The spectrum of a class of almost periodic operators (Q1415081)

For almost Mathieu operators which are defined by \[ (H(\alpha,\,\beta,\,\theta)\xi)_n =\xi_{n+1}+\xi_{n-1}+2\beta\cos(2\pi\alpha n+\theta)\xi_n,\quad \xi\in l^2(\mathbb Z), \] where \(\alpha,\,\beta\) and \(\theta\) are real parameters, the author proves that the occurrence of Cantor spectrum and the existence, for every point in the spectrum and suitable phase parameters, of at least one localized eigenfunction which decays exponentially, are inconsistent properties.