Metal-insulator transition for the almost Mathieu operator

We prove that for Diophantine ! and almost every µ; the almost Mathieu operator, (H!;‚;µ“)(n )=“ (n +1 ) +“ (ni 1) +‚ cos 2…(!n+µ)“(n), exhibits localization for ‚> 2 and purely absolutely continuous spectrum for ‚< 2:

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