On Boundary Value Problems for Operator Differential Equations

Горбачук, М. Л.; Горбачук, Василь

doi:10.1007/978-94-017-0693-3_25

Cited by 51 publications

(81 citation statements)

References 6 publications

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“…The theory of boundary value spaces (also known as boundary triplets) associated with symmetric operators has its origins in the work of Kočubeȋ [24] and Gorbachuk and Gorbachuk [14] with developments from many authors, (see [6,25,27,28,35,37,39,41]). In this context, the theory of the Weyl-M -function was developed by Derkach and Malamud [9,10], where spectral properties of the operator were investigated via the M -function and Kreȋn-type resolvent formulae were established.…”

Section: Introductionmentioning

confidence: 99%

M ‐functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems

Brown

Grubb

Wood

2009

Mathematische Nachrichten

140

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Key words Closed extension, M -function, abstract boundary spaces, boundary triplets, elliptic PDEs, pseudodifferential boundary operators, essential spectrum MSC (2000) 35J25, 35J30, 35J55, 35P05, 47A10, 47A11 Dedicated to the memory of Leonid R. VolevichIn this paper, we combine results on extensions of operators with recent results on the relation between the M -function and the spectrum, to examine the spectral behaviour of boundary value problems. M -functions are defined for general closed extensions, and associated with realisations of elliptic operators. In particular, we consider both ODE and PDE examples where it is possible for the operator to possess spectral points that cannot be detected by the M -function.

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Section: Introductionmentioning

confidence: 99%

M ‐functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems

Brown

Grubb

Wood

2009

Mathematische Nachrichten

140

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“…It is straightforward to verify that (C 2 , Γ 0 , Γ 1 ) performs a boundary triplet for S[V ] * , for definition see [16] and references therein, where Γ 0 , Γ 1 : dom(S[V ] * ) → C 2 are linear operators, given by…”

Section: Boundary Triplets Weyl Function and γ-Fieldmentioning

confidence: 99%

Scattering Matrix, Phase Shift, Spectral Shift and Trace Formula for One-dimensional Dissipative Schrödinger-type Operators

Neidhardt

Rehberg

2007

Integr. equ. oper. theory

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The paper is devoted to Schrödinger operators with dissipative boundary conditions on bounded intervals. In the framework of the LaxPhillips scattering theory the asymptotic behaviour of the phase shift is investigated in detail and its relation to the spectral shift is discussed. In particular, the trace formula and the Birman-Krein formula are verified directly. The results are exploited for dissipative Schrödinger-Poisson systems. Mathematics Subject Classification (2000). Primary 47A20; Secondary 47B44, 47A40.

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“…The theory of boundary triplets and their application to solving boundary control problems is now classical; see [13]. The concept of boundary triplets has been generalised to that of boundary relations by a group of authors in papers such as [10] and [12].…”

Section: Introductionmentioning

confidence: 99%

On Passive and Conservative State/Signal Systems

Kurula

2010

Integr. Equ. Oper. Theory

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This article is devoted to a study of continuous-time passive and conservative systems within the state/signal framework. The main idea of the state/signal approach is to not a priori distinguish between inputs and outputs, but rather to combine these two into a single external signal. The so-called node space is introduced as the product of two copies of the state space of the system and one copy of the space where the external signals of the system live. This node space is equipped with a sesquilinear product that makes it a Kreȋn space. A generating subspace is defined as a closed subspace of the node space, and the node space determines the trajectories of a state/signal system. One of the main results of this article is that a subspace of the node space generates a passive state/signal system if and only if it is a maximally nonnegative subspace of the node space and it satisfies a certain nondegeneracy condition. In this case the generating subspace can be interpreted as the graph of a scattering-passive input/state/output system node. Mathematics Subject Classification (2000). Primary 47A48, 93C25; Secondary 47N70, 46C20.

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On Boundary Value Problems for Operator Differential Equations

Cited by 51 publications

References 6 publications

M ‐functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems

M ‐functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems

Scattering Matrix, Phase Shift, Spectral Shift and Trace Formula for One-dimensional Dissipative Schrödinger-type Operators

On Passive and Conservative State/Signal Systems

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