The order problem for canonical systems

Canonical systems are differential equations of a specific form which frequently appear in natural sciences. For example in Hamiltonian mechanics, where they model the motion of a particle under the influence of a time- dependent potential, or as generalizations of Sturm-Liouville problems, e.g. in the study of a vibrating string with non-homogeneous mass distribution. A canonical system is given by a locally integrable function taking positive semidefinite real matrices as values, its Hamiltonian. A chain of Hilbert spaces of entire functions is closely connected with a canonical system, known as the associated chain of de Branges spaces. The theory of such spaces plays a prominent role in the investigation of the spectrum of canonical systems. For example it is the basis for the Inverse Spectral Theorem which states that a Hamiltonian is essentially uniquely determined by its spectral function. The elements of the de Branges spaces associated with a canoncial system are entire functions, and hence it is natural to pose the question how the growth (in particular order and type) of these functions is related to the Hamiltonian. With exception of some particular cases, the only known general result in this direction is that all functions are of order at most one and finite exponential type. In fact, the exponential type can be computed from the Hamiltonian by means of a simple formula. This project aims at establishing general quantitative relations between the Hamiltonian on the one hand, and growth referring to small exponential orders or to proximate orders (growth functions) on the other. Moreover, we will investigate the consequences of such relations for the spectral theory of canonical systems. The basic problem breaks into three tasks: Estimates for growth in terms of the Hamiltonian (direct problem), characterisation of those Hamiltonians whose associated de Branges spaces possess or do not exceed a prescribed growth (inverse problem), estimating or determining the growth in terms of the spectral measure associated with the canonical system. Answers to these basic questions will lead, in particular, to statements about the asymptotic distribution of eigenvalues of a system in case its spectrum is discrete. Several classes of canoncial systems are of particular interest. For example, Jacobi matrices and Schrödinger operators can be considered as canonical systems. The first are closely related to the power moment problem, the second are basic objects in quantum mechanics. The general results shall be applied to these situations. Methods of various areas of analysis will be employed to tackle the problem. These include: the theory of differential operators (Volterra integral equations, variational techniques, Levinson-type theorems), complex analysis (growth and zero-distibution, singular integrals, subharmonic functions), and functional analysis (reproducing kernel Hilbert spaces, Krein`s theory of entire operators, resolvent matrices).

Canonical systems are differential equations which frequently appear in natural sciences. They are described by one function: their Hamiltonian. In mechanics they model the motion of a particle under the influence of a time- dependent potential (and this potential determines the Hamiltonian), in the study of a vibrating string with non-homogeneous mass distribution (and this mass distribution determines the Hamiltonian). A discrete version appears in probability theory via the power moments of a probablity measure (and these moments determines the Hamiltonian).Eigenvalues of the system represent states of the modelled system which may decay or blow up when time passes, but otherwise remain stable. For a vibrating string, eigenvalues determine the fundamental frequency of the string and its overtones. The asymptotic distribution of the eigenvalues of the system is intimately related to growth properties of the solution. The projects aim was to establish general quantitative relations between the Hamiltonian and growth properties of its solution.The study of systems naturally branches in two directions.(1) Direct problems: we have full theoretic understanding about how the system works, and wish to predict how the state will change in the future from knowing the present state.(2) Inverse problems: we have some experience and measurements about how the systems behaves, and wish to recover the rules which the systems obeys.A large part of our research revolved around the afore mentioned discrete version dealing with the power moments of a measure. We were able to give estimates for the spectral asympotics in general and to exactly determine the asymptotics provided that the Hamiltonian of the system does not behave too wildly. Another important aspect was the development of a method to estimate growth by transforming the system to another system of an algebraically simpler form but giving rise to an indefinite inner product. Both of these aspects deal with the direct problem. In the regime of inverse problems, we developed perturbation methods which allow to control small shifts of eigenvalues and small additive components. Perturbation results are of particular relevance in applications, thinking of possible inaccuracies of measurements.