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Eigenoscillations in an angular domain and spectral properties of functional equations

Part of: Representations of solutions Partial differential equations

Published online by Cambridge University Press:  06 May 2021

M. A. LYALINOV Open the ORCID record for M. A. LYALINOV [Opens in a new window]
M. A. LYALINOV*
Affiliation:
Department of Mathematics and Mathematical Physics, Saint-Petersburg University, Universitetskaya nab.7/9, Saint-Petersburg, 199034, Russia emails: lyalinov@yandex.ru; m.lyalinov@spbu.ru
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Abstract

This work studies functional difference equations of the second order with a potential belonging to a special class of meromorphic functions. The equations depend on a spectral parameter. Consideration of this type of equations is motivated by applications in diffraction theory and by construction of eigenfunctions for the Laplace operator in angular domains. In particular, such eigenfunctions describe eigenoscillations of acoustic waves in angular domains with ‘semitransparent’ boundary conditions. For negative values of the spectral parameter, we study essential and discrete spectrum of the equations and describe properties of the corresponding solutions. The study is based on the reduction of the functional difference equations to integral equations with a symmetric kernel. A sufficient condition is formulated for the potential that ensures existence of the discrete spectrum. The obtained results are applied for studying the behaviour of eigenfunctions for the Laplace operator in adjacent angular domains with the Robin-type boundary conditions on their common boundary. At infinity, the eigenfunctions vanish exponentially as was expected. However, the rate of such decay depends on the observation direction. In particular, in a vicinity of some directions, the regime of decay is switched from one to another and such asymptotic behaviour is described by a Fresnel-type integral.

Keywords

Functional difference equationsspectrumasymptotic behavioureigenfunctions of the LaplacianMalyuzhinets’ functional equations

MSC classification

Primary: 35A22: Transform methods (e.g. integral transforms)
Secondary: 35C15: Integral representations of solutions 39B22: Equations for real functions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 3 , June 2022 , pp. 538 - 559
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Babich, V. M., Dement’ev, D. B., Samokish, B. A. & Smyshlyaev, V. P. (2000) On Evaluation of the diffraction coefficients for arbitrary ‘nonsingular’ directions of a smooth convex cone. SIAM J. Appl. Math. 60, 536–73.Google Scholar
Babich, V. M., Lyalinov, M. A. & Grikurov, V. E. (2008) Diffraction Theory. The Sommerfeld-Malyuzhinets Technique. Alpha Science Series on Wave Phenomena Alpha Science, Oxford, UK.Google Scholar
Behrndt, B., Exner, P. & Lotoreichik, V. (2013) Schrödinger operators with δ- and $$\delta '$$-interactions on Lipschitz surfaces and chromatic numbers of associated partitions. Rev. Math. Phys. 26(08). See doi: 10.1142/S0129055X14500159.CrossRefGoogle Scholar
Bernard, J.-M. L. (1997) Méthode analytique et transformées fonctionnelles pour la diffraction d’ondes par une singularité conique: équation intégrale de noyau non oscillant pour le cas d’impédance constante. Rapport CEA-R-5764. Editions Dist-Saclay, an extended version in Bernard, J.-M. L. (2014) Advanced Theory of Diffraction by a Semi-infinite Impedance Cone. Alpha Science Series on Wave Phenomena, Alpha Science, Oxford, UK.Google Scholar
Birman, M. Sh. & Solomjak, M. Z. (1987) Spectral Theory of Selfadjoint Operators in Hilbert Spaces, Holland, Dordrecht.Google Scholar
Budaev, B. V. (1995) Diffraction by Wedges, Pitman Research Notes in Mathematics Series, Vol. 322, Longman Scientific and Technical, Essex, UK.Google Scholar
Ehrenmark, U. T. (2018) Wave response to a submerged wedge-shaped rubble mound breakwater. Q. J. Mech. Appl. Math. 72(1), 25–50.CrossRefGoogle Scholar
Faddeev, L. D., Kashaev, R. & Volkov, A. (2001) Strongly coupled quantum discrete Liouville theory: algebraic approach and duality. Commun. Math. Phys. 219, 199–219.CrossRefGoogle Scholar
Fedoryuk, M. V. (1987) Asymptotics: Integrals and Series, Nauka, Moscow (in Russian).Google Scholar
Fedotov, A. A & Sandomirskiy, F. (2015) An exact renormalization formula for the Mariland model. Commun. Math. Phys. 334, 1083–99.CrossRefGoogle Scholar
Gradstein, I. S. & Ryzhik, I. M. (1980) Tables of Integrals, Series and Products, 4th ed., Academic Press, Orlando.Google Scholar
Grikurov, V. E. & Lyalinov, M. A. (2008) Diffraction of the surface H-polarized wave by an angular break of a thin dielectric slab. J. Math. Sci. 155(3), 390–396.Google Scholar
Jost, R. (1955) Mathematical analysis of a simple model for the strippng reaction. ZAMP VI, 316–26.Google Scholar
Kato, T. (1995) Perturbation Theory for Linear Operators, Springer-Verlag, Berlin.Google Scholar
Khalile, M. & Pankrashkin, K. (2018) Eigenvalues of Robin Laplacians in infinite sectors. Mathematische Nachrichten 291, 928–65. doi: 10.1002/mana.201600314.Google Scholar
Lawrie, J. B. & King, A. C. (1994) Exact solution to a class of the functional difference equations with application to a moving contact line flow. Euro. J. Appl. Math. 5, 141–57.Google Scholar
Lyalinov, M. A.& Zhu, N. Y. (2007) Acoutic scattering by a circular semi-transparent conical surface. J. Eng. Math. 59, 385–98.Google Scholar
Lyalinov, M. A., Zhu, N. Y. & Smyshlyaev, V. P. (2010) Scattering of a plane electromagnetic wave by a hollow circular cone with thin semi-transparent walls. IMA J. Appl. Math. 75, 676–719. (See doi: 10.1093/imamat/hxq030)CrossRefGoogle Scholar
Lyalinov, M. A. & Zhu, N. Y. (2012) Scattering of Waves by Wedges and Cones with Impedance Boundary Conditions, Mario Boella Series on Electromagnetism in Information & Communication, SciTech-IET, Edison, NJ.Google Scholar
Lyalinov, M. A. (2020) Functional difference equations and eigenfunctions of a Schrӧdinger operator with $$\delta '$$-interaction on a circular conical surface. Proc. R. Soc. A 476, 20200179. http://dx.doi.org/10.1098/rspa.2020.0179.Google Scholar
Maliuzhinets [Malyuzhinets], G. D. (1958) Excitation, reflection and emission of surface waves from a wedge with given face impedances. Soviet Phys. Doklady. 3, 752–755.Google Scholar
Pankrashkin, K. (2015) Variational proof of the existence of eigenvalues for star graphs. In: J. Dittrich and H. Kovarik (editors), Functional Analysis and Operator Theory for Quantum Physics, EMS Series of Congress Reports, Vol. 3, pp. 447–458. doi: 10.4171/175-1/22.Google Scholar
Roseau, M. (1958) Short waves parallel to the shore over a sloping beach.Comm. Pure Appl. Math. 11 A958, 433–93.CrossRefGoogle Scholar
Titchmarsh, E. C. 1937 Introduction to the Theory of Fourier Integrals, Oxford University Press, UK.Google Scholar
Williams, W. E. (1959) Diffraction of an E–polarised wave by an imperfectly conducting wedge. Proc. R. Soc. Lond. A252, 195–209.Google Scholar