Properties of the simplest class of self-similar potentials are analyzed.
Wave functions of the corresponding Schr\"odinger equation provide bases of
representations of the q-deformed Heisenberg-Weyl algebra. When the parameter
q is a root of unity the functional form of the potentials can be found
explicitly. The general q3=1 and the particular q4=1 potentials are given
by the equianharmonic and (pseudo)lemniscatic Weierstrass functions
respectively.Comment: 15 pp, Latex, to appear in Lett.Math.Phy