Bound states and decay dynamics in $N$-level Friedrichs model with factorizable interactions

Considering an $N$-level system interacting factorizably with a continuous spectrum, we derive analytical expressions for the bound states and the dynamical evolution within this single-excitation Friedrichs model by using the projection operator formalism. First, we establish explicit criteria to determine the number of bound states, whose existence suppresses the complete spontaneous decay of the system. Second, we derive the open system’s dissipative dynamics, which is naturally described by an energy-independent non-Hermitian Hamiltonian in the Markovian limit. As an example, we apply our framework to an atomic chain embedded in a photonic crystal waveguide, uncovering a rich variety of decay dynamics and realizing an anti-$\mathcal{PT}$-symmetric Hamiltonian in the system’s evolution.

💡 Research Summary

The paper studies a generalized single‑excitation Friedrichs model in which N non‑degenerate discrete levels are coupled to a continuum via a factorizable interaction of the form fₙ g(ω). By introducing the Feshbach projection operators Q (onto the discrete subspace) and P (onto the continuum), the authors eliminate the continuum degrees of freedom and obtain an exact energy‑dependent effective Hamiltonian
H_eff(E)=∑ₙεₙ|n⟩⟨n|+Σ(E)∑{n,n′}fₙf*{n′}|n⟩⟨n′|,
with the self‑energy Σ(E)=∫_{ω_low}^{ω_up} J(ω)/(E−ω) dω and spectral density J(ω)=|g(ω)|²ρ(ω).

Bound states correspond to real solutions of det