Negative eigenvalues of the Y-type chain of weakly coupled ball resonators
Abstract
Spectral properties of a system are strongly associated with its geometry. The spectral problem for the Y-bent chain of weakly-coupled ball resonators is investigated. The Y-bent system can be described as a central ball linking three chains consisting of balls of the same radius. There is a δ-coupling condition with parameter α at every contact point. Specifically, it is assumed that the axis passing through the center of each ball lies in the same plane and the centers of balls that are the closest to the central ball form an equilateral triangle. The transfer-matrix approach and the theory of extensions are employed to solve the spectral problem for this system. It is shown that such system with a certain value of parameter α has at most one negative eigenvalue in the case of δ-coupling in contact points.
Supporting agencies: This work was supported by FTP “Scientific and Educational Human Resources for Innovation-Driven Russia”(contract 16.740.11.0030), grant 11-08-00267 of RFBR, FTP “Researches and Development in the Priority Directions Developments of a Scientific and Technological Complex of Russia 2007-2013”(state contract 07.514.11.4146) and grants of the President of Russia (state contract 14.124.13.2045-MK and grant MK-1493.2013.1)
About the Author
A. S. Anikevich
St. Petersburg National Research University of Information Technologies, Mechanics and Optics
Russian Federation
Russian Federation
St. Petersburg
References
1. P. Duclos, P. Exner, O. Turek. On the spectrum of a bent chain graph. J. Phys. A: Math. Theor. 41, P. 415206 (2008).
2. N.I. Gerasimenko, B.S. Pavlov. Scattering problems on noncompact graphs. Theoret. Math. Phys. 74, P. 230–240 (1988).
3. P. Kuchment. Graph models for waves in thin structures. Wave in random media. 12 (4), R1-R24 (2002).
4. E. Korotyaev, I. Lobanov. Schr¨ odinger operators on zigzag graphs. Ann. Henri Poincar´e. 8 (6), P. 1151-1176 (2007).
5. I.Yu. Popov, P.I. Smirnov. Spectral problem for branching chain quantum graph. Phys. Lett. A. 377, P. 439–442 (2013).
6. I.Yu. Popov. The resonator with narrow slit and the model based on the operator extensions theory. J. Math. Phys. 33 (11), P. 3794–3801 (1992).
7. R. Brown, P.D. Hislop, A. Martinez. Eigenvalues and resonances for domains with tubes: Neumann boundary conditions. J. Diff. Equs. 115 (2), P. 458–476 (1995).
8. I.Yu. Popov. Model of point-like window for electromagnetic Helmholtz resonator. Zeitschrift fur Analysis und ihre Anwendungen. 32 (2), P. 155–162 (2013).
9. R.R. Gadylshin. On the vibration amplitude for the Helmholtz resonator. DAN SSSR. 310 (5), P. 1094-1097 (1990).
10. I.Yu. Popov. Operator extensions theory model for electromagnetic fieldelectron interaction. Journal of Mathematical Physics. 53 (6), P. 063505/1-8 (2012).
11. J. Br¨ uning, P. Exner, V. Geyler. Large gaps in point-coupled periodic systems of manifolds. J. Phys. A, Math. Gen. 36 (17), P. 4875–4890 (2003).
12. J. Br¨ uning, G. Martin, B. Pavlov. Calculation of the Kirchhoff coefficients for the Helmholtz resonator. Russian J. Math. Phys. 16 (2), P. 188–207 (2009).
13. A. Laurain, S.A. Nazarov, J. Sokolowski. Singular perturbations of curved boundaries in three dimensions. The spectrum of the Neumann Laplacian. Z. Anal. Anwend. 30 (2), P. 145–180 (2011).
14. N.I. Akhiezer, I.M. Glazman. Theory of Linear Operators in Hilbert Space. Vol. I. Translated from the third Russian edition by E.R. Daqson. Translation edited by W.N. Everitt. Monographs and Stidies in Mathematics, 9. London: Pitman (Advanced Publishing Program), 1981. 312 pp.
15. N. Bagraev, G. Martin, B.S. Pavlov, A. Yafyasov. Landau-Zener effect for a quasi-2D periodic sandwich. Nanosystems: Physics, Chemistry, Mathyematics, 2 (4), P. 32–50 (2011)
Review
For citations:
Anikevich A.S. Negative eigenvalues of the Y-type chain of weakly coupled ball resonators. Nanosystems: Physics, Chemistry, Mathematics. 2013;4(4):545-549.
Views:
2
Email this article 