On continuous spectrum of magnetic Schr\"odinger operators on periodic discrete graphs

Evgeny Korotyaev; Natalia Saburova

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arXiv · 2101.05571

On continuous spectrum of magnetic Schr\"odinger operators on periodic discrete graphs

Evgeny Korotyaev·Natalia Saburova

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Abstract

We consider Schr\"odinger operators with periodic electric and magnetic potentials on periodic discrete graphs. The spectrum of such operators consists of an absolutely continuous (a.c.) part (a union of a finite number of non-degenerate bands) and a finite number of eigenvalues of infinite multiplicity. We prove the following results: 1) the a.c. spectrum of the magnetic Schr\"odinger operators is empty for specific graphs and magnetic fields; 2) we obtain necessary and sufficient conditions under which the a.c. spectrum of the magnetic Schr\"odinger operators is empty; 3) the spectrum of the magnetic Schr\"odinger operator with each magnetic potential $t\alpha$, where $t$ is a coupling constant, has an a.c. component for all except finitely many $t$ from any bounded interval.

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BibTeX RIS

Evgeny Korotyaev, Natalia Saburova. 2021-01-14. On continuous spectrum of magnetic Schr\"odinger operators on periodic discrete graphs. https://arxiv.org/abs/2101.05571

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