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41
Homology of pseudodifferential operators on manifolds with corners I. Manifolds with boundary
, 1996
"... Respectfully dedicate to Professor M. Sato on the occasion of his 70th birthday Abstract. Let X be a compact manifold with boundary. Suppose that the boundary is fibred, φ: ∂X − → Y, and let x ∈ C ∞ (X) be a boundary defining function. This data fixes the space of ‘fibred cusp ’ vector fields, consi ..."
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Cited by 137 (26 self)
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Respectfully dedicate to Professor M. Sato on the occasion of his 70th birthday Abstract. Let X be a compact manifold with boundary. Suppose that the boundary is fibred, φ: ∂X − → Y, and let x ∈ C ∞ (X) be a boundary defining function. This data fixes the space of ‘fibred cusp ’ vector fields, consisting of those vector fields V on X satisfying V x = O(x 2) and which are tangent to the fibres of φ; it is a Lie algebra and C ∞ (X) module. This Lie algebra is quantized to the ‘small calculus ’ of pseudodifferential operators Ψ ∗ Φ (X). Mapping properties including boundedness, regularity, Fredholm condition and symbolic maps are discussed for this calculus. The spectrum of the Laplacian of an ‘exact fibred cusp ’ metric is analyzed as is the wavefront set associated to the calculus.
Traces on algebras of parameter dependent pseudodifferential operators and the eta–invariant
, 1999
"... ..."
Groupoids and an index theorem for conical pseudomanifolds
, 2006
"... We define an analytical index map and a topological index map for conical pseudomanifolds. These constructions generalize the analogous constructions used by Atiyah and Singer in the proof of their topological index theorem for a smooth, compact manifold M. A main ingredient is a noncommutative alge ..."
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Cited by 16 (5 self)
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We define an analytical index map and a topological index map for conical pseudomanifolds. These constructions generalize the analogous constructions used by Atiyah and Singer in the proof of their topological index theorem for a smooth, compact manifold M. A main ingredient is a noncommutative algebra that plays in our setting the role of C0(T ∗ M). We prove a Thom isomorphism between noncommutative algebras which gives a new example of wrong way functoriality in Ktheory. We then give a new proof of the AtiyahSinger index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds. We thus prove a topological index theorem for conical pseudomanifolds.
Trace functionals for a class of pseudodifferential operators in
 R n ”, Math. Phys. Anal. Geom
"... Abstract. In this paper we define trace functionals on the algebra of pseudodifferential operators with coneshaped exits to infinity. Furthermore, we improve the Weyl formula on the asymptotic distribution of eigenvalues and make use of it in order to establish inclusion relations between the inte ..."
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Cited by 14 (1 self)
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Abstract. In this paper we define trace functionals on the algebra of pseudodifferential operators with coneshaped exits to infinity. Furthermore, we improve the Weyl formula on the asymptotic distribution of eigenvalues and make use of it in order to establish inclusion relations between the interpolation normed ideals of compact operators in L2(Rn) and the above operator classes.
An index formula on manifolds with fibered cusp ends
"... Abstract. We consider a compact manifold X whose boundary is a locally trivial fiber bundle, and an associated pseudodifferential algebra that models fibered cusps at infinity. Using tracelike functionals that generate the 0dimensional Hochschild cohomology groups we first express the index of a f ..."
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Cited by 11 (6 self)
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Abstract. We consider a compact manifold X whose boundary is a locally trivial fiber bundle, and an associated pseudodifferential algebra that models fibered cusps at infinity. Using tracelike functionals that generate the 0dimensional Hochschild cohomology groups we first express the index of a fully elliptic fibered cusp operator as the sum of a local contribution from the interior of X answer to the index problem formulated in [11]. We give a more precise answer for firstorder differential operators when the base of the boundary fiber bundle is S1. In particular, for Dirac operators associated to a metric of the form gX = dx2 x4 + dθ2 x2 + gF near ∂X = {x = 0} with twisting bundle T we obtain index(A) =
Weyl laws on open manifolds
 Math. Annalen
"... Abstract. Under suitable invertibility hypothesis, the spectrum of the Dirac operator on certain open spin Riemannian manifolds is discrete, and obeys a growth law depending qualitatively on the (in)finiteness of the volume. 1. ..."
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Cited by 10 (5 self)
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Abstract. Under suitable invertibility hypothesis, the spectrum of the Dirac operator on certain open spin Riemannian manifolds is discrete, and obeys a growth law depending qualitatively on the (in)finiteness of the volume. 1.
Noncommutative residue, Dixmier’s trace, and heat trace expansions on manifolds with boundary
, 1999
"... For manifolds with boundary, we define an extension of Wodzicki’s noncommutative residue to boundary value problems in Boutet de Monvel’s calculus. We show that this residue can be recovered with the help of heat kernel expansions and explore its relation to Dixmier’s trace. ..."
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Cited by 10 (0 self)
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For manifolds with boundary, we define an extension of Wodzicki’s noncommutative residue to boundary value problems in Boutet de Monvel’s calculus. We show that this residue can be recovered with the help of heat kernel expansions and explore its relation to Dixmier’s trace.
RELATIVE PAIRING IN CYCLIC COHOMOLOGY AND DIVISOR FLOWS
, 2006
"... We show that Melrose’s divisor flow and its generalizations by Lesch and Pflaum are invariants of Ktheory classes for algebras of parametric pseudodifferential operators on a closed manifold. These invariants are obtained by means of pairing the relative Ktheory modulo the symbols with the cyclic ..."
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Cited by 8 (1 self)
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We show that Melrose’s divisor flow and its generalizations by Lesch and Pflaum are invariants of Ktheory classes for algebras of parametric pseudodifferential operators on a closed manifold. These invariants are obtained by means of pairing the relative Ktheory modulo the symbols with the cyclic cohomological characters of odd relative cycles, constructed out of the regularized operator trace together with its symbolic boundary. This representation gives a clear and conceptual explanation to all the essential features of the divisor flow – its homotopy nature, additivity and integrality. It also provides a cohomological formula for the spectral flow along a smooth path of selfadjoint elliptic first order differential operators, between any two invertible such operators on a closed manifold.
Pseudodifferential operators and regularized traces
, 2009
"... This is a survey on trace constructions on various operator algebras with an emphasis on regularized traces on algebras of pseudodifferential operators. For motivation our point of departure is the classical Hilbert space trace which is the unique semifinite normal trace on the algebra of bounded ..."
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Cited by 7 (0 self)
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This is a survey on trace constructions on various operator algebras with an emphasis on regularized traces on algebras of pseudodifferential operators. For motivation our point of departure is the classical Hilbert space trace which is the unique semifinite normal trace on the algebra of bounded operators on a separable Hilbert space. Dropping the normality assumption leads to the celebrated Dixmier traces. Then we give a leisurely introduction to pseudodifferential operators. The parameter dependent calculus is emphasized and it is shown how this calculus leads naturally to the asymptotic expansion of the resolvent trace of an elliptic differential operator. The Hadamard partie finie regularization of an integral is explained and used to extend the Hilbert space trace to the KontsevichVishik canonical trace on pseudodifferential operators of non–integral order. Then the stage is well prepared for the residue trace of WodzickiGuillemin and its purely functional analytic interpretation as a Dixmier trace by Alain Connes. We also discuss existence and uniqueness of traces for the algebra of parameter dependent pseudodifferential operators; the results are surprisingly different. Finally, we will discuss the analogue of the regularized traces on the symbolic level and study the de Rham cohomology of R n with coefficients being symbol functions. This generalizes a recent result of S. Paycha concerning the characterization of the Hadamard partie finie integral and the residue integral in light of the Stokes property.