Opuscula Mathematica
Opuscula Math. 35, no. 2 (2015), 199-234

Decisiveness of the spectral gaps of periodic Schrödinger operators on the dumbbell-like metric graph

Hiroaki Niikuni
Abstract. In this paper, we consider periodic Schrödinger operators on the dumbbell-like metric graph, which is a periodic graph consisting of lines and rings. Let one line and two rings be in the basic period. We see the relationship between the structure of graph and the band-gap spectrum.
Keywords: quantum graph, spectral gap, band structure, Hill operator.
Mathematics Subject Classification: 34L05, 34L15, 34B45.
Full text (pdf)
References ▼
  1. [1] G. Berkolaiko, P. Kuchment, Introduction to Quantum Graphs, AMS, Providence, R.I., 2012.
  2. [2] P. Duclos, P. Exner, O. Turek, On the spectrum of a bent chain graph, J. Phys. A: Math. Theor. 41 (2008), 415206, 18 pp.
  3. [3] J. Garnett, E. Trubowitz, Gaps and bands of one dimensional periodic Schrödinger operators, Comment. Math. Helv. 59 (1984), 258-312.
  4. [4] C. Gérard, F. Nier, The Mourre theory for analytically fibered operators, J. Funct. Anal. 152 (1998) 1, 202-219.
  5. [5] P. Kargaev, E. Korotyaev, Effective masses and conformal mappings, Comm. Math. Phys. 169 (1995), 597-625.
  6. [6] E. Korotyaev, The inverse problem for the Hill operator. I, Int. Math. Res. Not. 3 (1997), 113-125.
  7. [7] E. Korotyaev, I. Lobanov, Schrödinger operators on zigzag nanotubes, Ann. Henri Poincaré 8 (2007), 1151-1176.
  8. [8] V. Kostrykin, R. Schrader, Kirchhoff's rule for quantum wires, J. Phys. A: Math. Gen. 32 (1999), 595-630.
  9. [9] P. Kuchment, O. Post, On the spectra of carbon nano-structures, Comm. Math. Phys. 275 (2007), 805-826.
  10. [10] W. Magnus, S. Winkler, Hill's Equation, Wiley-Interscience, New York, 1966.
  11. [11] V. Marchenko, I. Ostrovski, A characterization of the spectrum of the Hill operator, Math. USSR Sb. 26 (1975), 493-554.
  12. [12] J. Poschel, E. Trubowitz, Inverse Spectral Theory, Academic Press, Orlando, 1987.
  13. [13] M. Reed, B. Simon, Methods of Modern Mathematical Physics, IV. Analysis of Operators, Academic Press, New York, 1978.
  14. [14] E. Titchmarsh, The Theory of Functions, Second ed., Univ. Press, London, 1975.
About the authors ▼
Hiroaki Niikuni
Maebashi Institute of Technology, 460-1 Kamisadori, Maebashi City, Gunma, 371-0816, Japan
About the article ▼
Communicated by P.A. Cojuhari.
Received: 2014-05-23.
Revised: 2014-07-22.
Accepted: 2014-07-26.
Published online: 2014-11-18.
Cite this article as ▼
Hiroaki Niikuni, Decisiveness of the spectral gaps of periodic Schrödinger operators on the dumbbell-like metric graph, Opuscula Math. 35, no. 2 (2015), 199-234, http://dx.doi.org/10.7494/OpMath.2015.35.2.199.
Download this article's citation as: