The spectrum of the almost Mathieu operator is a Cantor set for all irrational frequencies

Analysis 1 Minute

You need to know: Set {\mathbb Z} of integers, infinite sum \sum\limits_{n \in {\mathbb Z}} x_n, Cantor set.

Background: Let l^2(\mathbb Z) be the set of all infinite sequences x=(\dots, x_{-1}, x_0, x_1, \dots) such that \sum\limits_{n \in {\mathbb Z}} x_n^2 < \infty. A map T:l^2(\mathbb Z) \to l^2(\mathbb Z) is called invertible if for every y \in l^2(\mathbb Z) there exists a unique x \in l^2(\mathbb Z) such that T(x)=y. The almost Mathieu operator is the map H:l^2(\mathbb Z) \to l^2(\mathbb Z) mapping each x \in l^2(\mathbb Z) into (Hx)_n = x_{n+1} + x_{n-1} + 2 \lambda \cos 2\pi (\theta + n \alpha) x_n, \, n\in{\mathbb Z}, where \lambda\neq 0, \alpha, and \theta are real parameters, called coupling, frequency, and phase, respectively. The set of all t\in{\mathbb R} for which map H_t:l^2(\mathbb Z) \to l^2(\mathbb Z) given by (H_tx)_n = t x_n - (Hx)_n, n\in{\mathbb Z}, is not invertible is called the spectrum of H.

The Theorem: On 17th March 2005, Artur Avila and Svetlana Jitomirskaya submitted to arxiv a paper in which they proved that the spectrum of the almost Mathieu operator is a Cantor set for all irrational \alpha and for all \theta and all \lambda \neq 0.

Short context: The almost Mathieu operator and its spectrum arise from applications in physics. The Theorem confirms the conjecture proposed by Azbel in 1964. In 1981, Mark Kac offered ten martinis for anyone who could prove or disprove it, and since then the problem has been known as “The Ten Martini Problem”.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 9.6 of this book for an accessible description of the Theorem.

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Published by Bogdan Grechuk

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