Schrödinger operators on graphs: Symmetrization and Eulerian cycles

Karreskog, G.; Kurasov, P.; Trygg Kupersmidt, I.

doi:10.1090/proc12784

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