Weyl–Yurko Matrix Spectral Theory

Updated 23 October 2025

Weyl–Yurko matrix is a matrix-valued spectral invariant that encodes spectral data for complex, higher-order, and non-classical operator systems.
It is constructed using fundamental solution matrices and boundary conditions, generalizing the classical Weyl function to settings with rectangular potentials and distributional coefficients.
It facilitates direct and inverse spectral analysis for diverse applications including quantum graphs, discrete operators, and non-self-adjoint problems, enabling reconstruction and uniqueness results.

A Weyl–Yurko matrix is a matrix-valued spectral invariant designed to encode the spectral data of higher-order and/or non-classical differential, difference, and operator systems, particularly in contexts beyond classical scalar second-order Sturm–Liouville theory. This matrix generalization is central in direct and inverse spectral analysis for systems with rectangular matrix potentials, distributional coefficients, non-self-adjoint or discrete settings, and for quantum graphs and lattices. The Weyl–Yurko matrix, often denoted $M(\lambda)$ , is defined via relationships between fundamental solution matrices subject to prescribed boundary or normalization conditions, and its entries encode essential information for reconstructing operator coefficients, potentials, or graph parameters from spectral data. Its analytic and algebraic properties generalize the role of the scalar Weyl (or Titchmarsh–Weyl) function to highly non-classical settings.

1. Formal Definition and Construction

The Weyl–Yurko matrix is constructed as follows. Given a system—typically an $n$ -th order scalar or matrix differential operator (possibly with distributional coefficients)—apply a regularization or matrix-valued reformulation (often via quasi-derivatives) to rewrite the problem as a first-order matrix system: $Y'(x) = (F(x) + \Lambda)Y(x),$ where $F(x)$ is a regularization matrix depending on the potential or coefficients, and $\Lambda$ encodes the spectral parameter. The system admits a fundamental solution matrix $C(x,\lambda)$ normalized at an endpoint, and a set of Weyl solutions $\Phi_k(x,\lambda)$ meeting specific boundary or spectral conditions (density at infinity, decay properties, or boundary forms).

The Weyl–Yurko matrix $M(\lambda)$ is then defined by

$\Phi(x, \lambda) = C(x, \lambda) M(\lambda),$

with the columns of $\Phi$ comprising Weyl solutions. Typically, each column of $M(\lambda)$ admits a meromorphic structure, with entries expressed as ratios of characteristic determinants associated with distinct boundary value problems: $M_{j,k}(\lambda) = -\frac{\Delta_{j,k}(\lambda)}{\Delta_{k,k}(\lambda)}.$ Here, $\Delta_{j,k}(\lambda)$ represents characteristic determinants tied to mixed or separated boundary conditions, and the poles of $M(\lambda)$ coincide with eigenvalues of these boundary problems.

In non-classical scenarios, such as Dirac systems with rectangular potentials ( $m_1 \neq m_2$ ) (Fritzsche et al., 2011, Fritzsche et al., 2011), discrete Dirac systems (Fritzsche et al., 2012), higher-order distributional operators (Bondarenko, 2023, Guan et al., 28 Feb 2024, Bondarenko, 22 Oct 2025), and quantum graphs (Avdonin et al., 2023, Wu et al., 23 Aug 2024), this construction is generalized to account for matrix sizes, boundary conditions, and uniqueness up to lower-triangular gauge transformations.

2. Analytical Properties and Non-Classical Generalizations

In classical settings, the Weyl function is scalar, analytic in the upper half-plane, and encodes the direct and inverse spectral data for second-order problems. The Weyl–Yurko matrix extends this to matrix-valued (possibly rectangular) objects:

For Dirac systems with block structure and non-square potentials,

$V(x) = \begin{bmatrix} 0 & v(x) \ v(x)^* & 0 \end{bmatrix},$

with $v(x)$ $m_1\times m_2$ , the Weyl function (Yurko matrix) is a non-expansive analytic matrix in $\mathbb{C}_+$ satisfying

$\int_0^\infty \left[ u(x,z) \begin{pmatrix} I_{m_1} \ \varphi(z) \end{pmatrix} \right]^* \left[ u(x,z) \begin{pmatrix} I_{m_1} \ \varphi(z) \end{pmatrix} \right] dx < \infty.$

In higher-order and distributional settings, $M(\lambda)$ is typically lower-triangular with unity diagonal, and possible ambiguity up to lower-triangular matrix multiplication corresponds to different regularizations (see (Bondarenko, 2023)).
Discrete Dirac and Jacobi systems admit analogous objects, with connections to Szegő recurrence, Schur coefficients, and quantum control (Fritzsche et al., 2012, Eichinger et al., 5 Mar 2025, Mikhaylov et al., 23 Sep 2025).

These matrices admit Laurent expansions at singular values,

$M(\lambda) = \frac{M_{-1}}{\lambda - \lambda_0} + M_0 + O(\lambda - \lambda_0),$

with residues encoding spectral data (weights).

Analytic properties such as non-expansiveness, analyticity, and matrix-valued high-energy asymptotic expansions (typically of Fourier/Laplace type in $\lambda$ ) are crucial in inverse spectral analysis (Fritzsche et al., 2011, Sakhnovich, 2021).

3. Inverse Spectral Theory: Uniqueness and Reconstruction

The Weyl–Yurko matrix serves as the central spectral invariant for the recovery of operator coefficients, potentials, or graph parameters. The general workflow in inverse problems is:

Given spectral data (Weyl matrix, poles, residues, or boundary spectral values), reconstruct the Weyl–Yurko matrix.
Utilize this matrix to derive the operator or potential via direct inversion or spectral mapping. For rational (finite-zone) cases in Dirac systems,

$\varphi(z) = -i \Lambda_2^* \Sigma_0^{-1} (zI_n - a)^{-1} \Lambda_1,$

explicit recovery of potential parameters is possible via Riccati equations and minimal realization data (Fritzsche et al., 2011).

Establish uniqueness theorems: if two operators have the same Weyl–Yurko matrix (possibly up to lower-triangular gauge), their coefficients coincide. For example, in (Bondarenko, 22 Oct 2025), for third-order operators with distribution coefficients, the theorem asserts $\sigma(x) = \tilde{\sigma}(x)$ almost everywhere if Weyl–Yurko matrices coincide.

The method extends to Borg–Marchenko type results for unique recovery from high-energy asymptotics, Taylor expansions, or discrete matrix data (Fritzsche et al., 2011, Fritzsche et al., 2012, Eichinger et al., 5 Mar 2025, Guan et al., 28 Feb 2024). These results also cover cases with multiple spectra and allow for recovery from limited data, with poles/residues (discrete spectral data) sufficing in some contexts (Bondarenko, 2023, Guan et al., 28 Feb 2024).

4. Synthesis on Graphs, Lattices, and Quantum Networks

Recent developments exploit the modular nature of the Weyl–Yurko matrix for synthesis and analysis of quantum graphs and periodic lattices (Avdonin et al., 2023, Wu et al., 23 Aug 2024):

The matrix may be synthesized recursively by adding edges and solving local systems of linear equations. For a quantum tree, matrices for the augmented graph are obtained by updating the Weyl matrix of the subgraph using continuity and flow conditions at vertices.
Similar procedures exist for square lattices, where the transfer of local spectral information lets the matrix be efficiently constructed for large graphs.
The synthesized Weyl matrix is equivalent to the Dirichlet-to-Neumann map, foundational in control, inverse boundary value problems, and spectral geometry.
The approach generalizes to periodic lattices with cycles, such as hexagonal geometries, by lifting modular steps to account for local cycles and complex connectivity.

This synthesis is computationally efficient, scalable, and theoretically connects local changes in graph structure to global spectral properties.

5. Applications in Moment Problems, Matrix Balls, and Non-Self-Adjoint Settings

The Weyl–Yurko matrix is fundamental in parametrizing sets of solutions for matrix moment problems (e.g., truncated Hamburger moment problem) via the notion of matrix balls (Fritzsche et al., 2021). In these settings, canonical block decompositions, Schur-type algorithms, and orthogonal matrix polynomial representations provide explicit formulas for the center and semi-radii of the solution set, building a bridge to the earlier FMI methods by Potapov and explicit formulas by Kovalishina.

In non-self-adjoint Jacobi matrices (Eichinger et al., 5 Mar 2025, Mikhaylov et al., 23 Sep 2025), matrix-valued Weyl functions and the Weyl–Yurko matrix encode the spectral data (measures plus phase functions), and analysis is extended to unbounded operators. Properties such as bijective correspondence between operator and spectral data, local Borg–Marchenko theorems, and asymptotic formulas underpin recovery and stability in these non-self-adjoint and discrete settings.

6. Theoretical Impact and Open Problems

The Weyl–Yurko matrix paradigm generalizes classical spectral theory to systems with complex, non-classical operators, singular or distributional coefficients, and complex networks or graphs. It is central to direct and inverse spectral problems, boundary value analysis, spectral geometry, quantum network design, and moment problems. It provides:

Constructive algorithms for both direct and inverse problems, including explicit recovery formulas;
Unified treatment of regular, singular, and rectangular matrix potentials;
Analytical tools for uniqueness, stability, and modular synthesis;
Framework for handling non-self-adjoint, distributional, or discrete settings;
Deep connections to structured operator identities, transfer matrix representations, and factorization.

Current open problems involve:

Rigorous convergence in reconstruction formulas for singular potentials and distributional coefficients (Bondarenko, 22 Oct 2025);
Development of associated regularization matrices for general differential expressions involving multiple singular coefficients;
Extension to higher-order systems, complex boundary conditions, and quantized lattices with cycles;
Analysis of stability, sensitivity, and computational complexity in large-scale modular synthesis, particularly for lattices with cycles and higher genus graphs;
Full local and global solvability for inverse spectral problems with limited spectral data.

The Weyl–Yurko matrix thus stands as a central tool in contemporary spectral theory, supporting research from analytic foundations to practical algorithms for operator and system identification in mathematics and mathematical physics.