An almost periodic model for general reflectionless spectral data

A basic feature of second order almost periodic differential and finite-difference operators is the reflectionless property of their Weyl functions. Conversely, each regular enough pair of reflectionless Nevanlinna functions generate an almost periodic operator on the real line whose half-line Weyl functions coincide with the given pair. Until recently, for operators with unbounded spectra, this scheme worked under quite restrictive assumptions on the “quality” of the spectrum. We extend it to cover all homogeneous spectra, and, more generally, to all spectra satisfying Widom and DCT conditions.

This is a joint work with M. Lukic and P. Yuditskii.
The author is partially supported by the Russian Science Foundation grant 19-11-0005.