713 research outputs found
Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials
We investigate the asymptotic zero distribution of Heine-Stieltjes
polynomials - polynomial solutions of a second order differential equations
with complex polynomial coefficients. In the case when all zeros of the leading
coefficients are all real, zeros of the Heine-Stieltjes polynomials were
interpreted by Stieltjes as discrete distributions minimizing an energy
functional. In a general complex situation one deals instead with a critical
point of the energy. We introduce the notion of discrete and continuous
critical measures (saddle points of the weighted logarithmic energy on the
plane), and prove that a weak-* limit of a sequence of discrete critical
measures is a continuous critical measure. Thus, the limit zero distributions
of the Heine-Stieltjes polynomials are given by continuous critical measures.
We give a detailed description of such measures, showing their connections with
quadratic differentials. In doing that, we obtain some results on the global
structure of rational quadratic differentials on the Riemann sphere that have
an independent interest.Comment: 70 pages, 14 figures. Minor corrections, to appear in Comm. Math.
Physic
Effect of electron-lattice interaction on the phase separation in strongly correlated electron systems with two types of charge carriers
The effect of electron-lattice interaction is studied for a strongly
correlated electron system described by the two-band Hubbard model. A two-fold
effect of electron-lattice interaction is taken into account: in non-diagonal
terms, it changes the effective bandwidth, whereas in diagonal terms, it shifts
the positions of the bands and the chemical potential. It is shown that this
interaction significantly affects the doping range corresponding to the
electronic phase separation and can even lead to a jump-like transition between
states with different values of strains.Comment: 6 pages, 7 figures, submitted to Phys. Rev.
On asymptotic behavior of Heine-Stieltjes and Van Vleck polynomials
We investigate the strong asymptotics of Heine-Stieltjes polynomials -
polynomial solutions of a second order differential equations with complex
polynomial coefficients. The solution is given in terms of critical measures
(saddle points of the weighted logarithmic energy on the plane), that are
tightly related to quadratic differentials with closed trajectories on the
plane. The paper is a continuation of the research initiated in
[arXiv:0902.0193]. However, the starting point here is the WKB method, which
allows to obtain the strong asymptotics.Comment: 24 pages, 6 figure
LIGO End-to-End simulation Program
A time-domain simulation program has been developed to provide an accurate description of interferometric gravitational wave detectors. This is being utilized to build a model of LIGO with the aim of aiding in the shakedown and integration of the interferometer subsystems, and ultimately the optimization of detector sensitivity
Isolation of gravitational waves from displacement noise and utility of a time-delay device
Interferometers with kilometer-scale arms have been built for
gravitational-wave detections on the ground; ones with much longer arms are
being planned for space-based detection. One fundamental motivation for long
baseline interferometry is from displacement noise. In general, the longer the
arm length L, the larger the motion the gravitational-wave induces on the test
masses, until L becomes comparable to the gravitational wavelength. Recently,
schemes have been invented, in which displacement noises can be evaded by
employing differences between the influence of test-mass motions and that of
gravitational waves on light propagation. However, in these schemes, such
differences only becomes significant when L approaches the gravitational
wavelength, and shot-noise limited sensitivity becomes worse than that of
conventional configurations by a factor of at least (f L/c)^(-2), for f<c/L.
Such a factor, although can be overcome theoretically by employing high optical
powers, makes these schemes quite impractical. In this paper, we explore the
use of time delay in displacement-noise-free interferometers, which can improve
their shot-noise-limited sensitivity at low frequencies, to a factor of (f
L/c)^(-1) of the shot-noise-limited sensitivity of conventional configurations.Comment: 10 pages, 12 figures, a proceeding for the Spanish Relativity Meeting
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