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

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

Mott--Hubbard transition vs. Anderson localization of correlated, disordered electrons

The phase diagram of correlated, disordered electrons is calculated within dynamical mean--field theory using the geometrically averaged (''typical'') local density of states. Correlated metal, Mott insulator and Anderson insulator phases, as well as coexistence and crossover regimes are identified. The Mott and Anderson insulators are found to be continuously connected.Comment: 4 pages, 4 figure

Two-eigenfunction correlation in a multifractal metal and insulator

We consider the correlation of two single-particle probability densities $|\Psi_{E}({\bf r})|^{2}$ at coinciding points ${\bf r}$ as a function of the energy separation $\omega=|E-E'|$ for disordered tight-binding lattice models (the Anderson models) and certain random matrix ensembles. We focus on the models in the parameter range where they are close but not exactly at the Anderson localization transition. We show that even far away from the critical point the eigenfunction correlation show the remnant of multifractality which is characteristic of the critical states. By a combination of the numerical results on the Anderson model and analytical and numerical results for the relevant random matrix theories we were able to identify the Gaussian random matrix ensembles that describe the multifractal features in the metal and insulator phases. In particular those random matrix ensembles describe new phenomena of eigenfunction correlation we discovered from simulations on the Anderson model. These are the eigenfunction mutual avoiding at large energy separations and the logarithmic enhancement of eigenfunction correlations at small energy separations in the two-dimensional (2D) and the three-dimensional (3D) Anderson insulator. For both phenomena a simple and general physical picture is suggested.Comment: 16 pages, 18 figure

Anderson localization vs. Mott-Hubbard metal-insulator transition in disordered, interacting lattice fermion systems

We review recent progress in our theoretical understanding of strongly correlated fermion systems in the presence of disorder. Results were obtained by the application of a powerful nonperturbative approach, the Dynamical Mean-Field Theory (DMFT), to interacting disordered lattice fermions. In particular, we demonstrate that DMFT combined with geometric averaging over disorder can capture Anderson localization and Mott insulating phases on the level of one-particle correlation functions. Results are presented for the ground-state phase diagram of the Anderson-Hubbard model at half filling, both in the paramagnetic phase and in the presence of antiferromagnetic order. We find a new antiferromagnetic metal which is stabilized by disorder. Possible realizations of these quantum phases with ultracold fermions in optical lattices are discussed.Comment: 25 pages, 5 figures, typos corrected, references update

Markov evolutions and hierarchical equations in the continuum I. One-component systems

General birth-and-death as well as hopping stochastic dynamics of infinite particle systems in the continuum are considered. We derive corresponding evolution equations for correlation functions and generating functionals. General considerations are illustrated in a number of concrete examples of Markov evolutions appearing in applications.Comment: 47 page

A generalization of the Heine--Stieltjes theorem

We extend the Heine-Stieltjes Theorem to concern all (non-degenerate) differential operators preserving the property of having only real zeros. This solves a conjecture of B. Shapiro. The new methods developed are used to describe intricate interlacing relations between the zeros of different pairs of solutions. This extends recent results of Bourget, McMillen and Vargas for the Heun equation and answers their question on how to generalize their results to higher degrees. Many of the results are new even for the classical case.Comment: 12 pages, typos corrected and refined the interlacing theorem