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arXiv:1711.03950 [math-ph]AbstractReferencesReviewsResources

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

Leonid Parnovski, Roman Shterenberg

Published 2017-11-10Version 1

We consider the family of operators $H^{(\epsilon)}:=-\frac{d^2}{dx^2}+\epsilon V$ in ${\mathbb R}$ with almost-periodic potential $V$. We prove that the length of any spectral gap has a complete asymptotic expansion in natural powers of $\epsilon$ when $\epsilon\to 0$. We also study the behaviour of the integrated density of states (IDS) $N(H^{(\epsilon)};\lambda)$ when $\epsilon\to 0$ 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 $\epsilon$; these powers are either integer, or in some special cases half-integer. We also prove that when the potential is not quasi-periodic, there is an exceptional set $\mathcal S$ (which we call $\hbox{the super-resonance set}$) 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. These results are new even for periodic $V$.

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