Parametric Nonlinear Eigenvalue Problems

• Parametric nonlinear eigenvalue problems are systems where eigenvalues and eigenvectors vary with both the spectral parameter and additional parameters, critical in PDEs and matrix pencils.
• They exhibit intricate spectral behaviors including bifurcations, resonance, and multiplicity changes, analyzed using variational methods, critical groups, and analytic perturbation techniques.
• Robust computational techniques such as Taylor/Chebyshev expansions, contour integration, and surrogate-based methods enable efficient tracking and approximation of eigenpairs in high-dimensional settings.

Parametric nonlinear eigenvalue problems (PNEVPs) form a broad and active area in nonlinear analysis, spectral theory, and numerical computation. They encompass operator and matrix families dependent on both the spectral parameter and one or more additional parameters, such that the spectrum, eigenvectors, and their multiplicities vary in possibly intricate ways under changes of the parameter. This class includes nonlinear PDE eigenvalue problems, matrix-valued analytic or polynomial pencils, nonlinear operator perturbations, and eigenproblems with singular, superlinear, or sublinear nonlinearities. Analytical, topological, and computational frameworks have been developed for existence, multiplicity, bifurcation, and numerical approximation of parameter-dependent nonlinear spectra.

1. Core Definitions and Problem Types

Let $\mathcal{P} \subset \mathbb{R}^k$ be a parameter domain and $X$ a Banach (or Hilbert) space. A generic parametric nonlinear eigenvalue problem seeks, for each $\mu\in\mathcal{P}$, eigenvalue-eigenvector pairs $(\lambda(\mu), u(\mu))$ (or possibly curves $\lambda^i(\mu)$) solving

$$T(\lambda(\mu), \mu) u(\mu) = 0,\quad u(\mu) \neq 0,$$

where $T$ may be an operator-valued or matrix-valued nonlinear analytic function in both spectral argument $\lambda$ and parameter $\mu$. In PDE and variational contexts, such as semilinear and quasilinear elliptic operators, the prototypical parametric nonlinear eigenproblem takes the form (Papageorgiou et al., 2019, Bahn, 2023): $$\begin{cases} -\nabla\cdot(a(x)\nabla u(x,\mu)) + b(x,\mu) u(x,\mu) + \eta |u|^{p-1} u = \lambda(\mu) u(x,\mu), & x\in\Omega, \ u(x,\mu) = 0, & x\in\partial\Omega, \end{cases}$$ with $b(x,\mu)$ affinely or nonlinearly parameterized.

Discrete analogues model $n\times n$ matrix pencils or general analytic matrix families $A(\lambda,\mu)$, where the NEVP is $A(\lambda(\mu),\mu)x(\mu) = 0$ (Mach et al., 2023, Balicki et al., 4 Jan 2026). Notably, systems exhibiting eigenvector nonlinearities of the form $A(u)u = \lambda u$ or in orthonormal subspaces $A(V)V = V S$ (basis-invariant) are encompassed in this general scheme (Jarlebring et al., 2020).

Key subclasses include:

• Polynomial NEVPs: $P(\lambda; \mu) = \sum_{j=0}^m A_j(\mu) \lambda^j$ (Hochstenbach et al., 2018).
• Semilinear/Quasilinear PDE eigenproblems: nonlinearities of power, singular, or resonance type depending on spectral and parametric inputs (Papageorgiou et al., 2019, Papageorgiou et al., 2019, Bahn, 2023).
• Nonlocal and eigenvector-dependent NEVPs: spectral parameter enters both linear and nonlinear terms, coefficients dependent on invariant subspaces (Jarlebring et al., 2020).
• Multiparameter NEVPs: several parameters affect the operator or boundary conditions (Hochstenbach et al., 2018, Pradovera et al., 2023).

2. Existence, Multiplicity, and Bifurcation Structure

The analytical theory of PNEVPs focuses on under what parameter regimes solutions exist, the structure and multiplicity of eigenpairs, and qualitative features such as bifurcations, resonance, and threshold phenomena.

• Variational methods: For variationally-structured PNEVPs (e.g. $p$-Laplacian with nonlinear reaction) one frames the problem via a parametric energy functional $\phi_\lambda$ and seeks critical points as weak solutions. Existence and multiplicity often rely on verifying the Cerami (or Palais–Smale) compactness within $W^{1,p}$ or $H_0^1$ (Papageorgiou et al., 2019, Papageorgiou et al., 2018).
• Multiplicity via critical groups: For nonlinear Robin problems with resonance at a nonprincipal eigenvalue and competing $(p-1)$-sublinear/linear terms, Morse theory and computation of critical groups yield at least five nontrivial smooth solutions for all large $\lambda$. The presence of both sublinear and resonant linear terms, truncation, and strong comparison enables detection of sign-changing and multiple constant-sign solutions beyond classical cases (Papageorgiou et al., 2019).
• Bifurcation and turning points: In singular Dirichlet problems, bifurcation and threshold parameters (e.g. $\lambda^*$) are sharply characterized. As the parameter crosses critical values, the number and nature of solutions transition non-smoothly — e.g., a U- or S-shaped bifurcation curve in the $(\lambda,\|u\|_\infty)$-plane: below $\lambda^*$ there are two ordered positive solutions coalescing at $\lambda^*$, and none above (Papageorgiou et al., 2019).
• Monotonicity and minimal solutions: Pointwise minimal positive solutions $\overline{u}_\lambda$ exist for sublinear/superlinear NEVPs for $\lambda$ below threshold, with strict monotonicity and left-continuity of the map $\lambda \mapsto \overline{u}_\lambda$ (Papageorgiou et al., 2018).

3. Analytical Dependence and Uncertainty Quantification

Recent work establishes the full parametric analyticity of eigenpairs and their statistical quantification in high-dimensional or stochastic parameter spaces.

• Analyticity and mixed-derivative bounds: For semilinear elliptic PNEVPs with coefficients affinely parameterized by countably many variables, mixed derivatives of ground state eigenpairs admit factorial-weighted $\ell^q$-type bounds, ensuring not only smooth but holomorphic parameter dependence. Uniform spectral gaps and boundedness provide the required invertibility for an analytic implicit-function approach (Bahn, 2023).
• Dimension-independent UQ rates: These analyticity results empower high-dimensional uncertainty quantification by quasi-Monte Carlo (QMC) methods. Given suitable decay of parameterization (e.g., $\ell^1$-summability of the coefficient functions) and bounds on mixed partials, QMC achieves dimension-robust algebraic convergence rates in pathwise and expectation errors for eigenvalues and functionals of eigenvectors (Bahn, 2023).

4. Numerical Algorithms for Parametric NEVPs

The numerical solution and tracking of parametric nonlinear spectra utilize both direct and surrogate-based methods, adapted for analytic, polynomial, or PDE-based operators.

• Taylor and Chebyshev series expansions: For smoothly parameterized problems $A(\mu)$, parametric eigenvalues and eigenvectors are approximated either locally by iterative Taylor expansions (bordered systems for each coefficient, converging in the neighborhood of $\mu_0$), or globally on intervals by Chebyshev expansions refined via Newton iteration on large coupled nonlinear systems. Taylor methods are $O(n^3+m^2n^2)$ for order-$m$, effective locally; Chebyshev approximations are $O((pn)^3)$ per eigenbranch, but provide uniform global accuracy (Mach et al., 2023).
• Parametric Keldysh decomposition and contour integration: For matrix/operator pencils $T(z,\mu)$ analytic in $(z,\mu)$, the parametric Keldysh theorem enables a decomposition of the resolvent into a rational fraction whose poles are the sought eigenvalues $\lambda_i(\mu)$ (analytic in $\mu$) plus a remainder vanishing under contour integrals. Algorithms leverage this for rapid computation: an intensive offline phase constructs a rational surrogate in $(z,\mu)$ by contour quadrature and rational or Loewner interpolation at sampled $(z,\mu)$ pairs; subsequent eigenvalue extraction for any $\mu$ is reduced to cheap small-scale eigenproblems (Balicki et al., 4 Jan 2026). This framework is robust for both linear and nonlinear pencils, and handles analytic tracking up to branch points.
• Match-based global surrogate construction: Eigenvalue tracking over quoted parameter spaces leverages match-based strategies to stitch local non-parametric contour-based eigenvalue solves into global continuous curves $\lambda^i(\mu)$, even as eigenvalues enter/exit the computation domain or undergo bifurcations. Assignment problems (Hungarian method, min-cost flows) solve for optimal pairings; bifurcation detection uses cost-perturbation criteria, with roots of interpolated nodal polynomials supplying multifold branches (Pradovera et al., 2023).
• Divided-difference selection for multiparameter NEVPs: For large sparse NEVPs, especially with multiple parameters, divided-difference–based selection criteria prevent reconvergence to already found eigenpairs and facilitate consistent tracking of distinct branches, with variants for homogeneous coordinates (including infinite eigenvalues) and higher-dimensional parameter sets (Hochstenbach et al., 2018).
• Algorithms for eigenvector nonlinearities: When the matrix function $A(V)$ itself depends on its invariant subspace, two principal schemes are analyzed: the self-consistent field (SCF) iteration (each step is a standard linear eigenproblem for the current $A(V)$) and a new implicit-Jacobian-based iteration (each step uses the Newton Jacobian at the previous iterate), with the latter exhibiting local quadratic convergence under spectral gap assumptions. Iteration cost, local convergence rate, and suitability for high-precision parameter sweeps are compared (Jarlebring et al., 2020).

5. Special Phenomena: Resonance, Bifurcation, and Singularities

The nonlinear and parameter-dependent character of PNEVPs can result in intricate spectral phenomena.

• Resonance at nonprincipal eigenvalues: When the reaction term in a PDE NEVP is $(p-1)$-linear and tuned to a nonprincipal variational eigenvalue, classical bifurcation and existence results do not apply directly. Use of Morse theory, careful truncation, and computation of critical groups enables the detection of additional solutions, with the resonance structure leading to nontrivial multiplicity (Papageorgiou et al., 2019).
• Bifurcation and multiplicity diagrams: Parametric dependence often organizes solution sets into bifurcation diagrams with branches meeting at turning points, folds, or undergoing node/defect coalescence. These structures are quantitatively tractable in both analytic (Balicki et al., 4 Jan 2026) and PDE (Papageorgiou et al., 2019) contexts, and motivate adaptive sampling and bifurcation-detection in numerical solvers (Pradovera et al., 2023).
• Analytic branch points and singularities: Singularity structure (e.g., points of eigenvalue collision, defective matrix behavior) sharply limits the radius of convergence for Taylor series, or the interval of accuracy for global polynomial approximants. Eigenvalue/eigenvector branches may fail to be analytic beyond these loci, requiring careful localization in both analysis and computation (Mach et al., 2023).

6. Applications and Computational Impact

PNEVPs arise in nonlinear spectral analysis of PDEs (reaction-diffusion, elasticity, quantum mechanics), control and stabilization of parameter-dependent systems, electronic structure computations (mean-field, Hartree–Fock/Kohn–Sham), dynamical stability of large structures, and stochastic modeling with parametric uncertainty.

Recent algorithmic developments enable scalable high-accuracy computation of eigenbands, robust handling of bifurcations and migrations, and fast parametric sampling or UQ in high-dimensional spaces. The combination of parametric analyticity, spectral surrogates, and adaptive sampling delivers orders-of-magnitude acceleration for functional evaluations over parameter grids or Monte Carlo samples (Pradovera et al., 2023, Balicki et al., 4 Jan 2026, Mach et al., 2023, Bahn, 2023), with rigorous error control accessible in QMC-based statistical regimes (Bahn, 2023).

7. Representative Results and Open Directions

Reference	Type/Setting	Principal Results/Distinctive Methodology
(Papageorgiou et al., 2019)	Nonlinear PDE (Robin-type)	Multiplicity (≥5 solutions) under resonance at nonprincipal eigs via Morse theory
(Papageorgiou et al., 2019)	Singular $p$-Laplacian, Dirichlet	Two-branch bifurcation; sharp threshold; U/S-shaped diagram
(Papageorgiou et al., 2018)	(p,q)-type operator; Robin/Dirichlet	Minimal positive solution monotonicity across λ; sharp threshold
(Bahn, 2023)	Semilinear elliptic PDE	Parametric analyticity, factorial mixed-derivative bounds, QMC UQ
(Mach et al., 2023)	Matrix NEVP, analytic $A(\mu)$	Taylor/Chebyshev expansion methods, complexity and limitation analysis
(Balicki et al., 4 Jan 2026)	Analytic matrix pencil	Parametric Keldysh decomposition, offline/online spectral surrogate
(Pradovera et al., 2023)	General parametric NEVPs	Match-based global eigencurve construction, adaptive sampling, bifurcation handling
(Hochstenbach et al., 2018)	Large-scale (multi)param. NEVP	Selection/divided-difference criteria, Jacobi-Davidson variants
(Jarlebring et al., 2020)	Eigenvector-nonlinear NEVPs	Implicit (J-version/SCF) algorithms, local convergence rates

Active research directions include system and operator generalizations (anisotropic, nonlocal, or quasi-linear structures), rigorous complexity estimation for high-dimensional QMC or surrogate methods, extension to non-analytic or low-regularity parameterizations, and global geometric and topological classification of parametric nonlinear spectra.

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References: (Papageorgiou et al., 2019, Papageorgiou et al., 2019, Papageorgiou et al., 2018, Pradovera et al., 2023, Bahn, 2023, Balicki et al., 4 Jan 2026, Mach et al., 2023, Hochstenbach et al., 2018, Jarlebring et al., 2020)