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Metal-insulator transition for the almost Mathieu operator. (English) Zbl 0946.47018

Summary: We prove that for Diophantine \(\omega\) and almost every \(\theta\), the almost Mathieu operator, \((H_{\omega,\lambda,\theta}\Psi)(n)= \Psi(n+ 1)+ \Psi(n- 1)+ \lambda\cos 2\pi(\omega n+ \theta)\Psi(n)\), exhibits localization for \(\lambda> 2\) and purely absolutely continuous spectrum for \(\lambda< 2\). This completes the proof of (a correct version of) the Aubry-André conjecture.

MSC:

47B39 Linear difference operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
39A70 Difference operators
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
47N50 Applications of operator theory in the physical sciences
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