2,771 research outputs found
Matrix Models, Complex Geometry and Integrable Systems. I
We consider the simplest gauge theories given by one- and two- matrix
integrals and concentrate on their stringy and geometric properties. We remind
general integrable structure behind the matrix integrals and turn to the
geometric properties of planar matrix models, demonstrating that they are
universally described in terms of integrable systems directly related to the
theory of complex curves. We study the main ingredients of this geometric
picture, suggesting that it can be generalized beyond one complex dimension,
and formulate them in terms of the quasiclassical integrable systems, solved by
construction of tau-functions or prepotentials. The complex curves and
tau-functions of one- and two- matrix models are discussed in detail.Comment: 52 pages, 19 figures, based on several lecture courses and the talks
at "Complex geometry and string theory" and the Polivanov memorial seminar;
misprints corrected, references adde
Gravitational radiation reaction in the equations of motion of compact binaries to 3.5 post-Newtonian order
We compute the radiation reaction force on the orbital motion of compact
binaries to the 3.5 post-Newtonian (3.5PN) approximation, i.e. one PN order
beyond the dominant effect. The method is based on a direct PN iteration of the
near-zone metric and equations of motion of an extended isolated system, using
appropriate ``asymptotically matched'' flat-space-time retarded potentials. The
formalism is subsequently applied to binary systems of point particles, with
the help of the Hadamard self-field regularisation. Our result is the 3.5PN
acceleration term in a general harmonic coordinate frame. Restricting the
expression to the centre-of-mass frame, we find perfect agreement with the
result derived in a class of coordinate systems by Iyer and Will using the
energy and angular momentum balance equations.Comment: 28 pages, references added, to appear in Classical and Quantum
Gravit
Semi-spectral Chebyshev method in Quantum Mechanics
Traditionally, finite differences and finite element methods have been by
many regarded as the basic tools for obtaining numerical solutions in a variety
of quantum mechanical problems emerging in atomic, nuclear and particle
physics, astrophysics, quantum chemistry, etc. In recent years, however, an
alternative technique based on the semi-spectral methods has focused
considerable attention. The purpose of this work is first to provide the
necessary tools and subsequently examine the efficiency of this method in
quantum mechanical applications. Restricting our interest to time independent
two-body problems, we obtained the continuous and discrete spectrum solutions
of the underlying Schroedinger or Lippmann-Schwinger equations in both, the
coordinate and momentum space. In all of the numerically studied examples we
had no difficulty in achieving the machine accuracy and the semi-spectral
method showed exponential convergence combined with excellent numerical
stability.Comment: RevTeX, 12 EPS figure
Asymptotic Uniqueness of Best Rational Approximants to Complex Cauchy Transforms in of the Circle
For all n large enough, we show uniqueness of a critical point in best
rational approximation of degree n, in the L^2-sense on the unit circle, to
functions f, where f is a sum of a Cauchy transform of a complex measure \mu
supported on a real interval included in (-1,1), whose Radon-Nikodym derivative
with respect to the arcsine distribution on its support is Dini-continuous,
non-vanishing and with and argument of bounded variation, and of a rational
function with no poles on the support of \mu.Comment: 28 page
Hadamard regularization of the third post-Newtonian gravitational wave generation of two point masses
Continuing previous work on the 3PN-accurate gravitational wave generation
from point particle binaries, we obtain the binary's 3PN mass-type quadrupole
and dipole moments for general (not necessarily circular) orbits in harmonic
coordinates. The final expressions are given in terms of their ``core'' parts,
resulting from the application of the pure Hadamard-Schwartz (pHS) self-field
regularization scheme, and augmented by an ``ambiguous'' part. In the case of
the 3PN quadrupole we find three ambiguity parameters, xi, kappa and zeta, but
only one for the 3PN dipole, in the form of the particular combination
xi+kappa. Requiring that the dipole moment agree with the center-of-mass
position deduced from the 3PN equations of motion in harmonic coordinates
yields the relation xi+kappa=-9871/9240. Our results will form the basis of the
complete calculation of the 3PN radiation field of compact binaries by means of
dimensional regularization.Comment: 33 pages, to appear in Phys. Rev.
A fast and well-conditioned spectral method for singular integral equations
We develop a spectral method for solving univariate singular integral
equations over unions of intervals by utilizing Chebyshev and ultraspherical
polynomials to reformulate the equations as almost-banded infinite-dimensional
systems. This is accomplished by utilizing low rank approximations for sparse
representations of the bivariate kernels. The resulting system can be solved in
operations using an adaptive QR factorization, where is
the bandwidth and is the optimal number of unknowns needed to resolve the
true solution. The complexity is reduced to operations by
pre-caching the QR factorization when the same operator is used for multiple
right-hand sides. Stability is proved by showing that the resulting linear
operator can be diagonally preconditioned to be a compact perturbation of the
identity. Applications considered include the Faraday cage, and acoustic
scattering for the Helmholtz and gravity Helmholtz equations, including
spectrally accurate numerical evaluation of the far- and near-field solution.
The Julia software package SingularIntegralEquations.jl implements our method
with a convenient, user-friendly interface
Truncated linear statistics associated with the top eigenvalues of random matrices
Given a certain invariant random matrix ensemble characterised by the joint
probability distribution of eigenvalues , many
important questions have been related to the study of linear statistics of
eigenvalues , where is a known
function. We study here truncated linear statistics where the sum is restricted
to the largest eigenvalues: .
Motivated by the analysis of the statistical physics of fluctuating
one-dimensional interfaces, we consider the case of the Laguerre ensemble of
random matrices with . Using the Coulomb gas
technique, we study the limit with fixed. We show that the
constraint that is fixed drives an
infinite order phase transition in the underlying Coulomb gas. This transition
corresponds to a change in the density of the gas, from a density defined on
two disjoint intervals to a single interval. In this latter case the density
presents a logarithmic divergence inside the bulk. Assuming that
is monotonous, we show that these features arise for any random matrix ensemble
and truncated linear statitics, which makes the scenario described here robust
and universal.Comment: LaTeX, 30 pages, 20 pdf figures. Updated version: a typo has been
corrected in Eq. (3.30) and more details are provided in the Appendi
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