Differential operators of Fuchs type, conical singularities, and asymptotic methods

TL;DR

This work provides a comprehensive analysis of Fuchs type differential operators on manifolds with conical singularities, focusing on heat equation analysis, index theory, and spectral functions, with applications to stratified spaces.

Contribution

It offers a self-contained treatment of heat equation analysis, index theorems, and spectral functions for Fuchs type operators on singular manifolds, extending classical theories.

Findings

Development of short-time asymptotics for heat kernels

Formulation of index theorems for singular spaces

Analysis of spectral functions like η and ζ

Abstract

This text is a revised version of the authors Habilitationsschrift which was submitted to the University of Augsburg, 1993. Fuchs type differential operators are used to model the analysis on manifolds with cone--like singularities, or more general, stratified spaces. This book provides a self--contained treatment of the analysis of the heat equation and index theory for these operators. Major topics are short--time asymptotics, $\eta$-- and $\zeta$--functions, (relative) index theorems. Another chapter is devoted to the discussion of deficiency indices and Dirac Schr\"odinger operators.