A new exactly integrable hypergeometric potential for the Schr\"odinger equation

We introduce a new exactly integrable potential for the Schr\"odinger equation for which the solution of the problem may be expressed in terms of the Gauss hypergeometric functions. This is a potential step with variable height and steepness. We present the general solution of the problem, discuss the transmission of a quantum particle above the barrier, and derive explicit expressions for the reflection and transmission coefficients

WKB-approach for the 1D hydrogen atom

Taking into account results of WKB-approximation, we derive exact quantum energies and wave functions of even and odd states in the one-dimensional Coulomb potentia