Global h Fourier Integral Operators with Complex-Valued Phase Functions

The author considers globally defined h Fourier integral operators(h FIO) with complex-valued phase functions. Symbolic calculusof h FIO is considered and, using a new complex Gauss transform,the composition of h pseudodifferential operators (h PDO) andh FIO is considered. For a self-adjoint h PDO A(h) and h PDOP(h) and Q(h) with compactly supported symbols, the resultsare applied to approximate the kernel of the operator by a single, globally defined h-oscillatoryintegral. 2000 Mathematics Subject Classification 81Q20, 35S30.