Spectral properties of canonical systems: discreteness and distribution of eigenvalues (2504.00182v1)

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 $\sigma$ is discrete, and if it is, what is its density. Here we measure density by the growth of the counting function of $\sigma$ 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 However, full answers have been obtained only very recently -- in partly still unpublished work.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 Schr\"odinger 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.

Spectral Properties of Canonical Systems: Discreteness and Distribution of Eigenvalues

The paper of spectral properties of two-dimensional canonical systems occupies a central role in the field of operator theory, particularly regarding the distribution and discreteness of eigenvalues. This paper by Jakob Reiffenstein and Harald Woracek synthesizes both classical results and recent advances on this topic, presenting a thorough account focused on whether the spectrum is discrete and, if so, the density of this spectrum.

Canonical systems, expressed in a differential form involving a Hamiltonian, serve as a model for many self-adjoint operators, including those arising from Schrödinger and Sturm-Liouville equations. The system itself is characterized by two-dimensional matrices where conditions on the Hamiltonian determine various spectral properties. This paper does not delve into spectral asymptotics; rather, it emphasizes measures of density related to eigenvalue counting functions, contrasting with more common spectral asymptotic inquiries.

Discreteness Criterion and Independence Theorem

The discreteness criterion given by the authors is notably straightforward: the spectrum of a canonical system is discrete if and only if there exists a direction in which the Hamiltonian demonstrates integrability alongside a vanishing product condition towards the boundary of the interval. This finding refines earlier results by I.S. Kac and M.G. Krein, emphasizing the significance of rotational components of the Hamiltonian while notably excluding off-diagonal elements from affecting the result. The independence theorem asserts that the spectral properties depend primarily on the diagonal components, a finding robust enough to hold under perturbations described by the Matsaev property—a property characterizing certain symmetrically normed operator ideals.

Spectrum Density via Operator Ideals and Lorentz Ideals

The paper explores the density of the spectrum in terms of operator ideals, where the resolvent's membership in a specified ideal can indicate whether the canonical system's spectrum is dense compared to certain known distributions like integers. This approach utilizes complex analysis and functional calculus methods, providing nuanced characterizations of spectrum size in Lorentz ideal terms, less commonly discussed but crucial in finer spectral descriptions.

Limit Circle Case and Monodromy Matrices

For systems in the limit circle case, the monodromy matrix provides insight into the growth and distribution of eigenvalues. Here, the paper discusses connections between eigenvalue distribution and the growth properties of entire functions as defined within the Cartwright class. Notably, the Krein-de~Branges formula describes the type of growth as proportional to integrals over the Hamiltonian, providing an interweaving of classical complex analysis with operator theory.

Jacobi Matrices and Moment Problems

Sections related to Jacobi matrices and Hamburger moment problems highlight the interplay between discrete spectral methods and canonical systems. The authors provide multiple results, emphasizing how various conditions on Jacobi parameters lead to determinate or indeterminate problems, thereby influencing spectral characteristics. In particular, the connection between moment sequences and growth properties via Nevanlinna matrices is expanded—an area deepened by historical results of Livšic and Berezanskii which are here updated and integrated into the broader narrative of canonical systems.

Recent Developments and Illustrative Examples

Adding practical depth, the authors introduce illustrative examples, such as continuous Hamiltonians with known boundary behaviors, like the chirp function, for which they concretely evaluate spectral growth. Extensions into more nuanced or pathological types illustrate both the flexibility in theoretical abstraction and the ability to ground abstractions into concrete spectral phenomena.

Conclusion and Future Perspectives

The implications of the findings are profound, offering a clearer path to understanding spectral densities beyond conventional asymptotic studies. Methodologically diverse—from differential equation reformulations to complex function theory—the approaches suggested here propose not just new results but pathways to explore open questions in the spectral theory of canonical systems. As spectral theory continues to evolve, implications for physical models—from quantum mechanics to advanced oscillatory systems—remain a core motivation underscoring the need for such clear yet comprehensive examinations of operator spectra. Future developments, especially in refining numerical approaches and exploring non-standard Hamiltonian classes, seem a promising direction guided by the foundational work laid out in this paper.