Closed formulae for Green functions on fractal lattices

Closed form solutions are found for Schrödinger Green functions x (corner-to-same-corner) and y (corner-to-other-corner) on the 3-simplex, the 4-simplex and two other fractal lattices. It is elementary to derive the well known recursions of the form x → X(x, y), y → Y(x, y) relating generations n and n + 1 of the lattice. We have now obtained an infinite hierarchy of exact solutions to these recursion expressing x, y as closed formulae in terms of initial condition (energy) and generation number n. This amounts to constructing a hierarchy of orbits for the dynamics of the renormalization map. To our knowledge, no other such solutions have been found previously. For each of these solutions, y scales as a power of the lattice size, which is of interest in relation to conductance scaling in Anderson localization. One cannot study power-law scaling numerically using the recursions alone, since the asymptotic behavior of y at large length scale is chaotic with respect to the energy parameter. Thus, the chances of finding any power-law solution are measure zero in the initial conditions, most of which lead to superlocalized, stretched exponential behaviors of y with lattice size. In contrast, the exact solutions each connect a value of energy to an unambiguous asymptotic behavior.