Geodesic
On the number of quasimodes of whispering gallery type
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 13, Tome 128 (1983), pp. 152-157

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New two-sealing expansion for eigenfunctions of whispering gallery type and corresponding eigenvalues of Laplace operator with Dirichlet and Heumaan boundary conditions in the plane region is offered. Eigen functions localize in a vicinity of the boundary and are enumerated by two natural numbers $(q, p)$ where $q$ and $p$ are respectivly numbers of knots along the boundary and along the normal to it. The validity of this asymptotic expansion is ensured provided $0\leqslant p\leqslant{\rm const}\:q^{1-\varepsilon}$ for $\forall\varepsilon\in(0, 1]$ where $q\to\infty$. 
@article{ZNSL_1983_128_a16,
     author = {V. V. Skripnikov},
     title = {On the number of quasimodes of whispering gallery type},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {152--157},
     publisher = {mathdoc},
     volume = {128},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1983_128_a16/}
}
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V. V. Skripnikov. On the number of quasimodes of whispering gallery type. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 13, Tome 128 (1983), pp. 152-157. http://geodesic.mathdoc.fr/item/ZNSL_1983_128_a16/