On the characteristic polynomial of the almost Mathieu operator

Abstract:

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