On the convergence of some products of Fourier integral operators

Abstract

An approximation Ansatz for the operator solution, $U(z',z)$, of a hyperbolic first-order pseudodifferential equation, \d_z + a(z,x,D_x) with $\Re (a) \geq 0$, is constructed as the composition of global Fourier integral operators with complex phases. We prove a convergence result for the Ansatz to $U(z',z)$ in some Sobolev space as the number of operators in the composition goes to $\infty$, with a convergence of order $\alpha$, if the symbol $a(z,.)$ is in \Con^{0,\alpha} with respect to the evolution parameter $z$. We also study the consequences of some truncation approximations of the symbol $a(z,.)$ in the construction of the Ansatz