  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 $q^3=1$ and the particular $q^4=1$ potentials are given
by the equianharmonic and (pseudo)lemniscatic Weierstrass functions
respectively.
