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Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds
, 2004
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Aspects of locally covariant quantum field theory
, 2008
"... This thesis considers various aspects of locally covariant quantum field theory (see Brunetti et al., Commun. Math. Phys. 237 (2003), 31 68), a mathematical framework to describe axiomatic quantum field theories in curved spacetimes. Chapter 1 argues that the use of morphisms in this framework can b ..."
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This thesis considers various aspects of locally covariant quantum field theory (see Brunetti et al., Commun. Math. Phys. 237 (2003), 31 68), a mathematical framework to describe axiomatic quantum field theories in curved spacetimes. Chapter 1 argues that the use of morphisms in this framework can be seen as a model for modal logic. To our knowledge this is the first interpretative description of this aspect of the framework. Chapter 2 gives an exposition of locally covariant quantum field theory which differs from the original in minor details, notably in the new notion of nowhereclassicality and the sharpened timeslice axiom, which puts a restriction on the state space as well as the algebras. Chapter 3 deals with the wellstudied example of the free real scalar field and includes an elegant proof of the new general result that the commutation relations together with the Hadamard condition on the twopoint distribution of a state completely x the singularity structure of all npoint distributions. Chapter 4 describes the free Dirac field as a locally covariant quantum field, using a new representation independent
Uniqueness results for ill posed characteristic problems in curved spacetimes
, 2007
"... Abstract. We prove two uniqueness theorems concerning linear wave equations; the first theorem is in Minkowski spacetimes, while the second is in the domain of outer communication of a Kerr black hole. Both theorems concern illposed Cauchy problems on bifurcate, characteristic hypersurfaces. In th ..."
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Abstract. We prove two uniqueness theorems concerning linear wave equations; the first theorem is in Minkowski spacetimes, while the second is in the domain of outer communication of a Kerr black hole. Both theorems concern illposed Cauchy problems on bifurcate, characteristic hypersurfaces. In the case of the Kerr spacetime, the hypersurface is precisely the event horizon of the black hole. The uniqueness theorem in this case, based on two Carleman estimates, is intimately connected to our strategy to prove uniqueness of the Kerr black holes among smooth, stationary solutions of the Einsteinvacuum equations, as formulated in [14]. Contents
On the local structure of the KleinGordon field on curved spacetimes
, 2000
"... This paper investigates waveequations on spacetimes with a metric which is locally analytic in the time. We use recent results in the theory of the noncharacteristic Cauchy problem to show that a solution to a waveequation vanishing in an open set vanishes in the “envelope” of this set, which may ..."
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This paper investigates waveequations on spacetimes with a metric which is locally analytic in the time. We use recent results in the theory of the noncharacteristic Cauchy problem to show that a solution to a waveequation vanishing in an open set vanishes in the “envelope” of this set, which may be considerably larger and in the case of timelike tubes may even coincide with the spacetime itself. We apply this result to the real scalar field on a globally hyperbolic spacetime and show that the field algebra of an open set and its envelope coincide. As an example there holds an analog of Borchers ’ timelike tube theorem for such scalar fields and hence, algebras associated with world lines can be explicitly given. Our result applies to cosmologically relevant spacetimes.
On the Determination of Moving Boundaries for Hyperbolic Equations
, 902
"... Abstract. We consider wave equations in domains with timedependent boundaries (moving obstacles) contained in a fixed cylinder for all time. We give sufficient conditions for the determination of the moving boundary from the Cauchy data on part of the boundary of the cylinder. We also study the rel ..."
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Abstract. We consider wave equations in domains with timedependent boundaries (moving obstacles) contained in a fixed cylinder for all time. We give sufficient conditions for the determination of the moving boundary from the Cauchy data on part of the boundary of the cylinder. We also study the related problem of accessibility of the moving boundary by timelike curves from the boundary of the cylinder. In this article we study the possibility of determining a moving boundary from Cauchy data on a stationary boundary. The setting of this problem is as follows. Let Q be an exterior domain in Rn x ×Rt with smooth boundary ∂Q. We assume that the complement of Q is contained in the cylinder C = {(x, t) : x  < R, t ∈ R}, and
Maxwell’s Equations with Scalar Impedance: Direct and Inverse Problems
, 2002
"... Abstract: The article deals with electrodynamics in the presence of anisotropic materials having scalar wave impedance. Maxwell’s equations written for differential forms over a 3manifold are analysed. The system is extended to a Dirac type first order elliptic system on the Grassmannian bundle ove ..."
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Abstract: The article deals with electrodynamics in the presence of anisotropic materials having scalar wave impedance. Maxwell’s equations written for differential forms over a 3manifold are analysed. The system is extended to a Dirac type first order elliptic system on the Grassmannian bundle over the manifold. The second part of the article deals with the dynamical inverse boundary value problem of determining the electromagnetic material parameters from boundary measurements. By using the boundary control method, it is proved that the dynamical boundary data determines the electromagnetic travel time metric as well as the scalar wave impedance on the manifold. This invariant result leads also to a complete characterization of the nonuniqueness of the corresponding inverse problem in bounded domains of R 3.
RADIATION FIELDS ON ASYMPTOTICALLY EUCLIDEAN MANIFOLDS
"... Abstract. F.G. Friedlander introduced the notion of radiation elds for asymptotically Euclidean manifolds. Here we answer some of the questions he proposed and apply the results to give a unitary translation representation of the wave group, and to obtain the scattering matrix for such manifolds. ..."
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Abstract. F.G. Friedlander introduced the notion of radiation elds for asymptotically Euclidean manifolds. Here we answer some of the questions he proposed and apply the results to give a unitary translation representation of the wave group, and to obtain the scattering matrix for such manifolds. We also obtain a support theorem for the radiation elds. 1. Introduction and Statement
An Introduction to the Controllability of Partial Differential Equations
"... These notes are a written abridged version of a course that both authors have ..."
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These notes are a written abridged version of a course that both authors have
New examples of damped wave equations with gradientlike structure
"... Abstract: This article shows how to use perturbation methods to get new examples of evolutionary partial dierential equations with gradientlike structure. In particular, we investigate the case of the wave equation with a variable damping satisfying the geometric control condition only, and the cas ..."
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Abstract: This article shows how to use perturbation methods to get new examples of evolutionary partial dierential equations with gradientlike structure. In particular, we investigate the case of the wave equation with a variable damping satisfying the geometric control condition only, and the case of the wave equation with a damping of indenite sign.