Numerical estimates of the essential spectra of quantum graphs with delta-interactions at vertices

In this paper, we consider periodic metric graphs embedded in R-n, equipped by Schrodinger operators with bounded potentials q, and delta-type vertex conditions. Graphs are periodic with respect to a group G isomorphic to Z(m). Applying the limit operators method, we give a formula for the essential spectra of associated unbounded operators consisting of a union of the spectra of the limit operators defined by the potential q. We apply this formula and the spectral parameter power series (SPPS) method for the analysis of the essential spectral of Schrodinger operators with potentials q of the form q = q(0) + q(1), where q(0) is a periodic potential and q(1) is a slowly oscillating at infinity potential. The conjunction of both methods lead to an effective technique that can be used for performing numerical analysis as well. Several numerical examples demonstrate the effectiveness of our approach.