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Zeta and eta functions for Atiyah-Patodi-Singer operators

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Abstract

This paper concerns Dirac-type operatorsP on manifoldsX with boundary which are “product-type” near the boundary. That is,\(P = \sigma \left( {\frac{\partial }{{\partial x_n }} + A} \right)\) for a unitary morphism σ and a self-adjoint first-order operatorA onbdry(X);x n denotes the normal coordinate. For a realizationP B defined by a boundary operatorB of Atiyah-Patodi-Singer type, the paper gives a complete description of the singularities of the traces of the meromorphic continuations of Γ(s)D i )s and Γ(s)DP i )s where Δ1 =P * B P B , Δ2 =P B P * B , andD is any differential operator onX which is tangential and independent of 4x n nearbdry(X). This implies expansions for the associated heat kernels and resolvents, containing the usual powers (with both “local” and “global” coefficients) together with logarithmic terms.

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Work partially supported by NSF Grant DMS-9004655.

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Grubb, G., Seeley, R.T. Zeta and eta functions for Atiyah-Patodi-Singer operators. J Geom Anal 6, 31–77 (1996). https://doi.org/10.1007/BF02921566

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