Abstract
We consider the class of bounded self-adjoint Hankel operators $\bH$, 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 $\bH$, which represents $\bH$ as a direct integral of certain compact fiber operators. As a consequence, $\bH$ 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.
By analogy with the spectral theory of periodic Schr\"{o}dinger operators, we develop a Floquet-Bloch decomposition for this class of Hankel operators $\bH$, which represents $\bH$ as a direct integral of certain compact fiber operators. As a consequence, $\bH$ 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.
| Original language | English |
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| Journal | Duke mathematical journal |
| Publication status | Accepted/In press - 12 Jun 2024 |