Quantum graphs with the Bethe-Sommerfeld property
https://doi.org/10.17586/2220-8054-2017-8-3-305-309
Abstract
In contrast to the usual quantum systems which have at most a finite number of open spectral gaps if they are periodic in more than one direction, periodic quantum graphs may have gaps arbitrarily high in the spectrum. This property of graph Hamiltonians, being generic in a sense, inspires the question about the existence of graphs with a finite and nonzero number of spectral gaps. We show that the answer depends on the vertex couplings together with commensurability of the graph edges. A finite and nonzero number of gaps is excluded for graphs with scale invariant couplings; on the other hand, we demonstrate that graphs featuring a finite nonzero number of gaps do exist, illustrating the claim on the example of a rectangular lattice with a suitably tuned δ-coupling at the vertices.
Keywords
periodic quantum graphs,
gap number,
δ-coupling,
rectangular lattice graph,
scale-invariant coupling,
Bethe-Sommerfeld conjecture,
golden mean
Supporting agencies: The research was supported by the Czech Science Foundation (GACR) within the project 17-01706S.
About the Authors
P. Exner
Czech Technical University; Nuclear Physics Institute CAS
Czech Republic
Czech Republic
Doppler Institute for Mathematical Physics and Applied Mathematics; Department of Theoretical Physics
Bˇrehova 7, 11519 Prague; 25068 Reˇz near Prague
O. Turek
Nuclear Physics Institute CAS; Joint Institute for Nuclear Research; Kochi University of Technology
Czech Republic
Czech Republic
Department of Theoretical Physics; Bogoliubov Laboratory of Theoretical Physics; Laboratory for Unified Quantum Devices
25068 Reˇz near Prague; 141980 Dubna; Kochi 782-8502
References
1. Berkolaiko G., Kuchment P. Introduction to Quantum Graphs. AMS, Providence, R.I., 2013.
2. Exner P., Manko S. S. Spectra of magnetic chain graphs: coupling constant perturbations. J. Phys. A: Math. Theor., 2015, 48, P. 125302.
3. Sommerfeld A., Bethe H. Electronentheorie der Metalle. 2nd ed., Handbuch der Physik, Springer Verlag, Berlin, 1933.
4. Skriganov M. M. Proof of the Bethe-Sommerfeld conjecture in dimension two. Soviet Math. Dokl., 1979, 20, P. 956–959.
5. Dahlberg J., Trubowitz E. A remark on two dimensional periodic potentials. Comment. Math. Helvetici, 1982, 57, P. 130–134.
6. Skriganov M. M. The spectrum band structure of the threedimensional Schrodinger operator with periodic potential. ¨ Invent. Math., 1985, 80, P. 107–121.
7. Helffer B., Mohamed A. Asymptotic of the density of states for the Schrodinger operator with periodic electric potential. ¨ Duke Math. J., 1998, 92, P. 1–60.
8. Parnovski L. Bethe-Sommerfeld conjecture. Ann. Henri Poincare´, 2008, 9, P. 457–508.
9. Schenker J. H., Aizenman M. The creation of spectral gaps by graph decoration. Lett. Math. Phys., 2000, 53, P. 253–262.
10. Band R., Berkolaiko G. Universality of the momentum band density of periodic networks. Phys. Rev. Lett., 2013, 111, P. 130404.
11. Kostrykin V., Schrader R. Kirchhoff’s rule for quantum wires. J. Phys. A: Math. Gen., 1999, 32, P. 595–620.
12. Harmer M. Hermitian symplectic geometry and extension theory. J. Phys. A: Math. Gen., 2000, 33, P. 9193–9203.
13. Cheon T., Exner P., Turek O. Approximation of a general singular vertex coupling in quantum graphs. Ann. Phys. (NY), 2010, 325, P. 548–578.
14. Cheon T., Exner P., Turek O. Tripartite connection condition for a quantum graph vertex. Phys. Lett. A, 2010, 375, P. 113–118.
15. Exner P., Turek O. Periodic quantum graphs from the Bethe–Sommerfeld perspective, arXiv: 1705.07306 [math-ph]
16. Exner P. Lattice Kronig–Penney models. Phys. Rev. Lett. 1995, 74, P. 3503–3506.
17. Exner P. Contact interactions on graph superlattices. J. Phys. A: Math. Gen., 1996, 29, P. 87–102.
18. Exner P., Gawlista R. Band spectra of rectangular graph superlattices. Phys. Rev. B, 1996, 53, P. 7275–7286.
19. Khinchin Y. Ya. Continued Fractions. University of Chicago Press, 1964.
20. Cassels J. W. An introduction to Diophantine Approximation. Cambridge University Press, 1957.
21. Pelantova E., Starosta ´ S., Znojil M. Markov constant and quantum instabilities. ˇ J. Phys. A: Math. Theor., 2016, 49, P. 155201.
22. Hurwitz A. Uber die angen ¨ aherte Darstellung der Irrationalzahlen durch rationale Br¨uche. ¨ Math. Ann., 1891, 39, P. 279–284.
Review
For citations:
Exner P., Turek O. Quantum graphs with the Bethe-Sommerfeld property. Nanosystems: Physics, Chemistry, Mathematics. 2017;8(3):305-309. https://doi.org/10.17586/2220-8054-2017-8-3-305-309
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