Abstract
We consider Schrödinger operators Hh=(ih d+A)*(ih d+A) with the periodic magnetic field B=dA on covering spaces of compact manifolds. Using methods of a paper by Kordyukov, Mathai and Shubin [14], we prove that, under some assumptions on B, there are in arbitrarily large number of gaps in the spectrum of these operators in the semiclassical limit of the strong magnetic field h→0.
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Communicated by B. Simon
Acknowledgement I am very thankful to Bernard Helffer for bringing these problems to my attention and useful discussions and to Mikhail Shubin for his comments.
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Kordyukov, Y. Spectral Gaps for Periodic Schrödinger Operators with Strong Magnetic Fields. Commun. Math. Phys. 253, 371–384 (2005). https://doi.org/10.1007/s00220-004-1134-3
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DOI: https://doi.org/10.1007/s00220-004-1134-3