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 Izv. Akad. Nauk SSSR Ser. Mat., 1973, Volume 37, Issue 2, Pages 437–465 (Mi izv2256)

Asymptotics of the eigenvalues of the Laplacian and quasimodes. A series of quasimodes corresponding to a system of caustics close to the boundary of the domain

V. F. Lazutkin

Abstract: For a bounded convex domain in the plane, asymptotic formulas with error tending to zero are constructed for a certain series of eigenvalues of the Laplacian with zero boundary conditions. The boundary of the domain is assumed to be sufficiently smooth. It is proved that
$$\varliminf_{\lambda\to+\infty}N^*(\lambda)/N(\lambda)>0,$$
where $N(\lambda)$ is the number of eigenvalues (with multiplicities taken into account) less than $\lambda$ and $N^*(\lambda)$ is the number of those eigenvalues for which an asymptotic expansion has been found.

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English version:
Mathematics of the USSR-Izvestiya, 1973, 7:2, 439–466

Bibliographic databases:

UDC: 517.43
MSC: 35P20, 35J05, 47F05

Citation: V. F. Lazutkin, “Asymptotics of the eigenvalues of the Laplacian and quasimodes. A series of quasimodes corresponding to a system of caustics close to the boundary of the domain”, Izv. Akad. Nauk SSSR Ser. Mat., 37:2 (1973), 437–465; Math. USSR-Izv., 7:2 (1973), 439–466

Citation in format AMSBIB
\Bibitem{Laz73} \by V.~F.~Lazutkin \paper Asymptotics of the eigenvalues of the Laplacian and quasimodes. A~series of quasimodes corresponding to a~system of caustics close to the boundary of the domain \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1973 \vol 37 \issue 2 \pages 437--465 \mathnet{http://mi.mathnet.ru/izv2256} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=333485} \zmath{https://zbmath.org/?q=an:0255.35074} \transl \jour Math. USSR-Izv. \yr 1973 \vol 7 \issue 2 \pages 439--466 \crossref{https://doi.org/10.1070/IM1973v007n02ABEH001949} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. I. Shnirelman, “Ob asimptoticheskoi kratkosti spektra operatora Laplasa”, UMN, 30:4(184) (1975), 265–266
2. M V Berry, J Phys A Math Gen, 10:12 (1977), 2083
3. Vadim Zharnitsky, “Quasiperiodic Motion in the Billiard Problem with a Softened Boundary”, Phys. Rev. Lett, 75:24 (1995), 4393
4. P. Stefanov, G. Vodev, “Neumann resonances in linear elasticity for an arbitrary body”, Comm Math Phys, 176:3 (1996), 645
5. Martin Kunz, J Phys A Math Gen, 33:31 (2000), 5567
6. J. Math. Sci. (N. Y.), 128:2 (2005), 2680–2685
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