On the nature of the spectrum of the quasi-periodic Schrödinger operator.
\textit{E. I. Dinaburg} and \textit{Ya. G. Sinai} [Funct. Anal. Appl. 9, 279--289 (1975; Zbl 0333.34014)] were the first to prove the existence of a large class of quasi-periodic potentials \(q\) for which the spectrum of \(H(q):= -d^2/dt^2+ q\) is, in the high-energy regime, absolutely continuous and a Cantor set. \textit{J. Moser} [Comment. Math. Helv. 56, 198--224 (1981; Zbl 0477.34018)] constructed a limit-periodic \(q\) for which the spectrum of \(H(q)\) is a Cantor set and \textit{J. Avron} and \textit{B. Simon} [Commun. Math. Phys. 82, 101--120 (1981; Zbl 0484.35069)] and \textit{V. A. Chulaevskij} [Usp. Mat. Nauk 36, 143--144 (1981; Zbl 0487.34021)] established that this was the generic situation for limit-periodic potentials. Using results of Moser's paper, the authors show that, given \(r\in\mathbb{N}_0\), generically quasi-periodic potentials \(q\in C^r\) give rise to a Lyapunov exponent associated with the eigensolutions of \(H(q)\) which is positive on a set of locally positive measure. The authors' second theorem states that generically the spectrum of \(H(q)\) is a Cantor set for quasi-periodic potentials \(q\in C^0\). This result can already be found in [\textit{R. Johnson}, J. Differ. Equations 91, 88--110 (1991; Zbl 0734.34074)], but was ``not completely proved'' there, as the authors put it. Since further explanation is lacking and the proof given here draws heavily on this paper and is ``sketchy at some points, especially when the necessary arguments are to be found in other papers'', the presentation is less connected and transparent than one might ideally wish. Finally, the authors consider the example \[ q_0(t):= \cos t+ \cos\sqrt{2} t+ \cos\sqrt{3} t \] and provide numerical evidence for the conjecture that the spectrum of \(H(q_0)\) has at least four gaps in the low-energy regime.
scientific article; zbMATH DE number 1915788