Recovering quantum graphs from their Bloch spectrum

Ralf Rueckriemen[1]

  • [1] Dartmouth College, Kemeny Hall, Hanover, 03755 NH, USA

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 3, page 1149-1176
  • ISSN: 0373-0956

Abstract

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We define the Bloch spectrum of a quantum graph to be the map that assigns to each element in the deRham cohomology the spectrum of an associated magnetic Schrödinger operator. We show that the Bloch spectrum determines the Albanese torus, the block structure and the planarity of the graph. It determines a geometric dual of a planar graph. This enables us to show that the Bloch spectrum indentifies and completely determines planar 3 -connected quantum graphs.

How to cite

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Rueckriemen, Ralf. "Recovering quantum graphs from their Bloch spectrum." Annales de l’institut Fourier 63.3 (2013): 1149-1176. <http://eudml.org/doc/275536>.

@article{Rueckriemen2013,
abstract = {We define the Bloch spectrum of a quantum graph to be the map that assigns to each element in the deRham cohomology the spectrum of an associated magnetic Schrödinger operator. We show that the Bloch spectrum determines the Albanese torus, the block structure and the planarity of the graph. It determines a geometric dual of a planar graph. This enables us to show that the Bloch spectrum indentifies and completely determines planar $3$-connected quantum graphs.},
affiliation = {Dartmouth College, Kemeny Hall, Hanover, 03755 NH, USA},
author = {Rueckriemen, Ralf},
journal = {Annales de l’institut Fourier},
keywords = {quantum graphs; Schrödinger operators; spectrum; inverse spectral problem; quantum graph; Bloch spectrum},
language = {eng},
number = {3},
pages = {1149-1176},
publisher = {Association des Annales de l’institut Fourier},
title = {Recovering quantum graphs from their Bloch spectrum},
url = {http://eudml.org/doc/275536},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Rueckriemen, Ralf
TI - Recovering quantum graphs from their Bloch spectrum
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 3
SP - 1149
EP - 1176
AB - We define the Bloch spectrum of a quantum graph to be the map that assigns to each element in the deRham cohomology the spectrum of an associated magnetic Schrödinger operator. We show that the Bloch spectrum determines the Albanese torus, the block structure and the planarity of the graph. It determines a geometric dual of a planar graph. This enables us to show that the Bloch spectrum indentifies and completely determines planar $3$-connected quantum graphs.
LA - eng
KW - quantum graphs; Schrödinger operators; spectrum; inverse spectral problem; quantum graph; Bloch spectrum
UR - http://eudml.org/doc/275536
ER -

References

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  1. Ram Band, Ori Parzanchevski, Gilad Ben-Shach, The isospectral fruits of representation theory: quantum graphs and drums, J. Phys. A 42 (2009) Zbl1176.58019MR2539297
  2. Joachim von Below, Can one hear the shape of a network?, Partial differential equations on multistructures (Luminy, 1999) 219 (2001), 19-36, Dekker, New York Zbl0973.35143MR1824563
  3. Jens Bolte, Sebastian Endres, Trace formulae for quantum graphs, Analysis on graphs and its applications 77 (2008), 247-259, Amer. Math. Soc., Providence, RI Zbl1153.81493MR2459873
  4. Reinhard Diestel, Graph theory, 173 (2005), Springer-Verlag, Berlin Zbl1074.05001MR2159259
  5. Gregory Eskin, James Ralston, Eugene Trubowitz, On isospectral periodic potentials in R n . II, Comm. Pure Appl. Math. 37 (1984), 715-753 Zbl0582.35031MR762871
  6. Bernard Gaveau, Masami Okada, Differential forms and heat diffusion on one-dimensional singular varieties, Bull. Sci. Math. 115 (1991), 61-79 Zbl0722.58003MR1086939
  7. Carolyn S. Gordon, Pierre Guerini, Thomas Kappeler, David L. Webb, Inverse spectral results on even dimensional tori, Ann. Inst. Fourier (Grenoble) 58 (2008), 2445-2501 Zbl1159.58015MR2498357
  8. V. Guillemin, Inverse spectral results on two-dimensional tori, J. Amer. Math. Soc. 3 (1990), 375-387 Zbl0702.58075MR1035414
  9. Boris Gutkin, Uzy Smilansky, Can one hear the shape of a graph?, J. Phys. A 34 (2001), 6061-6068 Zbl0981.05095MR1862642
  10. Mark Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), 1-23 Zbl0139.05603MR201237
  11. Motoko Kotani, Toshikazu Sunada, Jacobian tori associated with a finite graph and its abelian covering graphs, Adv. in Appl. Math. 24 (2000), 89-110 Zbl1017.05038MR1748964
  12. Tsampikos Kottos, Uzy Smilansky, Periodic orbit theory and spectral statistics for quantum graphs, Ann. Physics 274 (1999), 76-124 Zbl1036.81014MR1694731
  13. Peter Kuchment, Quantum graphs: an introduction and a brief survey, Analysis on graphs and its applications 77 (2008), 291-312, Amer. Math. Soc., Providence, RI Zbl1210.05169MR2459876
  14. Bojan Mohar, Carsten Thomassen, Graphs on surfaces, (2001), Johns Hopkins University Press, Baltimore, MD Zbl0979.05002MR1844449
  15. Olaf Post, First order approach and index theorems for discrete and metric graphs, Ann. Henri Poincaré 10 (2009), 823-866 Zbl1206.81044MR2533873
  16. Jean-Pierre Roth, Le spectre du laplacien sur un graphe, Théorie du potentiel (Orsay, 1983) 1096 (1984), 521-539, Springer, Berlin Zbl0557.58023MR890375

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