R. Romanov, H. Woracek:

"Canonical systems with discrete spectrum";

in: "ASC Report 04/2019", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2019, ISBN: 978-3-902627-12-4, 1 - 33.

We study spectral properties of two-dimensional canonical systems

y0(t) = zJH(t)y(t), t 2 [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 suffciently fast. Making an analogy with complex analysis, this correponds 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.

canonical system, discrete spectrum, eigenvalue distribution, operator ideal, Volterra operator, de Branges space

http://www.asc.tuwien.ac.at/preprint/2019/asc04x2019.pdf

Created from the Publication Database of the Vienna University of Technology.