2001/02/08 by Boris Vaillant, Vaillant, Boris · 4 citations
Mathematics · #58G25 #58Gxx #Advanced Operator Algebra Research #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.DG #math.SP #msc:58G25 #msc:58Gxx
paper · pdf · doi:10.48550/arxiv.math/0102072
122 pages, 8 figures, doctoral thesis
arxiv created 2001/02/08 · openalex publication_date 2001/02/08 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 [exp(-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.