Estimates for the Hill Operator, I

Abstract

We consider the Hill operator T=−d2/dt2+q(t) in L2(R), 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 T⩾0. Introduce the sequences γ={|γn|}, h={hn}, g={|gn|}, ρ={ρn} and the norms ‖f‖2m=∑n⩾1(2πn)2mf2n, m⩾0. The following results are obtained: (i) double-sided estimates of ‖γ‖, ‖h‖1, ‖g‖1 in terms of ‖q‖2=∫10q(t)2dt, (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.