2015/10/25 by Hartung, Tobias, Scott, Simon
#35S30 #46F10 #58J40 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Operator Algebras (math.OA) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1510.07324
Based on Guillemin's work on gauged Lagrangian distributions, we will introduce the notion of a poly-log-homogeneous distribution as an approach to ζ-functions for a class of Fourier Integral Operators which includes cases of amplitudes with asymptotic expansion ∑k∈ℕamk where each amk is log-homogeneous with degree of homogeneity mk but violating \Re(mk)→-∞. We will calculate the Laurent expansion for the ζ-function and give formulae for the coefficients in terms of the phase function and amplitude as well as investigate generalizations to the Kontsevich-Vishik quasi-trace. Using stationary phase approximation, series representations for the Laurent coefficients and values of ζ-functions will be stated explicitly. Additionally, we will introduce an approximation method (mollification) for ζ-functions of Fourier Integral Operators whose symbols have singularities at zero by ζ-functions of Fourier Integral Operators with regular symbols.