On the spectrum of the periodic focusing Zakharov-Shabat operator

TL;DR

This paper analyzes the spectrum of the focusing Zakharov-Shabat operator on the circle, revealing its structure, bounds, and localization properties, especially in the semiclassical limit, with both theoretical proofs and illustrative examples.

Contribution

It provides new insights into the spectral properties of the focusing Zakharov-Shabat operator, including the resolvent set structure, spectral bounds, and localization behavior in the semiclassical regime.

Findings

Resolvent set consists of two connected components.

Derived bounds on Floquet and Dirichlet spectra, some depending on the semiclassical parameter.

Spectrum localizes to a 'cross' in the spectral plane as the semiclassical limit is approached.

Abstract

The spectrum of the focusing Zakharov-Shabat operator on the circle is studied, and its explicit dependence on the presence of a semiclassical parameter is also considered. Several new results are obtained. In particular: (i) it is proved that the resolvent set is comprised of two connected components, (ii) new bounds on the location of the Floquet and Dirichlet spectra are obtained, some of which depend explicitly on the value of the semiclassical parameter, (iii) it is proved that the spectrum localizes to a "cross" in the spectral plane in the semiclassical limit. The results are illustrated by discussing several examples in which the spectrum is computed analytically or numerically.