Exact results for infinite and finite Sierpinski gasket fractals: Extended electron states and transmission properties

We present an exact calculation to show that an infinite Sierpinski gasket fractal supports an infinite number of extended electron states. We work within the real space renormalization group (RSRG) scheme and show that by analysing the recursion relations for the Hamiltonian parameters one can extract the eigenvalues for an infinity of eigenstates that are of extended character. We also calculate the transmission coefficient for fractals of arbitrarily large generation. For the energy eigenvalues corresponding to the extended electron states, the transmission coefficient exhibits a novel feature. It turns out to be scale invariant with a value between zero and one depending upon the initial choice of the on-site potentials and the nearest-neighbour hopping integrals.