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