Perturbation Theory for Almost-Periodic Potentials I: One-Dimensional Case

Abstract

We consider the family of operators {H^{(varepsilon)}:=-frac{d^2}{dx^2}+varepsilon V} in {mathbb{R}} with almost-periodic potential V. We study the behaviour of the integrated density of states (IDS) {N(H^{(varepsilon)};lambda)} when {varepsilonto 0} and {lambda} is a fixed energy. When V is quasi-periodic (i.e. is a finite sum of complex exponentials), we prove that for each {lambda} the IDS has a complete asymptotic expansion in powers of {varepsilon}; these powers are either integer, or in some special cases half-integer. These results are new even for periodic V. We also prove that when the potential is neither periodic nor quasi-periodic, there is an exceptional set {mathcal{S}} of energies (which we call the super-resonance set) such that for any {sqrtlambdanotinmathcal{S}} there is a complete power asymptotic expansion of IDS, and when {sqrtlambdainmathcal{S}}, then even two-terms power asymptotic expansion does not exist. We also show that the super-resonant set {mathcal{S}} is uncountable, but has measure zero. Finally, we prove that the length of any spectral gap of {H^{(varepsilon)}} has a complete asymptotic expansion in natural powers of {varepsilon} when {varepsilon to 0}.