Boundary relations and their Weyl families

Derkach, Vladimir; Hassi, Seppo; Malamud, M. M.; Snoo, H.S.V. de

doi:10.1090/s0002-9947-06-04033-5

Cited by 134 publications

(338 citation statements)

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“…In addition, the spectral properties of these self-adjoint extensions can be described with the help of the Weyl function and the corresponding boundary parameter. We refer the reader to [12][13][14][15]28] and Section 2 for more details on boundary triples and their Weyl functions.…”

Section: Introductionmentioning

confidence: 99%

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

Behrndt¹,

Langer²,

Lotoreichik³

2016

Nanosystems: Phys. Chem. Math.

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The self-adjoint Schrödinger operator A δ,α with a δ-interaction of constant strength α supported on a compact smooth hypersurface C is viewed as a self-adjoint extension of a natural underlying symmetric operator S in L 2 (R n ). The aim of this note is to construct a boundary triple for S * and a self-adjoint parameter Θ δ,α in the boundary space L 2 (C) such that A δ,α corresponds to the boundary condition induced by Θ δ,α . As a consequence the well-developed theory of boundary triples and their Weyl functions can be applied. This leads, in particular, to a Krein-type resolvent formula and a description of the spectrum of A δ,α in terms of the Weyl function and Θ δ,α .

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Section: Introductionmentioning

confidence: 99%

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

Behrndt¹,

Langer²,

Lotoreichik³

2016

Nanosystems: Phys. Chem. Math.

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“…Other related work includes Staffans (2005) and Weiss (1994) who study the feedback theory for (regular) well-posed linear systems. Similar approaches for composing conservative linear systems can be found in Villegas (2007), Cervera et al (2007), Kurula et al (2010), and Derkach et al (2006).…”

Section: Introductionmentioning

confidence: 84%

Wave Propagation in Networks: a System Theoretic Approach

Aalto

Malinen

2011

IFAC Proceedings Volumes

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and jmalinen@cc.hut.fi).Abstract: We consider dynamical systems defined on graph structures. The dynamics on the edges are governed by partial differential equations that are interconnected at the graph vertices through algebraic conditions involving the boundary conditions of the PDEs. We show that a variety of such wave propagation problems on networks are solvable (forward in time) and energy passive or conservative -given that the governing PDEs are solvable on the separate edges. We treat these problems in the operator theoretic boundary control system framework.

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“…We only sketch the ideas here; for more details on boundary triples, we refer to [BGP06,DHMdS06] and the references therein.…”

Section: Boundary Triplesmentioning

confidence: 99%

“…[BGP06]). For a general treatment of boundary triples we refer to [BGP06,DHMdS06] and the references therein.Our main purpose here is not to characterise all self-adjoint extensions of a given symmetric operator, but to show that the concept of boundary triples can also be used in the PDE case, namely to Laplacians on a manifold with boundary. The standard theory of boundary triples does not directly apply in this case, since Green's formula…”

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confidence: 99%

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First-order operators and boundary triples

Post

2007

Russ. J. Math. Phys.

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Abstract. The aim of the present paper is to introduce a first order approach to the abstract concept of boundary triples for Laplace operators. Our main application is the Laplace operator on a manifold with boundary; a case in which the ordinary concept of boundary triples does not apply directly. In our first order approach, we show that we can use the usual boundary operators also in the abstract Green's formula. Another motivation for the first order approach is to give an intrinsic definition of the Dirichlet-to-Neumann map and intrinsic norms on the corresponding boundary spaces. We also show how the first order boundary triples can be used to define a usual boundary triple leading to a Dirac operator.In memoriam Vladimir A. Geyler (1943-2007 1. IntroductionThe concept of boundary triples, originally introduced in [V63], has successfully be applied to the theory of self-adjoint extensions of symmetric operators, for example on quantum graphs, singular perturbations or point interactions on manifolds (see e.g. [BGP06]). For a general treatment of boundary triples we refer to [BGP06,DHMdS06] and the references therein.Our main purpose here is not to characterise all self-adjoint extensions of a given symmetric operator, but to show that the concept of boundary triples can also be used in the PDE case, namely to Laplacians on a manifold with boundary. The standard theory of boundary triples does not directly apply in this case, since Green's formuladoes not extend to f, g in the maximal operator domain dom ∆ max = { f ∈ L 2 (X) | ∆ max f ∈ L 2 (X) (distributional sense) } (cf. Remark 4.2 for details). A solution to overcome this problem is either to modify the boundary operators (restriction of the function and the normal derivative onto ∂X) as e.g. in [BMNW07,Pc07], or to introduce the concept of quasi boundary triples as in [BL07] (cf. also the references therein for further treatments of boundary triples in the PDE case). Here, we use a different approach: we start with first order operators, namely the exterior derivative d taking functions (0-forms) to 1-forms and its adjoint, the divergence operator δ, mapping 1-forms into functions, since the first order operator domains are simpler. The Laplacian (on functions) is then defined as ∆ 0 := δd. Certainly, in our approach we do not cover all selfadjoint extensions of the minimal Laplacian.The abstract approach also allows to define the Dirichlet-to-Neumann map in an intrinsic manner, and also the norm of G 1/2 = H 1/2 (∂X) is defined intrinsicly. This might be a great advantage when dealing with parameter-depending manifolds, as it is the case for graph-like manifolds (see e.g. [EP07,P06]). We will treat this question in a forthcoming publication. Our approach is related to the recent works of Arlinskii [A00], Posilicano [Pc07] and Brown et al. [BMNW07], where also a PDE example is treated in the context of boundary triples.Date: January 15, 2018. To precise our idea of the first order approach we sketch the construction here. The given data are, where H...

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Boundary relations and their Weyl families

Abstract: Abstract. The concepts of boundary relations and the corresponding Weyl families are introduced. Let S be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space H, let H be an auxiliary Hilbert space, let

Cited by 134 publications

References 30 publications

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

Wave Propagation in Networks: a System Theoretic Approach

First-order operators and boundary triples

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