Self-similar analogues of Stark ladders: a path to fractal potentials

We treat the eigenvalue problem posed by self-similar potentials, i.e. homogeneous functions under a particular affine transformation, by means of symmetry techniques. We find that the eigenfunctions of such problems are localized, even when the potential does not rise to infinity in every direction. It is shown that the logarithm of the energy displays levels contained in families that are analogous to Wannier-Stark ladders. The position of each ladder is proved to be determined by the specific details of the potential and not by its transformation properties. This is done by direct computation of matrix elements. The results are compared with numerical solutions of the Schr\"odinger equation.