On Spectral Bands of Discrete Periodic Operators

We consider discrete periodic operator on ℤ^d with respect to lattices Γ⊂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 .