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Complex Band Structure and localisation transition for tridiagonal non-Hermitian k-Toeplitz operators with defects

by Y. De Bruijn and E.O. Hiltunen

(Report number 2025-17)

Abstract
Using the Bloch-Floquet theory, we propose an innovative technique to obtain the eigenvectors of tridiagonal k-Toeplitz operators. This method offers a more extensive and quantitative basis for describing localised eigenvectors beyond the non-trivial winding zone, yielding sharp decay bounds. The validity of our results is confirmed numerically in one-dimensional resonator chains, showcasing non-Hermitian skin localisation, bulk localisation, and tunnelling effects. We conclude the paper by analysing non-Hermitian tight binding Hamiltonians, illustrating the broad applicability of the complex band structure.

Keywords: Tridiagonal k-Toeplitz operator, block-Toeplitz operator, subwavelength resonances, evanescent modes, band gap, non-Hermitian skin effect, eigenmode condensation, non-Hermitian defected metamaterials, topological phase transition, non-Hermitian Hamiltonian, pseudospectra, Bloch theory, complex Brillouin zone

@Techreport{DH25_1138,
  author = {Y. De Bruijn and E.O. Hiltunen},
  title = {Complex Band Structure and localisation transition for tridiagonal non-Hermitian k-Toeplitz operators with defects},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2025-17},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2025/2025-17.pdf },
  year = {2025}
}

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