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 $H$. 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 $H$. We also provide versions of this result for Sturm–Liouville operators and Krein strings. Using classical Abelian–Tauberian theorems, we deduce characterizations of spectral properties such as integrability of a given comparison function with respect to 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). It turns out that this behavior of $q\_H(ir)$ imposes a substantial restriction on the growth of $|q\_H(ir)|$. Our results in this context are interesting also from a function theoretic point of view.