Physical relevance of time-independent scattering predictions in periodic $\mathcal{PT}$-symmetric chains

Abstract

Time-independent scattering methods are widely used to analyze transport in periodic $\mathcal{PT}$-symmetric systems. However, their predictions become unphysical when the system supports time-growing bound states (TGBSs), which manifest as $S$-matrix poles in the first quadrant of the complex wave-number plane. Here, we analytically delineate the region of physical relevance for a $\mathcal{PT}$-symmetric chain of $N$ unit cells with gain/loss strength $\gamma$. We derive the TGBS onset threshold $\gamma_c = 2\sin[\pi/(4N)]$, which scales as $\pi/(2N)$ for large $N$ and vanishes in the thermodynamic limit. Enlarging the structure thus enriches stationary scattering phenomenology but inevitably triggers TGBSs at weaker gain/loss. Time-dependent wave-packet simulations confirm this analytical boundary quantitatively. Applying this criterion, we show that many previously reported predictions of gain-loss-induced localization, reflectionless transport, and coherent perfect absorbers and lasers in large periodic structures fall outside the physically relevant regime. $S$-matrix pole analysis is therefore an indispensable prerequisite for interpreting time-independent scattering predictions in periodic non-Hermitian systems.