KAM THEORY IN MOMENTUM SPACE AND QUASI-PERIODIC SCHRODINGER-OPERATORS

We construct a family of quasiperiodic Schrodinger operators in dimension one and in the tight binding approximation, having purely absolutely continuous spectrum. We work in momentum space and use a superconvergent approximation scheme to construct a unitary transformation that diagonalizes these operators on L2(B), B = [-pi, pi) being the first Brillouin zone of the unperturbed part. The transformed operators are multiplications by a function E(infinity)(k) is-an-element-of L(infinity) (B) which might have a dense set of jump discontinuities and is the uniform limit as n --> infinity of functions E(n)(k) with a finite number of discontinuities. Our control on the functions E(n)(k) and its first two derivatives is good enough to ensure the absence of pure point and singular continuous spectrum.