10 research outputs found

    Abstract kinetic equations with positive collision operators

    Full text link
    We consider "forward-backward" parabolic equations in the abstract form Jdψ/dx+Lψ=0Jd \psi / d x + L \psi = 0, 0<x<τ 0< x < \tau \leq \infty, where JJ and LL are operators in a Hilbert space HH such that J=J=J1J=J^*=J^{-1}, L=L0L=L^* \geq 0, and kerL=0\ker L = 0. The following theorem is proved: if the operator B=JLB=JL is similar to a self-adjoint operator, then associated half-range boundary problems have unique solutions. We apply this theorem to corresponding nonhomogeneous equations, to the time-independent Fokker-Plank equation μψx(x,μ)=b(μ)2ψμ2(x,μ) \mu \frac {\partial \psi}{\partial x} (x,\mu) = b(\mu) \frac {\partial^2 \psi}{\partial \mu^2} (x, \mu), 0<x<τ 0<x<\tau, μR \mu \in \R, as well as to other parabolic equations of the "forward-backward" type. The abstract kinetic equation Tdψ/dx=Aψ(x)+f(x) T d \psi/dx = - A \psi (x) + f(x), where T=TT=T^* is injective and AA satisfies a certain positivity assumption, is considered also.Comment: 20 pages, LaTeX2e, version 2, references have been added, changes in the introductio

    Scattering theory for Klein-Gordon equations with non-positive energy

    Full text link
    We study the scattering theory for charged Klein-Gordon equations: \{{array}{l} (\p_{t}- \i v(x))^{2}\phi(t,x) \epsilon^{2}(x, D_{x})\phi(t,x)=0,[2mm] \phi(0, x)= f_{0}, [2mm] \i^{-1} \p_{t}\phi(0, x)= f_{1}, {array}. where: \epsilon^{2}(x, D_{x})= \sum_{1\leq j, k\leq n}(\p_{x_{j}} \i b_{j}(x))A^{jk}(x)(\p_{x_{k}} \i b_{k}(x))+ m^{2}(x), describing a Klein-Gordon field minimally coupled to an external electromagnetic field described by the electric potential v(x)v(x) and magnetic potential b(x)\vec{b}(x). The flow of the Klein-Gordon equation preserves the energy: h[f, f]:= \int_{\rr^{n}}\bar{f}_{1}(x) f_{1}(x)+ \bar{f}_{0}(x)\epsilon^{2}(x, D_{x})f_{0}(x) - \bar{f}_{0}(x) v^{2}(x) f_{0}(x) \d x. We consider the situation when the energy is not positive. In this case the flow cannot be written as a unitary group on a Hilbert space, and the Klein-Gordon equation may have complex eigenfrequencies. Using the theory of definitizable operators on Krein spaces and time-dependent methods, we prove the existence and completeness of wave operators, both in the short- and long-range cases. The range of the wave operators are characterized in terms of the spectral theory of the generator, as in the usual Hilbert space case

    A functional model, eigenvalues, and finite singular critical points for indefinite Sturm-Liouville operators

    Full text link
    Eigenvalues in the essential spectrum of a weighted Sturm-Liouville operator are studied under the assumption that the weight function has one turning point. An abstract approach to the problem is given via a functional model for indefinite Sturm-Liouville operators. Algebraic multiplicities of eigenvalues are obtained. Also, operators with finite singular critical points are considered.Comment: 38 pages, Proposition 2.2 and its proof corrected, Remarks 2.5, 3.4, and 3.12 extended, details added in subsections 2.3 and 4.2, section 6 rearranged, typos corrected, references adde

    Model Based Testing with Labelled Transition Systems

    No full text
    Model based testing is one of the promising technologies to meet the challenges imposed on software testing. In model based testing an implementation under test is tested for compliance with a model that describes the required behaviour of the implementation. This tutorial chapter describes a model based testing theory where models are expressed as labelled transition systems, and compliance is defined with the ‘ioco’ implementation relation. The ioco-testing theory, on the one hand, provides a sound and well-defined foundation for labelled transition system testing, having its roots in the theoretical area of testing equivalences and refusal testing. On the other hand, it has proved to be a practical basis for several model based test generation tools and applications. Definitions, underlying assumptions, an algorithm, properties, and several examples of the ioco-testing theory are discussed, involving specifications, implementations, tests, the ioco implementation relation and some of its variants, a test generation algorithm, and the soundness and exhaustiveness of this algorithm
    corecore