On the convergence of some products of Fourier integral operators

An approximation Ansatz for the operator solution, U(z′,z), of a hyperbolic first-order pseudodifferential equation, ∂_z+a(z,x,D_x) with Re(a)≥0, is constructed as the composition of global Fourier integral operators with complex phases. The symbol a(z,·) is assumed to have a regularity as low as Hölder, 𝒞^0,α, with respect to the evolution parameter z. 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 ∞, with a convergence of order α. We also study the consequences of some truncation approximations of the symbol a(z,·) in the construction of the Ansatz.