Schrodinger operators with sparse potentials: Asymptotics of the Fourier transform of the spectral measure

We study the pointwise behavior of the Fourier transform of the spectral measure for discrete one-dimensional Schrodinger operators with sparse potentials. We find a resonance structure which admits a physical interpretation in terms of a simple quasiclassical model. We also present an improved version of known results on the spectrum of such operators.