The Schrödinger equation with the $\text{sech}^2$ was first studied by Epstein in 1930 [1]. There is an extensive literature on this exactly solvable case. A recent reference is [2]. The 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. 


[1] P.S. Epstein, <A HREF="https://www.jstor.org/stable/85754
">Reflection of waves in an inhomogeneous absorbing me-
dium</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).