Maslov's canonical operator on a pair of Lagrangian manifolds and Green function type asymptotics

Maslov's canonical operator on a pair of Lagrangian manifolds and Green function type asymptotics
We consider differential and pseudodifferential equations with properties similar to those of the Helmholtz equation with spatially localized right hand side (in particular, the Dirac delta function). Our analysis uses ideas close to ones due to Keller, Babich, Kucherenko, Melrose, Uhlmann, Sternin, and Shatalov and shows that an asymptotic solution of such an equation can be represented via Maslov's canonical operator on an appropiate pair of Lagrangian manifolds. By way of example, a pseudodifferential equation arising in linear water wave theory is presented.
This work was done together with A.Anikin, V.Nazaikinskii, and M.Rouleux and was supported by RFBR grant 14-01-00521.