Canonical systems whose Weyl coefficients have dominating real part

TL;DR

This paper provides criteria to determine when the real part of the Weyl coefficient dominates its imaginary part for a class of canonical systems, linking spectral measure behavior to properties of the Hamiltonian.

Contribution

It introduces explicit analytic and geometric conditions characterizing when the spectral measure's singular integral dominates its Poisson integral in canonical systems.

Findings

Analytic criterion based on the primitive of the Hamiltonian

Geometric condition involving oscillations and rotation of the Hamiltonian

Explicit characterization of spectral measure dominance

Abstract

For a two-dimensional canonical system $y'(t)=zJH(t)y(t)$ on the half-line $(0,\infty)$ 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 $H\mapsto q_H$ is a bijection between Hamiltonians (suitably normalised) and Nevanlinna functions. The main result of the paper is a criterion when the singular integral of the spectral measure, i.e. Re $q_H(iy)$, dominates its Poisson integral Im $q_H(iy)$ for $y\to+\infty$. 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, but explicit in terms of $H$ itself. 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 $H$.