Parametrization of Periodic Weighted Operators in Terms of Gap Lengths

We consider the periodic weighted operator T y = ? ?2 (2 y 0) 0 in L 2 (R; (x) 2 dx), where is a 1-periodic real positive function with (0) = 1. We assume q = 0 == 2 L 2 (0; 1). The spectrum of T consists of intervals n = + n?1 ; ? n ] separated by gaps (? n ; + n); n > 1. Using essentially the square of the gap lengths, the centre of the gap and the Dirichlet eigenvalues on the unit interval we construct a gap length mapping q ! ` 2 ` 2 which provides a real analytic parametrization of the weight. For the proof we use nonlinear functional analysis in Hilbert space combined with sharp asymptotic estimates on the fundamental solution and the Lyapunov function in the high energy limit for complex q.