Periodic and limit-periodic discrete Schrödinger operators

Abstract

The theory of discrete periodic and limit-periodic Schrödinger 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.

References in corpus (2)

• Optimality of log Hölder continuity of the integrated density of states
• Spectral properties of a limit-periodic Schrödinger operator in dimension two