z-functions of Fourier Integral Operators

Abstract

Based on Guillemin’s work on gauged Lagrangian distributions, we will introduce the notion of a gauged 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 Σk∈N amk where each amk is log-homogeneous with degree of homogeneity mk but violating R(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 trace. Using stationary phase approximation, series representations for the Laurent coefficients and values of ζ-functions will be stated explicitly, and the kernel singularity structure will be studied. This will yield algebras of Fourier Integral Operators which purely consist of Hilbert-Schmidt operators and whose ζ-functions are entire, as well as algebras in which the generalized Kontsevich- Vishik trace is form-equivalent to the pseudo-differential operator case. Additionally, we will introduce an approximation method (mollification) for ζ-functions of Fourier Integral Operators whose amplitudes are poly-log-homogeneous at zero by ζ-functions of Fourier Integral Operators with “regular” amplitudes. In part II, we will study Bochner-, Lebesgue-, and Pettis integration in algebras of Fourier Integral Operators. The integration theory will extend the notion of parameter dependent Fourier Integral Operators and is compatible with the Atiyah-Jänich index bundle as well as the ζ-function calculus developed in part I. Furthermore, it allows one to emulate calculations using holomorphic functional calculus in algebras without functional calculus, and to consider measurable families of Fourier Integral Operators as they appear, for instance, in heat- and wave-traces of manifolds whose metrics are subject to random (possibly singular) perturbations.

Date of Award	2015
Original language	English
Awarding Institution	King's College London
Supervisor	Simon Scott (Supervisor)