<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi mathvariant="script">R</mml:mi><mml:mi mathvariant="script">T</mml:mi></mml:math>-Symmetric Laplace Operators on Star Graphs: Real Spectrum and Self-Adjointness

Astudillo, Maria; Kurasov, Pavel; Usman, Muhammad

doi:10.1155/2015/649795

Cited by 12 publications

(17 citation statements)

References 25 publications

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“…In what follows,  (1) indicates a sum over all subscripts such that the lowest possible value for subscripts is 1 and the highest possible value for subscripts is N, and  (2) indicates the same except that the highest possible value for subscripts is N − 1 instead.…”

Section: Appendix: Proofs Of Propositions 1 Andmentioning

confidence: 99%

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Quantum graphs: PT-symmetry and reflection symmetry of the spectrum

Kurasov

Garjani

2017

Journal of Mathematical Physics

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Not necessarily self-adjoint quantum graphs -differential operators on metric graphsare considered. Assume in addition that the underlying metric graph possesses an automorphism (symmetry) P. If the differential operator is PT -symmetric, then its spectrum has reflection symmetry with respect to the real line. Our goal is to understand whether the opposite statement holds, namely whether the reflection symmetry of the spectrum of a quantum graph implies that the underlying metric graph possesses a non-trivial automorphism and the differential operator is PT -symmetric. We give partial answer to this question by considering equilateral star-graphs. The corresponding Laplace operator with Robin vertex conditions possesses reflection-symmetric spectrum if and only if the operator is PT -symmetric with P being an automorphism of the metric graph.

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Section: Appendix: Proofs Of Propositions 1 Andmentioning

confidence: 99%

“…1 The main difference to the current paper is that Neumann conditions were assumed at the degree-one vertices, while one considered rotation-invariant conditions at the central vertex. Operators with the Robin conditions at degree-one vertices were considered by Znojil.…”

Section: Introductionmentioning

confidence: 97%

Quantum graphs: PT-symmetry and reflection symmetry of the spectrum

Kurasov

Garjani

2017

Journal of Mathematical Physics

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“…Proof. According Assumption 1, by ⃗ ( ) we denote the unique solution to operator equation (11). By ⃗ 1 , .…”

Section: Self-adjoint Restrictions Of Maximal Operator λmentioning

confidence: 99%

“…Among the recent publications devoted to various aspects of the theory on the inverse problems on graphs, we mention works [6]- [10]. In works [11]- [14], there were studied spectral properties of differential operators on equilateral quantum star-like graphs. In this paper the Lagrange formula was obtained for a differential operator defined on an arbitrary geometric graph, in contrast to work [12], where a similar relation was given for a simple stargraph.…”

Section: Introductionmentioning

confidence: 99%

Self-adjoint restrictions of maximal operator on graph

Кангужин¹,

Konyrkulzhaeva²

2017

Ufimskii Matematicheskii Zhurnal

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Abstract. In the work we study differential operators on arbitrary geometric graphs without loops. We extend the known results for differential operators on an interval to the differential operators on the graphs. In order to define properly the maximal operator on a given graph, we need to choose a set of boundary vertices. The non-boundary vertices are called interior vertices. We stress that the maximal operator on a graph is determined not only by the given differential expressions on the edges, but also by the Kirchhoff conditions at the interior vertices of the graph. For the introduced maximal operator we prove an analogue of the Lagrange formula. We provide an algorithm for constructing adjoint boundary forms for an arbitrary set of boundary conditions. In the conclusion of the paper, we present a complete description of all self-adjoint restrictions of the maximal operator.

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“…This observation is behind the boom of the so-called "PT-symmetric quantum mechanics" [5,41], which we use here as a source of interesting quasi-self-adjoint models. In this context, non-self-adjoint operators on metric graphs were previously considered in [4,47].…”

Section: Introductionmentioning

confidence: 99%

Non-self-adjoint graphs

Hussein

Krejčiřı́k

Siegl

2014

Trans. Amer. Math. Soc.

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On finite metric graphs we consider Laplace operators, subject to various classes of non-self-adjoint boundary conditions imposed at graph vertices. We investigate spectral properties, existence of a Riesz basis of projectors and similarity transforms to self-adjoint Laplacians. Among other things, we describe a simple way how to relate the similarity transforms between Laplacians on certain graphs with elementary similarity transforms between matrices defining the boundary conditions. * On a leave from b.

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RT-Symmetric Laplace Operators on Star Graphs: Real Spectrum and Self-Adjointness

Cited by 12 publications

References 25 publications

Quantum graphs: PT-symmetry and reflection symmetry of the spectrum

Quantum graphs: PT-symmetry and reflection symmetry of the spectrum

Self-adjoint restrictions of maximal operator on graph

Non-self-adjoint graphs

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