Cantor Spectrum and KDS Eigenstates

• DOI: 10.1007/s00220-006-0071-8
• Publication Date: 2006-08-18T11:45:07Z
• Authors: Joaquim Puig
• Keywords: Eigenfunction; Schrödinger's cat

In this note we consider KDS eigenstates of one-dimensional Schrodinger operators with ergodic potential, which are a class of generalized eigenfunctions including Bloch eigenstates. We show that if the spectrum, restricted to an interval, has zero Lyapunov exponents and is a Cantor set, then for a residual subset of energies, KDS eigenstates do not exist. In particular, we show that the quasi-periodic Schrodinger operators whose Schrodinger quasi-periodic cocycles are reducible for all energies have a limit band-type spectrum.