Henrik Winkler

Direct and inverse spectral problems for generalized strings

AbstractLet the functionQ be holomorphic in he upper half plane ℂ+ and such that ImQ(z ≥ 0 and ImzQ(z) ≥ 0 ifz ε ℂ+. A basic result of M.G. Krein states that these functionsQ are the principal Titchmarsh-Weyl coefficiens of a (regular or singular) stringS[L,m] with a (non-decreasing) mass distribution functionm on some interval [0,L) with a free left endpoint 0. This string corresponds to the eigenvalue problemdf +λfdm = 0; f′(0−) = 0. In this note we show that the set of functionsQ which are holomorphic in ℂ+ and such that the kernel

Almost Pontryagin Spaces

\frac{{Q(z) - \overline {Q(\zeta )} }}{{z - \bar \zeta }}

Extremal extensions for the sum of nonnegative selfadjoint relations

hasκ negative squares of ℂ+ and ImzQ(z) ≥ 0 ifz ε ℂ+ is the principal Titchmarsh-Weyl coefficient of a generalized string, which is described by the eigenvalue problemdf′ +λfdm +λ2fdD = 0 on [0,L),f′(0−) = 0. Hereκ is the number of pointsx whereD increases or 0 >m(x + 0) −m(x − 0) ≥ −∞; outside of these pointsx the functionm is locally non-decreasing and the functionD is constant.

On Transformation of Canonical Systems

We investigate the structure of a maximal chain of matrix functions whose Weyl coefficient belongs to \( \mathcal{N}_\kappa ^ + \) . It is shown that its singularities must be of a very particular type. As an application we obtain detailed results on the structure of the singularities of a generalized string which are explicitly stated in terms of the mass function and the dipole function. The main tool is a transformation of matrices, the construction of which is based on the theory of symmetric and semibounded de Branges spaces of entire functions. As byproducts we obtain inverse spectral results for the classes of symmetric and essentially positive generalized Nevanlinna functions.

Sesquilinear forms corresponding to a non-semibounded Sturm-Liouville operator

The purpose of this note is to provide an axiomatic treatment of a generalization of the Pontryagin space concept to the case of degenerated inner product spaces.

Generalized Friedrichs Extensions Associated with Interface Conditions for Sturm-Liouville Operators

We consider a singular two-dimensional canonical system Jy’ = −zHy on [0, L) such that at L Weyl’s limit point case holds. Here H is a real and nonnegative definite matrix function, the so-called Hamiltonian. From results of L. de Branges it follows that the correspondence between canonical systems (or their Hamiltonians H) and their Titchmarsh-Weyl coefficients is a bijection between the class of trace normed Hamiltonians H and the class of Nevanlinna functions. In this note we show that the Hamiltonian H of a canonical system with a semibounded spectrum has the property det H = 0 and that its components are functions of locally bounded variation. Further, a characterization of the class of Hamiltonians corresponding to canonical systems with a finite number of negative (or positive) eigenvalues is given.

Small perturbations of canonical systems

The sum A + B of two nonnegative selfadjoint relations (multivalued operators) A and B is a nonnegative relation. The class of all extremal extensions of the sum A + B is characterized as products of relations via an auxiliary Hilbert space associated with A and B. The so-called form sum extension of A+B is a nonnegative selfadjoint extension, which is constructed via a closed quadratic form associated with A and B. Its connection to the class of extremal extensions is investigated and a criterion for its extremality is established, involving a nontrivial dependence on A and B.

Non-semi-bounded closed symmetric forms associated with a generalized Friedrichs extension

In this note we consider transformations of a canonical system of the form