4+12t{z}^2{K_1}^2{K_3}^4{k}_2-12t{z}^2{K_1}^2{K_3}^4{k}_4-$
• $12t{z}^2{K_1}^2{K_4}^4{k}_2+12t{z}^2{K_1}^2{K_4}^4{k}_3+12t{z}^2{K_2}^4{K_3}^2{k}_1-12t{z}^2{K_2}^4{K_3}^2{k}_4-12t{z}^2{K_2}^4{K_4}^2{k}_1+$
• $12t{z}^2{K_2}^4{K_4}^2{k}_3-12t{z}^2{K_2}^2{K_3}^4{k}_1+12t{z}^2{K_2}^2{K_3}^4{k}_4+12t{z}^2{K_2}^2{K_4}^4{k}_1-12t{z}^2{K_2}^2{K_4}^4{k}_3+$
• $12t{z}^2{K_3}^4{K_4}^2{k}_1-12t{z}^2{K_3}^4{K_4}^2{k}_2-12t{z}^2{K_3}^2{K_4}^4{k}_1+12t{z}^2{K_3}^2{K_4}^4{k}_2-3z^2{K_1}^2{K_2}^2{k}_1{k}_3+$
• $3z^2{K_1}^2{K_2}^2{k}_1{k}_4+3z^2{K_1}^2{K_2}^2{k}_2{k}_3-3z^2{K_1}^2{K_2}^2{k}_2{k}_4+3z^2{K_1}^2{K_3}^2{k}_1{k}_2-3z^2{K_1}^2{K_3}^2{k}_1{k}_4-$
• $3z^2{K_1}^2{K_3}^2{k}_2{k}_3+3z^2{K_1}^2{K_3}^2{k}_3{k}_4+x^3{z}^2{K_1}^4{K_2}^2{k}_3-x^3{z}^2{K_1}^4{K_2}^2{k}_4-x^3{z}^2{K_1}^4{K_3}^2{k}_2+$
• $x^3{z}^2{K_1}^4{K_3}^2{k}_4+x^3{z}^2{K_1}^4{K_4}^2{k}_2-x^3{z}^2{K_1}^4{K_4}^2{k}_3-x^3{z}^2{K_1}^2{K_2}^4{k}_3+$
• $x^3{z}^2{K_1}^2{K_2}^4{k}_4+x^3{z}^2{K_1}^2{K_3}^4{k}_2-x^3{z}^2{K_1}^2{K_3}^4{k}_4){\mathrm{e}}^{ixz}$
• and
• $d(x,t)=z^2(x^3{K_1}^4{K_2}^2{k}_3-x^3{K_1}^4{K_2}^2{k}_4-x^3{K_1}^4{K_3}^2{k}_2+x^3{K_1}^4{K_3}^2{k}_4+x^3{K_1}^4{K_4}^2{k}_2-$
• $x^3{K_1}^4{K_4}^2{k}_3-x^3{K_1}^2{K_2}^4{k}_3+x^3{K_1}^2{K_2}^4{k}_4+x^3{K_1}^2{K_3}^4{k}_2-x^3{K_1}^2{K_3}^4{k}_4-x^3{K_1}^2{K_4}^4{k}_2+$
• $x^3{K_1}^2{K_4}^4{k}_3+x^3{K_2}^4{K_3}^2{k}_1-x^3{K_2}^4{K_3}^2{k}_4-x^3{K_2}^4{K_4}^2{k}_1+x^3{K_2}^4{K_4}^2{k}_3-x^3{K_2}^2{K_3}^4{k}_1+$
• $x^3{K_2}^2{K_3}^4{k}_4+x^3{K_2}^2{K_4}^4{k}_1-x^3{K_2}^2{K_4}^4{k}_3+x^3{K_4}^2{K_3}^4{k}_1-x^3{K_4}^2{K_3}^4{k}_2-x^3{K_4}^4{K_3}^2{k}_1+$
• $x^3{K_4}^4{K_3}^2{k}_2+12t{K_1}^4{K_2}^2{k}_3-12t{K_1}^4{K_2}^2{k}_4-12t{K_1}^4{K_3}^2{k}_2+12t{K_1}^4{K_3}^2{k}_4+$
• $12t{K_1}^4{K_4}^2{k}_2-12t{K_1}^4{K_4}^2{k}_3-12{K_1}^{2}t{K_2}^4{k}_3+12{K_1}^{2}t{K_2}^4{k}_4+12{K_1}^{2}t{K_3}^4{k}_2-$
• $12{K_1}^{2}t{K_3}^4{k}_4-12{K_1}^{2}t{K_4}^4{k}_2+12{K_1}^{2}t{K_4}^4{k}_3+12{K_2}^{4}t{K_3}^2{k}_1-12t{K_2}^4{K_3}^2{k}_4-$
• $12{K_2}^{4}t{K_4}^2{k}_1+12t{K_2}^4{K_4}^2{k}_3-12t{K_2}^2{K_3}^4{k}_1+12{K_2}^{2}t{K_3}^4{k}_4+12t{K_2}^2{K_4}^4{k}_1-$
• $12{K_2}^{2}t{K_4}^4{k}_3+12{K_3}^{4}t{K_4}^2{k}_1-12{K_3}^{4}t{K_4}^2{k}_2-12t{K_4}^4{K_3}^2{k}_1+12t{K_4}^4{K_3}^2{k}_2-$
• $3{K_1}^2{K_2}^2{k}_1{k}_3+3{K_1}^2{K_2}^2{k}_1{k}_4+3{K_1}^2{K_2}^2{k}_3{k}_2-3{K_1}^2{K_2}^2{k}_4{k}_2+3{K_1}^2{K_3}^2{k}_2{k}_1-$
• $3{K_1}^2{K_3}^2{k}_1{k}_4-3{K_1}^2{K_3}^2{k}_3{k}_2+3{K_1}^2{K_3}^2{k}_3{k}_4-3{K_1}^2{K_4}^2{k}_2{k}_1+3{K_1}^2{K_4}^2{k}_1{k}_3+$
• $3{K_1}^2{K_4}^2{k}_4{k}_2-3{K_1}^2{K_4}^2{k}_3{k}_4-3{K_2}^2{K_3}^2{k}_2{k}_1+3{K_2}^2{K_3}^2{k}_1{k}_3+3{K_2}^2{K_3}^2{k}_4{k}_2-$
• $3{K_2}^2{K_3}^2{k}_3{k}_4+3{K_2}^2{K_4}^2{k}_2{k}_1-3{K_2}^2{K_4}^2{k}_1{k}_4-3{K_2}^2{K_4}^2{k}_3{k}_2+3{K_2}^2{K_4}^2{k}_3{k}_4-$
• $3{K_3}^2{K_4}^2{k}_1{k}_3+3{K_3}^2{K_4}^2{k}_1{k}_4+3{K_3}^2{K_4}^2{k}_2{k}_3-3{K_3}^2{K_4}^2{k}_2{k}_4)$
• is a solution to the Schrödinger Equation (2),
• with the potential
• $u(x,t)={\displaystyle \frac{n_u(x,t)}{d_u(x,t)}},$
• where
• $n_u(x,t)=-6({K_1}^4{K_2}^2{k}_3-{K_1}^4{K_2}^2{k}_4-{K_1}^4{K_3}^2{k}_2+{K_1}^4{K_3}^2{k}_4+{K_1}^4{K_4}^2{k}_2-{K_1}^4{K_4}^2{k}_3-$
• ${K_1}^2{K_2}^4{k}_3+{K_1}^2{K_2}^4{k}_4+{K_1}^2{K_3}^4{k}_2-{K_1}^2{K_3}^4{k}_4-{K_1}^2{K_4}^4{k}_2+{K_1}^2{K_4}^4{k}_3+{K_2}^4{K_3}^2{k}_1-$
• ${K_2}^4{K_3}^2{k}_4-{K_2}^4{K_4}^2{k}_1+{K_2}^4{K_4}^2{k}_3-{K_2}^2{K_3}^4{k}_1+{K_2}^2{K_3}^4{k}_4+{K_2}^2{K_4}^4{k}_1-{K_2}^2{K_4}^4{k}_3+$
• ${K_3}^4{K_4}^2{k}_1-{K_3}^4{K_4}^2{k}_2-{K_4}^4{K_3}^2{k}_1+{K_4}^4{K_3}^2{k}_2)x(-x^3{K_1}^4{K_2}^2{k}_3+x^3{K_1}^4{K_2}^2{k}_4+$
• $x^3{K_1}^4{K_3}^2{k}_2-x^3{K_1}^4{K_3}^2{k}_4-x^3{K_1}^4{K_4}^2{k}_2+x^3{K_1}^4{K_4}^2{k}_3+x^3{K_1}^2{K_2}^4{k}_3-x^3{K_1}^2{K_2}^4{k}_4-$
• $x^3{K_1}^2{K_3}^4{k}_2+x^3{K_1}^2{K_3}^4{k}_4+x^3{K_1}^2{K_4}^4{k}_2-x^3{K_1}^2{K_4}^4{k}_3-x^3{K_2}^4{K_3}^2{k}_1+x^3{K_2}^4{K_3}^2{k}_4+$
• $x^3{K_2}^4{K_4}^2{k}_1-x^3{K_2}^4{K_4}^2{k}_3+x^3{K_2}^2{K_3}^4{k}_1-x^3{K_2}^2{K_3}^4{k}_4-x^3{K_2}^2{K_4}^4{k}_1+x^3{K_2}^2{K_4}^4{k}_3-$
• $x^3{K_4}^2{K_3}^4{k}_1+x^3{K_4}^2{K_3}^4{k}_2+x^3{K_4}^4{K_3}^2{k}_1-x^3{K_4}^4{K_3}^2{k}_2+24t{K_1}^4{K_2}^2{k}_3-24t{K_1}^4{K_2}^2{k}_4-$
• $24t{K_1}^4{K_3}^2{k}_2+24t{K_1}^4{K_3}^2{k}_4+24t{K_1}^4{K_4}^2{k}_2-24t{K_1}^4{K_4}^2{k}_3-24{K_1}^{2}t{K_2}^4{k}_3+$
• $24{K_1}^{2}t{K_2}^4{k}_4+24{K_1}^{2}t{K_3}^4{k}_2-24{K_1}^{2}t{K_3}^4{k}_4-24{K_1}^{2}t{K_4}^4{k}_2+24{K_1}^{2}t{K_4}^4{k}_3+$
• $24{K_2}^{4}t{K_3}^2{k}_1-24t{K_2}^4{K_3}^2{k}_4-24{K_2}^{4}t{K_4}^2{k}_1+24t{K_2}^4{K_4}^2{k}_3-24t{K_2}^2{K_3}^4{k}_1+$
• $24{K_2}^{2}t{K_3}^4{k}_4+24t{K_2}^2{K_4}^4{k}_1-24{K_2}^{2}t{K_4}^4{k}_3+24{K_3}^{4}t{K_4}^2{k}_1-24{K_3}^{4}t{K_4}^2{k}_2-$
• $24t{K_4}^4{K_3}^2{k}_1+24t{K_4}^4{K_3}^2{k}_2-6{K_1}^2{K_2}^2{k}_1{k}_3+6{K_1}^2{K_2}^2{k}_1{k}_4+6{K_1}^2{K_2}^2{k}_3{k}_2-$
• $6{K_1}^2{K_2}^2{k}_4{k}_2+6{K_1}^2{K_3}^2{k}_2{k}_1-6{K_1}^2{K_3}^2{k}_1{k}_4-6{K_1}^2{K_3}^2{k}_3{k}_2+6{K_1}^2{K_3}^2{k}_3{k}_4-$
• $6{K_1}^2{K_4}^2{k}_2{k}_1+6{K_1}^2{K_4}^2{k}_1{k}_3+6{K_1}^2{K_4}^2{k}_4{k}_2-6{K_1}^2{K_4}^2{k}_3{k}_4-6{K_2}^2{K_3}^2{k}_2{k}_1+$
• $6{K_2}^2{K_3}^2{k}_1{k}_3+6{K_2}^2{K_3}^2{k}_4{k}_2-6{K_2}^2{K_3}^2{k}_3{k}_4+6{K_2}^2{K_4}^2{k}_2{k}_1-6{K_2}^2{K_4}^2{k}_1{k}_4-$
• $6{K_2}^2{K_4}^2{k}_3{k}_2+6{K_2}^2{K_4}^2{k}_3{k}_4-6{K_3}^2{K_4}^2{k}_1{k}_3+6{K_3}^2{K_4}^2{k}_1{k}_4+6{K_3}^2{K_4}^2{k}_2{k}_3-6{K_3}^2{K_4}^2{k}_2{k}_4)$
• and
• $d_u(x,t)=(x^3{K_1}^4{K_2}^2{k}_3-x^3{K_1}^4{K_2}^2{k}_4-x^3{K_1}^4{K_3}^2{k}_2+x^3{K_1}^4{K_3}^2{k}_4+x^3{K_1}^4{K_4}^2{k}_2-$
• $x^3{K_1}^4{K_4}^2{k}_3-x^3{K_1}^2{K_2}^4{k}_3+x^3{K_1}^2{K_2}^4{k}_4+x^3{K_1}^2{K_3}^4{k}_2-x^3{K_1}^2{K_3}^4{k}_4-x^3{K_1}^2{K_4}^4{k}_2+$
• $x^3{K_1}^2{K_4}^4{k}_3+x^3{K_2}^4{K_3}^2{k}_1-x^3{K_2}^4{K_3}^2{k}_4-x^3{K_2}^4{K_4}^2{k}_1+x^3{K_2}^4{K_4}^2{k}_3-x^3{K_2}^2{K_3}^4{k}_1+$
• $x^3{K_2}^2{K_3}^4{k}_4+x^3{K_2}^2{K_4}^4{k}_1-x^3{K_2}^2{K_4}^4{k}_3+x^3{K_4}^2{K_3}^4{k}_1-x^3{K_4}^2{K_3}^4{k}_2-x^3{K_4}^4{K_3}^2{k}_1+$
• $x^3{K_4}^4{K_3}^2{k}_2+12t{K_1}^4{K_2}^2{k}_3-12t{K_1}^4{K_2}^2{k}_4-12t{K_1}^4{K_3}^2{k}_2+12t{K_1}^4{K_3}^2{k}_4+12t{K_1}^4{K_4}^2{k}_2-$
• $12t{K_1}^4{K_4}^2{k}_3-12{K_1}^{2}t{K_2}^4{k}_3+12{K_1}^{2}t{K_2}^4{k}_4+12{K_1}^{2}t{K_3}^4{k}_2-12{K_1}^{2}t{K_3}^4{k}_4-$
• $12{K_1}^{2}t{K_4}^4{k}_2+12{K_1}^{2}t{K_4}^4{k}_3+12{K_2}^{4}t{K_3}^2{k}_1-12t{K_2}^4{K_3}^2{k}_4-12{K_2}^{4}t{K_4}^2{k}_1+$
• $12t{K_2}^4{K_4}^2{k}_3-12t{K_2}^2{K_3}^4{k}_1+12{K_2}^{2}t{K_3}^4{k}_4+12t{K_2}^2{K_4}^4{k}_1-12{K_2}^{2}t{K_4}^4{k}_3+$
• $12{K_3}^{4}t{K_4}^2{k}_1-12{K_3}^{4}t{K_4}^2{k}_2-12t{K_4}^4{K_3}^2{k}_1+12t{K_4}^4{K_3}^2{k}_2-3{K_1}^2{K_2}^2{k}_1{k}_3+$
• $3{K_1}^2{K_2}^2{k}_1{k}_4+3{K_1}^2{K_2}^2{k}_3{k}_2-3{K_1}^2{K_2}^2{k}_4{k}_2+3{K_1}^2{K_3}^2{k}_2{k}_1-3{K_1}^2{K_3}^2{k}_1{k}_4-$
• $3{K_1}^2{K_3}^2{k}_3{k}_2+3{K_1}^2{K_3}^2{k}_3{k}_4-3{K_1}^2{K_4}^2{k}_2{k}_1+3{K_1}^2{K_4}^2{k}_1{k}_3+3{K_1}^2{K_4}^2{k}_4{k}_2-$
• $3{K_1}^2{K_4}^2{k}_3{k}_4-3{K_2}^2{K_3}^2{k}_2{k}_1+3{K_2}^2{K_3}^2{k}_1{k}_3+3{K_2}^2{K_3}^2{k}_4{k}_2-3{K_2}^2{K_3}^2{k}_3{k}_4+$
• $3{K_2}^2{K_4}^2{k}_2{k}_1-3{K_2}^2{K_4}^2{k}_1{k}_4-3{K_2}^2{K_4}^2{k}_3{k}_2+3{K_2}^2{K_4}^2{k}_3{k}_4-3{K_3}^2{K_4}^2{k}_1{k}_3+$
• $3{K_3}^2{K_4}^2{k}_1{k}_4+3{K_3}^2{K_4}^2{k}_2{k}_3-3{K_3}^2{K_4}^2{k}_2{k}_4$.

5. Conclusions

We succeeded in building different representations of the solutions to the Schrödinger equation. First, we proceed to express the degenerate $\theta$ function into an explicit Fredholm determinant. Then, we transform the Fredholm determinant into a Wronskian.

We obtain rational solutions to the Schrödinger equation with a KdV potential. So, we receive an infinite hierarchy of multi-parametric families of rational solutions to the Schrödinger equation as a quotient of the polynomials depending on real parameters.

The quasi-rational solutions obtained by the passage to the limit when one of the parameters tends towards zero are not obtained uniformly as in the construction of the solutions of the nonlinear Schrödinger equation [15]. We might have thought that using the parameter $l\left(j\right)={\sum }_{m=1}^N{k}_m{\left(je\right)}^{2m-2}$ could have led to a uniform result. That is not the case. In the quasi-rational solutions presented, the parameters depending on e were chosen in such a way as to obtain quasi-rational solutions of maximum degree in x and in t. This was performed for small orders from 1 to 4. It is important to continue this study for higher orders and to study the structure of the polynomials obtained. Unlike other equations, such as the NLS equation [15,16], there are no specific structures that emerge depending on the parameters.