On the characteristic polynomial of the almost Mathieu operator期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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On the characteristic polynomial of the almost Mathieu operator

Authors:	Michael P Lamoureux James A Mingo

Institution:	Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2T 1A1 ; Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, Canada K7L 3N6

Abstract:	Let $A_\theta$ be the rotation C-algebra for angle $\theta$ . For $\theta = p/q$ with and relatively prime, $A_\theta$ is the sub-C-algebra of $M_q(C(\mathbb{ T}^2))$ generated by a pair of unitaries and satisfying $uv = e^{2 \pi i \theta} v u$ . Let $\displaystyle h_{\theta, \lambda} = u + u^{-1} + \lambda/2(v + v^{-1})$ be the almost Mathieu operator. By proving an identity of rational functions we show that for even, the constant term in the characteristic polynomial of $h_{\theta, \lambda}$ is $(-1)^{q/2}(1 + (\lambda/2)^q) - (z_1^q + z_1^{-q} + (\lambda/2)^q(z_2^q + z_2^{-q}))$ .

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