Limit behavior of Weyl coefficients

The sets of radial or nontangential limit points towards $i\infty$ of a Nevanlinna function $q$ are studied. Given a nonempty, closed, and connected subset $\mathcal{L}$ of $\overline{\mathbb C_+}$, a Hamiltonian $H$ is constructed explicitly such that the radial and outer angular cluster sets towards $i\infty$ of the Weyl coefficient $q_H$ are both equal to $\mathcal{L}$. The method is based on a study of the continuous group action of rescaling operators on the set of all Hamiltonians.