Hankel operators with band spectra and elliptic functions

2023/07/18 by Pushnitski, Alexander, Sobolev, Alexander · 1 citation
#47B35 #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2307.09242

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.

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• Ergodic Hankel operators

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