Sir:

There are recursion formulae for many physically relevant functions solving the stationary Schrödinger equation. They connect stationary states with different energies. The energy law requires each system to have got a finite minimum energy. This distinguishes the recursion formula with minimum energy. For the linear harmonic oscillator, this leads to the Hermitean functions.

As a matter of fact, this approach represents a non-classical solution method of the stationary Schrödinger equation as envisaged by Schrödinger himself, ie, without the classical (!) boundary-valued method.

Thus, I would be interestesd to collaborate with you in this direction.