A class of exactly solvable potentials. I. One-dimensional Schrödinger equation

The one-dimensional Schrödinger equation is solved for a new class of potentials with varying depths and shapes. The energy eigenvalues are given in algebraic form as a function of the depth and shape of the potential. The eigenfunctions and scattering function are also given in closed form. For certain shapes these potentials resemble the mean field of an atomic nucleus.