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Boris Shapiro

Publications and source records attributed to Boris Shapiro.

At least 127 records · Page 7Linked to original sources
arXiv2008

Algebro-gemetric aspects of Heine-Stieltjes theory

The goal of this paper is to develop a Heine-Stieltjes theory for univariate linear differential operators of higher order. Namely, for a given given operator T=\sum_i Q_i(z)d^i/dz^i with polynomial coefficients Q_i(z) set r=max_i (deg Q_i(z)-i). Following the classical approach of Heine and Stieltjes we study the multiparameter spectral problem of finding all polynomial V(z) of degree at most r such that the equation: T(z)S(z)+V(z)S(z=0 has for a given positive integer n a polynomial solution S(z) of degree n. We show that under some mild non-degeneracy assumptions there exist exactly ((n+r) choose n) such polynomials V_n,i(z). We generalize a number of classically known results in this area and discuss occurring degeneracies.

math-ph↗
arXiv2008

Self-consistent study of Anderson localization in the Anderson-Hubbard model in two and three dimensions

We consider the change in electron localization due to the presence of electron-electron repulsion in the \HA model. Taking into account local Mott-Hubbard physics and static screening of the disorder potential, the system is mapped onto an effective single-particle Anderson model, which is studied within the self-consistent theory of electron localization. We find rich nonmonotonic behavior of the localization length $ξ$ in two-dimensional systems, including an interaction-induced exponential enhancement of $ξ$ for small and intermediate disorders although $ξ$ remains finite. In three dimensions we identify for half filling a Mott-Hubbard-assisted Anderson localized phase existing between the metallic and the Mott-Hubbard-gapped phases. For small $U$ there is re-entrant behavior from the Anderson localized phase to the metallic phase.

cond-mat.dis-nn↗
arXiv2008

Narayana numbers and Schur-Szego composition

In the present paper we find a new interpretation of Narayana polynomials N_n(x) which are the generating polynomials for the Narayana numbers N_{n,k} counting Dyck paths of length n and with exactly k peaks. Strangely enough Narayana polynomials also occur as limits as n->oo of the sequences of eigenpolynomials of the Schur-Szego composition map sending (n-1)-tuples of polynomials of the form (x+1)^{n-1}(x+a) to their Schur-Szego product, see below. As a corollary we obtain that every N_n(x) has all roots real and non-positive. Additionally, we present an explicit formula for the density and the distribution function of the asymptotic root-counting measure of the polynomial sequence {N_n(x)}.

math.CA↗
arXiv2008

Root asymptotics of spectral polynomials for the Lame operator

The study of polynomial solutions to the classical Lamé equation in its algebraic form, or equivalently, of double-periodic solutions of its Weierstrass form has a long history. Such solutions appear at integer values of the spectral parameter and their respective eigenvalues serve as the ends of bands in the boundary value problem for the corresponding Schrödinger equation with finite gap potential given by the Weierstrass $\wp$-function on the real line. In this paper we establish several natural (and equivalent) formulas in terms of hypergeometric and elliptic type integrals for the density of the appropriately scaled asymptotic distribution of these eigenvalues when the integer-valued spectral parameter tends to infinity. We also show that this density satisfies a Heun differential equation with four singularities.

math.CA↗
arXiv2008

Preserving positive polynomials and beyond

Following the classical approach of Pólya-Schur theory we initiate in this paper the study of linear operators acting on $\mathbb{R}[x]$ and preserving either the set of positive univariate polynomials or similar sets of non-negative and elliptic polynomials.

math.CA↗
arXiv2007

On eigenvalues of rectangular matrices

Given a $(k+1)$-tuple $A, B_1,...,B_k$ of $(m\times n)$-matrices with $m\le n$ we call the set of all $k$-tuples of complex numbers $\{\la_1,...,\la_k\}$ such that the linear combination $A+\la_1B_1+\la_2B_2+...+\la_kB_k$ has rank smaller than $m$ the {\it eigenvalue locus} of the latter pencil. Motivated primarily by applications to multi-parameter generalizations of the Heine-Stieltjes spectral problem, see \cite{He} and \cite{Vol}, we study a number of properties of the eigenvalue locus in the most important case $k=n-m+1$.

math.AG↗
arXiv2007

Statistics of Resonances in a Semi-infinite Disordered Chain

We study the average density of resonances (DOR) for a semi-infinite disordered chain, coupled to the outside world by a (semi-infinite) perfect lead. A set of equations is derived, which provides the general framework for calculating the average DOR, for an arbitrary disorder and coupling strength. These general equations are applied to the case of weak coupling and an asymptotically exact expression for the averaged DOR is derived, in the limit of small resonance width. This expression is universal, in the sense that it holds for any degree of disorder and everywhere in the (unperturbed) energy band.

cond-mat.dis-nn↗
arXiv2006

Zeros of eigenfunctions of some anharmonic oscillators

We study eigenfunctions of Schrodinger operators -y"+Py on the real line with zero boundary conditions, whose potentials P are real even polynomials with positive leading coefficients. For quartic potentials we prove that all zeros of all eigenfunctions belong to the union of the real and imaginary axes. Similar result holds for sextic potentials and their eigenfunctions with finitely many complex zeros. As a byproduct we obtain a complete classification of such eigenfunctions of sextic potentials.

math-ph↗
arXiv2006

Resonances in one-dimensional Disordered Chain

We study the average density of resonances, $<ρ(x,y)>$, in a semi-infinite disordered chain coupled to a perfect lead. The function $<ρ(x,y)>$ is defined in the complex energy plane and the distance $y$ from the real axes determines the resonance width. We concentrate on strong disorder and derive the asymptotic behavior of $<ρ(x,y)>$ in the limit of small $y$.

cond-mat.dis-nn↗
arXiv2006

Around matrix-tree theorem

Generalizing the classical matrix-tree theorem we provide a formula counting subgraphs of a given graph with a fixed 2-core. We use this generalization to obtain an analog of the matrix-tree theorem for the root system $D_n$ (the classical theorem corresponds to the $A_n$-case). Several byproducts of the developed technique, such as a new formula for a specialization of the multivariate Tutte polynomial, are of independent interest.

math.CO↗
arXiv2005

On rational approximation of algebraic functions

We construct a new scheme of approximation of any multivalued algebraic function $f(z)$ by a sequence $\{r_{n}(z)\}_{n\in \mathbb{N}}$ of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by $f(z)$. Compared to the usual Padé approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Padé Conjecture and Nuttall's Conjecture for the sequence $\{r_{n}(z)\}_{n\in \mathbb{N}}$ in the complement $\mathbb{CP}^1\setminus \D_{f}$, where $\D_{f}$ is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family $\{r_{n}(z)\}_{n\in \mathbb{N}}$. As an application we settle the so-called 3-conjecture of Egecioglu {\em et al} dealing with a 4-term recursion related to a polynomial Riemann Hypothesis.

math.CA↗
arXiv2004

Mystery of point charges

We discuss the problem of finding an upper bound for the number of equilibrium points of a potential of several fixed point charges in R^n. This question goes back to J.C.Maxwell and M.Morse. Using fewnomial theory we show that for a given number of charges there exists an upper bound independent on the dimension, and show it to be 12 for three charges. We conjecture the exact upper bound for a given configuration of nonnegative charges in terms of its Voronoi diagram, and prove it asymptotically.

math-ph↗
arXiv2004

Classifying real polynomial pencils

Let $\bP^n$ be the space of all homogeneous polynomials of degree $n$ in two variables with real coefficients. The standard discriminant $\D_{n+1}\subset \bP^n$ is Whitney stratified according to the number and the multiplicities of multiple real zeros. A real polynomial pencil, that is, a line $L\subset \bP^n$ is called generic if it intersects $\D_{n+1}$ transversally. Nongeneric pencils form the Grassmann discriminant $\D_{2,n+1}\subset \gtn$, where $\gtn$ is the Grassmannian of lines in $\bP^n$. We enumerate the connected components of the set $\widetilde \gtn=\gtn\setminus \D_{2,n+1}$ of all generic lines in $\bP^n$ and relate this topic to the Hawaii conjecture and the classical theorems of Obreschkoff and Hermite-Biehler.

math.AG↗
arXiv2003

Hyperbolic polynomials and spectral order

The spectral order on $\bR$ induces a partial ordering on the manifold $\calH_{n}$ of monic hyperbolic polynomials of degree $n$. We show that the semigroup $\tilde{\calS}$ generated by differential operators of the form $(1-\la \frac{d}{dx})e^{\la \frac{d}{dx}}$, $\la \in \bR$, acts on the poset $\calH_{n}$ in an order-preserving fashion. We also show that polynomials in $\calH_{n}$ are global minima of their respective $\tilde{\calS}$-orbits and we conjecture that a similar result holds even for complex polynomials. Finally, we show that only those pencils of polynomials in $\calH_{n}$ which are of logarithmic derivative type satisfy a certain local minimum property for the spectral order.

math.CA↗
arXiv2003

Trees, parking functions, syzygies, and deformations of monomial ideals

For a graph G, we construct two algebras, whose dimensions are both equal to the number of spanning trees of G. One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis elements correspond to G-parking functions that naturally came up in the abelian sandpile model. These ideals are instances of the general class of monotone monomial ideals and their deformations. We show that the Hilbert series of a monotone monomial ideal is always bounded by the Hilbert series of its deformation. Then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals. In some cases these resolutions coincide with Scarf resolutions. We prove several formulas for Hilbert series of monotone monomial ideals and investigate when they are equal to Hilbert series of deformations. In the appendix we discuss the sandpile model.

math.CO↗
arXiv2001

Topological classification of generic real rational functions

To any real rational function with generic ramification points we assign a combinatorial object, called a garden, which consists of a weighted labeled directed planar chord diagram and of a set of weighted rooted trees each corresponding to a face of the diagram. We prove that any garden corresponds to a generic real rational function, and that equivalent functions have equivalent gardens.

math.AG↗
arXiv2001

A Model for Addition Spectra in Quantum Dots

A simple model for addition spectra in quantum dots is proposed and studied. It is an extension of the standard charging model which assumes that the charge spreads uniformely over the entire dot. The proposed model attempts to account for a nonuniform distribution of the charge, by introducing an extra parameter U. When U increases, the distribution of the conductance peak spacings changes from the Wigner-Dyson shape towards a broader, more symmetric distribution.

cond-mat↗