Scattering Matrix and Analytic Torsion

TL;DR

This paper investigates the spectral and metric asymptotics of manifolds with cylindrical ends in an adiabatic limit, providing an analytic proof of the gluing formula for analytic torsion.

Contribution

It introduces an adiabatic limit approach to analyze spectral and metric asymptotics, and offers a new analytic proof of the gluing formula for analytic torsion.

Findings

Spectral asymptotics of Hodge-Laplacian in the adiabatic limit

Asymptotic behavior of the $L^2$-metric on de Rham cohomology

Analytic proof of the gluing formula for analytic torsion

Abstract

For a compact manifold, which has a part isometric to a cylinder of finite length, we consider an adiabatic limit procedure, in which the length of the cylinder tends to infinity. We study the asymptotic of the spectrum of Hodge-Laplacian and the asymptotic of the $L^2$-metric on de Rham cohomology. As an application, we give a pure analytic proof of the gluing formula for analytic torsion.