A quantitative formula for the imaginary part of a Weyl coefficient

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.