Solutions to the Schrödinger equation on some fractal lattices

The Schr\"odinger 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 ${N}_{n\ensuremath{-}1}$ sites, via a kind of decimation process. As $n\ensuremath{\rightarrow}\ensuremath{\infty}$ the resulting energy levels are discrete, very closely spaced, and highly degenerate. Smoothed densities of states have a wide variety of singularities.