1,172 research outputs found
Regularity Theory and Superalgebraic Solvers for Wire Antenna Problems
We consider the problem of evaluating the current distribution that is induced on a straight wire antenna by a time-harmonic incident electromagnetic field. The scope of this paper is twofold. One of its main contributions is a regularity proof for a straight wire occupying the interval . In particular, for a smooth time-harmonic incident field this theorem implies that , where is an infinitely differentiable function—the previous state of the art in this regard placed in the Sobolev space , . The second focus of this work is on numerics: we present three superalgebraically convergent algorithms for the solution of wire problems, two based on Hallén's integral equation and one based on the Pocklington integrodifferential equation. Both our proof and our algorithms are based on two main elements: (1) a new decomposition of the kernel of the form , where and are analytic functions on the real line; and (2) removal of the end-point square root singularities by means of a coordinate transformation. The Hallén- and Pocklington-based algorithms we propose converge superalgebraically: faster than and for any positive integer , where and are the numbers of unknowns and the number of integration points required for construction of the discretized operator, respectively. In previous studies, at most the leading-order contribution to the logarithmic singular term was extracted from the kernel and treated analytically, the higher-order singular derivatives were left untreated, and the resulting integration methods for the kernel exhibit convergence at best. A rather comprehensive set of tests we consider shows that, in many cases, to achieve a given accuracy, the numbers of unknowns required by our codes are up to a factor of five times smaller than those required by the best solvers previously available; the required number of integration points, in turn, can be several orders of magnitude smaller than those required in previous methods. In particular, four-digit solutions were found in computational times of the order of four seconds and, in most cases, of the order of a fraction of a second on a contemporary personal computer; much higher accuracies result in very small additional computing times
The State Space of Perturbative Quantum Field Theory in Curved Spacetimes
The space of continuous states of perturbative interacting quantum field
theories in globally hyperbolic curved spacetimes is determined. Following
Brunetti and Fredenhagen, we first define an abstract algebra of observables
which contains the Wick-polynomials of the free field as well as their
time-ordered products, and hence, by the well-known rules of perturbative
quantum field theory, also the observables (up to finite order) of interest for
the interacting quantum field theory. We then determine the space of continuous
states on this algebra. Our result is that this space consists precisely of
those states whose truncated n-point functions of the free field are smooth for
all n not equal to two, and whose two-point function has the singularity of a
Hadamard fundamental form. A crucial role in our analysis is played by the
positivity property of states. On the technical side, our proof involves
functional analytic methods, in particular the methods of microlocal analysis.Comment: 24 pages, Latex file, no figure
Plancherel formula for Berezin deformation of on Riemannian symmetric space
Consider the space B of complex matrces with norm <1. There
exists a standard one-parameter family of unitary representations of the
pseudounitary group U(p,q) in the space of holomorphic functions on B (i.e.
scalar highest weight representations). Consider the restriction of
to the pseudoorthogonal group O(p,q).
The representation of O(p,q) in on the symmetric space
is a limit of the representations in some
precise sence. Spectrum of a representation is comlicated and it depends
on .
We obtain the complete Plancherel formula for the representations for
all admissible values of the parameter . We also extend this result to
all classical noncompact and compact Riemannian symmetric spaces
The Regularized Siegel-Weil Formula (The Second Term Identity) and the Rallis Inner Product Formula
In this paper, we establish the second term identity of the Siegel-Weil
formula in full generality, and derive the Rallis inner product formula for
global theta lifts for any dual pair. As a corollary, we resolve the
non-vanishing problem of global theta lifts initiated by Steve Rallis
On -Whittaker functions
The -Whittaker functions are eigenfunctions of the modular -deformed
open Toda system introduced by Kharchev, Lebedev, and
Semenov-Tian-Shansky. Using the quantum inverse scattering method, the named
authors obtained a Mellin-Barnes integral representation for these
eigenfunctions. In the present paper, we develop the analytic theory of the
-Whittaker functions from the perspective of quantum cluster algebras. We
obtain a formula for the modular open Toda system's Baxter operator as a
sequence of quantum cluster transformations, and thereby derive a new modular
-analog of Givental's integral formula for the undeformed Whittaker
function. We also show that the -Whittaker functions are eigenvectors of the
Dehn twist operator from quantum higher Teichm\"uller theory, and obtain
-analogs of various integral identities satisfied by the undeformed
Whittaker functions, including the continuous Cauchy-Littlewood identity of
Stade and Corwin-O'Connell-Sepp\"al\"ainen-Zygouras. Using these results, we
prove the unitarity of the -Whittaker transform, thereby completing the
analytic part of the proof of the conjecture of Frenkel and Ip on tensor
products of positive representations of , as well as the
main step in the modular functor conjecture of Fock and Goncharov. We conclude
by explaining how the theory of -Whittaker functions can be used to derive
certain hyperbolic hypergeometric integral evaluations found by Rains.Comment: 36 pages, minor changes, references adde
Coherent States of the q--Canonical Commutation Relations
For the -deformed canonical commutation relations for in some Hilbert
space we consider representations generated from a vector
satisfying , where .
We show that such a representation exists if and only if .
Moreover, for these representations are unitarily equivalent
to the Fock representation (obtained for ). On the other hand
representations obtained for different unit vectors are disjoint. We
show that the universal C*-algebra for the relations has a largest proper,
closed, two-sided ideal. The quotient by this ideal is a natural -analogue
of the Cuntz algebra (obtained for ). We discuss the Conjecture that, for
, this analogue should, in fact, be equal to the Cuntz algebra
itself. In the limiting cases we determine all irreducible
representations of the relations, and characterize those which can be obtained
via coherent states.Comment: 19 pages, Plain Te
Noncommutative Spheres and Instantons
We report on some recent work on deformation of spaces, notably deformation
of spheres, describing two classes of examples. The first class of examples
consists of noncommutative manifolds associated with the so called
-deformations which were introduced out of a simple analysis in terms
of cycles in the -complex of cyclic homology. These examples have
non-trivial global features and can be endowed with a structure of
noncommutative manifolds, in terms of a spectral triple (\ca, \ch, D). In
particular, noncommutative spheres are isospectral
deformations of usual spherical geometries. For the corresponding spectral
triple (\cinf(S^{N}_\theta), \ch, D), both the Hilbert space of spinors \ch=
L^2(S^{N},\cs) and the Dirac operator are the usual ones on the
commutative -dimensional sphere and only the algebra and its action
on are deformed. The second class of examples is made of the so called
quantum spheres which are homogeneous spaces of quantum orthogonal
and quantum unitary groups. For these spheres, there is a complete description
of -theory, in terms of nontrivial self-adjoint idempotents (projections)
and unitaries, and of the -homology, in term of nontrivial Fredholm modules,
as well as of the corresponding Chern characters in cyclic homology and
cohomology.Comment: Minor changes, list of references expanded and updated. These notes
are based on invited lectures given at the ``International Workshop on
Quantum Field Theory and Noncommutative Geometry'', November 26-30 2002,
Tohoku University, Sendai, Japan. To be published in the workshop proceedings
by Springer-Verlag as Lecture Notes in Physic
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