865 research outputs found
Conformal classes of asymptotically flat, static vacuum data
We show that time-reflection symmetric, asymptotically flat, static vacuum
data which admit a non-trivial conformal rescaling which leads again to such
data must be axi-symmetric and admit a conformal Killing field. Moreover, it is
shown that there exists a 3-parameter family of such data.Comment: 23 page
On Vector Bundles of Finite Order
We study growth of holomorphic vector bundles E over smooth affine manifolds.
We define Finsler metrics of finite order on E by estimates on the holomorphic
bisectional curvature. These estimates are very similar to the ones used by
Griffiths and Cornalba to define Hermitian metrics of finite order. We then
generalize the Vanishing Theorem of Griffiths and Cornalba to the Finsler
context. We develop a value distribution theory for holomorphic maps from the
projectivization of E to projective space. We show that the projectivization of
E can be immersed into a projective space of sufficiently large dimension via a
map of finite order.Comment: version 2 has some typos corrected; to appear in Manuscripta
Mathematic
A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation
We give a short proof of asymptotic completeness and global existence for the
cubic Nonlinear Klein-Gordon equation in one dimension. Our approach to dealing
with the long range behavior of the asymptotic solution is by reducing it, in
hyperbolic coordinates to the study of an ODE. Similar arguments extend to
higher dimensions and other long range type nonlinear problems.Comment: To appear in Lett. Math. Phy
A Unique Continuation Result for Klein-Gordon Bisolutions on a 2-dimensional Cylinder
We prove a novel unique continuation result for weak bisolutions to the
massive Klein-Gordon equation on a 2-dimensional cylinder M. Namely, if such a
bisolution vanishes in a neighbourhood of a `sufficiently large' portion of a
2-dimensional surface lying parallel to the diagonal in the product manifold of
M with itself, then it is (globally) translationally invariant. The proof makes
use of methods drawn from Beurling's theory of interpolation. An application of
our result to quantum field theory on 2-dimensional cylinder spacetimes will
appear elsewhere.Comment: LaTeX2e, 9 page
Simple Non Linear Klein-Gordon Equations in 2 space dimensions, with long range scattering
We establish that solutions, to the most simple NLKG equations in 2 space
dimensions with mass resonance, exhibits long range scattering phenomena.
Modified wave operators and solutions are constructed for these equations. We
also show that the modified wave operators can be chosen such that they
linearize the non-linear representation of the Poincar\'e group defined by the
NLKG.Comment: 19 pages, LaTeX, To appear in Lett. Math. Phy
Strongly hyperbolic second order Einstein's evolution equations
BSSN-type evolution equations are discussed. The name refers to the
Baumgarte, Shapiro, Shibata, and Nakamura version of the Einstein evolution
equations, without introducing the conformal-traceless decomposition but
keeping the three connection functions and including a densitized lapse. It is
proved that a pseudo-differential first order reduction of these equations is
strongly hyperbolic. In the same way, densitized Arnowitt-Deser-Misner
evolution equations are found to be weakly hyperbolic. In both cases, the
positive densitized lapse function and the spacelike shift vector are arbitrary
given fields. This first order pseudodifferential reduction adds no extra
equations to the system and so no extra constraints.Comment: LaTeX, 16 pages, uses revtex4. Referee corections and new appendix
added. English grammar improved; typos correcte
Quantum ergodicity for restrictions to hypersurfaces
Quantum ergodicity theorem states that for quantum systems with ergodic
classical flows, eigenstates are, in average, uniformly distributed on energy
surfaces. We show that if N is a hypersurface in the position space satisfying
a simple dynamical condition, the restrictions of eigenstates to N are also
quantum ergodic.Comment: 22 pages, 1 figure; revised according to referee's comments. To
appear in Nonlinearit
Regularity properties of distributions through sequences of functions
We give necessary and sufficient criteria for a distribution to be smooth or
uniformly H\"{o}lder continuous in terms of approximation sequences by smooth
functions; in particular, in terms of those arising as regularizations
.Comment: 10 page
An absolute quantum energy inequality for the Dirac field in curved spacetime
Quantum Weak Energy Inequalities (QWEIs) are results which limit the extent
to which the smeared renormalised energy density of a quantum field can be
negative. On globally hyperbolic spacetimes the massive quantum Dirac field is
known to obey a QWEI in terms of a reference state chosen arbitrarily from the
class of Hadamard states; however, there exist spacetimes of interest on which
state-dependent bounds cannot be evaluated. In this paper we prove the first
QWEI for the massive quantum Dirac field on four dimensional globally
hyperbolic spacetime in which the bound depends only on the local geometry;
such a QWEI is known as an absolute QWEI
A general worldline quantum inequality
Worldline quantum inequalities provide lower bounds on weighted averages of
the renormalised energy density of a quantum field along the worldline of an
observer. In the context of real, linear scalar field theory on an arbitrary
globally hyperbolic spacetime, we establish a worldline quantum inequality on
the normal ordered energy density, valid for arbitrary smooth timelike
trajectories of the observer, arbitrary smooth compactly supported weight
functions and arbitrary Hadamard quantum states. Normal ordering is performed
relative to an arbitrary choice of Hadamard reference state. The inequality
obtained generalises a previous result derived for static trajectories in a
static spacetime. The underlying argument is straightforward and is made
rigorous using the techniques of microlocal analysis. In particular, an
important role is played by the characterisation of Hadamard states in terms of
the microlocal spectral condition. We also give a compact form of our result
for stationary trajectories in a stationary spacetime.Comment: 19pp, LaTeX2e. The statement of the main result is changed slightly.
Several typos fixed, references added. To appear in Class Quantum Gra
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