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209
The Cauchy Problem for Einstein Equations
 in B.G.Schmidt (ed.), Einstein’s Field Equations and their Physical Interpretation
, 2000
"... Abstract. Various aspects of the Cauchy problem for the Einstein equations are surveyed, with the emphasis on local solutions of the evolution equations. Particular attention is payed to giving a clear explanation of conceptual issues which arise in this context. The question of producing reduced sy ..."
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Cited by 33 (7 self)
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Abstract. Various aspects of the Cauchy problem for the Einstein equations are surveyed, with the emphasis on local solutions of the evolution equations. Particular attention is payed to giving a clear explanation of conceptual issues which arise in this context. The question of producing reduced systems of equations which are hyperbolic is examined in detail and some new results on that subject are presented. Relevant background from the theory of partial differential equations is also explained at some length. 1
Compression Techniques for Boundary Integral Equations  Optimal Complexity Estimates
, 2002
"... In this paper matrix compression techniques in the context of wavelet Galerkin schemes for boundary integral equations are developed and analyzed that exhibit optimal complexity in the following sense. The fully discrete scheme produces approximate solutions within discretization error accuracy offe ..."
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Cited by 30 (8 self)
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In this paper matrix compression techniques in the context of wavelet Galerkin schemes for boundary integral equations are developed and analyzed that exhibit optimal complexity in the following sense. The fully discrete scheme produces approximate solutions within discretization error accuracy offered by the underlying Galerkin method at a computational expense that is proven to stay proportional to the number of unknowns. Key issues are the second compression, that reduces the near field complexity significantly, and an additional aposteriori compression. The latter one is based on a general result concerning an optimal work balance, that applies, in particular, to the quadrature used to compute the compressed stiffness matrix with sufficient accuracy in linear time. The theoretical results are illustrated by a 3D example on a nontrivial domain.
Semiclassical Nonconcentration near Hyperbolic Orbits
"... Abstract. For a large class of semiclassical pseudodifferential operators, including Schrödinger operators, P(h) = −h 2 ∆g + V (x), on compact Riemannian manifolds, we give logarithmic lower bounds on the mass of eigenfunctions outside neighbourhoods of generic closed hyperbolic orbits. More precis ..."
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Cited by 29 (7 self)
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Abstract. For a large class of semiclassical pseudodifferential operators, including Schrödinger operators, P(h) = −h 2 ∆g + V (x), on compact Riemannian manifolds, we give logarithmic lower bounds on the mass of eigenfunctions outside neighbourhoods of generic closed hyperbolic orbits. More precisely we show that if A is a pseudodifferential operator which is microlocally equal to the identity near the hyperbolic orbit and microlocally zero away from the orbit, then ‖u ‖ ≤ C ( √ log(1/h)/h)‖P(h)u ‖ + C √ log(1/h)‖(I − A)u ‖. This generalizes earlier estimates of Colin de VerdièreParisse [CVP] obtained for a special case, and of BurqZworski [BuZw] for real hyperbolic orbits. 1.
Microlocal spectrum condition and Hadamard form . . .
, 2000
"... The characterization of Hadamard states in terms of a specific form of the wavefront set of their twopoint functions has been developed some years ago by Radzikowski for scalar fields on a fourdimensional globally hyperbolic spacetime, and initiated a major progress in the understanding of Hadam ..."
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Cited by 22 (4 self)
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The characterization of Hadamard states in terms of a specific form of the wavefront set of their twopoint functions has been developed some years ago by Radzikowski for scalar fields on a fourdimensional globally hyperbolic spacetime, and initiated a major progress in the understanding of Hadamard states and the further development of quantum field theory in curved spacetime. In the present work, the characterization of Hadamard states through a particular form of the wavefront set of their twopoint functions will be generalized from scalar fields to vector fields (sections in a vector bundle) which are subject to a waveequation and are quantized so as to fulfill the covariant canonical commutation relations, or which obey a Dirac equation and are quantized according to the covariant anticommutation relations, in any globally hyperbolic spacetime having dimension three or higher. In proving this result, a gap which is present in the published proof for the scalar field case will be removed. Moreover we determine the shortdistance scaling limits of Hadamard states for vectorbundle valued fields, finding them to coincide with the corresponding flatspace, massless vacuum states.
Adiabatic vacuum states on general spacetime manifolds: Defintion, construction, and physical properties
 ANN HENRI POINCARÉ
, 2002
"... Adiabatic vacuum states are a wellknown class of physical states for linear quantum fields on RobertsonWalker spacetimes. We extend the definition of adiabatic vacua to general spacetime manifolds by using the notion of the Sobolev wavefront set. This definition is also applicable to interacting f ..."
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Cited by 21 (0 self)
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Adiabatic vacuum states are a wellknown class of physical states for linear quantum fields on RobertsonWalker spacetimes. We extend the definition of adiabatic vacua to general spacetime manifolds by using the notion of the Sobolev wavefront set. This definition is also applicable to interacting field theories. Hadamard states form a special subclass of the adiabatic vacua. We analyze physical properties of adiabatic vacuum representations of the KleinGordon field on globally hyperbolic spacetime manifolds (factoriality, quasiequivalence, local definiteness, Haag duality) and construct them explicitly, if the manifold has a compact Cauchy surface.
Multiwavelets for Second Kind Integral Equations
, 1997
"... We consider a Galerkin method for an elliptic pseudodifferential operator of order zero on a twodimensional manifold. We use piecewise linear discontinuous trial functions on a triangular mesh and describe an orthonormal wavelet basis. Using this basis we can compress the stiffness matrix from N² t ..."
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Cited by 21 (12 self)
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We consider a Galerkin method for an elliptic pseudodifferential operator of order zero on a twodimensional manifold. We use piecewise linear discontinuous trial functions on a triangular mesh and describe an orthonormal wavelet basis. Using this basis we can compress the stiffness matrix from N² to O(N log N) nonzero entries and still obtain (up to log N terms) the same convergence rates as for the exact Galerkin method.
The Convergence of Spline Collocation for Strongly Elliptic Equations on Curves
, 1985
"... Most boundary element methods for twodimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions. Here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly grade ..."
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Cited by 21 (2 self)
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Most boundary element methods for twodimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions. Here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order. The equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines. The crucial assumption for the generalized boundary integral and integrodifferential operators is strong ellipticity. Our analysis is based on simple Fourier expansions. In particular, we extend results by J. Saranen and W.L. Wendland from constant to variable coefficient equations. Our results include the first convergence proof of midpoint collocation with piecewise constant functions, i.e., the panel method for solving systems of Cauchy singular integral equations.
On the asymptotic expansion of Bergman kernel
"... Abstract. We study the asymptotic of the Bergman kernel of the spin c Dirac operator on high tensor powers of a line bundle. 1. ..."
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Cited by 18 (7 self)
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Abstract. We study the asymptotic of the Bergman kernel of the spin c Dirac operator on high tensor powers of a line bundle. 1.
The Hadamard condition for Dirac fields and adiabatic states on RobertsonWalker spacetimes
 Commun. Math. Phys
, 2001
"... We characterise the homogeneous and isotropic gauge invariant and quasifree states for free Dirac quantum fields on RobertsonWalker spacetimes in any even dimension. Using this characterisation, we construct adiabatic vacuum states of order n corresponding to some Cauchy surface. We then show that ..."
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Cited by 16 (2 self)
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We characterise the homogeneous and isotropic gauge invariant and quasifree states for free Dirac quantum fields on RobertsonWalker spacetimes in any even dimension. Using this characterisation, we construct adiabatic vacuum states of order n corresponding to some Cauchy surface. We then show that any two such states (of sufficiently high order) are locally quasiequivalent. We propose a microlocal version of the Hadamard condition for spinor fields on arbitrary spacetimes, which is shown to entail the usual short distance behaviour of the twopoint function. The polarisation set of these twopoint functions is determined from the Dencker connection of the spinorial KleinGordon operator which we show to be equals the (pullback) of the spin connection. Finally it is demonstrated that adiabatic states of infinite order are Hadamard, and that those of order n correspond, in some sense, to a truncated Hadamard series and will therefore allow for a point splitting renormalisation of the expected stressenergy tensor.