## Abstract

For a two-dimensional canonical system

The main result of the paper is a criterion when the singular integral of the spectral measure, i.e.

*y'(t)=zJH(t)y(t*) on the half-line (0, ∞) whose Hamiltonian*H*is a.e. positive semi-definite, denote by_{q_H}its Weyl coefficient. De Branges' inverse spectral theorem states that the assignment*H*→*is a bijection between Hamiltonians (suitably normalised) and Nevanlinna functions.*_{q_H}The main result of the paper is a criterion when the singular integral of the spectral measure, i.e.

*Re*, dominates its Poisson integral Im_{q_H}(iy)*for*_{q_H}(iy)*y*→ + ∞. Two equivalent conditions characterising this situation are provided. The first one is analytic in nature, very simple, and explicit in terms of the primitive*M*of*H*. It merely depends on the relative size of the off-diagonal entries of*M*compared with the diagonal entries. The second condition is of geometric nature and technically more complicated. It involves the relative size of the off-diagonal entries of*H*, a measurement for oscillations of the diagonal of*H*, and a condition on the speed and smoothness of the rotation of*H*.Original language | English |
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Number of pages | 30 |

Journal | Journal d'Analyse Mathématique |

Publication status | Accepted/In press - 3 Aug 2022 |

## Keywords

- canonical system
- Weyl coefficient
- high-energy behaviour
- singular integral
- dominating real part