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Quantum Graphs: $ \mathcal{PT}$-symmetry and reflection symmetry of the spectrum
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics. Stockholm University, Faculty of Science, Department of Physics.
2017 (English)Report (Other academic)
Abstract [en]

Not necessarily self-adjoint quantum graphs – differential operators on metric graphs – are considered. Assume in addition that the underlying metric graph possesses an automorphism (symmetry) $ \mathcal P $. If the differential operator is $ \mathcal P \mathcal T$-symmetric, then its spectrum has reflection symmetrywith respect to the real line. Our goal is to understand whether the opposite statement holds, namely whether the reflection symmetry of the spectrum ofa quantum graph implies that the underlying metric graph possesses a non-trivial automorphism and the differential operator is $ \mathcal P \mathcal T$-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 $ \mathcal P \mathcal T$-symmetric with $ \mathcal P $ being an automorphism of the metric graph.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2017. , p. 22
Series
Research Reports in Mathematics, ISSN 1401-5617 ; 2
Keywords [en]
Quantum Graphs, PT-symmetric Operators
National Category
Mathematical Analysis
Research subject
Theoretical Physics
Identifiers
URN: urn:nbn:se:su:diva-138279OAI: oai:DiVA.org:su-138279DiVA, id: diva2:1066630
Projects
D0497301
Funder
Swedish Research Council, D0497301Swedish Research Council, VR1526102Available from: 2017-01-18 Created: 2017-01-18 Last updated: 2017-12-14Bibliographically approved

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Kurasov, PavelMajidzadeh Garjani, Babak
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CiteExportLink to record
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Citation style
  • apa
  • ieee
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