On spectral bands of discrete periodic operators

We consider discrete periodic operator on $\mathbb Z^d$ with respect to lattices $Γ\subset\mathbb Z^d$ of full rank. We describe the class of lattices $Γ$ for which the operator may have a spectral gap for arbitrarily small potentials. We also show that, for a large class of lattices, the dimensions of the level sets of spectral band functions at the band edges do not exceed $d-2$.

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