Exact solution of the Schrodinger equation for a potential well with a barrier and other potentials

A method is described, with which exactly solvable, one-dimensional, stationary Schrodinger equations can be derived from solved differential equations. The procedure is illustrated by the example of a Schrodinger equation for a potential well with a barrier of the form U(z)=v q 2 tanh 2 z+q 1 tanhz/coshz+q 0 . The eigenvalues and eigenfunctions of this potential are calculated exactly. The results are explicit, analytical expressions in closed form for the whole eigenvalue spectrum as well as for all the eigenfunctions.