29,306 research outputs found
On the convergence of some products of Fourier integral operators
An approximation Ansatz for the operator solution, , of a hyperbolic first-order pseudodifferential equation, \d_z + a(z,x,D_x) with , is constructed as the composition of global Fourier integral operators with complex phases. We prove a convergence result for the Ansatz to in some Sobolev space as the number of operators in the composition goes to , with a convergence of order , if the symbol is in \Con^{0,\alpha} with respect to the evolution parameter . We also study the consequences of some truncation approximations of the symbol in the construction of the Ansatz
Twisted convolution and Moyal star product of generalized functions
We consider nuclear function spaces on which the Weyl-Heisenberg group acts
continuously and study the basic properties of the twisted convolution product
of the functions with the dual space elements. The final theorem characterizes
the corresponding algebra of convolution multipliers and shows that it contains
all sufficiently rapidly decreasing functionals in the dual space.
Consequently, we obtain a general description of the Moyal multiplier algebra
of the Fourier-transformed space. The results extend the Weyl symbol calculus
beyond the traditional framework of tempered distributions.Comment: LaTeX, 16 pages, no figure
Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U_q(sl(2,R))
The decomposition of tensor products of representations into irreducibles is
studied for a continuous family of integrable operator representations of
. It is described by an explicit integral transformation involving
a distributional kernel that can be seen as an analogue of the Clebsch-Gordan
coefficients. Moreover, we also study the relation between two canonical
decompositions of triple tensor products into irreducibles. It can be
represented by an integral transformation with a kernel that generalizes the
Racah-Wigner coefficients. This kernel is explicitly calculated.Comment: 39 pages, AMS-Latex; V2: Added comments and references concerning
relation to Faddeev's modular double, minor corrections, version to be
published in CM
Schatten classes on compact manifolds: Kernel conditions
In this paper we give criteria on integral kernels ensuring that integral
operators on compact manifolds belong to Schatten classes. A specific test for
nuclearity is established as well as the corresponding trace formulae. In the
special case of compact Lie groups, kernel criteria in terms of (locally and
globally) hypoelliptic operators are also given.Comment: 22 page
Shocks, Superconvergence, and a Stringy Equivalence Principle
We study propagation of a probe particle through a series of closely situated
gravitational shocks. We argue that in any UV-complete theory of gravity the
result does not depend on the shock ordering - in other words, coincident
gravitational shocks commute. Shock commutativity leads to nontrivial
constraints on low-energy effective theories. In particular, it excludes
non-minimal gravitational couplings unless extra degrees of freedom are
judiciously added. In flat space, these constraints are encoded in the
vanishing of a certain "superconvergence sum rule." In AdS, shock commutativity
becomes the statement that average null energy (ANEC) operators commute in the
dual CFT. We prove commutativity of ANEC operators in any unitary CFT and
establish sufficient conditions for commutativity of more general light-ray
operators. Superconvergence sum rules on CFT data can be obtained by inserting
complete sets of states between light-ray operators. In a planar 4d CFT, these
sum rules express (a-c)/c in terms of the OPE data of single-trace operators.Comment: 93 pages plus appendice
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