Duality and singular continuous spectrum in the almost Mathieu equation

. We study the almost Mathieu operator (h ;ff;` u)(n) = u(n+ 1) +u(n \Γ 1) + cos(ßffn + `)u(n) on ` 2 (Z), and prove that the dual of point spectrum is absolutely continuous spectrum. We use this to show that for = 2 it has purely singular continuous spectrum for a.e. pairs (ff; `). The ff's for which we prove this are explicit. x1. Introduction Our main goal in this paper is to study the almost Mathieu operator on ` 2 (Z) defined by (h ;ff;` u)(n) = u(n + 1) + u(n \Γ 1) + cos(ßffn + `)u(n): (1.1) Our results in Section 2 on measurability of (normalizable) eigenfunctions may be of broader applicability. For background on (1.1), see [19,23]. Our main result here concerns (1.1) at the self-dual point = 2. Theorem 1. Let ff be an irrational so there exist q n !1 and p n in Z with q 2 n fi fi fi fi ff \Γ p n q n fi fi fi fi ! 0 (1.2) as n !1. Then for a.e. `, h =2;ff;` has purely singular continuous spectrum. Remarks. 1. (1.2) is used because for such ff, Las...