The Maslov canonical operator on a pair of Lagrangian manifolds and asymptotic solutions of stationary equations with localized right-hand sides

The problem of constructing the asymptotics of the Green function for the Helmholtz operator h2Δ + n2(x), x ∈ Rn, with a small positive parameter h and smooth n2(x) has been studied by many authors; see, e.g., [1, 2, 4]. In the case of variable coefficients, the asymptotics was constructed by matching the asymptotics of the Green function for the equation with frozen coefficients and a WKB-type asymptotics or, in a more general situation, the Maslov canonical operator. The paper presents a different method for evaluating the Green function, which does not suppose the knowledge of the exact Green function for the operator with frozen variables. This approach applies to a larger class of operators, even when the right-hand side is a smooth localized function rather than a δ-function. In particular, the method works for the linearized water wave equations.