GLOBAL $h$ FOURIER INTEGRAL OPERATORS WITH COMPLEX-VALUED PHASE FUNCTIONS

Abstract

The author considers globally defined $h$ Fourier integral operators $(h\, {\rm FIO})$ with complex-valued phase functions. Symbolic calculus of $h\, {\rm FIO}$ is considered and, using a new complex Gauss transform, the composition of $h$ pseudodifferential operators $(h\, {\rm PDO})$ and $h\, {\rm FIO}$ is considered. For a self-adjoint $h\, {\rm PDO}\, A(h)$ and $h\, {\rm PDO}\, P(h)$ and $Q(h)$ with compactly supported symbols, the results are applied to approximate the kernel of the operator \[ U_{P,Q}(t;h):=P(h)e^{-ih^{-1}tA(h)}Q(h)^{\ast},\quad t\in {\bb R},\;h>0, \] by a single, globally defined $h$ -oscillatory integral.