On the convergence of some products of Fourier integral operators

Jerome Le Rousseau (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités

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.

Mots clés: Fourier integral operators; convergence; infinite products; first order hyperbolic equations

Domaines: Equations aux dérivées partielles [math.AP]

Jerome Le Rousseau. On the convergence of some products of Fourier integral operators. 2006. ⟨hal-00019924v1⟩