Spectral properties of canonical systems: discreteness and distribution of eigenvalues
Jakob Reiffenstein, Harald Woracek
TL;DR
This survey addresses when the spectrum of a two-dimensional canonical system is discrete and how densely the eigenvalues are distributed, without pursuing spectral asymptotics. It unifies operator-theoretic, complex-analytic, and moment-problem methods, introducing Weyl disks, the monodromy matrix, and the Weyl coefficient, and then develops both discrete-spectrum criteria and density bounds. A central theme is the dichotomy between dense versus sparse spectra, with explicit discreteness criteria and density bounds obtained via operator ideals, Krein–de Branges formulas, and regular variation techniques; key results include Romanov–Woracek bounds, Livšic-type estimates, Berezanskii-type theorems, and algorithmic growth evaluation. The work emphasizes connections to Hamburger moment problems and Jacobi matrices, providing a self-contained framework and practical tools (algorithms and bounds) applicable to a broad readership across spectral theory and mathematical physics.
Abstract
In this survey paper we review classical results and recent progress about a certain topic in the spectral theory of two-dimensional canonical systems. Namely, we consider the questions whether the spectrum $σ$ is discrete, and if it is, what is its density. Here we measure density by the growth of the counting function of $σ$ in the sense of integrability or $\limsup$-conditions relative to suitable comparison functions. These questions have been around for many years. However, full answers have been obtained only very recently -- in partly still unpublished work. The way we measure density of eigenvalues must be clearly distinguished from spectral asymptotics, where one asks for an asymptotic expansion of eigenvalues. We explicitly and on purpose do not go into this direction and do not present any results about spectral asymptotics. We understand this survey as a focussed presentation of results revolving around the initially stated questions, and not as an exhaustive account on the literature which is in the one or other way related to the area. It does not contain any proofs, but we do comment on proof methods. The theorems we present are very diverse. They rely on different methods from operator theory, complex analysis, and classical analysis, and beautifully invoke the interplay between these areas. Besides the fundamental theorems solving the problem and several selected additions, we decided to include a number of results devoted to the spectral theory of Jacobi matrices, in other words, to the Hamburger moment problem. The connection between the theories of canonical systems and moment problems gives rise to important new insights, yet seems to be less widely known than other connections, e.g., with Krein-Feller or Schrödinger operators. We compile all necessary prerequisites from the general theory to make the exposition as self-contained as possible. It is our aim that this survey can be read and enjoyed by a broad community, including non-specialist readers.