Parametric Keldysh Decomposition

• Parametric Keldysh Decomposition is a framework that splits analytic matrix functions with parameter dependence into a rational pole-residue part and an analytic remainder.
• It extends the classical Keldysh decomposition by incorporating parametric effects, enabling efficient algorithms for handling nonlinear eigenvalue problems and nonequilibrium quantum systems.
• The method supports offline/online algorithms using rational approximation and Loewner reduction, with applications ranging from delay-differential stability to advanced quantum field theory analyses.

The parametric Keldysh decomposition provides a principled framework for dissecting analytic matrix-valued functions of spectral and auxiliary parameters, with direct algorithmic and structural implications for nonlinear eigenvalue problems and nonequilibrium quantum field theory. It extends the classical Keldysh decomposition, which separates a matrix function into a rational part capturing its spectral structure and an analytic remainder, by introducing parametric dependence and thus enabling efficient algorithms for parametric contour integral eigenvalue solvers and illuminating dualities in diagrammatic expansions.

1. Formal Definition and Core Properties

Let $T(z,\mu)$ be an $n \times n$ matrix-valued function, analytic jointly in $(z,\mu)$ on $$\Omega \times \Pi \subset \mathbb{C} \times \mathbb{C}$$, with $\Omega \subset \mathbb{C}$ a bounded, simply connected spectral region and $\Pi \subset \mathbb{C}$ a parameter domain. Suppose for each $\mu \in \Pi$ that $\det T(\cdot,\mu)$ is not identically zero and admits exactly $m$ zeros (counting algebraic multiplicity) in $\Omega$, and that this multiplicity is constant across $\Pi$.

The unique parametric Keldysh decomposition is given by

$T(z,\mu)^{-1} = H(z,\mu) + N(z,\mu),$

where:

• $N(\cdot,\mu)$ is analytic in $\Omega$ for each $\mu$, ensuring its contour-integral moments on $\partial\Omega$ vanish by Cauchy’s theorem.
• $H(z,\mu) = R(z,\mu)/u(z,\mu)$, with $u(z,\mu)$ a monic degree-$m$ polynomial in $z$ whose roots $\{\lambda_i(\mu)\}_{i=1}^m$ are precisely the eigenvalues of $T(\cdot,\mu)$ in $\Omega$. $R(z,\mu)$ is analytic in $(z,\mu)$ and $\deg_z R < m$.

Upon selecting a local eigenbasis, for simple eigenvalues, $H(z,\mu)$ admits the pole-residue expansion:

$$H(z,\mu) = \sum_{i=1}^m \frac{v_i(\mu) w_i(\mu)^*}{z - \lambda_i(\mu)},$$

with $T(\lambda_i(\mu),\mu) v_i(\mu) = 0$, $w_i(\mu)^* T(\lambda_i(\mu),\mu) = 0$, and normalized such that $w_i(\mu)^* T'_z(\lambda_i(\mu),\mu) v_i(\mu)=1$.

Under any analytic $f$,

$$(2\pi i)^{-1} \oint_{\partial\Omega} f(z) T(z,\mu)^{-1} dz = \sum_{i=1}^m f(\lambda_i(\mu)) v_i(\mu) w_i(\mu)^*.$$

This decomposition is unique and generalizes the classical (non-parametric) Keldysh theorem to analytic parameter families (Balicki et al., 4 Jan 2026).

2. Theoretical Foundations and Proof Structure

The parametric Keldysh decomposition relies fundamentally on the two-variable Weierstrass Preparation and Division theorems:

• $T^{-1} = \mathrm{adj}\,T/\det T$ is well-defined off the vanishing locus.
• $\det T(z,\mu) = u(z,\mu) h(z,\mu)$, where $u$ is the Weierstrass polynomial, and $h$ is nonvanishing and analytic on $\Omega \times \Pi$.
• Define $P(z,\mu) = \mathrm{adj}\,T(z,\mu)/h(z,\mu)$. For each entry, Weierstrass Division enables

$P(z,\mu) = N(z,\mu) u(z,\mu) + R(z,\mu),$

with $N, R$ analytic and $\deg_z R < m$.

Unicity follows from analytic continuation and the monicity of $u(z,\mu)$. The specific identification of the pole-residue part in terms of eigenvectors is obtained by matching Laurent expansions, paralleling the residue formula in the non-parametric case.

Analytic dependence of eigenvalues and eigenvectors in $\mu$ (for simple spectra) is ensured by the Implicit Function Theorem. Higher-order Jordan structures and multiple eigenvalues require additional care for local analytic parametrization (Balicki et al., 4 Jan 2026).

3. Parametric Contour Integral Algorithms

The decomposition is exploited in the parametric multipoint Loewner framework for rapid offline/online algorithms:

Offline phase:

• Sample $\{p_j\}_{j=1}^q \subset \Pi$ and apply quadrature on $\partial\Omega$ at nodes $\{z_t, w_t\}_{t=1}^N$.
• For each $(p_j,z_t)$ and random probe vectors ${r_k}, {\ell_k}$, solve linear systems $T(z_t,p_j) x = r_k$ and $T(z_t,p_j)^* y = \ell_k$.
• Form quadrature approximants for the action of $H(s_i, p_j)$ on $r_k$ and $\ell_k$ at a selection of offline sample points.
• Apply vector-valued multivariate rational approximation (e.g., p-AAA) to construct surrogates $L_k(z, \mu) \approx \ell_k^* H(z, \mu)$ and $R_k(z, \mu) \approx H(z, \mu) r_k$, analytic in $(z, \mu)$.

Online phase (for new $\mu$):

• Evaluate $L_k, R_k$ on interlaced sample sets $\{\theta_i\}, \{\sigma_j\}$ to assemble Loewner matrices.
• Rank-$m$ SVD reduction and solution of the resulting generalized eigenproblem yields the eigenvalue diagonal matrix $J(\mu)$.
• Recovery of residue data via standard Loewner formulas.

The table below summarizes the phases:

Phase	Main Computation	Complexity
Offline	Linear solves + rational approx.	$O(Nq)$ solves, $O(qr^3)$ rational
Online	Loewner reduction + SVD	$O(r^3)$, no dependence on $n$

If $H(z, \mu)$ is rational in $(z, \mu)$, the p-AAA approximation recovers the eigenpairs exactly up to numerical precision. If $H$ is merely analytic (e.g., due to branch points), the rational approximation still achieves high accuracy within the region of approximation (Balicki et al., 4 Jan 2026).

4. Applications, Examples, and Practical Considerations

The parametric Keldysh decomposition supports a spectrum of nonlinear eigenvalue problems with parameter dependence, including:

• Linear test case (3×3): Eigenvalues $\pm \sqrt{1-p}$ with branch point at $p=1$, demonstrating accurate rational and analytic recovery depending on whether the branch point is within the parameter range.
• Delay-differential stability: $T(z,p) = (z + 0.01 e^{-pz}) I + E$ with infinite eigenvalues; four rightmost eigenvalues are reliably computed for $p \in [30,35]$ and extrapolated to $p=20,50$.
• Damped-string model: $T(z,p)$ involving $\sqrt{z^2 + 2pz}$; both eigenvalue coalescence and smooth spectral optimization are handled, including across coalescence points, with high-precision residuals.

Practical guidance:

• Contour $\partial\Omega$ must be chosen to enclose exactly $m$ eigenvalues for all parameter values, avoiding the introduction of branch or essential singularities.
• Probing vectors should statistically oversample the rank ($r > m$) to ensure numerical stability.
• Rational approximation denominators must avoid vanishing in $\Omega \times \Pi$; approximation tolerance must balance accuracy and offline computational burden.
• Ill-conditioning of Loewner pencils and numerical issues with defective spectra may necessitate precision enhancement or algorithmic deflation.
• For optimal extrapolation, parameter sampling should cover the intended domain, but rational models may fail near singularities or branch points (Balicki et al., 4 Jan 2026).

5. Relations to Quantum Field Theory and Large-$N$ Decompositions

A structurally related notion of parametric decomposition arises in nonequilibrium quantum field theory via the Keldysh rotation. In the Schwinger–Keldysh formalism, matrix fields defined on the “$+$” and “$-$” parts of the time contour $(M_+, M_-)$ are linearly recombined into “classical” and “quantum” fields:

$$M_{\text{cl}} = \frac{1}{2}(M_+ + M_-), \quad M_\text{qu} = M_+ - M_-.$$

The propagators in the new basis decompose into retarded ($G_R$), advanced ($G_A$), and Keldysh ($G_K$) components, with three nonzero correlators and a vanishing $\langle M_\text{qu} M_\text{qu} \rangle$ due to sum rules.

Notably, in large-$N$ matrix theories, the Feynman diagram expansion dualized to ribbon graphs and string worldsheets gives rise—post Keldysh rotation—to a two-fold (“parametric”) decomposition:

$\Sigma = \Sigma^{\rm cl} \cup \Sigma^{\rm qu},$

where the classical foundation $\Sigma^{\rm cl}$ and the quantum embellishment $\Sigma^{\rm qu}$ each carry their own independent genus expansions, and the total Euler characteristic splits as $$\chi(\Sigma) = \chi(\Sigma^{\rm cl}) + \chi(\Sigma^{\rm qu})$$. This structure supports refined ’t Hooft expansions and fresh insights into the interplay of dynamics (retarded/advanced propagators) and statistical data (Keldysh sector) (Horava et al., 2020).

6. Impact, Limitations, and Outlook

The parametric Keldysh decomposition provides a unified analytic-spectral tool for parametric nonlinear eigenvalue analysis and is foundational for rational contour integral algorithms with efficient offline/online decomposition. Its robustness for both rationally and more generally analytically parameter-dependent matrix functions underpins a variety of application domains, from stabilization of delay systems to damping optimization in mechanical models.

Key limitations include:

• Sensitivity to ill-conditioning in the rational approximation.
• Algorithmic fragility near spectral coalescences or branch points, which may require refined sampling or high-precision arithmetic.
• Exact recovery of multiple or defective eigenvalues is theoretically covered but is prone to numerical instability.

A plausible implication is that further research may refine approaches to handle branch singularities and improve rational-analytic hybrid approximation schemes, as well as extend the two-fold worldsheet decomposition perspective to additional classes of nonequilibrium systems and topological expansions (Balicki et al., 4 Jan 2026, Horava et al., 2020).