On the number of quasimodes of the “bouncing ball” type

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 $p\to+\infty$.