Hankel operators with band spectra and elliptic functions

TL;DR

This paper develops a Floquet-Bloch theory for a class of Hankel operators with band spectra, revealing flat and non-flat bands, and connects their spectral properties to elliptic functions.

Contribution

It introduces a Floquet-Bloch decomposition for Hankel operators with band spectra and establishes a novel link between their spectral properties and elliptic functions.

Findings

Hankel operators have a band spectrum with flat and non-flat bands.

The secular determinant's analytic continuation is an elliptic function.

Explicit examples demonstrate coexistence of flat and non-flat bands.

Abstract

We consider the class of bounded self-adjoint Hankel operators $\mathbf H$, realised as integral operators on the positive semi-axis, that commute with dilations by a fixed factor. By analogy with the spectral theory of periodic Schr\"{o}dinger operators, we develop a Floquet-Bloch decomposition for this class of Hankel operators $\mathbf H$, which represents $\mathbf H$ as a direct integral of certain compact fiber operators. As a consequence, $\mathbf H$ has a band spectrum. We establish main properties of the corresponding band functions, i.e. the eigenvalues of the fiber operators in the Floquet-Bloch decomposition. A striking feature of this model is that one may have flat bands that co-exist with non-flat bands; we consider some simple explicit examples of this nature. Furthermore, we prove that the analytic continuation of the secular determinant for the fiber operator is an elliptic function; this link to elliptic functions is our main tool.