Non-Hermitian extension of the nonlinear decomposition

Develop a rigorous treatment of the nonlinear decomposition for open quantum systems governed by explicitly specified effective non-Hermitian Hamiltonians, including the derivation from first principles of the corresponding imaginary local Hamiltonian component e_I and the resulting density dynamics; in particular, analyze complex optical potentials V(r) = V_re(r) + iV_im(r) with V_im(r) < 0 in absorbing regions.

Background

The paper derives the relation between the imaginary local Hamiltonian component e_I and the density evolution for closed Hermitian quantum systems. In that setting, unitarity forces the density-weighted integral of e_I to vanish, although e_I may remain nonzero pointwise and act as a local source or sink for probability flow.

The authors identify open quantum systems with effective non-Hermitian Hamiltonians as an unresolved extension. For such systems, the density-weighted integral of e_I need not vanish, potentially describing decay in absorbing or finite-lifetime resonance states and growth in amplifying or unstable resonance modes. The paper specifies that addressing this issue requires choosing a concrete non-Hermitian Hamiltonian, such as one containing a complex optical potential, and deriving the corresponding e_I within the nonlinear decomposition framework.