We consider two-dimensional canonical systems $y'(t) = zJH(t)y(t)$ on an interval $(0,L)$ where $J\colon={\tiny\begin{pmatrix} 0 \\ 1 & 0\end{pmatrix}}$, $z\in\mathbb C$, and where the Hamiltonian $H\colon(0,L)\to\mathbb R^{2\times 2}$ is locally integrable on $[0,L)$ with $H(t)\geq 0$, ${\rm tr} H(t)>0$, and $\int_0^L {\rm tr} H(t)\,dt=\infty$. Such systems have an operator model, which consists of a Hilbert space $L^2(H)$, a self-adjoint operator (or linear relation), and a boundary map.

Given a Hamiltonian $H$, Weyl's nested discs method produces a function $q_H$ called its Weyl coefficient. It is a Nevanlinna function, i.e.analytic in the open upper half-plane with ${\rm Im}\,q_H(z)\geq 0$ (or $q_H\equiv\infty$).The operator $A_H$ has simple spectrum, and a spectral measure $\sigma_H$ is obtained from the Herglotz integral representation of $q_H$. This makes it possible to investigate the spectrum of a canoncial system via the analytic function $q_H$.

The behaviour of the Weyl coefficient $q_H(z)$ when $z$ approaches $+i\infty$ is often named its high-energy behaviour. By classical Abelian-Tauberian theorems, it corresponds to the behaviour of the spectral measure at $\pm\infty$.

In this talk we discuss some direct and some inverse spectral theorems which relate the high-energy behaviour of $q_H$ to properties of the Hamiltonian $H$. Among them regularly varying asymptotics, radial cluster sets, and dominating real part.

This talk is based on the arXiv preprints 210607391v1, 210604167v1, 210810162v1, and some manuscripts in preparation.