Geodesic
Eigenfunctions of the essential spectrum of the Laplace operator in an angle with the Robin–Neumann boundary conditions
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 52, Tome 516 (2022), pp. 121-134 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

This work studies eigenfunction problem of the Laplace operator in the angular domain with the Robin-type boundary condition on the upper side of the angle and the Neumann-type boundary condition on the bottom side of the angle. From the physical point of view, such eigenfunctions describe waves over sloping beach. Negative values of the spectral parameter were considered. We obtained the eigenfunction of the essential spectrum and studied a special case of eigenfunction, which are elementary functions. The Sommerfeld integral representation of an eigenfunction of the negative part of the essential spectrum of the Laplace operator was obtained. Moreover, we calculated it's asymptotic far away from the angle's vertex. It is bounded on the top side of the angle and vanishes exponentially in the angle's interior with its bottom side. So, the eigenfunction of essential spectrum behaves like a surface wave. 
@article{ZNSL_2022_516_a5,
     author = {M. A. Lyalinov and N. S. Fedorov},
     title = {Eigenfunctions of the essential spectrum of the {Laplace} operator in an angle with the {Robin{\textendash}Neumann} boundary conditions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {121--134},
     year = {2022},
     volume = {516},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2022_516_a5/}
}
TY  - JOUR
AU  - M. A. Lyalinov
AU  - N. S. Fedorov
TI  - Eigenfunctions of the essential spectrum of the Laplace operator in an angle with the Robin–Neumann boundary conditions
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2022
SP  - 121
EP  - 134
VL  - 516
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2022_516_a5/
LA  - ru
ID  - ZNSL_2022_516_a5
ER  - 
%0 Journal Article
%A M. A. Lyalinov
%A N. S. Fedorov
%T Eigenfunctions of the essential spectrum of the Laplace operator in an angle with the Robin–Neumann boundary conditions
%J Zapiski Nauchnykh Seminarov POMI
%D 2022
%P 121-134
%V 516
%U http://geodesic.mathdoc.fr/item/ZNSL_2022_516_a5/
%G ru
%F ZNSL_2022_516_a5
M. A. Lyalinov; N. S. Fedorov. Eigenfunctions of the essential spectrum of the Laplace operator in an angle with the Robin–Neumann boundary conditions. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 52, Tome 516 (2022), pp. 121-134. http://geodesic.mathdoc.fr/item/ZNSL_2022_516_a5/

[1] F. Ursell, “Edge Waves on a Sloping Beach”, Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, 214:1116 (1952), 79–97 | MR

[2] V. M. Babich, M. A. Lyalinov, V. E. Grikurov, Diffraction Theory: The Sommerfeld-Malyuzhinets Technique, Alpha Science International, 2008

[3] N. Kuznetsov, V. Maz'ya, B. Vainberg, Linear Water Waves: A Mathematical Approach, 2002 | MR

[4] M. Khalile, K. Pankrashkin, “Eigenvalues of Robin Laplacians in infinite sectors”, Mathematische Nachrichten, 291:5–6 (2018), 928–965 | DOI | MR

[5] M. A. Lyalinov, “Kommentarii o sobstvennykh funktsiyakh i sobstvennykh chislakh operatora Laplasa v ugle s kraevymi usloviyami Robena”, Zap. nauchn. semin. POMI, 483, 2019, 116–127

[6] M. A. Lyalinov, “Sobstvennye funktsii otritsatelnogo spektra operatora Shrëdingera v poluploskosti s singulyarnym potentsialom na luche i s usloviem Neimana na granitse”, Zap. nauchn. semin. POMI, 493, 2020, 232–258 | MR

[7] B. A. Packham, “A note on generalized edge waves on a sloping beach”, The Quarterly Journal of Mechanics and Applied Mathematics, 42:3 (1989), 441–446 | DOI | MR

[8] M. A. Lyalinov, “Eigenoscillations in an angular domain and spectral properties of functional equations”, Euro. J. Appl Math., 33 (2021), 538–559 | DOI | MR

[9] A. A. Tuzhilin, “Teoriya funktsionalnykh uravnenii Malyuzhintsa. I. Odnorodnye funktsionalnye uravneniya, obschie svoistva reshenii, chastnye sluchai”, Differents. uravneniya, 6:4 (1970), 692–704

[10] A. V. Osipov, “Vychislenie funktsii Malyuzhintsa v kompleksnoi oblasti”, Akusticheskii zh., 36:1 (1990), 116–121

[11] I. S. Gradstein, I. M. Ryzhik, Table of integrals, series, and products, Academic Press, 2007 | MR