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 ERE 200