Reducibility of one-dimensional quasi-periodic Schrodinger equations

Consider the time-dependent linear Schrodinger equation i(q) over dot(n) = is an element of(q(n+1) + q(n-1)) + V(x + n omega)q(n) + delta Sigma(m is an element of z) a(mn) (theta+xi t)q(m), n is an element of Z, where V is a nonconstant real-analytic function on T, omega satisfies a certain Diophantine condition and a(mn) (theta) is real-analytic on T-b, b is an element of Z(+), decaying with vertical bar m vertical bar and vertical bar n vertical bar. We prove that, if epsilon and delta 5 are sufficiently small, then for a.e. x is an element of T and "most" frequency vectors xi is an element of T-b, it can be reduced to an autonomous equation. Moreover, for this non-autonomous system, "dynamical localization" is maintained in a quasi-periodic time-dependent way. (C) 2015 Elsevier Masson SAS. All rights reserved.