Estimates for the Hill operator, I

We consider the Hill operator T = -d(2)/dt(2) + q(t) in L-2 (R), where q is an element of L-2 (0, 1) is a 1-periodic real potential. The spectrum of T consists of intervals sigma(n) = [lambda(n-1)(-), lambda(n)(+)] separated by gaps gamma(n) = (lambda(n)(-), lambda(n)(+)), n greater than or equal to 1, with the lengths \gamma(n)\greater than or equal to 0, and we assume lambda(0)(+) = 0. Let h(n) be a height of the corresponding slit in the quasimomentum domain and let rho(n) = pi(2) (2n -1) - \sigma(n)\ > 0 be the band shrinkage. We also have the gap g(n), n greater than or equal to 1, with the length \g(n)\, of the operator root T greater than or equal to 0. Introduce the sequences gamma = {\gamma(n)\}, h = {h(n)}, g = {\g(n)\}, rho = {rho(n)} and the norms parallel to f parallel to(m)(2) = Sigma(n greater than or equal to 1) (2 pi n)(2m) f(n)(2), m greater than or equal to 0. The following results are obtained: (i) double-sided estimates of parallel to gamma parallel to, parallel to h parallel to(1), parallel to q parallel to(1) in terms of parallel to q parallel to(2) = f(0)(1)q(t)(2) dt, (ii) estimates of parallel to rho parallel to in terms of parallel to gamma parallel to, parallel to h parallel to(1), parallel to g parallel to(1), parallel to q parallel to, 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. (C) 2000 Academic Press.