A quantitative formula for the imaginary part of a Weyl coefficient

TL;DR

This paper derives a formula for the imaginary part of the Weyl coefficient in two-dimensional canonical systems, linking spectral properties to the behavior of the coefficient along the imaginary axis.

Contribution

It provides a new explicit formula for the imaginary part of the Weyl coefficient and explores its implications for spectral analysis and function theory in canonical systems.

Findings

The imaginary part of the Weyl coefficient can be determined up to constants.

Spectral properties relate to the asymptotic behavior of the Weyl coefficient.

Oscillatory behavior of the Weyl coefficient affects spectral measure growth.

Abstract

We investigate two-dimensional canonical systems $y'=zJHy$ on an interval, with positive semi-definite Hamiltonian $H$, such that limit circle case prevails at the left endpoint and limit point case at the right . Let $q_H$ be the Weyl coefficient of the system. We prove a formula that determines the imaginary part of $q_H$ along the imaginary axis up to multiplicative constants, which are independent of $H$. Using classical Abelian-Tauberian theorems, we deduce characterizations of spectral properties such as integrability of a given comparison function w.r.t. the spectral measure $\mu_H$, and boundedness of the distribution function of $\mu_H$ relative to a given comparison function. We study in depth Hamiltonians for which $\arg q_H(ir)$ approaches $0$ or $\pi$ (at least on a subsequence). We show that tangential behavior of $q_H(ir)$ imposes a substantial restriction on the growth of $|q_H(ir)|$. An example is provided where $\arg q_H(ir)$ heavily oscillates. Our results in this context are interesting also from a function theoretic point of view.