RELATIVE INDEX THEOREMS AND ANALYTIC TORSION FOR OPEN MANIFOLDS

JURGEN EICHHORN

Abstract: For non-negative elliptic differential operators on closed manifolds, we have a well-established index theory and the notion of analytic torsion. On open manifolds, this becomes wrong since such operators must not have a finite dimensional kernel and cokernel and the spectrum must not be purely discrete. But there is a possibility to establish a similar theory for pairs of operators D, D′, where D′ is an appropriate perturbation of D. We consider pairs D^{2}, D^{′2}, where D, D′ are generalized Dirac operators (e.g. Laplace operators) and D′ is an appropriate perturbation of D. Doing this, there arise two canonical questions:

(1) what are appropriate perturbations and

(2) is e^{−tD^{2}} −e^{−tD̃′^{2}} a trace class operator, where D̃′is a certain transformation of D′?

We answer these two questions and establish in fact a very general relative index and analytic torsion theory.