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 Schrödinger
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.
This is a joint work with A. Pushnitski.
