Abstract
A new two-scale expansion is proposed for eigenfunctions of “bouncing-ball” type and the corresponding eigenvalues of the Laplace operator with the Dirichlet condition in a domain of the plane. The eigenfunctions are concentrated in a neighborhood of a stable diameter of the domain and are numbered with two indices (p, q) where p is the number of longitudinal nodes and q the number of nodes in a direction orthogonal to the diameter. The validity of the asymptotic expansions is guaranteed for 0 ⩽ q ⩽ const pɛ−1 and ∀ɛ>0 as p → +∞.
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Literature cited
V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Problems of the Diffraction of Short Waves [in Russian], Moscow (1972).
V. F. Lazutkin, The Convex Billiard and Eigenfunctions of the Laplace Operator [in Russian], Leningrad (1981).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 172–182, 1981.
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Lazutkin, V.F., Terman, D.Y. Number of quasimodes of “bouncing-ball” type. J Math Sci 24, 373–379 (1984). https://doi.org/10.1007/BF01086997
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DOI: https://doi.org/10.1007/BF01086997