The transfer matrices of the self-similar fractal potentials on the Cantor set

On the basis of an exact formalism, a system of functional equations for the tunnelling parameters of self-similar fractal potentials (SSFPs) is obtained. Three different families of solutions are found for these equations, two of them having one parameter and one being free of parameters. Both one-parameter solutions are shown to be described, in the long-wave limit, by a fractal dimension. At the same time, the third solution yields transfer matrices which are analytical in this region, similar to the case of structures with the `Euclidean geometry'. We have revealed some manifestations of scale invariance in the physical properties of SSFPs. Nevertheless, in the common case these potentials do not possess, strictly speaking, this symmetry. The point is that SSFPs in the common case are specified, in contrast to the Cantor set, by two length scales but not one. A particular case when SSFPs are exactly scale invariant to an electron with well defined energy is found.