J. Van Casteren

A note on semilinear abstract functional differential and integrodifferential equations with infinite delay

In this note, we obtain some new existence and uniqueness theorems for mild solutions of the Cauchy problems for semilinear abstract functional differential and integrodifferential equations with infinite delay.

Time dependent Desch-Schappacher type perturbations of Volterra integral equations

In this paper, we study time dependent multiplicative perturbations and unbounded additive perturbations of the Volterra integral equations. Some Desch-Schappacher type perturbation theorems, which generalize previous related results, are established by new and concise approaches.

On martingales and feller semigroups

AbstractLet E be a second countable locally compact Hausdorff space and let L be a linear operator with domain D(L) and range R(L) in C0(E). Suppose that D(L) is dense in E. The following assertions are equivalent: (a)For L the martingale problem is uniquely solvable and L is maximal for this property(b)The operator L generates a Feller semigroup in C0(E).

Completeness of scattering systems with obstacles of finite capacity

Let K 0 be a free Feller operator and let KΣ: be the corresponding operator with Dirichlet boundary conditions on Γ = ℝn \ Σ. The scattering system established by k 0,KΣ} is complete, i.e., the wave operators exist and are complete if the singularity region Γ has finite capacity. One consequence is the stability of the absolutely continuous spectra of K0 and KΣ respectively. Such sets can be unbounded, which yields a non-local freedom of perturbation in the scattering theory. Kato-Feller potentials can be included.

Stochastic Regularization Method for Backward Cauchy Problem in Banach Spaces

We consider a stochastic regularization method for solving the backward Cauchy problem in Banach spaces. An order of convergence is obtained on sourcewise representative elements.

Finite Capacities in Spectral Theory

The notion of capacity comprises a natural link between the spectrum of selfadjoint Feller operators in L 2 (Σ), Σ \(\subseteq\) ℝd, with Dirichlet boundary conditions on ∂Σ and geometric properties of the region Σ. Here we describe two complementary results. On changing the boundary of Σ, the lowest eigenvalue (ground state) turns out to be shifted if and only if the capacity of the difference set is positive. On the other hand the absolutely continuous spectra of Feller operators with Dirichlet conditions are not affected by changing the perturbations arbitrarily on sets of finite capacities, because the corresponding scattering systems turn out to be complete.

Perturbations of Generalized Schrödinger Operators in Stochastic Spectral Analysis

The objective of stochastic spectral analysis is explained. It is used to study regular perturbations for a general class of generators of Feller semigroups, also called generalized Schrodinger operators. Upon introducing the Kato-Feller norm, the asymptotic behaviour of several spectral data can be studied. In the present article mainly the convergence of scattering matrices is considered.

On non–symmetric generalized schrodinger semigroup

In this paper some general phenomena are described for not necessarily systemeric so–called generalized Schrodinger semigroups (or generalized absorption/exciatation semigroups). These results are also applicable in case we consider Schrodinger semigroups on R v. In particular we describe some results on integral kernels: continuity, pointwise inequalities, ultracontractivity etc.For these inequalities we use a kind of stochastic bridge measure. The operator H is a closed linear extension of the operator H 0 + V in the space C 0(E) Here E is a locally compact second countable Hausdorff space and –H 0 is supposed to generate a Feller semigroup in C 0(E). Results in Lp (E,m) are also availale. Some examples are given