LAUSR

We consider the family of operators $${H^{(\varepsilon)}:=-\frac{d^2}{dx^2}+\varepsilon V}$$H(ε):=-d2dx2+εV in $${\mathbb{R}}$$R with almost-periodic potential V. We study the behaviour of the integrated density of states (IDS) $${N(H^{(\varepsilon)};\lambda)}$$N(H(ε);λ) when $${\varepsilon\to 0}$$ε→0 and $${\lambda}$$λ is a fixed energy. When V is quasi-periodic (i.e. is a finite sum of complex exponentials), we prove that for each $${\lambda}$$λ the IDS has a complete asymptotic expansion in powers of $${\varepsilon}$$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 $${\mathcal{S}}$$S of energies (which we call the super-resonance set) such that for any $${\sqrt\lambda\not\in\mathcal{S}}$$λ∉S there is a complete power asymptotic expansion of IDS, and when $${\sqrt\lambda\in\mathcal{S}}$$λ∈S, then even two-terms power asymptotic expansion does not exist. We also show that the super-resonant set $${\mathcal{S}}$$S is uncountable, but has measure zero. Finally, we prove that the length of any spectral gap of $${H^{(\varepsilon)}}$$H(ε) has a complete asymptotic expansion in natural powers of $${\varepsilon}$$ε when $${\varepsilon \to 0}$$ε→0.