Physical relevance of time-independent scattering calculations in non-Hermitian systems: The role of time-growing bound states

Time-independent scattering methods are widely employed to analyze transport in non-Hermitian systems. Their application, however, rests on a critical yet often overlooked assumption: that an incident wave is a pure superposition of scattering states. In practice, any physically realistic, spatially localized wave packet will generally have a nonzero overlap with the system's bound states, thereby violating this premise. While this violation is inconsequential in Hermitian systems, it can invalidate the conventional scattering picture in their non-Hermitian counterparts. The underlying cause is the emergence of time-growing bound states, which manifest as poles of the scattering matrix ($S$ matrix) in the first quadrant of the complex wave-number plane. Any initial overlap with these states becomes exponentially amplified, eventually dominating the long-time dynamics. Consequently, the actual evolution of a wave packet diverges dramatically from the conventional scattering picture, rendering the transmission and reflection coefficients derived from time-independent scattering methods unphysical. Using tight-binding models with non-Hermiticity introduced via imaginary on-site potentials or asymmetric hopping, we demonstrate that parameter regimes supporting such growing states are common. We therefore conclude that an analysis of $S$-matrix poles is an indispensable step to confirm the physical relevance of time-independent scattering calculations in non-Hermitian systems.