Surface spectra of discrete Laplacians

It is well known that the spectra of Laplacians on periodic graphs consist of a finite number of non-degenerate bands and eigenvalues of infinite multiplicity. We consider Laplacians on periodic graphs with boundaries. Under some conditions on the boundary the spectrum of the Laplacian has the so-called surface part, i.e., the spectrum corresponding to waves localized near the boundary. The surface spectrum is of particular interest due to its connection with the study of thermal transport, propagation of electromagnetic and acoustic waves. In this work we describe this spectrum.