Abstract
We consider a 2-dimensional discrete operator which we call the Discrete Magnetic Laplacian (DML); it is an analogue of the magnetic Schrödinger operator. It follows from well known arguments that DML has the same spectrum (as a subset inR) as the Almost Mathieu operator (AM). They also have the same Integrated Density of States (IDS) which is known to be continuous. We show that DML is an element in a II1-factor and its IDS can be expressed through the trace in the II1-factor. It follows that DML never has anyL 2-eigenfunctions (i.e. has no point spectrum). Then we formulate a natural algebraic conjecture which implies that the spectrum of DML (hence the spectrum of AM) is a Cantor set.
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Communicated by B. Simon
Supported by NSF grant DMS-9222491
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Shubin, M.A. Discrete Magnetic Laplacian. Commun.Math. Phys. 164, 259–275 (1994). https://doi.org/10.1007/BF02101702
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DOI: https://doi.org/10.1007/BF02101702