Estimates for periodic Zakharov-Shabat operators

TL;DR

This paper provides a priori estimates for spectral data of periodic Zakharov-Shabat operators, linking gap lengths, effective masses, and slit heights through conformal mapping analysis.

Contribution

It introduces new a priori bounds for spectral parameters of Zakharov-Shabat operators using conformal mapping techniques.

Findings

Derived estimates for gap lengths, effective masses, and slit heights.

Established bounds in weighted l^p-norms for spectral sequences.

Applied conformal mapping analysis to spectral theory.

Abstract

We consider the periodic Zakharov-Shabat operators on the real line. The spectrum of this operator consists of intervals separated by gaps with the lengths $|g_n|\ge 0, n\in \Z$. Let $\m_n^\pm$ be the corresponding effective masses and let $h_n$ be heights of the corresponding slits in the quasimomentum domain. We obtain a priori estimates of sequences $g=(|g_n|)_{n\in \Z},\m^\pm=(\m_n^\pm)_{n\in \Z}, h=(h_n)_{n\in \Z}$ in terms of weighted $\ell^p-$norms at $p\ge 1$. The proof is based on the analysis of the quasimomentum as the conformal mapping.