Relation between OBC and periodic spectra in time‑modulated mechanical lattices

Determine the precise relationship between the spectrum of time‑modulated dynamic mechanical systems with open boundary conditions and the spectrum of the corresponding spatially periodic lattices in the complex quasifrequency plane, clarifying how these spectra are associated for systems whose stiffness is periodically modulated in time.

Background

Time‑modulated discrete mechanical systems (e.g., mass–spring chains with time‑periodic stiffness) are non‑Hermitian under the temporal Floquet formulation, leading to complex spectra and phenomena such as non‑reciprocity and non‑Hermitian skin effects. In Hermitian settings, Bloch band theory reliably links the spectra of large finite systems with open boundaries to those of infinite periodic lattices, but this correspondence can fail in non‑Hermitian regimes.

The paper highlights that, prior to their paper, the connection between the spectrum under open boundary conditions and the Bloch‑type spectrum of periodic lattices for time‑modulated dynamic mechanical systems had not been clarified. This motivates the development of a non‑Bloch band framework using a generalized Brillouin zone to characterize finite‑system spectra despite non‑Hermitian skin effects.