Eigenfunctions , transfer matrices , and absolutely continuous spectrum of one - dimensional SchroÈ dinger operators

…hu†…n† ˆ u…n‡ 1† ‡ u…nÿ 1† ‡ V …n†u…n† …1:1D† on `2…Z† (and the half-line problem, h‡, on `2…fn 2 Zjn > 0g† `2…Z‡†) with u…0† ˆ 0 boundary conditions. We will also discuss the continuum analog …Hu†…x† ˆ ÿu00…x† ‡ V …x†u…x† …1:1C† on L2…R† (and its half-line problem, H‡, on L2…0;1† with u…0† ˆ 0 boundary conditions). We will focus on a new approach to the absolutely continuous spectrum rac…h† and, more generally, Rac…h†, the essential support of the a.c. part of the spectral measures. What is new in our approach is that it relies on estimates on the transfer matrix, that is, the 2 2 matrix TE…n;m† which takes u…m‡ 1† u…m† to u…n‡ 1† u…n† for solutions u of hu ˆ Eu (in the contin-