Creating and controlling band gaps in periodic media with small resonators


  • Andrii Khrabustovskyi Department of Physics, Faculty of Science, University of Hradec Králové, Rokitanského 62, Hradec Králové, 50003, Czech Republic
    Department of Theoretical Physics, Nuclear Physics Institute of the Czech Academy of Sciences, Hlavní 130, Husinec-Řež, 25068, Czech Republic
  • Evgen Khruslov B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine


Ключові слова:

перiодичнi середовища, резонатори, лапласiан Неймана, спектральнi лакуни


Дослiджуються спектральнi властивостi лапласiана Неймана ${\mathcal A}_\varepsilon$ на перiодичнiй необмеженiй областi ${\Omega}_\varepsilon$ , яка залежить вiд малого параметра $\varepsilon>0$.  Область ${\Omega}_\varepsilon$ отримується шляхом видалення з ${\mathbb R}^n$ $m\in{\mathbb N}$ сiмейств $\varepsilon$-перiодично розташованих малих резонаторiв. Доведено, що спектр ${\mathcal A}_\varepsilon$ має принаймнi $m$ лакун. Першi $m$ лакун прямують при $\varepsilon\to 0$ до деяких iнтервалiв, розташуванням i довжиною яких можна керувати шляхом певного вибору резонаторiв; iншi лакуни (якщо вони є) прямують до нескiнченностi. Обговорюються застосування до теорiї фотонних кристалiв.

Mathematical Subject Classification 2020: 35B27, 35P05, 47A75


J. Arrieta, J. Hale and Q. Han, Eigenvalue problems for nonsmoothly perturbed domains, J. Differ. Equations 91 (1991), No. 1, 24--52,

D. Barseghyan and A. Khrabustovskyi, Gaps in the spectrum of a periodic quantum graph with periodically distributed δ'-type interactions, J. Phys. A, Math. Theor. 48 (2015), No. 25, 255201.

J. Behrndt and A. Khrabustovskyi, Construction of self-adjoint differential operators with prescribed spectral properties, Math. Nachr. 295 (2022), No. 6, 1063--1095.

G. Cardone and A. Khrabustovskyi, Spectrum of a singularly perturbed periodic thin waveguide, J. Math. Anal. Appl. 454 (2017), No. 2, 673--694.

Y. Colin de Verdiére, Construction de laplaciens dont une partie finie du spectre est donnée, Ann. Sci. École Norm. Sup. (4) 20 (1987), No. 4, 599--615.

R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. 1, Wiley-Interscience, New York, 1953.

E.B. Davies, Spectral Theory and Differential Operators, Cambridge University Press, Cambridge, 1995.

W. Dörfler, A. Lechleiter, M. Plum, G. Schneider and C. Wieners, Photonic Crystals. Mathematical Analysis and Numerical Approximation, Springer, Berlin, 2011.

M. Eastham, The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, Edinburg, 1973.

P. Exner and A. Khrabustovskyi, Gap control by singular Schrödinger operators in a periodically structured metamaterial, J. Math. Phys. Anal. Geom. 14 (2018), No. 3, 270--285.

F. Ferraresso and J. Taskinen, Singular perturbation Dirichlet problem in a double periodic perforated plane, Ann. Univ. Ferrara 61 (2015), 277--290.

A. Figotin and P. Kuchment, Band-gap structure of the spectrum of periodic dielectric and acoustic media. I. Scalar model, SIAM J. Appl. Math. 56 (1996), No. 1, 68--88.

A. Figotin and P. Kuchment, Band-gap structure of the spectrum of periodic dielectric and acoustic media. II. Two-dimensional photonic crystals, SIAM J. Appl. Math. 56 (1996), No. 6, 1561--1620.

E.L. Green, Spectral theory of Laplace-Beltrami operators with periodic metrics, J. Differ. Equ. 133 (1997), No. 1, 15--29.

R. Hempel, T. Kriecherbauer and P. Plankensteiner, Discrete and Cantor spectrum for Neumann Laplacians of combs, Math. Nachr. 188 (1997), No. 1, 141--168.

R. Hempel and K. Lienau, Spectral properties of periodic media in the large coupling limit, Commun. Partial Differ. Equations 25 (2000), No. 7--8, 1445--1470.

R. Hempel and O. Post, Spectral gaps for periodic elliptic operators with high contrast: an overview, Progress in Analysis, Proceedings of the 3rd International ISAAC Congress Berlin 2001, Vol. 1, 577-587, 2003.

R. Hempel, L.A. Seco, and B. Simon, The essential spectrum of Neumann Laplacians on some bounded singular domains, J. Funct. Anal. 102 (1991), No. 2, 448--483.

H. Hönl, A.W. Maue, and K. Westpfahl, Theorie der Beugung [Theory of Diffraction], in: Encyclopedia of Physics, 5/25/1 , Springer-Verlag, Berlin, 1961, 218--573 (German).

G.A. Iosif'yan, O.A. Oleinik, and A.S. Shamaev, On the limiting behaviour of the spectrum of a sequence of operators defined on different Hilbert spaces, Russ. Math. Surv. 44 (1989), No. 3, 195--196.

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1966.

A. Khrabustovskyi, Periodic Riemannian manifold with preassigned gaps in spectrum of Laplace-Beltrami operator, J. Differ. Equations 252 (2012), No. 3, 2339--2369.

A. Khrabustovskyi, Homogenization of the spectral problem on the Riemannian manifold consisting of two domains connected by many tubes, Proc. R. Soc. Edinb., Sect. A, Math. 143 (2013), No. 6, 1255--1289.

A. Khrabustovskyi, Periodic elliptic operators with asymptotically preassigned spectrum, Asymptotic Anal. 82 (2013), No. 1--2, 1--37.

A. Khrabustovskyi, Opening up and control of spectral gaps of the Laplacian in periodic domains, J. Math. Phys. 55 (2014), No. 12, 121502.

A. Khrabustovskyi, Periodic quantum graphs with predefined spectral gaps, J. Phys. A, Math. Theor. 53 (2020), No. 40, 405202.

A. Khrabustovskyi and E. Khruslov, Gaps in the spectrum of the Neumann Laplacian generated by a system of periodically distributed traps, Math. Methods Appl. Sci. 38 (2015), No. 1, 11--26.

A. Kostenko and N. Nicolussi, Laplacians on Infnite Graphs, EMS Press, Berlin, 2022.

P. Kuchment, Floquet Theory For Partial Differential Equations, Birkhauser, Basel, 1993.

P. Kuchment, An overview of periodic elliptic operators, Bull. Am. Math. Soc., New Ser. 53 (2016), No. 3, 343--414.

P. Kuchment and J. Taskinen, Dispersion relations and spectra of periodically perforated structures, Pure Appl. Funct. Anal. 7 (2022), No. 2, 721--731.

A. Lamacz and B. Schweizer, Effective acoustic properties of a meta-material consisting of small helmholtz resonators, Discrete Contin. Dyn. Syst. Ser. S 10 (2017), No. 4, 815--835.

V.A. Marchenko and E.Ya. Khruslov, Homogenization of Partial Differential Equations, Birkhäuser, Boston, 2006.

S.A. Nazarov, K. Ruotsalainen and J. Taskinen, Spectral gaps in the Dirichlet and Neumann problems on the plane perforated by a double-periodic family of circular holes, J. Math. Sci. (N.Y.) 181 (2012), No. 2, 164--222.

O.A. Oleinik, A.S. Shamaev, and G.A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992.

B. Schweizer, The low-frequency spectrum of small Helmholtz resonators, Proc. A, R. Soc. Lond. 471 (2015), No. 2174, 20140339.

B. Schweizer, Resonance meets homogenization. construction of meta-materials with astonishing properties, Jahresber. Dtsch. Math.-Ver. 119 (2017), 31--51.

V.V. Zhikov, On spectrum gaps of some divergent elliptic operators with periodic coefficients, St. Petersb. Math. J. 16 (2005), No. 5, 773--790.


Як цитувати

Khrabustovskyi, A.; Khruslov, E. Creating and controlling band gaps in periodic media with small resonators. Журн. мат. фіз. анал. геом. 2023, 19, 456–481.





Дані завантаження ще не доступні.