The direct spectral problem for indefinite canonical systems

Abstract: For indefinite (Pontryagin space) canonical systems that contain an inner singularity we prove the existence of generalised boundary values at the singularity, which are used to formulate interface conditions. With the help of such interface conditions we construct the monodromy matrix of the canonical system and write it as a product of matrices, which separates the contributions of the Hamiltonian function and the finitely many discrete parameters that are associated with the singularity.

• The paper provides explicit construction of generalized boundary values for indefinite canonical systems with inner singularities.
• It formulates clear interface conditions using unimodular matrix polynomials to match solutions across singular points.
• The analysis yields a precise factorization of the monodromy matrix and establishes a finite-rank perturbation model linking operator theory to spectral data.

The Direct Spectral Problem for Indefinite Canonical Systems

Introduction and Context

The study focuses on two-dimensional canonical systems described by the first-order differential equation

$y'(t) = z J H(t) y(t), \quad t \in (s_-, s_+),$

where $z$ is the spectral parameter, $J$ is the symplectic matrix, and $H$ is a measurable, locally integrable, positive semi-definite Hamiltonian. In the classical positive-definite setting, the spectral theory of such systems is well-developed, notably via the monodromy matrix and the associated spaces of solutions. Operator-theoretic models are unitarily equivalent to reproducing kernel Hilbert spaces of entire functions or $L^2$-spaces, facilitating a deep connection between direct and inverse spectral problems.

In the indefinite case—where the reproducing kernel has a finite number of negative squares (Pontryagin space framework)—the direct spectral theory is significantly less transparent. Notably, canonical systems may contain inner singularities: isolated points where the Hamiltonian degenerates or exhibits non-integrable growth. Prior operator-theoretic constructions by Krein, Langer, and further developed by Kaltenbäck and Woracek, allowed encoding the spectral data of such systems via the monodromy matrix and a finite set of discrete parameters associated to the singularities. However, those constructions were non-constructive and lacked explicit realization in terms of boundary/interface conditions at singularities.

Main Contributions

This work resolves a critical gap by providing an explicit construction and analysis of the direct spectral problem for indefinite canonical systems with inner singularities:

• Existence and Construction of Generalized Boundary Values: The authors rigorously prove the existence of regularized (generalized) boundary values at each inner singularity for indefinite canonical systems. These boundary values are constructed via limits of specific functionals involving the fundamental system of solutions and play a pivotal role in formulating matching (interface) conditions.
• Formulation of Interface Conditions: For an elementary indefinite Hamiltonian (the essential building block), the discrete data associated to singularities is encoded in a polynomial $p(z)$, explicitly linking spectral characteristics to the system parameters. The interface conditions elegantly couple solutions on either side of a singularity via the relation

$\Gamma^+(z)\hat f = R(z) \Gamma^-(z)\hat f,$

where $R(z)$ is a unimodular matrix depending only on the polynomial $p(z)$. This condition gives a constructive recipe for propagating solutions across a singular point, thereby enabling an explicit concatenation of localized solutions.

• Matrix Factorization of the Monodromy Matrix: The monodromy matrix, a central spectral invariant, is shown to admit a product factorization that delineates the contributions of the Hamiltonian function and the discrete singularity parameters. Specifically, for a solution system $V(t, z)$ on the right of a singularity, the canonical monodromy is given by

$z$0

Here, $z$1 and $z$2 encode generalised boundary data for the left and right intervals, respectively, and $z$3 encapsulates the entirety of the discrete singularity data.

• Operator Model as a Finite-Rank Perturbation: The natural operator model corresponding to indefinite canonical systems with inner singularities is explicitly realized as a finite-dimensional perturbation of the maximal operator induced by the formal differential expression. This precise structure allows for a direct transfer of operator-theoretic spectral information to function-theoretic terms.
• Parametric Dependence and Comparison Results: The work establishes a robust framework for analyzing how changes in the singularity parameters impact the spectral data. Via explicit formulae, the difference between monodromy matrices for two Hamiltonians sharing the same underlying $z$4 but differing parameters is realized through the "jump matrix" $z$5 pre-multiplied by the difference of the polynomials $z$6.

Strong Claims and Main Results

• Explicit Existence and Uniqueness of Regularized Boundary Values: For arbitrary prescribed boundary data $z$7 and spectral parameter $z$8, there exists a unique locally absolutely continuous solution achieving those (generalized) boundary values on either side of the singularity, for all indefinite Hamiltonians built from the elementary construction.
• Constructive Realization of Monodromy via Explicit Matrices: The universally valid factorization delineates bulk Hamiltonian effects and localized point interaction effects (singularities), allowing for explicit computation of the monodromy matrix from resolved local data.

Theoretical and Practical Implications

Theoretical:

• The treatment provides a canonical (no pun intended) structure to the spectral theory of indefinite systems, akin to the definitizable case, and clarifies long-standing ambiguities in the analytic continuation and patching of solutions.
• The explicit interface condition (in terms of matrix polynomials) may serve as a foundation for a full inverse spectral theory where both continuous and discrete spectral data are encoded transparently.

Practical/Future Directions:

• The methods establish a clear blueprint for numerical implementation: interface conditions can be encoded as algebraic constraints, facilitating the computation of spectral invariants for systems with finitely many singularities.
• Future developments may leverage these explicit interface conditions to study stability, perturbation, and inverse problems for indefinite systems in mathematical physics, e.g., in the analysis of quadratic operator pencils, indefinite Krein chains, or quantum graphs with sign-indefinite weights.
• The structure of the monodromy matrix and its parametric dependence on edge polynomials opens up a route for spectral optimization and control in applied indefinite systems.

Conclusion

The paper delivers a constructive, operator-theoretically transparent, and functionally explicit solution to the direct spectral problem for indefinite canonical systems with inner singularities. Regularized boundary values and their associated interface conditions provide the foundational basis for describing how local and discrete data control global spectral features. These developments resolve previously open issues and lay groundwork for further analysis and applications in indefinite spectral theory, with clear implications for operator models, system concatenation, and inverse spectral problems.

For the detailed mathematical framework, proofs, and explicit algorithmic recipes, see "The direct spectral problem for indefinite canonical systems" (2604.00541).