Almost periodic Schrödinger operators: A Review

Abstract

We review the recent rigorous literature on the one-dimensional Schrödinger equation, H = −d²/dx² + V(x) with V(x) almost periodic and the discrete (= tight binding) analog, i.e., the doubly infinite Jacobi matrix, h_(ij) = δ_(i,j+1) + δ_(i,j−1) + V_iδ_(i,j) with V_n almost periodic on the integers. Two themes dominate. The first is that the gaps in the spectrum tend to be dense, so that the spectrum is a Cantor set. We describe intuitions for this from the point of view of where gaps open, and from the point of view of anomalous long time behavior. We give a theorem of Avron and Simon, Chulaevsky, and Moser that for a generic sequence with Σ|a_n| < ∞, the continuum operator with V(x) = Σa_ncos(x/2^n) has a Cantor spectrum. The second theme involves unusual spectral types that tend to occur. We describe recurrent absolutely continuous spectrum, and show it occurs in some examples of the type just discussed. We give an intuition for dense point spectrum to occur, and some theorems on the occurrence of point spectrum. We sketch the proof of Avron and Simon, that for the discrete case with V_n = λ cos(2παn + ϑ), if λ > 2 and α is a Liouville number, then for a.e. ϑ, h has purely singular continuous spectrum.