najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2021-12-4-462-471najo-494Research ArticlePHYSICSФИЗИКАResonance asymptotics for quantum waveguides with semitransparent multi-perforated wallАсимптотика резонансов для квантовых волноводов с полупрозрачной перфорированной стенкой.VorobievA. M.VorobievA. M.

49, Kronverkskiy, St. Petersburg, 197101.

lenden31@yandex.ruITMO UniversityRussian Federation202104082025124462471Copyright © Vorobiev A.M., 20252025Vorobiev A.M.Vorobiev A.M.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/494

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.

Рассмотрена пара связанных квантовых волноводов с общей полупрозрачной стенкой. В стенке имеется конечное число малых отверстий. Рассматриваются резонансные состояния, сосредоточенные около каждого отверстия. Наличие нескольких отверстий требует описания их взаимного влияние в отличие от случай одного отверстия. Используя метод согласования асимптотических разложений, мы выводим формулы для резонансов и резонансных состояний.

резонансрезонансное состояниеволноводасимптотикаполупрозрачностьresonanceresonance statewaveguideasymptoticssemitransparencyReferencesLord Rayleigh O.M. The theory of Helmholtz Resonator. Proceeding of Royal Society, 1916, 638, P. 265–275.Lord Rayleigh O.M. The theory of Helmholtz Resonator. Proceeding of Royal Society, 1916, 638, P. 265–275.Morse F.M. Feshbach G. Methods of Theoretical Physics, V. 2. Foreign Literature Publishing House, 1960, 986 p.Morse F.M. Feshbach G. Methods of Theoretical Physics, V. 2. Foreign Literature Publishing House, 1960, 986 p.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.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.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.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.Borisov D.I. Discrete spectrum of an asymmetric pair of waveguides coupled through a window. Sb. Math., 2006, 197(4), P. 475–504.Borisov D.I. Discrete spectrum of an asymmetric pair of waveguides coupled through a window. Sb. Math., 2006, 197(4), P. 475–504.Achilleos V., Richoux O., et.al. Acoustic solitons in waveguides with Helmholtz resonators: Transmission line approach. Phys.Rev. E., 2015, 91, P. 023204.Achilleos V., Richoux O., et.al. Acoustic solitons in waveguides with Helmholtz resonators: Transmission line approach. Phys.Rev. E., 2015, 91, P. 023204.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.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.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.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.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.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.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).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).Popov I. Asymptotics of bound states and bands for laterally coupled waveguides and layers. J. Math. Phys., 2002, 43, P. 215–234.Popov I. Asymptotics of bound states and bands for laterally coupled waveguides and layers. J. Math. Phys., 2002, 43, P. 215–234.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.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.Briet Ph., Dittrich J., Soccorsi E. Scattering through a straight quantum waveguide with combined boundary conditions. J. Math. Phys., 2014, 55, P. 112104.Briet Ph., Dittrich J., Soccorsi E. Scattering through a straight quantum waveguide with combined boundary conditions. J. Math. Phys., 2014, 55, P. 112104.Ilyin A.M. Matching of the asymptotic expansions of solutions. Science, Moscow, 1989, 336 p.Ilyin A.M. Matching of the asymptotic expansions of solutions. Science, Moscow, 1989, 336 p.Exner P., Kreicirik D. Waveguides coupled through a semitransparent barrier: a Birman-Schwinger analysis. Rev. Math. Phys., 2001, 13, P. 307–334.Exner P., Kreicirik D. Waveguides coupled through a semitransparent barrier: a Birman-Schwinger analysis. Rev. Math. Phys., 2001, 13, P. 307–334.Exner P., Kreicirik D. Quantum waveguides with a lateral semitransparent barrier: spectral and scattering properties. J. Phys. A, 1999, 32, P. 4475.Exner P., Kreicirik D. Quantum waveguides with a lateral semitransparent barrier: spectral and scattering properties. J. Phys. A, 1999, 32, P. 4475.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.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.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.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.Popov I.Yu. The extension theory and the opening in semitransparent surface. J. Math. Phys., 1992. 33(5), P. 1685–1689.Popov I.Yu. The extension theory and the opening in semitransparent surface. J. Math. Phys., 1992. 33(5), P. 1685–1689.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.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.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.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.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.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.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.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.Behrndt J., Exner P., et.al. Approximation of Schroedinger operators with delta-interactions supported on hypersurfaces. Math. Nachr., 2017, 290, P. 12151248.Behrndt J., Exner P., et.al. Approximation of Schroedinger operators with delta-interactions supported on hypersurfaces. Math. Nachr., 2017, 290, P. 12151248.Popov I.Yu. The operator extension theory, semitransparent surface and short range potential. Math. Proc. Cambridge Phil. Soc., 1995, 118, P. 555–563.Popov I.Yu. The operator extension theory, semitransparent surface and short range potential. Math. Proc. Cambridge Phil. Soc., 1995, 118, P. 555–563.Tikhonov A.N., Samarskii A.A. Equations of Mathematical Physics. M.: Science, 1972, 531 p.Tikhonov A.N., Samarskii A.A. Equations of Mathematical Physics. M.: Science, 1972, 531 p.

The authors declare that there are no conflicts of interest present.