Periodic and limit-periodic discrete Schr\"odinger operators

The theory of discrete periodic and limit-periodic Schr\"odinger operators is developed. In particular, the Floquet--Bloch decomposition is discussed. Furthermore, it is shown that an arbitrarily small potential can add a gap for even periods. In dimension two, it is shown that for coprime periods small potential terms don't add gaps thus proving a Bethe--Sommerfeld type statement. Furthermore limit-periodic potentials whose spectrum is an interval are constructed.