Estimates for the Hill Operator, I

We consider the Hill operator T=−d2/dt2+q(t) in L2( ), where q L2(0, 1) is a 1-periodic real potential. The spectrum of T consists of intervals σn=[λ−n−1, λ+n] separated by gaps γn=(λ−n, λ+n), n 1, with the lengths γn 0, and we assume λ+0=0. Let hn be a height of the corresponding slit in the quasimomentum domain and let ρn=π2 (2n−1)−σn>0 be the band shrinkage. We also have the gap gn, n 1, with the length gn, of the operator 0. Introduce the sequences γ={γn}, h={hn}, g={gn}, ρ={ρn} and the norms f 2m=∑n 1 (2πn)2m f2n, m 0. The following results are obtained: (i) double-sided estimates of γ , h 1, g 1 in terms of q 2=∫10 q(t)2 dt, (ii) estimates of ρ in terms of γ , h 1, g 1, q , and (iii) a generalization of (i) and (ii) for more general potentials. The proof is based on the analysis of the quasimomentum as the conformal mapping, the embedding theorems and the identities.