Research reports

Years: 2025 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Spectra and pseudo-spectra of tridiagonal k-Toeplitz matrices and the topological origin of the non-Hermitian skin effect

by H. Ammari and S. Barandun and Y. De Bruijn and P. Liu and C. Thalhammer

(Report number 2024-05)

Abstract
We establish new results on the spectra and pseudo-spectra of tridiagonal k-Toeplitz operators and matrices. In particular, we prove the connection between the winding number of the eigenvalues of the symbol function and the exponential decay of the associated eigenvectors (or pseudo-eigenvectors). Our results elucidate the topological origin of the non-Hermitian skin effect in general one-dimensional polymer systems of subwavelength resonators with imaginary gauge potentials, proving the observation and conjecture in [H. Ammari et al., arXiv:2307.13551]. We also numerically verify our theory for these systems.

Keywords: Tridiagonal k-Toeplitz operator, block-Toeplitz operator, Tridiagonal k-Laurent operator, pseudospectra, Coburn's lemma, non-Hermitian skin effect, gauge capacitance matrix, eigenmode condensation

@Techreport{ABDLT24_1087,
  author = {H. Ammari and S. Barandun and Y. De Bruijn and P. Liu and C. Thalhammer},
  title = {Spectra and pseudo-spectra of tridiagonal k-Toeplitz matrices and the topological origin of the non-Hermitian skin effect},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2024-05},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2024/2024-05.pdf },
  year = {2024}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).