Order and type of 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. A canonical system is given by a locally integrable function taking positive semidefinite real matrices as values, its Hamiltonian. Several types of scalar second- order equations can be rewritten as two-dimensional canonical systems. Among them Schrödinger equations, Krein-Feller operators, or Jacobi operators. With a two-dimensional canonical system there is associated a chain of reproducing kernel Hilbert spaces of entire functions, its de Branges chain. This construction is of outstanding importance in the spectral theory 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 being entire functions naturally brings up the task to relate the growth (in particular order and type) of these functions to properties of the Hamiltonian H of the system. A classical result is the Krein-de Branges formula which evaluates the maximal exponential type of functions in the de Branges chain as the integral of the square root of the determinant of H. This formula can be used to determine spectral asymptotics (in the limit circle case where the spectrum is discrete), or to determine the type of the spectral measure (in the limit point case where the spectral measure will in general be non-discrete). There is a large variety of examples where the determinant of H vanishes identically. Such occur for instance from Krein-Feller operators on fractal subsets of the real line, from birth-and-death processes in probability theory, or from special functions in number theory and orthogonal polynomials. For these systems the Krein-de Branges formula does not give any significant information. For example (thinking of the limit circle case) it cannot be distinguished whether the spectrum behaves asymptotically like n3 or en; the first corresponding to order 1/3, the latter to order 0. This project aims at establishing relations between the Hamiltonian and growth of functions in the de Branges chain referring to orders different than 1. These relations shall be exploited to obtain information about concrete spectral problems, notably such occurring in probability or number theory. In the context of direct spectral problems we shall be interested in spectral asymptotics, in the context of inverse problems we shall focus on the `order problem for a measure` which we pose as a generalisation of the type of a measure, studied in harmonic analysis, to orders less than 1. Methods of various fields will come into play, including: classical complex analysis (growth vs. zero-distribution, estimates of canonical products), differential operators (growth estimates for solutions, Levinson-type theorems), operator theory (reproducing kernel Hilbert spaces, Krein`s theory of entire operators, indefinite inner product spaces). The present proposal is a follow-up of the Joint project `The Order Problem for Canonical Systems` funded by the FWF (I-1536-N25) and the RFBR (13-01-91002-ANF).

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. A canonical system is given by a locally integrable function taking positive semidefinite real matrices as values, its Hamiltonian. Several types of scalar second- order equations can be rewritten as two-dimensional canonical systems. Among them Schrödinger equations, Krein-Feller operators, or Jacobi operators. With a two-dimensional canonical system there is associated a chain of reproducing kernel Hilbert spaces of entire functions and a Nevanlinna function: its de Branges chain and monodromy matrices, and its Weyl coefficient. These constructions are of outstanding importance in the spectral theory of canonical systems. The spectrum of the differential operator underlying the system can be read off from the monodromy matrix in the limit circle case, and from the Weyl coefficient in the limit point case. We aim at understanding the case that the spectrum is discrete. Then it is meaningful to ask for asymptotic distribution of density of eigenvalues. A classical result is the Krein-de Branges formula which evaluates the maximal exponential type of functions in the de Branges chain as the integral of the square root of the determinant of H. This formula determines spectral asymptotics compared to the power r (in the limit circle case and provided the determinant of H does not vanish identically). Our most important achievements are: (1) A characterisation that the spectrum is discrete. (2) Characterisations of finite spectral density measured by either weighted summability or boundedness w.r.t. a weight function, for weight functions growing faster than r. (3) An in depth study of the high-energy behaviour of the Weyl coefficient which leads to characterisations of weighted integrability of tails of the spectral measure, and will lead to characterisations as in the above item for weight functions growing slower than r. Our research brings together methods of various fields: classical complex analysis (growth vs. zero- distribution, estimates of canonical products), differential operators (growth estimates for solutions, Levinson-type theorems), operator theory (reproducing kernel Hilbert spaces, Krein's theory of entire operators, indefinite inner product spaces).