Jeroen S.W. Lamb and F. Wijnands:

From multi-site to on-site transfer matrix models for selfsimilar chains

Abstract:

We show how transfer matrix models on chains that are
selfsimilar (renormalizable) with respect to a substitution rule can be
transformed from multi-site models in which transfer matrices depend on
the nature of a finite number of neighbouring sites, to on-site models in
which transfer matrices depend on the nature of one site only. We present
sufficient conditions and show that these conditions are satisfied in the
case of quasiperiodic chains of two symbols that are renormalizable with
respect to an invertible substitution rule.

We illustrate the application of our results to tight binding Schr\"odinger
equations modeling the electronic behaviour of selfsimilar chains of
atoms and to models describing the transmission of light through selfsimilarly
stacked multilayers.