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Lipshitz continuity of gap boundaries for Hofstadter-like spectra

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Abstract

We consider an effective HamiltonianH representing the motion of a single-band-two-dimensional electron in a uniform magnetic field. ThenH belongs to the rotation algebra, namely the algebra of continuous functions over a non-commutative 2-torus. We define a non-commutative analog of smooth functions by mean of elements of classC l,n, wherel andn characterize respectively the degree of differentiability with respect to the magnetic field and the torus variables. We show that ifH is of classC 1,3+ε, the gap boundaries of the spectrum ofH are Lipshitz continuous functions of the magnetic field at each point for which the gap is open.

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  1. Laboratoire Physique Quantique, Université Paul Sabatier, F-31062, Toulouse Cedex, France

    J. Bellissard

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  1. J. Bellissard
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Communicated by H. Araki

URA 505, CNRS

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Bellissard, J. Lipshitz continuity of gap boundaries for Hofstadter-like spectra. Commun.Math. Phys. 160, 599–613 (1994). https://doi.org/10.1007/BF02173432

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  • DOI: https://doi.org/10.1007/BF02173432

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