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

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## 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 \({\varepsilon\to 0}\) 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 \({\varepsilon}\); 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 \({\sqrt\lambda\not\in\mathcal{S}}\) 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 \({H^{(\varepsilon)}}\) has a complete asymptotic expansion in natural powers of \({\varepsilon}\) when \({\varepsilon \to 0}\).

## Notes

### Acknowledgements

We are grateful to Alexander Sobolev for reading the preliminary version of this manuscript and making useful suggestions. We are grateful to the referee for useful comments. The research of the first author was partially supported by the EPSRC Grant EP/J016829/1.

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