Solutions to the Schrödinger equation on some fractal lattices

The Schrödinger equation is solved on a variety of fractal lattices using a recursive technique. In this method, the energy levels and wave functions on a lattice with $N_n$ sites is calculated in terms of the corresponding quantities on a smaller lattice with $N_{n-1\mathrm{}}$ sites, via a kind of decimation process. As $n\to \infty$ the resulting energy levels are discrete, very closely spaced, and highly degenerate. Smoothed densities of states have a wide variety of singularities.