Periodic and limit-periodic discrete Schrödinger operators

Details Periodic and limit-periodic discrete Schrödinger operators Periodic and limit-periodic discrete Schrödinger operators

2011-08-07

arxiv.org/abs/1108.1584v1

Mathematics

Spectral Theory

Scientific paper

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.

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