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The direct spectral problem for indefinite canonical systems

Matthias Langer, Harald Woracek

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 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.

Paper Structure

This paper contains 13 sections, 11 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Reminder about canonical systems I. The positive definite case
  3. Canonical systems with positive Hamiltonian
  4. The operator model
  5. The conditions (I) and (HS)
  6. $H$-polynomials
  7. Reminder about canonical systems II. The indefinite case
  8. Elementary indefinite Hamiltonians of kind (A)
  9. The operator model
  10. Solution using regularised boundary values
  11. Regularised boundary values
  12. Factorisation of the monodromy matrix
  13. An example

Key Result

Lemma 2.13

Let $H\in{\mathbb{H}}_{\sf cp}(s_-,s_+)$ and assume that $H$ satisfies (I) and (HS). Then there exists a unique sequence $(\rho_n)_{n=0}^\infty$ of numbers $\rho_n\in{\mathbb{C}}$ such that $\rho_0=0$ and $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (50)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Remark 2.9
  • Definition 2.10
  • ...and 40 more