Anomalous transmission in a hierarchical lattice

We present an analytical method of studying "extended" electronic eigenstat es of a diamond hierarchical lattice, which may be taken as the simplest of the hierarchical models recently proposed for stretched polymers. We use i ntuitive arguments and a renormalization-group method to determine the dist ribution of amplitudes of the wave functions corresponding to some of these ''extended" eigenstates. An exact analysis of the end-to-end transmission property of arbitrarily large finite lattices reveals an anomalous behavior . It is seen that while for a special value of the energy the lattice, howe ver large, becomes completely transparent to an incoming electron, for the other energy eigenvalues the transmission decreases with system size. For o ne such energy eigenvalue we analytically obtain the precise scaling form o f the transmission coefficient. The same method can easily be adopted for o ther energies.