Abstract—
We consider a two-dimensional quantum system consisting of a resonator with the Neumann condition, to which N small similar resonators are attached along the boundary through small holes. The main purpose is to study the limiting case for \(N \to \infty \). It is shown that in this limit the problem is reduced to a boundary value problem with an energy-dependent boundary condition of the Robin type. Asymptotic methods are used, as well as approximations of the eigenstates of the system obtained using the pinhole model. The results are compared with numerically obtained values.
Similar content being viewed by others
REFERENCES
R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley-Interscience, New York, 1953; ГТТИ, 1933), Vol. 1.
J. M. Arrieta, J. K. Hale, and Q. J. Han, “Eigenvalue problems for nonsmoothly perturbed domains,” Differential Equations 91, 24–52 (1991).
E. Sanchez-Palencia, Nonhomogeneous Media and Vibration Theory (Springer, Berlin, 1980).
I. Yu. Popov, “Extension theory and localization of resonances for domains of trap type,” Mat. Sb. 181, 1366–1390 (1990).
I. Yu. Popov, “The resonator with narrow slit and the model based on the operator extensions theory,” J. Math. Phys. 33, 3794–3801 (1992).
B. S. Pavlov, “The theory of extensions and explicitly-soluble models,” Russ. Math. Surveys 42, 127—168 (1987).
R. R. Gadyl’shin, “Existence and asymptotics of poles with small imaginary part for the Helmholtz resonator,” Russ. Math. Surveys 52, 1—72 (1997).
A. M. Ilyin, Matching of Asymptotic Expansions of Boundary Value Problems (Nauka, Moscow, 1989) [in Russian].
I. Yu. Popov, “Waveguides coupled via apertures: asymptotic form of the eigenvalue,” Tech. Phys. Lett. 25, 57—59 (1999).
E. S. Trifanova, “Resonance phenomena in curved quantum waveguides coupled via windows,” Tech. Phys. Lett. 35, 180—182 (2009).
D. Borisov and P. Exner, “Distant perturbation asymptotics in window-coupled waveguides. I. The nonthreshold case,” J. Math. Phys. 47, 113502 (2006).
A. M. Vorobiev, A. S. Bagmutov, and A. I. Popov, “On formal asymptotic expansion of resonance for quantum waveguide with perforated semitransparent barrier,” Nanosystems: Phys. Chem., Math. 10, 415—419 (2019).
A. Khrabustovskyi, “Homogenization of eigenvalue problem for Laplace–Beltrami operator on Riemannian manifold with complicated ‘bubble-like’ microstructure,” Math. Meth. Appl. Sci. 32, 2123—2137 (2009).
G. Cardone and A. Khrabustovskyi, “Neumann spectral problem in a domain with very corrugated boundary,” J. Differential Equations 259, 2333—2367 (2015).
I. Yu. Popov, I. V. Blinova, and A. I. Popov, “A model of a boundary composed of the Helmholtz resonators,” Complex Var. Elliptic Eq. 66, 1256—1263 (2021).
X. Ni, K. Chen, M. Weiner, D. J. Apigo, C. Prodan, A. Alu, E. Prodan, and A. B. Khanikaev, “Observation of Hofstadter butterfly and topological edge states in reconfigurable quasi-periodic acoustic crystals,” Commun. Phys. 2, 55 (2019).
S. Huang, X. Fang, X. Wang, Assouar Badreddine, Cheng Qian, and Li Yong, “Acoustic perfect absorbers via Helmholtz resonators with embedded apertures,” J. Acoust. Soc. Am. 145, 254 (2019).
M. S. Birman and M. Z. Solomyak, Spectral Theory of Self-adjoint Operators in Hilbert Space (D. Reidel, Dordrecht, 1986).
R. C. McCann, R. D. Hazlett, and D. K. Babu, “Highly accurate approximations of Green’s and Neumann functions on rectangular domains,” Proc. R. Soc. Lond. A 457, 767—772 (2001).
Funding
The study was supported by the Russian Science Foundation (project no. 22-11-00046).
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors declare that they have no conflicts of interest.
Additional information
Translated by G. Dedkov
Rights and permissions
About this article
Cite this article
Bagmutov, A.S., Trifanova, E.S. & Popov, I.Y. Resonator with a Сorrugated Boundary: Numerical Results. Phys. Part. Nuclei Lett. 20, 96–99 (2023). https://doi.org/10.1134/S1547477123020103
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1547477123020103