Estimates for the Hill Operator, I

Inverse Spectral Problem for the Periodic Camassa-Holm Equation

“…Thus, the mapping k (and, therefore, h) allows to recover the sequences {z ± n }, {m n } from {h n } and inversely {h n } from {z ± n }, {m n }. In addition, the quasimomentum k is crucial to derive the double-sided estimates in assertion i) (see [22], [27], [29]). …”

“…Two-sided estimates for various parameters of the Hill operator (the norm of a periodic potential, effective masses, gap lengths, height of slits |h n | etc.) were obtained in [27,28,29]. The results for the Hill operator [34,35], [23,24] were extended in [25] to the case of singular potentials V with V ∈ L 2 (T).…”

Estimates for periodic Zakharov–Shabat operators

“…Then we should study the metric properties of a conformal mapping from C + onto a "comb" K + (h). A similar analysis was done partially in [11][12][13][14][15][16]. In the present paper we use an approach, based on the identities for the Dirichlet integral (1.2) from [11] and the estimates from Theorem 2.8 and Lemma 3.1.…”

“…Remark that Korotyaev obtained the two-sided estimates for the 2 ω -norms, where ω n = 1 + (2πn) 2m , m 0, for the even case h −n = h n , n ∈ Z [15][16][17], and for 2 1 -norms without symmetry [14,19] (in all these estimates the factor c 0 = e h ∞ /π is absent).…”

“…There are no a priori estimates for the case V (m) ∈ L 2 (0, 1), m 2. The proof of a priori estimates in [14][15][16][17][18][19] is based on the analysis of the quasi-momentum as the conformal mapping.…”