### Abstract

We study spectral properties of two-dimensional canonical systems y^{′}(t)=zJH(t)y(t), t∈[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.

Original language | English |
---|---|

Article number | 108318 |

Journal | Journal of Functional Analysis |

Volume | 278 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Mar 2020 |

### Scopus subject areas

- Analysis

## Fingerprint Dive into the research topics of 'Canonical systems with discrete spectrum'. Together they form a unique fingerprint.

## Cite this

*Journal of Functional Analysis*,

*278*(4), [108318]. https://doi.org/10.1016/j.jfa.2019.108318