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We study the structure and asymptotic behavior of the resolvent of elliptic cone pseudodifferential operators acting on weighted Sobolev spaces over a compact manifold with boundary. We obtain an asymptotic expansion of the resolvent as the spectral parameter tends to infinity, and use it to derive corresponding heat trace and zeta function expansions as well as an analytic index formula.

The goal of this paper is to establish a geometric program to study elliptic pseudodifferential boundary problems which arise naturally under cutting and pasting of geometric and spectral invariants of Dirac-type operators on manifolds with corners endowed with multi-cylindrical, or b-type, metrics and ‘b-admissible’ partitioning hypersurfaces. We show that the Cauchy data space of a Dirac operator on such a manifold is Lagrangian for the self-adjoint case, the corresponding Calderon projector is a b-pseudodifferential operator of order 0, characterize Fredholmness, prove relative index formulae, and solve the Bojarski conjecture.

Combining elements of the b-calculus and the theory of elliptic boundary value problems, we solve the gluing problem for b-determinants of Dirac type operators on manifolds with cylindrical ends. As a corollary of our proof, we derive a gluing formula for the b-eta invariant and also a relative invariant formula relating the b-spectral invariants on a manifold with cylindrical end to the spectral invariants with the augmented APS boundary condition on the corresponding compact manifold with boundary.

In this note, we announce gluing and comparison formulas for the spectral invariants of Dirac type operators on compact manifolds and manifolds with cylindrical ends. We also explain the central ideas in their proofs. 1. The gluing problem for the spectral invariants Since their inception, the eta invariant and the ζ-determinant of Dirac type operators have influenced mathematics and physics in innumerable ways. Especially with the development of quantum field theory, the behavior of these spectral invariants under gluing of the underlying manifold has become an increasingly important topic. However, the gluing formula for the ζ-determinant of a Dirac Laplacian has remained an open question due to the nonlocal nature of this invariant. In fact, Bleecker and Booss-Bavnbek stated that [2, p. 89] “no precise pasting formulas are obtained but only adiabatic ones.” In [23, 24], we give precise gluing formulas for ζ-determinants of Dirac type operators on compact manifolds and manifolds with cylindrical ends, respectively, and moreover we present new and unified derivations of the gluing formulas for both invariants. The purpose of this note is to announce these gluing formulas for the spectral invariants and to indicate the main ideas in their proofs. We also announce a relative invariant formula proved in [25, 24]. We begin with describing the gluing problem for compact manifolds. Let D be a Dirac type operator acting on C∞(M,S) where M is a closed compact Riemannian manifold of arbitrary dimension and S is a Clifford bundle over M . Let Y be an embedded hypersurface in M and let M = M− ∪ M+ be decomposition of M into manifolds with boundary such that ∂M− = ∂M+ = Y . We assume that all geometric structures are of product type over a tubular neighborhood N = [−1, 1]×Y of Y where the Dirac operator takes the product form D = G(∂u+DY ), where G is a unitary operator on S0 := S|Y and DY is a Dirac type operator over Y satisfying G = −Id and DY G = −GDY . Recall that the eta function of D and the zeta function of D2 are defined through the heat operator e−tD2 via (1.1) ηD(s) = 1 Γ( s+1 2 ) (∫ 1

We construct the complex powersAz for an elliptic cone (or Fuchs type) differential operatorA on a manifold with boundary. We show thatAz exists as an entire family ofb-pseudodifferential operators. We also examine the analytic structure of the Schwartz kernel ofAz, both on and off the diagonal. Finally, we study the meromorphic behavior of the zeta function Tr(Az).

We show that the resolvent kernel of an elliptic b -pseudodifferential operator on a compact manifold with corners (of arbitrary codimension) is a polyhomogeneous, or classical, function on a certain manifold with corners. The singularities of the resolvent kernel are shown to localize near the diagonal as the resolvent parameter goes to infinity. Explicit descriptions of the expansions, including logarithmic terms, are given. In particular, the asymptotics of the resolvent restricted to the diagonal follows as a corollary. Applications to the asymptotic behavior of the heat kernel and to the analysis of the poles of complex powers are also given.

In this paper we study the asymptotic behavior (in the sense of meromorphic functions) of the zeta function of a Laplace-type operator on a closed manifold when the underlying manifold is stretched in the direction normal to a dividing hypersurface, separating the manifold into two manifolds with infinite cylindrical ends. We also study the related problem on a manifold with boundary as the manifold is stretched in the direction normal to its boundary, forming a manifold with an infinite cylindrical end. Such singular deformations fall under the category of "analytic surgery", developed originally by Hassell, Mazzeo and Melrose \cite{mazz95-5-14,hass95-3-115,hass98-6-255} in the context of eta invariants and determinants., Comment: 33 pages, 12 figures

On a spin manifold with conformal cusps, we prove under an invertibility condition at infinity that the eta function of the twisted Dirac operator has at most simple poles and is regular at the origin. For hyperbolic manifolds of finite volume, the eta function of the Dirac operator twisted by any homogeneous vector bundle is shown to be entire., Comment: 22 pages, 2 figures

In this paper we combine elements of the b-calculus and elliptic boundary problems to solve the decomposition problem for the (regularized) �-determinant of the Laplacian on a manifold with cylin- drical end into the �-determinants of the Laplacians with Dirichlet con- ditions on the manifold with boundary and on the half infinite cylinder. We also compute all the contributions to this formula explicitly. We investigate the 'Mayer-Vietoris' or 'cut and paste' decomposition for- mula of the �-determinant for a Laplacian on a manifold with cylindrical end into the �-determinants of the Laplacians with Dirichlet conditions on the manifold with boundary and on the half infinite cylinder. We also introduce a new method to attack such surgery problems by comparing the problem to a corresponding model problem. This approach works for compact manifolds as well as manifolds with cylindrical ends and will be used to solve related decomposition problems for the spectral invariants of Dirac type operators in (12), (13). We remark that the noncompactness of the underlying manifold introduces many new facets and obstacles not found in the compact case, as we will explain later. We begin with a brief account of zeta determinants. The �-determinant of a Laplace type operator was pioneered in the seminal paper (21) by Ray and Singer. They were seeking an analytic version of the so- called Reidemeister torsion, a combinatorial-topological invariant introduced by Reidemeister (22) and Franz (6). They conjectured that their analytic in- variant was the same as the Reidemeister torsion. Later, this conjecture was proved independently by Cheeger (4) and Muller (17). The �-determinants have also been of great use in quantum field theory where they are being used to develop rigorous models for Feynman path integral techniques (10). Because of their use in differential topology and quantum field theory, much work has been done on understanding the nature of�-determinants, especially their behavior under 'cutting and pasting' of manifolds. This was initiated

In this paper we give a new perspective on the Cauchy integral and transform and Hardy spaces for Dirac-type operators on manifolds with corners of codimension two. Instead of considering Banach or Hilbert spaces, we use polyhomogeneous functions on a geometrically “blown-up” version of the manifold called the total boundary blow-up introduced by Mazzeo and Melrose [R.R. Mazzeo, R.B. Melrose, Analytic surgery and the eta invariant, Geom. Funct. Anal. 5 (1) (1995) 14–75]. These polyhomogeneous functions are smooth everywhere on the original manifold except at the corners where they have a “Taylor series” (with possible log terms) in polar coordinates. The main application of our analysis is a complete Fredholm theory for boundary value problems of Dirac operators on manifolds with corners of codimension two.