Electromagnetic Schrödinger Operators on Periodic Graphs with General Conditions at Vertices

Abstract

The main aim of the paper is the study of the Fredholm property and essential spectra of electromagnetic Schrodinger operators $$\mathcal{H}$$ on graphs periodic with respect to a group $$\mathbb{G}$$ isomorphic to ℤk. We consider the Schrodinger operators with nonperiodic electric and magnetic potentials and with general nonperiodic conditions on the vertices.We associate with $$\mathcal{H}$$ a family Lim( $$\mathcal{H}$$ ) of limit operators $$\mathcal{H}$$ h generated by sequences h: $$\mathbb{G}$$ ∍hm → ∞. The main results of the paper are: (i) $$\mathcal{H}$$ is a Fredholm operator if and only if all limit operators $$\mathcal{H}$$ h of $$\mathcal{H}$$ are invertible, (ii) 1 $$s{p_{ess}}H\, = \,\bigcup\limits_{{H^h} \in Lim\left( H \right)} {sp{H^h}} $$ where spess $$\mathcal{H}$$ is the essential spectrum of $$\mathcal{H}$$ . Formula (1) is applied to the study of the essential spectrum of Schrodinger operators whose potentials are perturbations of periodic magnetic and electric potentials by slowly oscillating terms.