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Discrete magnetic Laplacian. (English) Zbl 0811.46079

Summary: 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 in \(\mathbb{R}\)) 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 \(\text{II}_ 1\)- factor and its IDS can be expressed through the trace in the \(\text{II}_ 1\)-factor. It follows that DML never has any \(L^ 2\)- 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.

MSC:

46N50 Applications of functional analysis in quantum physics
46L60 Applications of selfadjoint operator algebras to physics
47N50 Applications of operator theory in the physical sciences
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