Index and Spectral Theory for Manifolds with Generalized Fibred Cusps

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Generalizing work of W. Müller we investigate the spectral theory for the Dirac operator D on a noncompact manifold X with generalized fibred cusps C(M)=M× [A,∞[ᵣ, g= d r²+ ϕ*gY+ e-2crgZ, at infinity. Here ϕ:Mʰ⁺ᵛ→ Yʰ is a compact fibre bundle with fibre Z and a distinguished horizontal space HM. The metric gZ is a metric in the fibres and gY is a metric on the base of the fibration. We also assume that the kernel of the vertical Dirac operator at infinity forms a vector bundle over Y. Using the ``ϕ-calculus'' developed by R. Mazzeo and R. Melrose we explicitly construct the meromorphic continuation of the resolvent G(λ) of D for small spectral parameter as a special ``conormal distribution''. From this we deduce a description of the generalized eigensections and of the spectral measure of D. Complementing this, we perform an explicit construction of the heat kernel [(-tD²)] for finite and small times t, corresponding to large spectral parameter λ. Using a generalization of Getzler's technique, due to R. Melrose, we can describe the singular terms in the heat kernel expansion and prove an index formula for D, calculating the extended L²-index of D in terms of the usual local expression, the family eta invariant for the family of vertical Dirac operators at infinity and the eta invariant for the horizontal ``Dirac'' operator at infinity.