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, Reflection of waves in an inhomogeneous absorbing me-
dium (1930).
[2] J. Lekner, Reflectionless eigenstates of the $\text{sech}^2$ potential (2007).