Resonant-State Approach in Open Systems

• Resonant-State Approach is a framework that defines resonances as solutions to wave equations with outgoing boundary conditions, encoding both resonance positions and decay widths.
• It employs computational methods like complex scaling, optimized Rayleigh–Ritz, and resonant-state expansion to derive complex eigenvalues for open systems.
• The approach underpins applications in quantum scattering, nanophotonics, and photonic crystal analysis, enabling efficient simulations and precise physical predictions.

A resonant-state approach encompasses a collection of rigorous mathematical and computational methods for analyzing and computing the resonant (quasinormal) modes of open physical systems, particularly in quantum mechanics, wave physics, and electrodynamics. Resonant states correspond to the discrete, complex eigenvalues of the underlying operator or Hamiltonian, subject to purely outgoing (Siegert–Gamow) boundary conditions. The resulting complex eigenenergies encode both the resonance position and decay width (lifetime), and their residues govern the pole structure of Green’s functions and observable quantities such as scattering amplitudes, transmission, and conductance.

1. Mathematical Foundations and Definitions

In the resonant-state framework, resonances are formalized as solutions to linear wave equations (e.g., Schrödinger, Maxwell) with outgoing boundary conditions at infinity. For quantum systems, this is articulated via the stationary Schrödinger equation,

$\hat{H}\,\psi(\mathbf{r}) = E\,\psi(\mathbf{r}),$

with $\psi(\mathbf{r}) \sim e^{+ik_n r}$ as $r\to\infty$ and complex $k_n$ ($\Im k_n<0$), leading to complex eigenvalues $E_n = E_r - i\,\Gamma/2$, where $E_r$ is the resonance position and $\Gamma$ the width (inverse lifetime) (Hatano, 2021, García-Calderón et al., 2019). In open optical systems, solutions to Maxwell’s equations with outgoing (radiative) boundary conditions similarly yield complex resonant frequencies (Doost et al., 2014, Fischbach et al., 5 Nov 2025).

The normalization of these non-Hermitian (typically non-square-integrable) eigenfunctions is achieved via flux–volume or analytic normalization conditions, such that the biorthogonality and completeness of the resonant-state set is preserved. In particular, the spectral (Mittag–Leffler) expansion of Green’s functions takes the form,

$$G(\mathbf{r}, \mathbf{r}'; E) = \sum_n \frac{\psi_n(\mathbf{r})\psi_n(\mathbf{r}')}{E-E_n}$$

(for Schrödinger systems), with analogous representations for electromagnetic fields (Doost, 2015, Doost et al., 2014).

2. Computational Methods for Resonant States

Two major computational paradigms exist: direct complex-scaling (or absorbing potential) approaches, and spectral or variational expansions.

Rayleigh–Ritz with Optimized Nonlinear Parameters: The Rayleigh–Ritz method constructs approximate eigenstates as linear combinations of basis functions $\{\phi_n(x; \alpha)\}$ with tunable (possibly complex) nonlinear parameters $\alpha$. For resonances, $\alpha$ is allowed to be complex; the trace of the effective Hamiltonian is rendered stationary with respect to $\alpha$, yielding the optimal parameter without state-specific iterative tuning. Diagonalization of the resulting (possibly complex symmetric) matrix at the optimal parameter provides all resonance energies and widths simultaneously (Kuroś et al., 2012). This method demonstrates rapid convergence without the need for state-by-state minimization.

Resonant-State Expansion (RSE): The RSE generalizes Brillouin–Wigner–type non-Hermitian perturbation theory for open systems. Here, the field or wavefunction of a perturbed system is expanded over the discrete set of resonant states of an analytically tractable unperturbed system. This leads to a complex-symmetric matrix eigenproblem whose dimension equals the truncated basis size. The RSE framework is directly applicable to Schrödinger, electromagnetic, and acoustic systems in one, two, and three dimensions, including dispersive (frequency-dependent) media (Doost, 2015, Doost et al., 2014, Armitage et al., 2013, Muljarov et al., 2015).

Generalized Virial Theorem and Invariance Criteria: Identification of true resonances in numerical schemes employing regularization (complex scaling, complex absorbing potentials, smooth exterior scaling) is achieved via invariance operators. The expectation value of the derivative of the regularized Hamiltonian with respect to the scaling parameter must vanish on a physical resonance state, providing a robust criterion for resonance identification independent of the regularization scheme (Genovese et al., 2015).

3. Physical Interpretation and Operator Structure

The non-Hermiticity in the resonant-state approach is physically rooted in the coupling of a localized system to an environment with a continuum of states. Two major, algebraically equivalent perspectives are prevalent:

Siegert Boundary Condition: Seeking solutions of the full (scattering or wave) problem subject to outgoing-wave boundary conditions at infinity directly yields the discrete set of complex resonance eigenvalues (Hatano, 2021, Hatano, 2014). This approach is universal for scalar and vector wave equations.

Feshbach Projector Formalism: The total Hilbert space is partitioned into a "system" (P subspace) and an "environment" (Q subspace). Elimination of the environment degrees of freedom yields a non-Hermitian, energy-dependent effective Hamiltonian $H_{\mathrm{eff}}(E) = PHP + PHQ(E - QHQ)^{-1}QHP$ whose discrete eigenvalues are precisely the resonance poles of the original problem (Hatano, 2014, Hatano et al., 2019). The explicit non-Hermiticity reflects the irreversible loss of probability/current into the reservoir.

Spectral expansions of Green's functions, observables, and response functions fully in terms of the resonant-state basis, possibly supplemented by a (discretized) cut integral for continuous branch cuts, enable both analytic continuation and efficient numerical computation (Doost et al., 2013, Muljarov et al., 2015).

4. Resonant-State Expansion in Electrodynamical and Wave Systems

The RSE framework has been systematically developed for open optical and photonic structures, including one-dimensional slabs, planar waveguides, two-dimensional cylinders, three-dimensional spheres, photonic crystal slabs, and non-uniform waveguides (Doost et al., 2011, Armitage et al., 2013, Doost et al., 2014, Doost et al., 2013, Neale et al., 2019, Lobanov et al., 2016). The central components include:

• Representation of fields as expansions in the RS basis, with analytic or numerically computed matrix elements for perturbations (permittivity, permeability changes).
• Treatment of both discrete resonant states (poles) and continuum contributions (cuts or branch points) for completeness and convergence.
• Generalization to dispersive materials, with the RSE remaining a linear eigenproblem if all physical poles of the material dispersion are incorporated in the basis (Muljarov et al., 2015).
• Accurate calculation of strong and weak scattering, transmission, Fano resonances, and even bound states in the continuum (Pankin et al., 2020, Hatano et al., 2019).

RSE-based computational approaches achieve rapid, systematic convergence (typically error scaling as $N^{-3}$ with the number of basis states N) and outperform traditional finite-element or finite-difference time-domain solvers, especially when localized basis subsets are employed (Doost et al., 2014).

5. Resonant States in Quantum Scattering and Open Systems

The resonant-state expansion plays a central role in quantum scattering, transport, and decay:

• Green's Function and Born Approximations: The RSE Born approximation expresses scattering amplitudes and Green’s functions as convergent series over RSs, automatically capturing the full pole structure and normalization without explicit background integration over the continuum (Doost, 2015, Tanimu et al., 2019).
• Fano Resonances and Asymmetry: Decomposition of observable transmission or conductance through open quantum-dot and mesoscopic systems into contributions from discrete resonances, anti-resonances, and bound states yields explicit, microscopic expressions for Fano asymmetry parameters connected to interference between discrete state pairs (Hatano et al., 2019).
• Unitarity and Non-Hermitian Formalism: The non-Hermitian analytic expansion involving the complete set of RSs and continuum scattering states exactly preserves unitarity; spatial divergence of individual RSs is canceled in physical wave packets evolving in time, maintaining probability conservation at all times (García-Calderón et al., 2019).
• Effective Hamiltonians: Energy-dependent non-Hermitian effective Hamiltonians $H_{\mathrm{eff}}(E)$, arising from the Feshbach (projection-operator) formalism, admit direct identification of resonance poles with Siegert boundary-value solutions. This equivalence has been rigorously established for paradigmatic open systems (Hatano, 2014, Hatano, 2021).

6. Advanced Applications: Nanophotonics and Light–Matter Coupling

The resonant-state approach unifies first-principles modeling of complex open photonic environments:

• Light–Matter Hybridization: Rigorous derivation of effective non-Hermitian Hamiltonians incorporating both photonic resonant states (quasinormal modes) and multiple material resonances yields analytically the hybrid polaritonic eigenfrequencies of coupled light–matter systems. Hybridization induces both off-diagonal couplings and Lamb/Radiative frequency shifts, allowing unambiguous identification of strong coupling regimes directly in the complex-frequency plane (Fischbach et al., 5 Nov 2025).
• Photonic Crystals and Metamaterials: The RSE enables accurate, physical decomposition of complicated mode spectra in periodic, aperiodic, and defect-engineered dielectric structures, revealing the role of bound states in the continuum, symmetry protection, and Fano interference in their transmission spectra (Neale et al., 2019, Pankin et al., 2020).

7. Practical Implementation and Numerical Considerations

Efficient implementation of the resonant-state approach involves:

• Analytical or semi-analytical construction of the unperturbed RS basis (including discretization of cuts/continua as needed).
• Pre-computation of analytic matrix elements for perturbations, exploiting symmetry and closure relations.
• Solution of complex-symmetric (possibly nonlinear) eigenvalue problems, with dimensions set by basis truncation.
• Use of trace stationarity or invariance criteria to automatically optimize nonlinear parameters or identify true resonance solutions.
• Basis localization and targeted subspace methods to accelerate computation for selected subsets of the total spectrum (Doost et al., 2014).
• Robust error estimation and polynomial extrapolation schemes to systematically enforce convergence and quantify error with increasing basis dimension (Doost et al., 2011, Armitage et al., 2013).

The resonant-state approach thus provides a rigorous, general, and computationally robust paradigm for the analysis and calculation of resonant phenomena across quantum, electromagnetic, and wave systems. Its mathematical structure underlies fundamental aspects of open-system theory, from non-Hermitian eigenproblems to the analytic properties of scattering matrices and physical observables.