C. V. M. van der Mee

Analytic operator functions with compact spectrum. I. Spectral nodes, linearization and equivalence

This paper arose from an attempt to classify analytic operator functions modulo equivalence in terms of their linearizations and to use the linearization as a tool to obtain spectral factorizations. In this first part spectral linearizations and spectral nodes are introduced to provide a general framework to deal with problems concerning the uniqueness of a linearization and the existence of analytic divisors. Two analytic operator functionsW1(.) andW2(.) with compact spectrum are shown to have similar spectral linearizations if and only if for some Banach spaceZ the functionsW1(.) ⨁IZ andW2(.) ⨁IZ are equivalent. In parts II and III of this paper spectral nodes will be used intensively to deal with a number of factorization problems. In particular, in part III for Hilbert spaces and bounded domains a full solution of the inverse problem will be given, which will be used to construct spectral factorizations explicitly and to solve the problem of spectrum displacement.

Analytic operator functions with compact spectrum. II. Spectral pairs and factorization

Using the technique introduced in the first part of this paper, various problems concerning factorization and divisibility of analytic operator functions with compact spectrum are studied in terms of spectral pairs of operators. The basic properties of such pairs are derived. Using these properties, stability of spectral divisors is proved and necessary and sufficient conditions (in terms of moments of the inverse function) are given in order that an analytic operator function with compact spectrum admits a generalized Wiener-Hopf factorization.

Transport theory inLp-spaces

In this article boundary value problems of linear transport theory are studied inLp-spaces (1≤p<+∞). It is shown that the results valid inL2-space can also be derived inLp-space (1≤p<+∞). For a non-multiplying medium formal expressions for the solutions are obtained.

Abstract Kinetic Equations Relevant to Supercritical Media

Abstract The abstract Hilbert space equation (Tƒ)′(x) = −(Aƒ)(x) , x∈ R +, is studied with a partial range boundary condition (Q + ƒ)(0) = ƒ + ϵ Ran Q + . Here T is bounded, injective and self-adjoint, A is Fredholm and self-adjoint, with finite-dimensional negative part, and Q+ is the orthogonal projection onto the maximal T-positive T-invariant subspace. This models half-space stationary transport problems in supercritical media. A complete existence and uniqueness theory is developed.

The number of bound states of the Coulomb plus Yamaguchi potential

It is shown that certain assertions on the number of bound states of a Coulomb plus Yamaguchi potential which Zachary [J. Math. Phys. 12, 1379 (1971); 14, 2018 (1973)] claims to have proved are incorrect. We prove that there are always infinitely many bound states if the Coulomb part of the potential is attractive and that, in case the Coulomb part of the potential is repulsive, there is one bound state only if the Yamaguchi potential is sufficiently attractive.

Transport equation on a finite domain.I. Reflection and transmission operators and diagonalization

In this article we study the time-independent linear transport equation in a finite homogeneous non-multiplying medium with anisotropic scattering. For a polynomial phase function the solution is expressed in finitely many auxiliary functions. A diagonalization of an operator associate to the equation is established. Reflection and transmission operators are introduced.

Stationary transport processes with unbounded collision operators

An abstract Hilbert space equation is studied, which models many of the stationary, one-dimensional transport equations with partial-range boundary conditions. In particular, the collision term may be unbounded and nondissipative. A complete existence and uniqueness theory is presented.

Unitary equivalence of linear transport models

Abstract One simple unitary transformation is provided between the linear transport model of R. Beals and a model presented else-where. Special attention is paid to similar relationships in electron transport theory.