ИСТИНА

The problem of constructing the high frequency asymptotics of the Green function for the Helmholtz equation has been studied by numerous authors. An even more important problem is that of finding asymptotics for equations of the Helmholtz type in which the right-hand side is a smooth localized function rather than a delta function. We develop a theory of Maslov’s canonical operator on a Lagrangian intersection (a pair of Lagrangian manifold satisfying certain conditions), which can be viewed as a semiclassical counterpart of the theory of Fourier integral operators on a Lagrangian intersection developed by R. Melrose and G. Uhlmann in the case of homogeneous Lagrangian manifolds. We show how our theory, combined with the new integral formulas recently developed by the author in collaboration with S. Dobrokhotov and A. Shafarevich for Maslov’s canonical operator on Lagrangian manifolds, produces efficient formulas for the asymptotic solutions of specific problems with localized right-hand sides and compare our results with earlier ones. In particular, the method works for the linearized water wave equations.