Surgery of Graphs: M-Function and Spectral Gap

• Authors: P. Kurasov
• Languages of publication: EN

Abstracts

EN

We discuss behaviour of the spectral gap for quantum graphs when two metric graphs are glued together. It appears that precise answer to this question can be given using a natural generalisation of the Titchmarsh-Weyl M-functions.

Keywords

EN

quantum graphs; spectral theory

• Publisher: Institute of Physics, Polish Academy of Sciences
• Journal: Acta Physica Polonica A
• Year: 2017
• Volume: 132
• Issue: 6
• Pages: 1666-1671

Dates

published

2017-12

Contributors

author

P. Kurasov

• Dept. of Mathematics, Stockholm Univ., 106 91, Stockholm, Sweden

References

• [1] P. Kurasov, Ark. Mat. 46, 95 (2008), doi: 10.1007/s11512-007-0059-4
• [2] P. Kurasov, J. Func. Anal. 254, 934 (2008), doi: 10.1016/j.jfa.2007.11.007
• [3] M. Aizenman, H. Schanz, U. Smilansky, S. Warzel, arXiv: 1710.07958 http://arXiv.org/abs/1710.07958
• [4] R. Band, G. Lévy, Ann. Henri Poincaré 18, 3269 (2017), doi: 10.1007/s00023-017-0601-2
• [5] G. Berkolaiko, J. Kennedy, P. Kurasov, D. Mugnolo, J. Phys. A 50, 365201 (2017), doi: 10.1088/1751-8121/aa8125
• [6] G. Berkolaiko, W. Liu, J. Math. Anal. Appl. 445, 803 (2017), doi: 10.1016/j.jmaa.2016.07.026
• [7] N.T. Do, P. Kuchment, B. Ong, On resonant spectral gaps in quantum graphs. Functional analysis and operator theory for quantum physics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich 2017, p. 213, doi: 10.4171/175-1/11
• [8] J. Kennedy, P. Kurasov, G. Malenova, D. Mugnolo, Ann. Henri Poincaré 17, 2439 (2016), doi: 10.1007/s00023-016-0460-2
• [9] P. Kurasov, G. Malenová, S. Naboko, J. Phys. A 46, 275309 (2013), doi: 10.1088/1751-8113/46/27/275309
• [10] P. Kurasov, S. Naboko, J. Spectral Theory 4, 211 (2014), doi: 10.4171/JST/67
• [11] J. Rohleder, Proc. Am. Math. Soc. 145, 2119 (2017), doi: 10.1090/proc/13403
• [12] P. Kurasov, S. Naboko, Surgery of graphs and spectral gap: Titchmarsh-Weyl operator-function approach, preprint 2017
• [13] G. Berkolaiko, P. Kuchment, Introduction to Quantum Graphs. Mathematical Surveys and Monographs, Vol. 186, American Mathematical Society, Providence (RI), 2013
• [14] P. Kurasov, Quantum graphs: spectral theory and inverse problems, to appear in Brikhäuser, 2018
• [15] O. Post, Spectral Analysis on Graph-Like Spaces, Lecture Notes in Mathematics, Vol. 2039, Springer, Heidelberg 2012, doi: 10.1007/978-3-642-23840-6
• [16] M. Fiedler, Czechoslov. Math. J. 23 (98), 298 (1973) http://zbmath.org/?q=an:0265.05119
• [17] S. Nicaise, Bull. Sci. Math. 111, 401 (1987)
• [18] L. Friedlander, Ann. Inst. Fourier 55, 199 (2005), doi: 10.5802/aif.2095
• [19] H. Weyl, Math. Ann. 71, 441 (1912) (in German), doi: 10.1007/BF01456804
• [20] W.N. Everitt, J. Comput. Appl. Math. 171, 185 (2004), doi: 10.1016/j.cam.2004.01.010
• [21] F. Gesztesy, E. Tsekanovskii, Math. Nachr. 218, 61 (2000), doi: 10.1002/1522-2616(200010)218:1<61::AID-MANA61>3.0.CO;2-D
• [22] S. Albeverio, P. Kurasov, Singular Perturbations of Differential Operators. Solvable Schrödinger Type Operators, London Mathematical Society Lecture Note Series, Vol. 271, Cambridge University Press, Cambridge 2000, doi: 10.1017/CBO9780511758904
• [23] R. Band, P. Kurasov, On Courant Theorem for Quantum Graphs, in preparation
• [24] J. Behrndt, A. Luger, J. Phys. A 43, 474006 (2010), doi: 10.1088/1751-8113/43/47/474006

Identifiers

DOI

10.12693/APhysPolA.132.1666