Hankel operators with band spectra and elliptic functions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916286423891968 |
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| author | Pushnitski, Alexander Sobolev, Alexander |
| author_facet | Pushnitski, Alexander Sobolev, Alexander |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_09242 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hankel operators with band spectra and elliptic functions Pushnitski, Alexander Sobolev, Alexander Spectral Theory 47B35 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. |
| title | Hankel operators with band spectra and elliptic functions |
| topic | Spectral Theory 47B35 |
| url | https://arxiv.org/abs/2307.09242 |