# Canonical systems with discrete spectrum

## TL;DR

This paper investigates the spectral properties of two-dimensional canonical systems with specific Hamiltonians, providing explicit conditions for the discreteness and distribution of the spectrum, and characterizes Hamiltonians related to de Branges spaces.

## Contribution

It offers explicit criteria for the discreteness and asymptotic distribution of spectra based solely on Hamiltonian diagonal entries and characterizes Hamiltonians of de Branges spaces.

## Key findings

- Spectrum is discrete under certain conditions on H
- Asymptotic distribution depends only on diagonal entries of H
- Complete characterization of Hamiltonians for de Branges spaces

## Abstract

We study spectral properties of two-dimensional canonical systems $y'(t)=zJH(t)y(t)$, $t\in[a,b)$, where the Hamiltonian $H$ is locally integrable on $[a,b)$, positive semidefinite, and Weyl's limit point case takes place at $b$. We answer the following questions explicitly in terms of $H$: Is the spectrum of the associated selfadjoint operator discrete ? If it is discrete, what is its asymptotic distribution ? Here asymptotic distribution means summability and limit superior conditions relative to comparison functions growing sufficiently fast. Making an analogy with complex analysis, this corresponds to convergence class and type w.r.t.\ proximate orders having order larger than $1$. It is a surprising fact that these properties depend only on the diagonal entries of $H$. In 1968 L.de~Branges posed the following question as a fundamental problem: Which Hamiltonians are the structure Hamiltonian of some\\ de~Branges space ? We give a complete and explicit answer.