Spectral properties of limit-periodic Schrödinger operators

Abstract

We survey results concerning the spectral properties of limit-periodic operators. The main focus is on discrete one-dimensional Schrödinger operators, but other classes of operators, such as Jacobi and CMV matrices, continuum Schrödinger operators and multi-dimensional Schrödinger operators, are discussed as well.We explain that each basic spectral type occurs, and it does so for a dense set of limit-periodic potentials. The spectrum has a strong tendency to be a Cantor set, but there are also cases where the spectrum has no gaps at all. The possible regularity properties of the integrated density of states range from extremely irregular to extremely regular. Additionally, we present background about periodic Schrödinger operators and almost-periodic sequences.In many cases we outline the proofs of the results we present.

Schrödinger operators with dynamically defined potentials

“…There is another dense set of sampling functions, where interesting spectral phenomena occur; compare [2,69]. Indeed, all statements in Theorems 5.6 and 5.7 except for singular continuity were shown by Avila in [2], and a proof of singular continuity was added by Damanik and Gan in [69,70]. The main reason why sampling functions in H are of interest is that they provide counterexamples to a conjecture of Simon, who had conjectured that positive Lyapunov exponents imply positive-measure spectrum.…”

Spectral properties of a limit-periodic Schrödinger operator in dimension two

“…Integrated density of states is investigated in [11]- [14]. Spectral properties of Schrödinger operators in l 2 (Z) with limit-periodic potentials are recently investigated in [15]. It regards such potentials as generated by continuous sampling along the orbits of a minimal translation of a Cantor group.…”

Spectral homogeneity of discrete one-dimensional limit-periodic operators

“…As soon as V departs from the class of periodic potentials, the spectral characteristics of H V become significantly more subtle and elusive. In this paper, we focus on the class of limit-periodic operators, that is, operators of the form (1) for which the potential can be written as an ℓ ∞ -limit of periodic sequences; see [1,5,6,7,12,13]. A typical example of such a potential is furnished by…”