Opuscula Math. 35, no. 2 (2015), 199-234
http://dx.doi.org/10.7494/OpMath.2015.35.2.199
Opuscula Mathematica
Decisiveness of the spectral gaps of periodic Schrödinger operators on the dumbbell-like metric graph
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.
- G. Berkolaiko, P. Kuchment, Introduction to Quantum Graphs, AMS, Providence, R.I., 2012.
- P. Duclos, P. Exner, O. Turek, On the spectrum of a bent chain graph, J. Phys. A: Math. Theor. 41 (2008), 415206, 18 pp.
- J. Garnett, E. Trubowitz, Gaps and bands of one dimensional periodic Schrödinger operators, Comment. Math. Helv. 59 (1984), 258-312.
- C. Gérard, F. Nier, The Mourre theory for analytically fibered operators, J. Funct. Anal. 152 (1998) 1, 202-219.
- P. Kargaev, E. Korotyaev, Effective masses and conformal mappings, Comm. Math. Phys. 169 (1995), 597-625.
- E. Korotyaev, The inverse problem for the Hill operator. I, Int. Math. Res. Not. 3 (1997), 113-125.
- E. Korotyaev, I. Lobanov, Schrödinger operators on zigzag nanotubes, Ann. Henri Poincaré 8 (2007), 1151-1176.
- V. Kostrykin, R. Schrader, Kirchhoff's rule for quantum wires, J. Phys. A: Math. Gen. 32 (1999), 595-630.
- P. Kuchment, O. Post, On the spectra of carbon nano-structures, Comm. Math. Phys. 275 (2007), 805-826.
- W. Magnus, S. Winkler, Hill's Equation, Wiley-Interscience, New York, 1966.
- V. Marchenko, I. Ostrovski, A characterization of the spectrum of the Hill operator, Math. USSR Sb. 26 (1975), 493-554.
- J. Poschel, E. Trubowitz, Inverse Spectral Theory, Academic Press, Orlando, 1987.
- M. Reed, B. Simon, Methods of Modern Mathematical Physics, IV. Analysis of Operators, Academic Press, New York, 1978.
- E. Titchmarsh, The Theory of Functions, Second ed., Univ. Press, London, 1975.
- Hiroaki Niikuni
- Maebashi Institute of Technology, 460-1 Kamisadori, Maebashi City, Gunma, 371-0816, Japan
- Communicated by P.A. Cojuhari.
- Received: 2014-05-23.
- Revised: 2014-07-22.
- Accepted: 2014-07-26.
- Published online: 2014-11-18.
