Periodic “weighted” operators

Consider the periodic weighted operator Ty=−ρ−2(ρ2y′)′ in , where the real function ρ is 1-periodic positive, and let q=ρ′/ρ L2(0,1). The spectrum of T consists of intervals separated by gaps γn,n 1, with the lengths γn. Let hn be a height of the corresponding slit in the quasimomentum domain and let gn,n 1, be the gap with the length gn of the operator . The following results are obtained: (i) the quasimomentum k for the weighted operator T is constructed and the basic properties of k are studied, (ii) two-sided estimates of ℓ2 norms of the sequences {γn/n}1∞,{hn}1∞,{gn}1∞, in terms of , (iii) the asymptotics of the gap length γn as n→∞, are determined. The proofs are based on the analysis of the quasimomentum as a conformal mapping, embedding theorems and the identities between the quasimomentum and the potential. In order to prove these results the asymptotics of the fundamental solutions and the Lyapunov function are obtained at high energy.