Magnetic Schrödinger operators on periodic discrete graphs

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• 作者：Evgeny Korotyaev^a ; ^{korotyaev@gmail.com} ; ^{e.korotyaev@spbu.ru} ; Natalia Saburova^b ; ^{n.saburova@gmail.com} ; ^{n.saburova@narfu.ru}
• 关键词：Spectral bands ; Flat bands ; Discrete magnetic Schr&ouml ; dinger operator ; Periodic graph
• 刊名：Journal of Functional Analysis
• 出版年：2017
• 出版时间：15 February 2017
• 年：2017
• 卷：272
• 期：4
• 页码：1625-1660
• 全文大小：626 K
• 卷排序：272

文摘

We consider magnetic Schrödinger operators with periodic magnetic and electric potentials on periodic discrete graphs. The spectrum of the operators consists of an absolutely continuous part (a union of a finite number of non-degenerate bands) plus a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. We estimate the Lebesgue measure of the spectrum in terms of the Betti numbers and show that these estimates become identities for specific graphs. We estimate a variation of the spectrum of the Schrödinger operators under a perturbation by a magnetic field in terms of magnetic fluxes. The proof is based on Floquet theory and a precise representation of fiber magnetic Schrödinger operators constructed in the paper.