The direct spectral problem for indefinite canonical systems

TL;DR

This paper investigates indefinite canonical systems with singularities, establishing boundary values and interface conditions to analyze the monodromy matrix and its components.

Contribution

It introduces generalized boundary values and interface conditions for indefinite canonical systems with singularities, enabling a structured analysis of the monodromy matrix.

Findings

Existence of generalized boundary values at singularities.

Construction of the monodromy matrix as a product of matrices.

Separation of Hamiltonian and discrete parameters in the monodromy matrix.

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.