Some properties of the spectrum of the Sierpinski gasket in a magnetic field

Abstract

The spectrum of the Sierpinski gasket in a magnetic field is discussed using a synthetic Green's-function technique. This directly relates the spectrum of an ($n+1$)-stage gasket to that of its $n$-stage components and allows effective use of the implicit symmetry. It is found that the ($n+1$)-stage spectrum is nested with three eigenvalues belonging to the three different representations between any two consecutive stage-$n$ eigenvalues. For the special points where the eigenvalues for stage $n$ and $n+1$ coincide we provide proofs for the two Rammal-Toulouse [Phys. Rev. Lett. 49, 1194 (1982)] nesting properties, derive explicit expressions for the evolution of the degeneracies, and construct the eigenfunctions. Some of the implications and remaining problems are also discussed.