Pontryagin spaces of entire functions I

Dependence of the Weyl coefficient on singular interface conditions

“…It was also proved in [27,Lemma 3.5] that for any chain ω ∈ M <∞ we have lim t σ0 ω(t) = I. Moreover, by (W3), it is non-decreasing.…”

“…The definition of T m may seem a little ad hoc, but one should bear in mind that the same transformation has already been successfully applied in [27] in order to study the local structure of singularities in matrix chains. In § 2 we recall the definition of general Hamiltonians and maximal chains of matrices, and some results from earlier work which are needed in the present considerations.…”

Resolvent Matrices in Degenerated Inner Product Spaces

“…If P is a nondegenerated dB -space, it is shown in [KW4], Section 10, that the space P − can be identified with the set of associated functions for P. The notion of triplet spaces in the degenerated situation, as introduced in the previous sections, enables us to supplement this result by proving that also if P is a degenerated dB -space, one can identify P − with the set of associated functions for P. In the following let P be a dB -space and assume that dim P • = ∆ > 0. For simplicity we assume moreover that for all w ∈ C there exists a function F ∈ P with F (w) = 0.…”

“…The first aim of this note is to introduce an appropriate notion of generalized elements for a space P satisfying (D1) and (D2) and a closed symmetric relation S ⊆ P 2 with defect index (1, 1), and to derive a formula of the type (1.1) for a generalized element u. Secondly, a characterization of those matrices W (z) shall be given which can be represented as u -resolvent matrices in this setting, i. e. with a relation S in a space P which is degenerated dim P • > 0 . Finally, we consider inner product spaces of entire functions which satisfy certain additional axioms (compare [dB], [KW4]) and show that for such spaces the set of generalized elements can be identified with a set of entire functions known as the set of associated functions. This supplements the results of [KW4], Section 10.…”

Bessel-Type Operators with an Inner Singularity

“…Introduction 1.1. During the last 20 years a number of papers on Sturm-Liouville operators with 'point interactions' or '(strongly) singular potentials' has appeared: [2,[9][10][11][12]18,23,33] and further papers quoted there, see also [22,29,32]. By some more or less abstract constructions, e.g.…”