Estimates for the Hill operator, II

Abstract

Consider the Hill operator Ty = - y ″ + q ( t ) y in L 2 ( R ) , where the real potential q is 1-periodic and q , q ′ ∈ L 2 ( 0 , 1 ) . The spectrum of T consists of spectral bands separated by gaps γ n , n ⩾ 1 with length | γ n | ⩾ 0 . We obtain two-sided estimates of the gap lengths ∑ n 2 | γ n | 2 in terms of ∫ 0 1 q ′ ( t ) 2 dt . Moreover, we obtain the similar two-sided estimates for spectral data (the height of the corresponding slit on the quasimomentum domain, action variables for the KdV equation and so on). In order prove this result we use the analysis of a conformal mapping corresponding to quasimomentum of the Hill operator. That makes it possible to reformulate the problems for the differential operator as the problems of the conformal mapping theory. Then the proof is based on the analysis of the conformal mapping, the embedding theorems and the identities. Furthermore, we obtain the similar two-sided estimates for potentials which have p ⩾ 2 derivatives.