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Eigenoscillations in an angular domain and spectral properties of functional equations

Part of: Partial differential equations Representations of solutions

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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