Titlebook: Spectral Methods for Operators of Mathematical Physics; Jan Janas,Pavel Kurasov,Sergei Naboko Conference proceedings 2004 Springer Basel A

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Daphne J. Gilbert,Bernard J. Harris,Suzanne M. Riehl

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Spectral Methods for Operators of Mathematical Physics

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A Remark on Spectral Meaning of the Symmetric Functional Model,n . of . coincides with the transmission coefficient of the outgoing waves. A general construction based on the self-adjoint delation and an example of the Lax-Phillips Semigroup for the 1-D wave equation on the infinite string with a bounded non-negative potential supported by semi-axis are considered.

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,Bounds for the Points of Spectral Concentration of One-dimensional Schrödinger Operators,Dirichlet .-function, from which the existence and properties of derivatives of the corresponding spectral functions can be established. An incidental outcome in the case of long range potentials is that upper bounds for embedded singular spectrum can also be obtained.

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The Problem of the Finiteness of the Point Spectrum for Self-adjoint Operators. Perturbations of Wiy, sufficient conditions for the finiteness of the point spectrum of some self-adjoint operators are established. The problem is treated for the general case of abstract operators and then, as applications, some concrete classes of operators are investigated. In particular, operators associated to J

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Singular Perturbations as Range Perturbations in a Pontryagin Space,fted to an exit Pontryagin space ℌ by means of an operator model, they become ordinary range perturbations of a self-adjoint operator .. in ℌ ⊃ ℌ.:..= ..-Ω..Ω*. Here . is a mapping from ℂ. into some scale space ℌ (..), ℕ, of generalized elements associated with .., while Ω is a mapping from into ℂ.

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Minimal Realizations of Scalar Generalized Nevanlinna Functions Related to Their Basic Factorizatiooduct of a Nevanlinna function . and of a rational function r . r, which collects the generalized poles and generalized zeros of . that are not of positive type. The key tool are reproducing kernel Pontryagin spaces.

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,Bounds for the Points of Spectral Concentration of One-dimensional Schrödinger Operators,ble classes of short range and long range potentials, we outline systematic procedures which enable numerical estimates of upper bounds for points of spectral concentration to be obtained. Our approach involves use of the Riccati equation to construct appropriate convergent series for a generalised

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