Resources on the stationary Schrödinger equation with the soliton potential

Question

I am studying the following Lamé equation in the Jacobi form \begin{equation} -\frac{d^2 v}{dx^2} - \left(2k^2 \operatorname{cn}^{2}\left [x\;|\;k\right ]\right)v = \lambda v, \end{equation} where $k\in(0, 1)$ is parameter related to the elliptic modulus of the Jacobi elliptic functions. In order to better understand this equation I am first taking the soliton limit when $k\rightarrow 1$ to obtain the following differential equation \begin{equation} -\frac{d^2 v}{dx^2} - 2\operatorname{sech}^2{(x)}v = \lambda v, \end{equation} for which analytic solutions exist depending on the values of the parameter $\lambda$. For different behaviour of eigenfunction $v$ is observed for $\lambda=-1,$ $\lambda\in(-\infty,-1)\cup(-1,0)$ and $\lambda\geq 0.$ I believe I obtained the correct solutions; however, I am having hard time finding them explicitly in the literature. Does anyone know good resources on the latter equation, or if it is known under a special name? My ultimate goal is to use these solutions in the context of the KdV equation, which is a compatibility condition for the stationary Schrödinger equation \begin{equation}\label{SE} -\frac{d^2 v}{dx^2} - u(x,t) v = \lambda v \end{equation}

Answer 1

The Schrödinger equation with the $\text{sech}^2$ potential (sometimes called the Pöschl–Teller potential) was first studied by Epstein in 1930 [1]. There is an extensive literature on this exactly solvable case. Two recent references are [2,3]. A special feature of this potential is that it is reflectionless, it admits unit transmission at all energies. For that reason it has also found many real-world applications, in particular in photonics.

1. P.S. Epstein, Reflection of waves in an inhomogeneous absorbing medium (1930).
2. J. Lekner, Reflectionless eigenstates of the $\text{sech}^2$ potential (2007).
3. C.S. Park, Transmission time of a particle in the reflectionless sech-squared potential (2011).