Canonical systems with discrete spectrum

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 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.AMS MSC 2010: 37J05, 34L20, 45P05, 46E22

Generalized indefinite strings with purely discrete spectrum

“…This connection will play a crucial role in our approach to obtain discreteness criteria for generalized indefinite strings. However, let us mention that the discreteness criteria as well as the criteria when the spectrum σ of a generalized indefinite string satisfies (1.2) can also be obtained by using the recent results of [30], as there is a bijective correspondence between generalized indefinite strings and 2 × 2 canonical systems; see [12,Section 6]. On the other hand, one may look at our results as an alternative way to obtain discreteness criteria for 2 × 2 canonical systems.…”

“…Removing the positivity assumption makes the corresponding considerations much more complicated. For instance, despite its fundamental importance, a discreteness criterion for 2 × 2 canonical systems was found only recently by R. Romanov and H. Woracek [30] (see also [29]). Surprisingly enough (at least to the authors), a discreteness criterion for indefinite strings (the case when the measure υ vanishes identically and ω is a real-valued Borel measure on [0, L)) has essentially been available since the 1970s, when C. A. Stuart [33] established a compactness criterion for integral operators in the Hilbert space L 2 [0, ∞) of the form J : f → ∞ 0 q(max( • , t))f (t)dt, (1.3) for a function q in L 2 loc [0, ∞).…”

Canonical systems whose Weyl coefficients have dominating real part

“…For illustration, let us mention two examples of such theorems. It is possible to explicitly characterise those Hamiltonians H for which q H has an analytic continuation to C \ [0, ∞), see [17], or those Hamiltonians for which q H has a meromorphic continuation to all of C, see [15]. The first result characterises that the differential operator associated with (1.1) is non-negative, the second one that it has discrete spectrum.…”

Generalised canonical systems related to matrix string equations: corresponding structured operators and high-energy asymptotics of the Weyl functions

“…Canonical systems are important objects of analysis, being perhaps the most important class of the one-dimensional Hamiltonian systems and including (as subclasses) several classical equations. They have been actively studied in many already classical as well as in various recent works (see, e.g., [1,5,7,12,13,18,20,[22][23][24][25][30][31][32][36][37][38]41,42] and numerous references therein).…”