The isospectral torus of quasi-periodic Schrodinger operators via periodic approximations

We study the quasi-periodic Schrodinger operator -psi ''(x) + V(x)psi(x) = E psi(x), x is an element of R in the regime of "small" V(x) = Sigma(m is an element of Zv) c(m) exp(2 pi im omega x), omega = (omega(1), ... , omega(v)) is an element of R-v, vertical bar c(m)vertical bar <= epsilon exp(-kappa(0)vertical bar m vertical bar). We show that the set of reflectionless potentials isospectral with V is homeomorphic to a torus. Moreover, we prove that any reflectionless potential Q isospectral with V has the form Q(x) = Sigma(m is an element of Zv) d(m) exp(2 pi im omega x), with the same omega and with vertical bar d(m)vertical bar <= root 2 epsilon exp(-kappa(0)/2 vertical bar m vertical bar). Our derivation relies on the study of the approximation via Hill operators with potentials (V) over tilde (x) = Sigma(m is an element of Zv) c(m)exp(2 pi im (omega) over tildex), where (omega) over tilde is a rational approximation of.. It turns out that the multi-scale analysis method of Damanik and Goldstein ( Publ Math Inst Hautes Etudes Sci 119: 217-401, 2014) applies to these Hill operators. Namely, in Damanik et al. (Trans AmMath Soc, to appear, arXiv:1409.2147, 2016) we developed the multi-scale analysis for the operators dual to the Hill operators in question. Themain estimates obtained inDamanik et al. ( TransAmMath Soc, to appear, arXiv:1409.2147, 2016) allowus here to establish estimates for the gap lengths and the Fourier coefficients in a form that is considerably stronger than the estimates known in the theory of Hill operators with analytic potentials in the general setting. Due to these estimates, the approximation procedure for the quasi-periodic potentials is effective, despite the fact that the rate of approximation vertical bar omega - (omega) over tilde vertical bar similar to (T) over tilde (-delta), 0 < delta < 1/ 2 is slow on the scale of the period (T) over tilde of the Hill operator.