Markus Klein

On the Born-Oppenheimer Expansion for Polyatomic Molecules

We consider the Schrödinger operatorP(h) for a polyatomic molecule in the semiclassical limit where the mass ratioh2 of electronic to nuclear mass tends to zero. We obtain WKB-type expansions of eigenvalues and eigenfunctions ofP(h) to all orders inh. This allows to treat the splitting of the ground state energy of a non-planar molecule. Our class of potentials covers the physical case of the Coulomb interaction. We use methods ofh-pseudodifferential operators with operator valued symbols, which by use of appropriate coordinate changes in local coordinate patches covering the classically accessible region become applicable even to our class of singular potentials.

Power-law corrections to the Kubo formula vanish in quantum Hall systems

In first order perturbation theory conductivity is given by the Kubo formula, which in a Quantum Hall System equals the first Chern class of a vector bundle. We apply the adiabatic theorem to show that these topological constraints quantize the averaged conductivity to all orders of perturbation theory. Hence the Kubo formula is valid to all orders.

On the mathematical theory of predissociation

We investigate spectral properties of molecular Schrodinger operators in the adiabatic aproximation. A mathematical theory of predissociation is presented in terms of the corresponding matrix Schrodinger operator and its resonances. Under a suitable nontrapping condition we give a convergent tunneling expansion for the location of resonances exponentially close to the real axis.

A Generic Component Framework for System Modeling

The aim of this paper is to present a generic component framework for system modeling which is especially useful for a large class of graph- and net-based modeling techniques. Moreover, the framework is also flexible with respect to a hierarchical connection of components, providing a compositional semantics of components. This means more precisely that the semantics and internal correctness of a system can be inferred from the semantics of its components. In contrast to constructor-based component concepts for data type specification techniques, our component framework is based on a generic notion of transformations. Refinements and transformations are used to express intradependencies, between the export interface and the body of a component, and interdependencies, between the import and the export interfaces of different components. This is shown by a small case study on modeling Java threads by high-level Petri nets in this paper.

A component framework for system modeling based on high-level replacement systems

The aim of this paper is to present a generic component framework for system modeling that satisfies main requirements for component-based development in software engineering. In this sense, we have defined a framework that can be used, by providing an adequate instantiation, in connection with a large class of semi-formal and formal modeling techniques. Moreover, the framework is also flexible with respect to the connection of components, providing a compositional semantics of components. This means more precisely that the semantics of a system can be inferred from the semantics of its components. In contrast to other component concepts for data type specification techniques, our component framework is based on a generic notion of transformations. In particular, refinements and transformations are used to express intradependencies, between the export interface and the body of a component, and interdependencies, between the import and the export interfaces of different components. The generic component framework generalizes module concepts for different kinds of Petri nets and graph transformation systems proposed in the literature, and seems to be also suitable for visual modeling techniques, including parts of the UML, if these techniques provide a suitable refinement or transformation concept. In this paper the generic approach is instantiated in two steps. First to high-level replacement systems generalizing the transformation concept of graph transformations. In a second step it is further instantiated to low-level and high-level Petri nets. To show applicability we present sample components from a case study in the domain of production automation as proposed in a priority program of the German Research Council (DFG).

Semantical Integration of Object-Oriented Viewpoint Specification Techniques

Complex systems have many heterogeneous aspects, which can be specified comprehensibly and adequately by viewpoint specification techniques dealing only with a suitable subset of these aspects. A methodology for the formal integration of collections of such viewpoint specification techniques is introduced and applied to object-oriented systems. As a main result, it is shown, how the semantical consistency of viewpoint specification techniques can be checked in this framework.

Breit-Wigner Formula for the Scattering Phase in the Stark Effect

Near resonance energy, we study the asymptotic behavior of the derivative of the scattering phase as the applied electric field tends to zero. We obtain the leading asymptotics of the spectral function near a simple resonance, and as an application we rigorously prove the Breit-Wigner formula which relates the width of resonances to the time delay of particles in a homogeneous electric field.

On the Born–Oppenheimer approximation of diatomic wave operators. II. Singular potentials

This paper is a continuation of an earlier paper [Commun. Math. Phys. 152, 73–95 (1993)]. Its purpose is to extend the quantitative estimates given previously for regular potentials to the case of potentials admitting some Coulomb-type singularities. More precisely, considering the two-cluster wave operators in diatomic-scattering, we approximate them by so-called “adiabatic’’ wave operators as the mass of the nuclei tends to infinity.

A Component Framework Based on High-Level Replacement Systems

Abstract This paper is based on two general concepts. The first one is a generic component framework for system modeling presented at FASE 2002, which is especially useful for graph- and net-based modeling techniques. The second one is the concept of high-level replacement systems, which has been studied within the last decade as an abstraction of the DPO-approach for graph transformation systems in a categorical framework, with instantiations to a large class of different modeling techniques. In this contribution both concepts are combined in the sense that the generic transformation concept - essentially used in the component framework - is instantiated by high-level replacement systems. As the main result we show how the properties for transformations required in the component framework can be shown in the case of high-level replacement systems. Moreover, some interesting extensions concerning multiple interfaces, union, and operational semantics of components are proposed.

Towards Multiple Access in Generic Component Architectures

Abstract The paper introduces an abstract framework for the specification of components with multiple require and provide interfaces that allows the specification of multiple access to a single provide interface. This framework can be regarded as a generalization of abstract hierarchical and connector-based component specification approaches. The main ideas are clarified in a sample specification, a component architecture for a web browser suite. For this, elementary nets are applied and are shown to be an instantiation of the abstract framework.