Electromagnetic Schrödinger Operators on Periodic Graphs with General Conditions at Vertices

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Abstract

The main aim of the paper is the study of the Fredholm property and essential spectra of electromagnetic Schrödinger operators H on graphs periodic with respect to a group G isomorphic to ℤk. We consider the Schrödinger operators with nonperiodic electric and magnetic potentials and with general nonperiodic conditions on the vertices.We associate with H a family Lim(H) of limit operators Hh generated by sequences h: G ∍hm → ∞. The main results of the paper are: (i) H is a Fredholm operator if and only if all limit operators Hh of H are invertible, (ii) (Formula Presented.) where spessH is the essential spectrum of H. Formula (1) is applied to the study of the essential spectrum of Schrödinger operators whose potentials are perturbations of periodic magnetic and electric potentials by slowly oscillating terms.

Original languageEnglish
Pages (from-to)185-205
Number of pages21
JournalRussian Journal of Mathematical Physics
Volume26
Issue number2
DOIs
StatePublished - 1 Apr 2019

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