5 research outputs found

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

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    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

    Abstract kinetic equations with positive collision operators

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    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