Canonical systems with discrete spectrum

Roman Romanov, Harald Woracek

Research output

2 Citations (Scopus)

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 languageEnglish
Article number108318
JournalJournal of Functional Analysis
Volume278
Issue number4
DOIs
Publication statusPublished - 1 Mar 2020

Scopus subject areas

  • Analysis

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