In this paper the scattering matrix of a scattering system consisting of two selfadjoint operators with finite dimensional resolvent difference is expressed in terms of a matrix Nevanlinna function. The problem is embedded into an extension theoretic framework and the theory of boundary triplets and associated Weyl functions for (in general nondensely defined) symmetric operators is applied. Th...

Weighted composition operators appear in the study of dynamical systems and also in characterizing isometries of some classes of Banach spaces. One of the most important generalizations of weighted composition operators, are generalized weighted composition operators which in special cases of their inducing functions give different types of well-known operators like: weighted composition operat...

The spectral shift function for pairs of selfadjoint operators was introduced in the paper by I.M. Lifshits [17]. In the same paper a trace formula for the difference of functions of the perturbed operator and the unperturbed operator was established. Ideas by Lifshits were developed in the paper by M.G. Krein [14], in which the spectral shift function ξ in L1(R) was defined for arbitrary pairs...

A collocation method is presented for numerical solution of a typical integral equation Rh := R D R(x, y)h(y)dy = f(x), x ∈ D of the class R, whose kernels are of positive rational functions of arbitrary selfadjoint elliptic operators defined in the whole space R, and D ⊂ R is a bounded domain. Several numerical examples are given to demonstrate the efficiency and stability of the proposed meth...

The classical concept of Q-functions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the Dirichlet-to-Neumann map in the theory of elliptic differential equations can be interpreted as a generalized Q-function. For couplings of uniformly elliptic second order differential expression on bounded and unbounded domains explicit Krein ...

The main objective of this paper is to give a rigorous treatment of Wigner’s and Eisenbud’s R-matrix method for scattering matrices of scattering systems consisting of two selfadjoint extensions of the same symmetric operator with finite deficiency indices. In the framework of boundary triplets and associated Weyl functions an abstract generalization of the R-matrix method is developed and the ...

Let A,B be two selfadjoint operators whose difference B −A is trace class. Krĕın proved the existence of a certain function ξ ∈ L1(R) such that tr[f (B)−f(A)] = ∫ Rf (x)ξ(x)dx for a large set of functions f . We give here a new proof of this result and discuss the class of admissible functions. Our proof is based on the integral representation of harmonic functions on the upper half plane and a...