(DVI file)

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 $e^{-tD^2} - e^{-t\tilde{D'}^2}$ a trace class operator,

where $\tilde{D'}$is a certain transformation of $D'$? We answer these two questions and establish in fact a very general relative index and analytic torsion theory.

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