The almost Mathieu operator is the discrete Schrödinger operator Hα,β,θ on `2(Z) defined via (Hα,β,θf)(k) = f(k+1)+f(k−1)+β cos(2παk+θ)f(k). We derive explicit estimates for the eigenvalues at the edge of the spectrum of the finite-dimensional almost Mathieu operator H α,β,θ. We furthermore show that the (properly rescaled) m-th Hermite function φm is an approximate eigenvector of H α,β,θ, and ...

For almost Mathieu operators, it is shown that the occurrence of Cantor spectrum and the existence, for every point in the spectrum and suitable phase parameters, of at least one localized eigenfunction which decays exponentially are inconsistent properties.

We classify the complex Laurent polynomials with the property that their powers have no constant term. The result confirms a conjecture of Mathieu for the case of tori. (A different case would imply Keller’s Jacobian Conjecture.)

We prove that the spectrum of the almost Mathieu operator is absolutely continuous if and only if the coupling is subcritical. This settles Problem 6 of Barry Simon’s list of Schrödinger operator problems for the twenty-first century.

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In symbolic manipulation packages such as MATHEMATICA it is possible to substitute the built-in function for integration by a user-programmed specific integration function and symbolically evaluate exact high-order Picard iterates for systems of linear differential equations with time-periodic parameter-dependent coefficients. With this technique we get excellent approximations in feasible CPU ...

Kicked Harper operators and on–resonance double kicked rotor operators model quantum systems whose semiclassical limits exhibit chaotic dynamics. Recent computational studies indicate a striking resemblance between the spectrums of these operators. In this paper we apply C–algebra methods to explain this resemblance. We show that each pair of corresponding operators belong to a common rotation ...