The harmonic oscillator and Coulomb potentials - Two exceptions from the point of view of a function theory

We show that the Coulomb potential, one- and three-dimensional harmonic oscillator potentials are the only potentials belonging to a certain class which have the following property. The Schrodinger equation has infinitely many eigenvalues belonging to the discrete spectrum with the eigenfunctions having only a finite number of zeros in a complex plane. We show that it is due to this fact that these two potentials are the only potentials, belonging to the class defined in the paper, for which the semiclassical quantization gives an energy spectrum coinciding with the results of an exact quantum mechanical treatment.