Jitomirskaya, Svetlana Ya. Metal-insulator transition for the almost Mathieu operator. (English) Zbl 0946.47018 Ann. Math. (2) 150, No. 3, 1159-1175 (1999). 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. Cited in 4 ReviewsCited in 122 Documents 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 Keywords:metal-insulator transition; Mathieu operator; localization; purely absolutely continuous spectrum; Aubry-André conjecture PDFBibTeX XMLCite \textit{S. Ya. Jitomirskaya}, Ann. Math. (2) 150, No. 3, 1159--1175 (1999; Zbl 0946.47018) Full Text: DOI arXiv EuDML Link