Non-self-adjoint graphs

Hussein, Amru; Krejčiřı́k, David; Siegl, Petr

doi:10.1090/s0002-9947-2014-06432-5

Cited by 34 publications

(35 citation statements)

References 41 publications

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“…The existence of such a process is in fact elementary and it can be constructed by piecing together Brownian motions in a rather direct way. The problem (1.1) can be also understood as a spectral problem for a non-self-adjoint graph with regular boundary conditions [9].…”

Section: Introductionmentioning

confidence: 99%

Spectral analysis of the diffusion operator with random jumps from the boundary

Kolb

Krejčiřı́k

2016

Math. Z.

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Using an operator-theoretic framework in a Hilbert-space setting, we perform a detailed spectral analysis of the one-dimensional Laplacian in a bounded interval, subject to specific non-self-adjoint connected boundary conditions modelling a random jump from the boundary to a point inside the interval. In accordance with previous works, we find that all the eigenvalues are real. As the new results, we derive and analyse the adjoint operator, determine the geometric and algebraic multiplicities of the eigenvalues, write down formulae for the eigenfunctions together with the generalised eigenfunctions and study their basis properties. It turns out that the latter heavily depend on whether the distance of the interior point to the centre of the interval divided by the length of the interval is rational or irrational. Finally, we find a closed formula for the metric operator that provides a similarity transform of the problem to a self-adjoint operator.

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Section: Introductionmentioning

confidence: 99%

Spectral analysis of the diffusion operator with random jumps from the boundary

Kolb

Krejčiřı́k

2016

Math. Z.

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“…For a survey on this actively developing field and references we refer the reader to the monograph [32] and the survey articles [31,104,106]. In the present section we consider the Laplacian on a finite, not necessarily compact metric graph, equipped with δ or more general non-self-adjoint vertex couplings; for further recent work on non-self-adjoint quantum graphs see [89,90,126]. Furthermore, for the treatment of quantum graphs via boundary triples and similar techniques we refer to, e.g.…”

Section: Quantum Graphs With δ-Type Vertex Couplingsmentioning

confidence: 99%

Spectral enclosures for non-self-adjoint extensions of symmetric operators

Behrndt

Langer²,

Lotoreichik

et al. 2018

Journal of Functional Analysis

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The spectral properties of non-self-adjoint extensions A [B] of a symmetric operator in a Hilbert space are studied with the help of ordinary and quasi boundary triples and the corresponding Weyl functions. These extensions are given in terms of abstract boundary conditions involving an (in general non-symmetric) boundary operator B. In the abstract part of this paper, sufficient conditions for sectoriality and m-sectoriality as well as sufficient conditions for A [B] to have a non-empty resolvent set are provided in terms of the parameter B and the Weyl function. Special attention is paid to Weyl functions that decay along the negative real line or inside some sector in the complex plane, and spectral enclosures for A [B] are proved in this situation. The abstract results are applied to elliptic differential operators with local and non-local Robin boundary conditions on unbounded domains, to Schrödinger operators with δ-potentials of complex strengths supported on unbounded hypersurfaces or infinitely many points on the real line, and to quantum graphs with non-self-adjoint vertex couplings.

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“…To obtain our main theorem in its most general form we use similarity transformations and bounded perturbations. In this way we are able to generalize the boundary conditions for non-self adjoint and non-compact graphs given in [27,24], see Example 2.12, as well as the general boundary conditions in terms of "boundary subspaces" presented in [36,Sect. 6.5], see Subsection 3.6.…”

Section: Introductionmentioning

confidence: 96%

Waves and diffusion on metric graphs with general vertex conditions

Engel¹,

Fijavž²

2019

Evolution Equations &Amp; Control Theory

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We prove well-posedness for general linear wave-and diffusion equations on compact or non-compact metric graphs allowing various conditions in the vertices. More precisely, using the theory of strongly continuous operator semigroups we show that a large class of (not necessarily self-adjoint) second order differential operators with general (possibly non-local) boundary conditions generate cosine families, hence also analytic semigroups, on2010 Mathematics Subject Classification. 47D06, 35R02, 35G46, 35K05, 35L05. 1 xn ⊺ .Using this notation we define the transformationsJ ϕ e f e ∶= f e ○ ϕ e for f e ∈ X e , Jφi ∈ L(X i ),Then J ϕ e and Jφi are invertible with bounded inverses J −1 ϕ e = J (ϕ e ) −1 ∈ L(X e ) and J −1 ϕ i = J (φ i ) −1 ∈ L(X i ). These maps will be used in Lemma 2.4 to transform space dependent diffusion coefficients into constant ones.

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Non-self-adjoint graphs

Cited by 34 publications

References 41 publications

Spectral analysis of the diffusion operator with random jumps from the boundary

Spectral analysis of the diffusion operator with random jumps from the boundary

Spectral enclosures for non-self-adjoint extensions of symmetric operators

Waves and diffusion on metric graphs with general vertex conditions

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