Indefinite Generalizations of Canonical Systems

A classical canonical system in this project is a special symmetric first order system of two ordinary linear differential equations which has locally summable coefficients and depends linearly on the spectral parameter. The spectral properties of such a system are completely determined by its Titchmarsh-Weyl coefficient, which is an analytic function mapping the upper half plane into itself, or, equivalently, for which a certain kernel is positive definite. The inverse spectral problem consists in the reconstruction of the canonical system from a given Titchmarsh-Weyl coefficient. In the classical situation its solution is contained in the work of de Branges. During the last 30 years also systems have been considered, whose coefficients have singularities or which depend nonlinearly on the eigenvalue parameter. They lead to Titchmarsh-Weyl coefficients for which this kernel is not positive definite but has a finite number of negative squares. The aim of the present project is to give a description of a general class of canonical systems, which have a Titchmarsh-Weyl coefficient of this type. These systems are closely related to Sturm-Liouville operators with singular potentials and floating singularities, and also to strings with indefinite mass distribution and dipoles.

Inverse spectral problems, that is, reconstructing the parameters of a system from observation data, play a central role in several applications of mathematics, like, e.g. in quantum theory, geophysics and hydrodynamics. Such problems are of the following type: find a differential operator (e.g. a Schrödinger operator or a canonical system) such that a given distribution function becomes its spectral function. Pioneering work in this area has been done in the middle of the last century by I. Gelfand, B. Levitan, M. Krein and V. Marchenko. Often - especially since the fundamental study of L. de Branges - the Stieltjes transform of the density function is chosen as starting point. It is a so-called Nevanlinna function (that is a holomorphic function which maps the complex upper half plane into itself); for differential operators this function becomes the so-called Titchmarsh-Weyl coefficient. Motivated by numerous examples studied in the last decade, within this project we have been dealing with a corresponding question for generalized Nevanlinna functions. These are functions for which a certain kernel, which for Nevanlinna functions is positive definite, can be indefinite, that is, a certain quadratic form has finitely many negative squares. Typically the corresponding "indefinite" canonical systems or differential operators have certain singularities, where, e.g. the potential is concentrated at one point (point interaction), is not summable, or where special interface or boundary conditions have to be imposed. A main task of the project was to clarify the analytic structure of generalized Nevanlinna functions, including their operator representations or realizations; in particular, the above mentioned differential operators are such realizations. Concrete models were given for a special subclass, which appears in connection with singular perturbations, and the structure of the elements of this subclass could be described completely. Other main results concern differential operators with singularities, in particular, operators of Bessel type. For them an operator model of the corresponding canonical system was described, approximating models were studied, and the connection between singular perturbations and the existence of a scalar spectral function (in case of two singular endpoints) was shown.