Generalized Wigner–Smith Operators

• Generalized Wigner–Smith operators are parametric logarithmic derivatives of scattering matrices that quantify sensitivities to arbitrary control parameters in complex systems.
• They extend the standard time-delay operator by differentiating with respect to non-frequency parameters, providing insights into resonance shifts and non-Hermitian behaviors.
• Applications include optimizing wavefront control, predicting complex pole shifts, and analyzing modal degeneracies such as exceptional points in photonics and electromagnetics.

Searching arXiv for relevant papers on generalized Wigner-Smith operators and related perturbation theory. Generalized Wigner–Smith operators are parametric logarithmic derivatives of a scattering or response matrix. They extend the standard Wigner–Smith time-delay construction from frequency differentiation to differentiation with respect to an arbitrary control parameter, thereby converting a delay operator into a sensitivity operator for geometry, material properties, gain, target position, or other perturbations. In unitary scattering at real frequency, this extension is closely related to the conventional Hermitian time-delay matrix; in open and non-Hermitian settings, the analytically continued form $$\mathbf{Q}_\xi=-i\,\mathbf{S}^{-1}\partial_\xi\mathbf{S}$$ becomes the central object for resonance perturbation theory, including complex pole motion, scattering zeros, and degenerate non-Hermitian singularities such as exceptional points. Complementary generalizations also arise from electromagnetic port formulations with radiating or evanescent channels, where the standard $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$ relation must be modified (Ambichl et al., 2017, Patel et al., 2020, Mao et al., 2022, Byrnes et al., 2024, Byrnes et al., 2024, Wang et al., 7 Aug 2025).

1. Definition and formal scope

The standard Wigner–Smith operator is a frequency derivative of the scattering matrix. In the electromagnetic convention adopted for lossless reciprocal systems with $e^{j\omega t}$, it is written as

$$\mathbf{Q}=j\,\mathbf{S}^\dagger \frac{\partial \mathbf{S}}{\partial \omega},$$

and its eigenvalues are group delays (Patel et al., 2020). In the generalized formulation developed for arbitrary perturbation parameters, the derivative is replaced by $\partial/\partial \alpha$ or, more generally, $\partial/\partial \xi$: $$\mathbf{Q}_\alpha=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \alpha},\qquad \mathbf{Q}_\xi=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \xi}.$$ This definition appears explicitly in the perturbative scattering-resonance literature, where $\xi$ may be $\omega$, a refractive-index perturbation, temperature, geometry, gain coefficient, ambient index, or another control variable (Byrnes et al., 2024).

The distinction between $\mathbf{S}^\dagger$ and $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$0 is not merely notational. In unitary scattering at real frequency, $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$1, so the Hermitian time-delay form and the logarithmic-derivative form coincide. In open, lossy, amplifying, or otherwise non-Hermitian problems, the perturbative theory is formulated explicitly in terms of $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$2, because the resonance problem is organized by poles and zeros of the analytically continued response rather than by unitary real-frequency scattering alone (Byrnes et al., 2024).

A complementary use of the term “generalized” concerns the channel setting rather than the derivative parameter. For systems with mixtures of propagating and evanescent modes, the conventional $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$3 no longer captures the correct frequency sensitivity, and a modified relationship is required. That extension preserves the Wigner–Smith program while changing the metric structure induced by the port basis (Mao et al., 2022).

2. Physical interpretation and modal content

In its classical form, the Wigner–Smith matrix packages the frequency sensitivities of all scattering amplitudes into a Hermitian operator. For a multiport system, the diagonal entries represent power-weighted average delays, and the eigenvectors define WS modes: coherent input superpositions that produce outputs with well-defined group delay and remain decoupled to first order under frequency variation (Patel et al., 2020).

The generalized operator inherits that first-order invariance property, but with respect to a parameter $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$4 rather than frequency. If

$j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$5

then the corresponding output is invariant to first order under $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$6 up to a scalar factor. In the unitary case, the operator can be written as

$j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$7

with $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$8, and its expectation value satisfies

$j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$9

For a global positional shift, the conjugate observable is momentum; for a local displacement of a selected scatterer, the eigenvalue is interpreted as local momentum transfer at that target (Ambichl et al., 2017).

When only a transmission matrix $e^{j\omega t}$0 is accessible, one may define

$e^{j\omega t}$1

This operator is generally non-Hermitian. Its eigenvalues are then complex, with the real part associated with the derivative of the transmitted phase and the imaginary part associated with the change in transmitted intensity under the perturbation. The exact momentum-transfer interpretation is weakened because the reflection block is omitted, but the ordering by interaction strength survives operationally (Ambichl et al., 2017).

For mixed propagating/evanescent channel sets, the full generalized matrix does not have a uniform physical interpretation across all blocks. The physically meaningful delay operator is the corrected propagating-sector block

$e^{j\omega t}$2

which remains Hermitian and supports a WS-mode interpretation after the evanescent correction terms are included (Mao et al., 2022).

3. Scattering poles, logarithmic derivatives, and perturbation theory

A major recent development is the use of generalized Wigner–Smith operators as a perturbation theory for resonances of open non-Hermitian systems. In this setting, resonances are poles $e^{j\omega t}$3 of the scattering matrix in the complex frequency plane, and perturbations shift those poles to $e^{j\omega t}$4. The core observation is that the pole shift is encoded in the residue of the logarithmic derivative of the scattering matrix (Byrnes et al., 2024).

For

$e^{j\omega t}$5

the logarithmic derivative identity gives

$e^{j\omega t}$6

Taking residues at a simple pole yields the central formula

$e^{j\omega t}$7

The same result can be written in limit form as

$e^{j\omega t}$8

This formulation is scattering-based, requires only simple poles and a small perturbation, and directly predicts complex shifts of resonance frequency and linewidth (Byrnes et al., 2024).

The arbitrary-$e^{j\omega t}$9 extension replaces the conjugated factorization by a transpose-based factorization,

$$\mathbf{Q}=j\,\mathbf{S}^\dagger \frac{\partial \mathbf{S}}{\partial \omega},$$0

which yields the unconjugated field bilinears appropriate to open resonances. In this framework, the first-order pole shift may also be expressed as

$$\mathbf{Q}=j\,\mathbf{S}^\dagger \frac{\partial \mathbf{S}}{\partial \omega},$$1

using $$\mathbf{Q}=j\,\mathbf{S}^\dagger \frac{\partial \mathbf{S}}{\partial \omega},$$2. For localized permittivity perturbations and weak field variation under perturbation, this reproduces the unconjugated open-system perturbation rule

$$\mathbf{Q}=j\,\mathbf{S}^\dagger \frac{\partial \mathbf{S}}{\partial \omega},$$3

whereas the high-$$\mathbf{Q}=j\,\mathbf{S}^\dagger \frac{\partial \mathbf{S}}{\partial \omega},$$4 conjugated version reduces to the familiar cavity-perturbation overlap with $$\mathbf{Q}=j\,\mathbf{S}^\dagger \frac{\partial \mathbf{S}}{\partial \omega},$$5 (Byrnes et al., 2024).

The same residue logic extends to any meromorphic response quantity $$\mathbf{Q}=j\,\mathbf{S}^\dagger \frac{\partial \mathbf{S}}{\partial \omega},$$6 with the relevant pole structure. One defines

$$\mathbf{Q}=j\,\mathbf{S}^\dagger \frac{\partial \mathbf{S}}{\partial \omega},$$7

and uses $$\mathbf{Q}=j\,\mathbf{S}^\dagger \frac{\partial \mathbf{S}}{\partial \omega},$$8 in place of $$\mathbf{Q}=j\,\mathbf{S}^\dagger \frac{\partial \mathbf{S}}{\partial \omega},$$9. This permits pole tracking via $\partial/\partial \alpha$0, $\partial/\partial \alpha$1, a Mie coefficient $\partial/\partial \alpha$2, or zero tracking via $\partial/\partial \alpha$3 or $\partial/\partial \alpha$4 (Byrnes et al., 2024).

4. Degenerate resonances: diabolic points and exceptional points

The simple-pole theory does not suffice at spectral degeneracies. A residue-based extension addresses degenerate non-Hermitian resonances, distinguishing diabolic points (DPs) from exceptional points (EPs) by algebraic and geometric multiplicity (Wang et al., 7 Aug 2025).

At $\partial/\partial \alpha$5, suppose $\partial/\partial \alpha$6 has a repeated eigenvalue $\partial/\partial \alpha$7 with algebraic multiplicity $\partial/\partial \alpha$8. If $\partial/\partial \alpha$9, the degeneracy is a DP. If $\partial/\partial \xi$0, there is a single $\partial/\partial \xi$1 Jordan block and the degeneracy is an EP of order $\partial/\partial \xi$2. Intermediate $\partial/\partial \xi$3 are hybrid cases. The scattering matrix is written in Mahaux–Weidenmüller form,

$\partial/\partial \xi$4

and the perturbation operator is

$\partial/\partial \xi$5

For a generic perturbation lifting an EP of order $\partial/\partial \xi$6, the resonance splitting obeys a Puiseux series

$\partial/\partial \xi$7

and the coefficient $\partial/\partial \xi$8 is determined from scattering data. Under the condition that

$\partial/\partial \xi$9

is analytic and nonzero at the EP, the central formula is

$$\mathbf{Q}_\alpha=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \alpha},\qquad \mathbf{Q}_\xi=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \xi}.$$0

This result uses only scattering data, reproduces the $$\mathbf{Q}_\alpha=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \alpha},\qquad \mathbf{Q}_\xi=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \xi}.$$1-fold symmetric EP splitting, and reduces to the known non-degenerate pole-shift formula when $$\mathbf{Q}_\alpha=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \alpha},\qquad \mathbf{Q}_\xi=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \xi}.$$2 (Wang et al., 7 Aug 2025).

DPs require a different treatment because the repeated eigenvalue remains diagonalizable and the perturbed branches generally split independently with ordinary Taylor expansions. The collective EP trace formula cannot recover all distinct DP shifts. Instead, one isolates scalar scattering functions containing one pole each and applies the non-degenerate formula branchwise. If $$\mathbf{Q}_\alpha=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \alpha},\qquad \mathbf{Q}_\xi=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \xi}.$$3, the scattering eigenvalues are

$$\mathbf{Q}_\alpha=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \alpha},\qquad \mathbf{Q}_\xi=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \xi}.$$4

and the scalar operators

$$\mathbf{Q}_\alpha=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \alpha},\qquad \mathbf{Q}_\xi=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \xi}.$$5

yield the individual shifts. If $$\mathbf{Q}_\alpha=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \alpha},\qquad \mathbf{Q}_\xi=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \xi}.$$6 is not proportional to identity but $$\mathbf{Q}_\alpha=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \alpha},\qquad \mathbf{Q}_\xi=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \xi}.$$7 is known, the resolvent can be reconstructed via

$$\mathbf{Q}_\alpha=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \alpha},\qquad \mathbf{Q}_\xi=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \xi}.$$8

which unmixes the poles. Thus EP response is collective and non-analytic, while DP response is branch-resolved and generically linear in $$\mathbf{Q}_\alpha=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \alpha},\qquad \mathbf{Q}_\xi=-i\,\mathbf{S}^{-1}\frac{\partial \mathbf{S}}{\partial \xi}.$$9 (Wang et al., 7 Aug 2025).

The same reasoning applies to degenerate scattering zeros by working with $\xi$0 instead of $\xi$1. This places coherent-perfect-absorption EPs and other zero singularities within the same residue framework (Wang et al., 7 Aug 2025).

5. Electromagnetic formulations and channel-generalized variants

Electromagnetic Wigner–Smith theory supplies a field-theoretic foundation for generalized operators. For guiding, scattering, and radiating systems excited through ports, the entries of the frequency-based WS matrix are energy-like overlap integrals of the electric and magnetic fields generated by port excitations. For guiding systems with TEM ports, the WS matrix reduces directly to the overlap matrix; for non-TEM ports, additional impedance-derivative correction terms appear; for free-space scattering and radiating systems, a renormalization subtracts the free-space traversal contribution (Patel et al., 2020).

This field interpretation underlies later perturbative generalizations. In waveguide-coupled non-Hermitian systems, the generalized operator can be factorized as

$\xi$2

and both $\xi$3 and $\xi$4 admit volume-integral representations. In the arbitrary-$\xi$5 extension, the transpose-based factorization

$\xi$6

replaces conjugated overlaps by unconjugated bilinears, which is the correct structure for open resonances and quasi-normal-mode perturbation beyond the high-$\xi$7 regime (Byrnes et al., 2024, Byrnes et al., 2024).

A separate electromagnetic generalization addresses ports supporting propagating and evanescent modes. There the generalized WS relation becomes

$\xi$8

with

$\xi$9

The matrices $\omega$0 and $\omega$1 encode the modal-character and impedance corrections required when evanescent channels are included. The physically interpretable object is the corrected propagating block

$\omega$2

whose eigenvectors are generalized WS modes and whose eigenvalues remain interpretable as time delays (Mao et al., 2022).

The mixed-mode theory also supports subsystem composition. For a cascade of subsystems $\omega$3 and $\omega$4, the expensive volume-integral matrix obeys

$\omega$5

after which the full generalized matrix $\omega$6 is obtained by adding the same port correction terms. This formulation is particularly relevant when internal interfaces carry evanescent coupling and direct port relocation would be computationally undesirable (Mao et al., 2022).

6. Operational uses and validated applications

Generalized Wigner–Smith operators have been used for wavefront control inside disordered media. When the parameter is the position of a target scatterer embedded in the medium, diagonalization of the corresponding operator produces states ordered by interaction strength with that target. In the reported microwave realization, eigenstates with the largest $\omega$7 focused on the designated brass scatterer, whereas eigenstates with the smallest $\omega$8 largely bypassed it. The protocol used the scattering or transmission matrix and its derivative with respect to target displacement, and it operated without optimization or phase-conjugation (Ambichl et al., 2017).

In open non-Hermitian resonator problems, the residue formulation has been validated on random photonic networks, where pole trajectories predicted from generalized WS residues closely matched direct pole tracking under refractive-index perturbations. The same framework was used for spatially selective pumping: by computing $\omega$9 for each candidate pumped link $\mathbf{S}^\dagger$0, one can rank perturbation channels by how efficiently they move a chosen pole toward the real axis, identifying advantageous pump locations without first computing the modal intensity profile (Byrnes et al., 2024).

The arbitrary-$\mathbf{S}^\dagger$1 extension has been applied to a multilayer nanoresonator sensor and to scattering zeros. For a silica-core/gold-shell nanosphere in water, the residue-based generalized WS sensitivity $\mathbf{S}^\dagger$2 agreed with direct pole tracking across the design space $\mathbf{S}^\dagger$3. The same analysis applied to a zero of a Mie coefficient by choosing $\mathbf{S}^\dagger$4, and the study found much larger sensitivity for the zero than for the localized plasmon pole in that example (Byrnes et al., 2024).

Degenerate-theory validations include both analytic Hamiltonian models and numerical electromagnetic simulations. For a non-Hermitian $\mathbf{S}^\dagger$5 dimer at an EP, the residue formula exactly reproduced the square-root splitting $\mathbf{S}^\dagger$6. For a DP realized in the same model, the scalar branchwise construction exactly reproduced the linear splitting $\mathbf{S}^\dagger$7. A metallic ellipsoid example treated the polarizability tensor $\mathbf{S}^\dagger$8 as the response matrix; for a gold sphere of radius $\mathbf{S}^\dagger$9 nm with a localized surface plasmon pole near $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$00, the GWS residue prediction agreed well with direct root tracking and finite-element COMSOL simulations for elongations from $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$01 to $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$02 nm. Transformation-optics-designed plasmonic nanowires supporting EP$j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$03, EP$j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$04, and EP$j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$05 showed good agreement between residue-based predictions and direct numerical pole tracking over six orders of magnitude in $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$06, with the expected slopes $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$07, $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$08, and $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$09 in log–log plots of $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$10 versus $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$11 (Wang et al., 7 Aug 2025).

A practical implication recurring across these applications is that the necessary input is external response data rather than explicit diagonalization of an internal non-Hermitian Hamiltonian. This suggests a route to precision tuning, inverse design, and perturbation ranking in systems where modal structure is difficult to access directly but scattering data are accessible (Byrnes et al., 2024, Byrnes et al., 2024, Wang et al., 7 Aug 2025).

7. Limitations, misconceptions, and terminological boundaries

The available formulations are perturbative and problem-dependent. The simple-pole residue theory assumes poles and zeros of $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$12 are simple and the perturbation is small enough that higher-order terms can be neglected. Step sizes may therefore need to be reduced when $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$13 is large and the pole is highly sensitive (Byrnes et al., 2024).

The arbitrary-$j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$14 electromagnetic connection is broader than the high-$j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$15 cavity formula, but the simplified overlap expression still assumes weak field variation under perturbation. The high-$j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$16 conjugated overlap with $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$17 is only strictly valid for infinite quality factor resonances, whereas the transpose-based residue formalism is designed for arbitrary-$j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$18 open systems (Byrnes et al., 2024).

The degenerate theory imposes additional conditions. The EP formula assumes a generic perturbation so that the usual $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$19 Puiseux splitting holds. The derivation also requires $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$20 to be analytic and nonzero at the EP; this can fail in dark-mode situations. DP analysis is less universal than the EP trace formula because it depends on separating poles, either through scattering eigenvalues under simple coupling or by resolvent reconstruction when $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$21 is known. Hybrid cases with $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$22 are only suggested to be treated Jordan-block by Jordan-block rather than developed as a full general formalism (Wang et al., 7 Aug 2025).

In electromagnetic mixed-mode theory, only the corrected propagating block has a clean delay interpretation; the blocks $j\mathbf{S}^\dagger \partial_\omega \mathbf{S}$23 do not appear to have a direct physical interpretation. Sufficient evanescent truncation is essential for convergence, especially in subsystem composition formulas (Mao et al., 2022).

A recurrent misconception concerns notation. “Generalized Wigner operator” in relativistic field theory is a different object from a generalized Wigner–Smith operator. The former is an intertwining integral transform from Wigner wave functions to local relativistic fields and is unrelated to scattering matrices, time delay, or parametric logarithmic derivatives. The similarity is terminological rather than conceptual (Buchbinder et al., 2023).

Taken together, these results establish “generalized Wigner–Smith operators” as a family of closely related constructions rather than a single formula. In unitary scattering they organize parameter-stable input states and conjugate-observable shifts; in open non-Hermitian resonance theory they provide residue formulas for complex pole and zero motion; in degenerate settings they separate collective EP sensitivity from branchwise DP splitting; and in electromagnetics they admit field-overlap and mixed-channel formulations adapted to realistic ports and open-system perturbations (Ambichl et al., 2017, Patel et al., 2020, Mao et al., 2022, Byrnes et al., 2024, Byrnes et al., 2024, Wang et al., 7 Aug 2025).