On traces of Fourier integral operators localized at a finite set of points

Abstract

Given a smooth embedding of manifolds $i: X \hookrightarrow M$ and a Fourier integral operator $\Phi$ acting on $M$, obtained by quantization of a canonical transformation, consider its trace $i^!(\Phi)$ on $X$ (in the sense of relative theory). We discuss the situation when $i^!(\Phi)$ has the form of a Fourier--Mellin operator and, in particular, is localized at a finite set of points.