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2017 | 132 | 6 | 1699-1703
Article title

Edge Switching Transformations of Quantum Graphs

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EN
Abstracts
EN
Discussed here are the effects of basics graph transformations on the spectra of associated quantum graphs. In particular it is shown that under an edge switch the spectrum of the transformed Schrödinger operator is interlaced with that of the original one. By implication, under edge swap the spectra before and after the transformation, denoted by {Eₙ}^{∞}ₙ₌₁ and {Ẽₙ}^{∞}ₙ₌₁ correspondingly, are level-2 interlaced, so that Eₙ-₂ ≤ Ẽₙ ≤ Eₙ₊₂. The proofs are guided by considerations of the quantum graphs' discrete analogs.
Keywords
EN
Year
Volume
132
Issue
6
Pages
1699-1703
Physical description
Dates
published
2017-12
References
  • [1] L. Pauling, J. Chem. Phys. 4, 673 (1936), doi: 10.1063/1.1749766
  • [2] T. Kottos, U. Smilansky, Phys. Rev. Lett. 79, 4794 (1997), doi: 10.1103/PhysRevLett.79.4794
  • [3] T. Kottos, U. Smilansky, Ann. Phys. 274, 76 (1999), doi: 10.1006/aphy.1999.5904
  • [4] S. Gnutzmann, Uzy Smilansky, Adv. Phys. 55, 527 (2006), doi: 10.1080/00018730600908042
  • [5] P. Kuchment, Analysis on Graphs, its Applications, Proc. Symp. Pure. Math. 77, AMS 2008, p. 291, doi: 10.1090/pspum/077
  • [6] G. Berkolaiko, P. Kuchment, Introduction to Quantum Graphs, AMS Mathematical Surveys and Monographs, vol. 186, 2012, doi: 10.1090/surv/186
  • [7] O. Post, Spectral analysis on graph-like spaces, Lecture Notes in Mathematics 2039, Springer 2012, doi: 10.1007/978-3-642-23840-6
  • [8] J.J. Seidel, in: Proc. Fifth Southeastern Conference on Combinatorics, Graph Theory, Computing Congressus Numerantium X, Utilitas Math, Winnipeg, MB (1974), p.125 http://catalog.hathitrust.org/Record/000500884
  • [9] B. Gutkin, U. Smilansky, J. Phys A. 31, 6061 (2001), doi: 10.1088/0305-4470/34/31/301
  • [10] M. Kac, Am. Math. Month. 73, 1 (1966), doi: 10.2307/2313748
  • [11] B. Simon, in: CRM Proceedings and Lecture Notes, Vol. 8, 1995, p. 109
  • [12] S. Albeverio, K. Pankrashkin, J. Phys A 38, 4859 (2005), doi: 10.1088/0305-4470/38/22/010
  • [13] J. Behrndt, M.M. Malamud, H. Neidhardt, Proc. Lond. Math. Soc. 97, 568 (2008), doi: 10.1112/plms/pdn016
  • [14] H. Weyl, Math. Ann. 71, 441 (1912), doi: 10.1007/BF01456804
  • [15] D. Hundertmark, E.H. Lieb, L.E. Thomas, Adv. Theor. Math. Phys. 2, 719 (1998), doi: 10.4310/ATMP.1998.v2.n4.a2
  • [16] F.J. Dyson, J. Math. Phys. 3, 140 (1962), doi: 10.1063/1.1703773
  • [16a] F.J. Dyson, J. Math. Phys. 3, 157 (1962), doi: 10.1063/1.170377
  • [16] F.J. Dyson, J. Math. Phys. 3, 166 (1962), doi: 10.1063/1.170377
  • [17] C.H. Joyner, U. Smilansky, J. Phys. A: Math. Theor. 48 (2015), doi: 10.1088/1751-8113/48/25/255101
  • [18] R. Band, G. Berkolaiko, H. Raz, U. Smilansky, Comm. Math. Phys. 311, 815 (2012), doi: 10.1007/s00220-011-1384-9
  • [19] O. Hul, S. Bauch, P. Pakoński, N. Savytskyy, K. Zyczkowski, L. Sirko, Phys. Rev. E 69, 056205 (2004), doi: 10.1103/PhysRevE.69.056205
  • [20] P. Kurasov, G. Malenová, S. Naboko, J. Phys. A 46, 275309 (2013), doi: 10.1088/1751-8113/46/27/275309
Document Type
Publication order reference
YADDA identifier
bwmeta1.element.bwnjournal-article-appv132n6p09kz
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