The Schrödinger equation with the $\text{sech}^2$ potential (sometimes called the <A HREF="https://en.wikipedia.org/wiki/Pöschl–Teller_potential">Pöschl–Teller potential</A>) 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, <A HREF="https://www.jstor.org/stable/85754">Reflection of waves in an inhomogeneous absorbing medium</A>
    (1930).    
 2. J. Lekner, <A HREF="http://www.physics.smu.edu/scalise/P6335fa21/notes/Reflectionless_eigenstates.pdf">Reflectionless
    eigenstates of the $\text{sech}^2$ potential</A> (2007).
 3. C.S. Park, <A HREF="https://www.sciencedirect.com/science/article/pii/S0375960111009340">Transmission time of a particle in the reflectionless sech-squared potential</A> (2011).