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arXiv · 2602.09734

Mathematical Foundation for the Generalised Brillouin zone of m-banded Toeplitz operators

Yannick de Bruijn·Erik Orvehed Hiltunen
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Abstract

We show that the spectrum of the open-boundary limit of banded Toeplitz matrices is real whenever the associated symbol function is real-valued along a closed polar curve. Building on this result, we develop both analytical and numerical methods to symmetrise a class of banded non-Hermitian Toeplitz matrices whose asymptotic spectra are real. Finally, we provide a rigorous mathematical foundation for the generalised Brillouin zone, a concept widely used in non-Hermitian physics, by proving that it coincides with the polar curve on which the symbol function takes real values.

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Yannick de Bruijn, Erik Orvehed Hiltunen. 2026-02-10. Mathematical Foundation for the Generalised Brillouin zone of m-banded Toeplitz operators. https://arxiv.org/abs/2602.09734

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