Hankel operators with band spectra and elliptic functions

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.