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BOUNDARY VALUE PROBLEMS AND LAYER POTENTIALS ON MANIFOLDS WITH CYLINDRICAL ENDS
, 2002
"... We extend the method of layer potentials to manifolds with boundary and cylindrical ends. To obtain this extension along the classical lines, we have to deal with several technical difficulties due to the noncompactness of the boundary, which prevents us from using the standard characterization of F ..."
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We extend the method of layer potentials to manifolds with boundary and cylindrical ends. To obtain this extension along the classical lines, we have to deal with several technical difficulties due to the noncompactness of the boundary, which prevents us from using the standard characterization of Fredholm and compact (pseudo)differential operators between Sobolev spaces. Our approach, which involves the study of layer potentials depending on a parameter on compact manifolds as an intermediate step, yields the invertibility of the relevant boundary integral operators in the global, noncompact setting, which is rather unexpected. As an application, we prove the wellposedness of the nonhomogeneous Dirichlet problem on manifolds with boundary and cylindrical ends. We also prove the existence of the DirichlettoNeumann map, which we show to be a pseudodifferential operator in the calculus of pseudodifferential operators that are “almost translation invariant at infinity,” a calculus that we study in this paper. The proof of the convergence of the layer potentials and of the existence of the DirichlettoNeumann map are based on a good understanding of resolvents of elliptic operators that are translation invariant at infinity.
On Some Analytical Index Formulas Related To OperatorValued Symbols
"... . For several classes of pseudodifferential operators with operatorvalued symbol analytic index formulas are found. The common feature is that usual index formulas are not valid for these operators. Applications are given to pseudodifferential operators on singular manifolds. 1. Introduction A ..."
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. For several classes of pseudodifferential operators with operatorvalued symbol analytic index formulas are found. The common feature is that usual index formulas are not valid for these operators. Applications are given to pseudodifferential operators on singular manifolds. 1. Introduction Analytical index formulas play an important part in the study of topological characteristics of elliptic operators. They complement index formulas expressed in topological and algebraical terms, and often enter in these formulas as an ingredient. For elliptic pseudodifferential operators on compact manifolds, such formulas were found by Fedosov [F1]; later, analytical index formulas for elliptic boundary value problems were obtained in [F2]. These formulas have a common feature: they involve an integral, with integrand containing analytical expressions for the classical characteristic classes entering into the cohomological formulas. In 90s a systematic study started of topological charact...
On the noncommutative spectral flow
 2007) 135–187. MR2333742 (2008e:58032), Zbl 1136.58014
"... We define and study the noncommutative spectral flow for paths of regular selfadjoint Fredholm operators on a Hilbert C ∗module. We give an axiomatic description and discuss some applications. One of them is the definition of a noncommutative Maslov index for paths of Lagrangians, which appears in ..."
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We define and study the noncommutative spectral flow for paths of regular selfadjoint Fredholm operators on a Hilbert C ∗module. We give an axiomatic description and discuss some applications. One of them is the definition of a noncommutative Maslov index for paths of Lagrangians, which appears in a splitting formula for the spectral flow. Analogously we study the spectral flow for odd operators on a Z/2graded module. MSC 2000: 58J30 (19K56; 46L80) 1
Adiabatic limits of eta and zeta functions of elliptic operators
 MATHEMATISCHE ZEITSCHRIFT
, 2008
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INDEX AND HOMOLOGY OF PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH BOUNDARY
"... Abstract. We prove a local index formula for cusppseudodifferential operators on a manifold with boundary. This is known to be equivalent to an index formula for manifolds with cylindrical ends, and hence we obtain a new proof of the classical AtiyahPatodiSinger index theorem for Dirac operators ..."
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Abstract. We prove a local index formula for cusppseudodifferential operators on a manifold with boundary. This is known to be equivalent to an index formula for manifolds with cylindrical ends, and hence we obtain a new proof of the classical AtiyahPatodiSinger index theorem for Dirac operators on manifolds with boundary, as well as an extension of Melrose’s bindex theorem. Our approach is based on an unpublished paper by Melrose and Nistor “Homology of pseudodifferential operators I. Manifolds with boundary ” [39]. We therefore take the opportunity to review some of the results from that paper from the perspective of subsequent research on the Hochschild and cyclic homologies of algebras of pseudodifferential operators and of their applications to index theory.
The spectrum of Schrödinger operators and Hodge Laplacians on conformally cusp manifolds
"... We describe the spectrum of the kform Laplacian on conformally cusp Riemannian manifolds. The essential spectrum is shown to vanish precisely when the k and k − 1 de Rham cohomology groups of the boundary vanish. We give Weyltype asymptotics for the eigenvaluecounting function in the purely disc ..."
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We describe the spectrum of the kform Laplacian on conformally cusp Riemannian manifolds. The essential spectrum is shown to vanish precisely when the k and k − 1 de Rham cohomology groups of the boundary vanish. We give Weyltype asymptotics for the eigenvaluecounting function in the purely discrete case. In the other case we analyze the essential spectrum via positive commutator methods and establish a limiting absorption principle. This implies the absence of the singular spectrum for a wide class of metrics. We also exhibit a class of potentials V such that the Schrödinger operator has compact resolvent, although V tends to −∞ in most of the infinity. We correct a statement from the literature regarding the essential spectrum of the Laplacian on forms on hyperbolic manifolds of finite volume, and we propose a conjecture about the existence of such manifolds in dimension four whose cusps are rational homology spheres.
ETA FORMS AND THE ODD PSEUDODIFFERENTIAL FAMILIES INDEX
"... Abstract. Let A(t) be an elliptic, producttype suspended (which is to say parameterdependant in a symbolic way) family of pseudodifferential operators on the fibres of a fibration φ with base Y. The standard example is A + it where A is a family, in the usual sense, of first order, selfadjoint an ..."
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Abstract. Let A(t) be an elliptic, producttype suspended (which is to say parameterdependant in a symbolic way) family of pseudodifferential operators on the fibres of a fibration φ with base Y. The standard example is A + it where A is a family, in the usual sense, of first order, selfadjoint and elliptic pseudodifferential operators and t ∈ R is the ‘suspending ’ parameter. Let πA: A(φ) − → Y be the infinitedimensional bundle with fibre at y ∈ Y consisting of the Schwartzsmoothing perturbations, q, making Ay(t) + q(t) invertible for all t ∈ R. The total eta form, ηA, as described here, is an even form on A(φ) which has basic differential which is an explicit representative of the odd Chern character of the index of the family: (*) dηA = π ∗ A γA, Ch(ind(A)) = [γA] ∈ H odd (Y). The 1form part of this identity may be interpreted in terms of the τ invariant (exponentiated eta invariant) as the determinant of the family. The 2form part of the eta form may be interpreted as a Bfield on the Ktheory gerbe for the family A with (*) giving the ‘curving ’ as the 3form part of the Chern character of the index. We also give ‘universal ’ versions of these constructions over a classifying space for odd Ktheory.
Gravitational and axial anomalies for generalized Euclidean TaubNUT metrics
 J. Phys. A – Math. Gen
"... The gravitational anomalies are investigated for generalized Euclidean TaubNUT metrics which admit hidden symmetries analogous to the RungeLenz vector of the Keplertype problem. In order to evaluate the axial anomalies, the index of the Dirac operator for these metrics with the APS boundary condi ..."
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The gravitational anomalies are investigated for generalized Euclidean TaubNUT metrics which admit hidden symmetries analogous to the RungeLenz vector of the Keplertype problem. In order to evaluate the axial anomalies, the index of the Dirac operator for these metrics with the APS boundary condition is computed. The role of the KillingYano tensors is discussed for these two types of quantum anomalies. Pacs: 04.62.+v 1
Periodicity and the determinant bundle
 Comm. Math. Phys
"... Abstract. The infinite matrix ‘Schwartz ’ group G− ∞ is a classifying group for odd Ktheory and carries Chern classes in each odd dimension, generating the cohomology. These classes are closely related to the Fredholm determinant on G−∞. We show that while the higher (even, Schwartz) loop groups of ..."
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Abstract. The infinite matrix ‘Schwartz ’ group G− ∞ is a classifying group for odd Ktheory and carries Chern classes in each odd dimension, generating the cohomology. These classes are closely related to the Fredholm determinant on G−∞. We show that while the higher (even, Schwartz) loop groups of G−∞, again classifying for odd Ktheory, do not carry multiplicative determinants generating the first Chern class, ‘dressed ’ extensions, corresponding to a star product, do carry such functions. We use these to discuss Bott periodicity for the determinant bundle and the eta invariant. In so doing we relate two distinct extensions of the eta invariant, to selfadjoint elliptic operators and to elliptic invertible suspended families and show that the corresponding τ