Geodesic
On the number of quasimodes of the ``bouncing ball'' type
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 12, Tome 117 (1981), pp. 172-182

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New two-scaling expansion for eigenfunctions of “bouncing ball” type and corresponding eigenvalues of Laplacian operator with Dirichlet boundary condition in the region in the plane has been offered. Eigen functions localized in the neighborhood of a stable diameter of the region and are numbered by two natural numbers $(p, q)$, where $p$ – number of knots in longitudinal and $q$ – in perpendicular to the diameter direction. The truth of this asimptotic expansion is ensured provided $0\leqslant q\leqslant \mathrm{const}\,p^{1-\varepsilon}$ for $\forall\varepsilon>0$, where $p\to+\infty$. 
@article{ZNSL_1981_117_a11,
     author = {V. F. Lazutkin and D. Ya. Terman},
     title = {On the number of quasimodes of the ``bouncing ball'' type},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {172--182},
     publisher = {mathdoc},
     volume = {117},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1981_117_a11/}
}
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V. F. Lazutkin; D. Ya. Terman. On the number of quasimodes of the ``bouncing ball'' type. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 12, Tome 117 (1981), pp. 172-182. http://geodesic.mathdoc.fr/item/ZNSL_1981_117_a11/