Zero measure spectrum for the almost Mathieu operator. (English) Zbl 0814.11040

The almost Mathieu (or Harper’s) operator on \(\ell^ 2(\mathbb{Z})\) is given by \[ (H_{\alpha,\lambda,\theta}u)(n)= u(n + 1) + u(n -1) + \lambda \cos (2\pi\alpha n + \theta) u(n) \] and is related to Schrödinger’s operator and to the rotation number. Upper and lower estimates for the Lebesgue measure of the spectrum of the operator are obtained for each \(\lambda\), \(\theta\), \(\alpha\). The measure is deduced to be \(| 4- | \lambda||\) for all \(\lambda\), \(\theta\) and for \(\alpha\) with unbounded partial quotients, thus for all \(\lambda\), \(\theta\) and for almost all \(\alpha\). When \(| \lambda| = 2\), the spectrum is a Cantor set of measure 0. In an interesting mix of operator and number theory, it is further shown that the Hausdorff dimension of the spectrum is at most 1/2 for irrational \(\alpha\) such that for any \(c > 0\), there are infinitely many rationals \(p_ n/q_ n\) satisfying \[ \left | \alpha - {p_ n\over q_ n}\right | < {c\over q_ n^ 4}, \] a set of dimension 1/2 (for \(\alpha\) with unbounded quotients, the denominator on the RHS is \(q^ 2_ n)\).


11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
34L05 General spectral theory of ordinary differential operators
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70H99 Hamiltonian and Lagrangian mechanics
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