Hankel operators with band spectra and elliptic functions

We consider the class of bounded self-adjoint Hankel operators H, realized as integral operators on the positive semiaxis, that commute with dilations by a fixed factor. By analogy with the spectral theory of periodic Schrödinger operators, we develop a Floquet–Bloch decomposition for this class of Hankel operators H, which represents H as a direct integral of certain compact fiber operators. As a consequence, H has a band spectrum. We establish main properties of the corresponding band functions, that is, 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 coexist with nonflat 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.