Perturbation theory for almost-periodic potentials I. One-dimensional case

We consider the family of operators $H^{(ϵ)}:=-\frac{d^2}{d{x}^2}+ϵV$ in $\mathbb{R}$ with almost-periodic potential $V$. We study the behaviour of the integrated density of states (IDS) $N(H^{(ϵ)};\lambda )$ when $ϵ\to 0$ 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 $ϵ$; 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 $\mathcal{S}$ of energies (which we call $\text{the super-resonance set}$) such that for any $\sqrt{\lambda }\in ̸\mathcal{S}$ there is a complete power asymptotic expansion of IDS, and when $\sqrt{\lambda }\in \mathcal{S}$, 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 $H^{(ϵ)}$ has a complete asymptotic expansion in natural powers of $ϵ$ when $ϵ\to 0$.