Essential Spectrum of Schrödinger Operators on Periodic Graphs

We give a description of the essential spectra of unbounded operators $\mathcal{H}_{q}$ on $L^{2}(\Gamma)$ determined by the Schrödinger operators $-d^{2}/dx^{2}+q(x)$ on the edges of $\Gamma$ and general vertex conditions. We introduce a set of limit operators of $\mathcal{H}_{q}$ such that the essential spectrum of $\mathcal{H}_{q}$ is the union of the spectra of limit operators. We apply this result to describe the essential spectra of the operators $\mathcal{H}_{q}$ with periodic potentials perturbed by terms slowly oscillating at infinity.