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

Abstract

We consider the family of operators H(ε):=-d2dx2+εV in R with almost-periodic potential V. We study the behaviour of the integrated density of states (IDS) N(H (ε) ; λ) when ε→ 0 and λ is a fixed energy. When V is quasi-periodic (i.e. is a finite sum of complex exponentials), we prove that for each λ the IDS has a complete asymptotic expansion in powers of ε; 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 S of energies (which we call the super-resonance set) such that for any λ∉S there is a complete power asymptotic expansion of IDS, and when λ∈S, then even two-terms power asymptotic expansion does not exist. We also show that the super-resonant set S is uncountable, but has measure zero. Finally, we prove that the length of any spectral gap of H (ε) has a complete asymptotic expansion in natural powers of ε when ε→ 0.