Lower bound on loss required for Coherent Perfect Absorption

Ascertain whether a lower bound exists for the uniform absorption strength η required for an S-matrix eigenvalue to reach zero (Coherent Perfect Absorption) in generic non-Hermitian scattering systems, and, if so, derive this bound explicitly from the trajectories of the S-matrix eigenvalues in the complex plane.

Background

The paper shows that CPAs require finite loss to move an S-matrix eigenvalue from the unit circle to zero. In random-matrix simulations using the same effective Hamiltonian, no CPAs are found at very small loss (e.g., η=0.0006), while a pair of CPAs appears at η=0.006. The authors argue there must be a critical loss ηc at which the number of CPAs is maximized, but the minimal threshold is unknown and potentially computable from eigenvalue trajectories.