Schrödinger operators on graphs: Symmetrization and Eulerian cycles
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- by G. Karreskog, P. Kurasov and I. Trygg Kupersmidt PDF
- Proc. Amer. Math. Soc. 144 (2016), 1197-1207 Request permission
Abstract:
Spectral properties of the Schrödinger operator on a finite compact metric graph with delta-type vertex conditions are discussed. Explicit estimates for the lowest eigenvalue (ground state) are obtained using two different methods: Eulerian cycle and symmetrization techniques.References
- Gregory Berkolaiko, Robert Carlson, Stephen A. Fulling, and Peter Kuchment (eds.), Quantum graphs and their applications, Contemporary Mathematics, vol. 415, American Mathematical Society, Providence, RI, 2006. MR 2279143, DOI 10.1090/conm/415
- G. Berkolaiko and P. Kuchment, Dependence of the spectrum of a quantum graph on vertex conditions and edge lengths, Spectral geometry, Proc. Sympos. Pure Math., vol. 84, Amer. Math. Soc., Providence, RI, 2012, pp. 117–137. MR 2985312, DOI 10.1090/pspum/084/1352
- Gregory Berkolaiko and Peter Kuchment, Introduction to quantum graphs, Mathematical Surveys and Monographs, vol. 186, American Mathematical Society, Providence, RI, 2013. MR 3013208, DOI 10.1090/surv/186
- L. Euler, Solutio problematis ad geometriam situs pertinentis, Comment. Academiae Sci. I. Petropolitanae 8 (1736), 128-140.
- Pavel Exner and Michal Jex, On the ground state of quantum graphs with attractive $\delta$-coupling, Phys. Lett. A 376 (2012), no. 5, 713–717. MR 2880105, DOI 10.1016/j.physleta.2011.12.035
- Leonid Friedlander, Extremal properties of eigenvalues for a metric graph, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 1, 199–211 (English, with English and French summaries). MR 2141695, DOI 10.5802/aif.2095
- Carl Hierholzer and Chr Wiener, Ueber die Möglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren, Math. Ann. 6 (1873), no. 1, 30–32 (German). MR 1509807, DOI 10.1007/BF01442866
- Peter Kuchment, Quantum graphs. I. Some basic structures, Waves Random Media 14 (2004), no. 1, S107–S128. Special section on quantum graphs. MR 2042548, DOI 10.1088/0959-7174/14/1/014
- Peter Kuchment, Quantum graphs. II. Some spectral properties of quantum and combinatorial graphs, J. Phys. A 38 (2005), no. 22, 4887–4900. MR 2148631, DOI 10.1088/0305-4470/38/22/013
- Pavel Kurasov, Graph Laplacians and topology, Ark. Mat. 46 (2008), no. 1, 95–111. MR 2379686, DOI 10.1007/s11512-007-0059-4
- Pavel Kurasov, Schrödinger operators on graphs and geometry. I. Essentially bounded potentials, J. Funct. Anal. 254 (2008), no. 4, 934–953. MR 2381199, DOI 10.1016/j.jfa.2007.11.007
- P. Kurasov, Quantum graphs: spectral theory and inverse problems, to appear in Birkhäuser.
- P. Kurasov, G. Malenová, and S. Naboko, Spectral gap for quantum graphs and their edge connectivity, J. Phys. A 46 (2013), no. 27, 275309, 16. MR 3081922, DOI 10.1088/1751-8113/46/27/275309
- Pavel Kurasov and Sergey Naboko, Rayleigh estimates for differential operators on graphs, J. Spectr. Theory 4 (2014), no. 2, 211–219. MR 3232809, DOI 10.4171/JST/67
- Serge Nicaise, Spectre des réseaux topologiques finis, Bull. Sci. Math. (2) 111 (1987), no. 4, 401–413 (French, with English summary). MR 921561
- Olaf Post, Spectral analysis on graph-like spaces, Lecture Notes in Mathematics, vol. 2039, Springer, Heidelberg, 2012. MR 2934267, DOI 10.1007/978-3-642-23840-6
Additional Information
- G. Karreskog
- Affiliation: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden
- P. Kurasov
- Affiliation: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden
- MR Author ID: 265224
- I. Trygg Kupersmidt
- Affiliation: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden
- Received by editor(s): February 3, 2015
- Received by editor(s) in revised form: March 9, 2015
- Published electronically: July 8, 2015
- Communicated by: Varghese Mathai
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 1197-1207
- MSC (2010): Primary 34L25, 81U40; Secondary 35P25, 81V99
- DOI: https://doi.org/10.1090/proc12784
- MathSciNet review: 3447672