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38
Pseudodifferential Operators on Manifolds with A LIE STRUCTURE AT INFINITY
, 2003
"... Several interesting examples of noncompact manifolds M0 whose geometry at infinity is described by Lie algebras of vector fields V ⊂ Γ(M; T M) (on a compactification of M0 to a manifold with corners M) were studied for instance in [28, 31, 46]. In [1], the geometry of manifolds described by Lie alg ..."
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Cited by 28 (13 self)
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Several interesting examples of noncompact manifolds M0 whose geometry at infinity is described by Lie algebras of vector fields V ⊂ Γ(M; T M) (on a compactification of M0 to a manifold with corners M) were studied for instance in [28, 31, 46]. In [1], the geometry of manifolds described by Lie algebras of vector fields – baptised “manifolds with a Lie structure at infinity ” there – was studied from an axiomatic point of view. In this paper, we define and study the algebra Ψ ∞ 1,0,V (M0), which is an algebra of pseudodifferential operators canonically associated to a manifold M0 with the Lie structure at infinity V ⊂ Γ(M; T M). We show that many of the properties of the usual algebra of pseudodifferential operators on a compact manifold extend to Ψ ∞ 1,0,V (M0). We also consider the algebra Diff ∗ V (M0) of differential operators on M0 generated by V and C ∞ (M), and show that Ψ ∞ 1,0,V (M0) is a “microlocalization” of Diff ∗ V (M0). We also define and study semiclassical and “suspended ” versions of the algebra Ψ ∞ 1,0,V (M0). Thus, our constructions solves a conjecture of Melrose [28].
Determinants Of Laplacians And Isopolar Metrics On Surfaces Of Infinite Area
 DUKE MATH. J
, 2001
"... We construct a determinant of the Laplacian for infinitearea surfaces which are hyperbolic near infinity and without cusps. In the case of a convex cocompact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with ..."
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Cited by 15 (4 self)
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We construct a determinant of the Laplacian for infinitearea surfaces which are hyperbolic near infinity and without cusps. In the case of a convex cocompact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the Laplacian. In the hyperbolic near infinity case the determinant is analyzed through the zetaregularized relative determinant for a conformal metric perturbation. We establish that this relative determinant is a ratio of entire functions of order two with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics. These results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances.
Schrödinger operators with complexvalued potentials and no resonances
 Duke Math Jour
"... Abstract. In dimension d ≥ 3, we give examples of nontrivial, compactly supported, complexvalued potentials such that the associated Schrödinger operators have no resonances. If d = 2, we show that there are potentials with no resonances away from the origin. These Schrödinger operators are isophas ..."
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Cited by 10 (6 self)
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Abstract. In dimension d ≥ 3, we give examples of nontrivial, compactly supported, complexvalued potentials such that the associated Schrödinger operators have no resonances. If d = 2, we show that there are potentials with no resonances away from the origin. These Schrödinger operators are isophasal and have the same scattering phase as the Laplacian on R d. In odd dimensions d ≥ 3 we study the fundamental solution of the wave equation perturbed by such a potential. If the space variables are held fixed, it is superexponentially decaying in time. 1.
GLUING SEMICLASSICAL RESOLVENT ESTIMATES VIA PROPAGATION OF SINGULARITIES
"... Abstract. We use semiclassical propagation of singularities to give a general method for gluing together resolvent estimates. As an application we prove estimates for the analytic continuation of the resolvent of a Schrödinger operator for certain asymptotically hyperbolic manifolds in the presence ..."
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Cited by 10 (5 self)
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Abstract. We use semiclassical propagation of singularities to give a general method for gluing together resolvent estimates. As an application we prove estimates for the analytic continuation of the resolvent of a Schrödinger operator for certain asymptotically hyperbolic manifolds in the presence of trapping which is sufficiently mild in one of several senses. As a corollary we obtain local exponential decay for the wave propagator and local smoothing for the Schrödinger propagator. 1.
On the geometry of Riemannian manifolds with a Lie structure at infinity
, 2003
"... A manifold with a “Lie structure at infinity” is a noncompact manifold M0 whose geometry is described by a compactification to a manifold with corners M and a Lie algebraV of vector fields on M subject to constraints only on M �M0. This definition recovers several classes of noncompact manifolds ..."
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Cited by 8 (7 self)
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A manifold with a “Lie structure at infinity” is a noncompact manifold M0 whose geometry is described by a compactification to a manifold with corners M and a Lie algebraV of vector fields on M subject to constraints only on M �M0. This definition recovers several classes of noncompact manifolds that were studied before: manifolds with cylindrical ends, manifolds that are Euclidean at infinity, conformally compact manifolds, and others. It hence provides a unified setting for the study of these classes of manifolds and of their geometric differential operators. The Lie structure at infinity on M0 determines a complete metric on M0 up to biLipschitz equivalence. This leads to the natural problem of understanding the Riemannian geometry of these manifolds, which is the main question addressed in this paper. We prove, for example, that on a manifold with a Lie structure at infinity the curvature tensor and its covariant derivatives are bounded, by extending the LeviCivita connection to an A ∗valued connection where the bundle A is uniquely determined by the Lie algebra V. We study a generalization of the geodesic spray
Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds
, 2004
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GROUPOIDS AND AN INDEX THEOREM FOR CONICAL PSEUDOMANIFOLDS
, 2006
"... Abstract. 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 noncommut ..."
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Cited by 7 (3 self)
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Abstract. 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. Contents
Weighted Sobolev spaces and regularity for polyhedral domains
, 2006
"... We prove a regularity result for the Poisson problem −∆u = f, u∂P = g on a polyhedral domain P ⊂ R3 using the Babuˇska–Kondratiev spaces Km a (P). These are weighted Sobolev spaces in which the weight is given by the distance to the set of edges [4, 29]. In particular, we show that there is no loss ..."
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Cited by 7 (5 self)
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We prove a regularity result for the Poisson problem −∆u = f, u∂P = g on a polyhedral domain P ⊂ R3 using the Babuˇska–Kondratiev spaces Km a (P). These are weighted Sobolev spaces in which the weight is given by the distance to the set of edges [4, 29]. In particular, we show that there is no loss of Km a –regularity for solutions of strongly elliptic systems with smooth coefficients. We also establish a “trace theorem ” for the restriction to the boundary of the functions in Km a (P).
Boundary value problems and regularity on polyhedral domains
, 2004
"... We prove a wellposedness result for second order boundary value problems in weighted Sobolev spaces on curvilinear polyhedral domains in R^n with Dirichlet boundary conditions. Our typical weight is the distance to the set of singular boundary points. ..."
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Cited by 5 (5 self)
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We prove a wellposedness result for second order boundary value problems in weighted Sobolev spaces on curvilinear polyhedral domains in R^n with Dirichlet boundary conditions. Our typical weight is the distance to the set of singular boundary points.