Unbounded quantum graphs with unbounded boundary conditions

Lenz, Daniel; Schubert, Carsten; Veselić, Ivan

doi:10.1002/mana.201200135

Cited by 14 publications

(17 citation statements)

References 16 publications

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“…Furthermore, for the treatment of quantum graphs via boundary triples and similar techniques we refer to, e.g. [46,59,61,109,120,123]. Let G be a finite graph consisting of a finite set V of vertices and a finite set E of edges, where we allow infinite edges, i.e.…”

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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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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“…Under the assumption that the direct sum triplet (1.4.1) is a boundary triplet for S * , we have that ∞ n=0 M n (λ 0 ), R and R −1 are bounded and we obtain the following special case of Theorem 1.4.2. For quantum graphs with edge length bounded from below, this result was also obtained in [22]. Corollary 1.4.1 Assume that the triplet {G, Γ 0 , Γ 1 } given by (1.4.1) is a boundary triplet for S * , then S loc L has the following properties:…”

Section: Consider the Index Setmentioning

confidence: 78%

“…Graphs with an infinite number of edges but with finite vertex degree were considered in [25], under the assumption that (e) = 1 for all e ∈ E, and assuming that inf e∈E (e) > 0 in [27]. The study of the operators S L was carried out in [2] for star-graphs and for quantum graphs satisfying inf e∈E (e) > 0 in [22].…”

Section: Introductionmentioning

confidence: 99%

Locally finite extensions and Gesztesy–Šeba realizations for the Dirac operator on a metric graph

Gernandt¹,

Trunk²

2021

Operator Theory

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We study extensions of direct sums of symmetric operators S = ⊕ n∈N S n. In general there is no natural boundary triplet for S * even if there is one for every S * n , n ∈ N. We consider a subclass of extensions of S which can be described in terms of the boundary triplets of S * n and investigate the self-adjointness, the semi-boundedness from below and the discreteness of the spectrum. Sufficient conditions for these properties are obtained from recent results on weighted discrete Laplacians. The results are applied to Dirac operators on metric graphs with point interactions at the vertices. In particular, we allow graphs with arbitrarily small edge length.

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“…Then Q is a Dirichlet form and −L generates a positivity preserving semigroup (see, e.g., [17,31]; we refer to [22,32] for more general boundary conditions). Proof.…”

Section: 2mentioning

confidence: 99%

Note on basic features of large time behaviour of heat kernels

Keller

Lenz

Vogt

et al. 2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Large time behaviour of heat semigroups (and, more generally, of positive selfadjoint semigroups) is studied. Convergence of the semigroup to the ground state and of averaged logarithms of kernels to the ground state energy is shown in the general framework of positivity improving selfadjoint semigroups. This framework encompasses all irreducible semigroups coming from Dirichlet forms as well as suitable perturbations thereof. It includes, in particular, Laplacians on connected manifolds, metric graphs and discrete graphs.

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Unbounded quantum graphs with unbounded boundary conditions

Cited by 14 publications

References 16 publications

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

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

Locally finite extensions and Gesztesy–Šeba realizations for the Dirac operator on a metric graph

Note on basic features of large time behaviour of heat kernels

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