ИСТИНА

Let $X$ be a compact $C^\infty$ manifold with smooth boundary $\partial X$, and let $D(x)$ be a smooth function on $X$ such that $D(x)>0$ in $X^\circ=X\setminus\partial X$, $D(x)=0$ on $\partial X$, and $\nabla D(x)\ne0$ on $\partial X$. Consider an operator $L$ on $X$ of the form $L=-\langle\nabla,D(x)A(x)\nabla\rangle$ in local coordinates, where the matrix $A={}^T\!A=(A_{jk})_{j,k=1}^{n}\in C^\infty(X)$ is real and positive definite up to the boundary. We consider the Cauchy problem for the wave equation $u_{tt}+Lu=0$ degenerating on the boundary as well as the eigenvalue problem $Lu=\omega^2u$ for the operator $L$. In applications (e.g., to tsunami waves generated by a localized source, waves trapped by the coast, or seiches), these problems contain a small (or large) parameter, and so it is natural to use the semiclassical approximation. However, the standard scheme of Maslov’s canonical operator does not apply here because of the degeneration, and therefore, the author and his colleagues have earlier constructed asymptotic solutions of these problems by introducing a nonstandard phase space $\Phi$ and a modified canonical operator. Here we suggest a completely different approach to the construction of semiclassical asymptotics in the above-mentioned problems based on an idea resembling that of Leray’s uniformization for differential equations on complex-analytic manifolds. Namely, we construct a closed manifold $M$ with an action of the group $S^1$ and a smooth projection $\pi :M\to X\simeq M/{{S}^{1}}$. This projection permits one to lift the problems in question to $M$, thus obtaining problems whose asymptotic solutions can be written out by standard methods. The solutions of the original problems are just the fiberwise constant solutions of these new problems. The nonstandard phase space $\Phi$ arises in this approach as the simplest version of the Marsden–Weinstein symplectic reduction of $T^*M$ by the action of $S^1$. The surprisingly simple implementation of this approach provides a complete analysis of asymptotic solutions of the original problems and simple efficient formulas for these solutions.