The Maslov Canonical Operator on a Pair of Lagrangian Manifolds and Asymptotic Solutions of Stationary Equations with Localized Right-Hand Sides

Abstract—The problem of constructing the asymptotics of the Green function for the Helmholtz operator ${{h}^{2}}\Delta + {{n}^{2}}(x)$, $x \in {{{\mathbf{R}}}^{n}}$, with a small positive parameter h and smooth ${{n}^{2}}(x)$ 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 $\delta $-function. In particular, the method works for the linearized water wave equations.