On formal asymptotic expansion of resonance for quantum waveguide with perforated semitransparent barrier
https://doi.org/10.17586/2220-8054-2019-10-4-415-419
Abstract
A quantum waveguide with a semitransparent barrier, placed across it, is considered. It is assumed that the barrier has a small window. This local perturbation of the waveguide causes the appearance of resonance states localized near the barrier with the window. The asymptotics (in small parameter – the window width) of the resonances (quasi-bound states) is obtained. The procedure of construction of full formal asymptotic expansion is described. The first two terms of the asymptotic expansion are obtained explicitly. These terms describe the shift of the resonance from the threshold and the life time of the corresponding resonance state.
Supporting agencies: This work was partially financially supported by the Government of the Russian Federation (grant 08-08), by grant 16-11-10330 of Russian Science Foundation.
About the Authors
A. M. Vorobiev
ITMO University
Russian Federation
Russian Federation
Kronverkskiy, 49, Staint Petersburg, 197101
A. S. Bagmutov
ITMO University
Russian Federation
Russian Federation
Kronverkskiy, 49, Staint Petersburg, 197101
A. I. Popov
ITMO University
Russian Federation
Russian Federation
Kronverkskiy, 49, Staint Petersburg, 197101
References
1. Lord Rayleigh O.M. The theory of Helmholtz Resonator. Proceeding of Royal Society, 1916, 638, P. 265–275.
2. Morse F.M., Feshbach G. Methods of theoretical physics, V. 2. .: Foreign Literature Publishing House, 1960, 986 p.
3. Kiselev A.A., Pavlov B.S. The eigenfrequencies and eigenfunctions of the Laplace operator of the Neumann problem in a system of two coupled resonators. Theor. Math. Phys., 1994, 100(3), P. 354–366.
4. Gadylshin R.R. The existence and asymptotics of poles with a small imaginary part for the Helmholtz resonator. Uspekhi of Mathematical Sciences, 1997, 52(313), P. 71–72.
5. Borisov D.I. Discrete spectrum of an asymmetric pair of waveguides coupled through a window. Sb. Math., 2006, 197(4), P. 475–504.
6. Achilleos V., Richoux O., et.al. Acoustic solitons in waveguides with Helmholtz resonators: Transmission line approach. Phys.Rev. E., 2015, 91, P. 023204.
7. Martinez A., Ned´ elec L. Optimal lower bound of the resonance widths for a Helmoltz tube-shaped resonator.´ J. Spectral Th., 2012, 2, P. 203– 223.
8. Gadyl’shin R.R. A two-dimensional analogue of the Helmholtz resonator with ideally rigid walls. Diff. Uravn., 1994, 30(2), P. 221–229. Translation in Diff. Eq., 1994, 30(2), P. 201–209.
9. Gadyl’shin R.R. Influence of the position of the opening and its shape on the properties of a Helmholtz resonator. Theor. Math. Phys., 1992, 93, P. 1151–1159.
10. Borisov D., Exner P. Distant perturbation asymptotics in window-coupled waveguides. I. The non-threshold case. J. Math. Phys., 2006, 47(11), P. 113502(1-24).
11. Exner P., Seba P., Tater M., Vanek D. Bound states and scattering in quantum waveguides coupled laterally through a boundary window. J. Math. Phys., 1996, 37, P. 4867–4887.
12. Exner P., Vugalter S.A. Asymptotic estimates for bound states in quantum waveguides coupled laterally through a narrow window. Ann. Inst. H. Poincare, 1996, 65, P. 109–123.
13. Popov I.Y. Asymptotics of bound state for laterally coupled waveguides. Rep. Math. Phys. 1999, 43, P. 427–437.
14. Borisov D.I., Gadyl’shin R.R. On the spectrum of the Laplacian with frequently alternating boundary conditions. Theor. Math. Phys., 1999, 118(3), P. 272–277.
15. Botman S.A., Leble S.B. Kinetic model of electron transport in cylindrical nanowire with rough surface. Nanosystems: Phys. Chem. Math., 2018, 9(2), P. 206–211.
16. Popov I.Yu. The operator extension theory, semitransparent surface and short range potential. Math. Proc. Cambridge Phil. Soc., 1995, 118, P. 555–563.
17. Popov I.Yu. The extension theory and the opening in semitransparent surface. J. Math. Phys., 1992, 33(5), P. 1685–1689.
18. Ikebe T., Shimada S. Spectral and scattering theory for the Scgrodinger operator with penetrable wall potentials.¨ J.Math. Kyoto Univ., 1991, 31(1), P. 219–258.
19. Popov I.Yu. The extension theory, domains with semitransparent surface and the model of quantum dot. Proc. Royal Soc. London A., 1996, 452, P. 1505–1515.
20. Exner P., Kreicirik D. Waveguides coupled through a semitransparent barrier: a Birman-Schwinger analysis. Rev. Math. Phys., 2001, 13, P. 307–334.
21. Behrndt J., Langer M., Lotoreichik V. Boundary triples for Schrodinger operators with singular interactions on hypersurfaces.¨ Nanosystems: Phys. Chem. Math., 2016, 7(2), P. 290–302.
22. Mantile A., Posilicano A. Laplacians with singular perturbations supported on hypersurfaces. Nanosystems: Phys. Chem. Math., 2016, 7(2), P. 315–323.
23. Exner P., Kondej S., Lotoreichik V. Asymptotics of the bound state induced by delta-interaction supported on a weakly deformed plane. J. Math. Phys., 2018, 59, P. 013051.
24. Behrndt J., Exner P., et.al. Approximation of Schroedinger operators with delta-interactions supported on hypersurfaces. Math. Nachr., 2017, 290, P. 12151248.
25. Popov I.Yu. The operator extension theory, semitransparent surface and short range potential. Math. Proc. Cambridge Phil. Soc., 1995, 118, P. 555–563.
26. Tikhonov A.N., Samarskii A.A. Equations of Mathematical Physics. M.: Science, 1972, 531 p.
27. Frolov S.V., Popov I.Yu. Resonances for laterally coupled quantum waveguides. J. Math. Phys., 2000, 41, P. 4391–4405.
28. Gadyl’shin R.R. Surface potentials and the method of matching asymptotic expansions in the Helmholtz resonator problem. Algebra i Analiz, 1992, 4(2), P. 88–115; translation in St. Petersburg Math. J., 1993, 4(2), P. 273–296.
Review
For citations:
Vorobiev A.M., Bagmutov A.S., Popov A.I. On formal asymptotic expansion of resonance for quantum waveguide with perforated semitransparent barrier. Nanosystems: Physics, Chemistry, Mathematics. 2019;10(4):415–419. https://doi.org/10.17586/2220-8054-2019-10-4-415-419
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