Preview

Nanosystems: Physics, Chemistry, Mathematics

Advanced search

Resonance asymptotics for quantum waveguides with semitransparent multi-perforated wall

https://doi.org/10.17586/2220-8054-2021-12-4-462-471

Abstract

A pair of coupled quantum waveguides with a common semitransparent wall is considered. The wall has a finite number of small windows. We consider resonance states localized near each window. The presence of several windows forces one to describe their common influence differently from that of the single-window case. Using the “matching of asymptotic expansions” method, we derive formulas for resonances and resonance states.

About the Author

A. M. Vorobiev
ITMO University
Russian Federation

49, Kronverkskiy, St. 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., Nedelec 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(10), P. 113502(1-24).

11. Popov I. Asymptotics of bound states and bands for laterally coupled waveguides and layers. J. Math. Phys., 2002, 43, P. 215–234.

12. 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.

13. Briet Ph., Dittrich J., Soccorsi E. Scattering through a straight quantum waveguide with combined boundary conditions. J. Math. Phys., 2014, 55, P. 112104.

14. Ilyin A.M. Matching of the asymptotic expansions of solutions. Science, Moscow, 1989, 336 p.

15. Exner P., Kreicirik D. Waveguides coupled through a semitransparent barrier: a Birman-Schwinger analysis. Rev. Math. Phys., 2001, 13, P. 307–334.

16. Exner P., Kreicirik D. Quantum waveguides with a lateral semitransparent barrier: spectral and scattering properties. J. Phys. A, 1999, 32, P. 4475.

17. Popov I.Yu. The extension theory, domains with semitransparent surface and the model of quantum dot. Proc. Royal Soc. London A., 1996, 452(1950), P. 1505–1515.

18. Popov I.Y. Model of a quantum point as a cavity with semitransparent boundary. Physics of the Solid State, 1994, 36(7), P. 1918–1922.

19. Popov I.Yu. The extension theory and the opening in semitransparent surface. J. Math. Phys., 1992. 33(5), P. 1685–1689.

20. Ikebe T., Shimada S. Spectral and scattering theory for the Scrodinger operator with penetrable wall potentials. J.Math. Kyoto Univ., 1991, 31(1), P. 219–258.

21. Popov I.Y., Trifanova E.S., Vorobiev A.M. 2D waveguides asymptotics of eigenvalue induced by a window in a semitransparent separating wall. Bulletin of the Transilvania University of Brasov, Series III: Mathematics, Informatics, Physics, 2021, 63(1), P. 221–230.

22. Vorobiev A.M., Popov I.Y. Resonances in two-dimensional quantum waveguides separated by semitransparence wall with small window. Technical Physics Letters, 2021, 47(9), P. 847–849.

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.


Review

For citations:


Vorobiev A.M. Resonance asymptotics for quantum waveguides with semitransparent multi-perforated wall. Nanosystems: Physics, Chemistry, Mathematics. 2021;12(4):462-471. https://doi.org/10.17586/2220-8054-2021-12-4-462-471

Views: 3


Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.


ISSN 2220-8054 (Print)
ISSN 2305-7971 (Online)