Thresholdless dynamical instability in transverse $\mathcal{PT}$-symmetric scattering systems: The hidden role of bound states in the continuum

Abstract

The stationary scattering properties of transverse parity-time ($\mathcal{PT}$) symmetric systems have been extensively studied, yet their dynamical stability, a prerequisite for any stationary description, remains largely unexplored. Here we uncover a thresholdless dynamical instability in such systems, driven by symmetry-protected bound states in the continuum (BICs). Without gain and loss, the up-down mirror symmetry of the transverse geometry generically protects a BIC, which manifests as the coalescence of an $S$-matrix pole and zero on the real axis of the complex wave-number plane. A Hermitian symmetry-breaking perturbation shifts the poles into the lower half-plane, converting the BIC into a resonance with a Fermi-golden-rule decay width. An anti-Hermitian $\mathcal{PT}$-symmetric perturbation instead reverses the sign of the second-order energy shift, driving the poles into the upper half-plane and producing a time-growing bound state with $\operatorname{Im}E=\gamma^{2}\Gamma_{V}/2+O(\gamma^{3})$, where $\gamma$ is the gain-loss strength and $\Gamma_{V}$ is the golden-rule coupling of the BIC to the continuum. The instability therefore sets in at arbitrarily small $\gamma$ whenever $\Gamma_{V}>0$. We confirm this mechanism in a three-site side-coupled model that is unstable despite possessing a unitary scattering matrix, and in a four-site rhombic model where transverse and longitudinal gain-loss placements yield vanishing and finite thresholds, respectively. These results establish a microscopic stability criterion and a design principle for stable $\mathcal{PT}$-symmetric scattering devices.