Limit behaviour of Weyl coefficients

TL;DR

This paper constructs Hamiltonians with prescribed limit sets for Weyl coefficients, advancing understanding of their boundary behaviors using group actions on Hamiltonians.

Contribution

It provides explicit constructions of Hamiltonians with specific limit sets for Weyl coefficients, utilizing group actions on Hamiltonians.

Findings

Explicit Hamiltonians with prescribed limit sets for Weyl coefficients

Method based on rescaling group actions on Hamiltonians

Both radial and angular limits can be controlled

Abstract

We study the sets of radial or nontangential limit points towards $i\infty$ of a Nevanlinna function q. Given a nonempty, closed, and connected subset L of $C_+$ , we explicitly construct a Hamiltonian H such that the radial- and outer angular cluster sets towards $i\infty$ of the Weyl coefficient q H are both equal to L. Our method is based on a study of the continuous group action of rescaling operators on the set of all Hamiltonians.