Georgia Institute of Technology College of Sciences

We offer several perspectives on the behavior at infinity of solutions of discrete Schroedinger equations. First we study pairs of discrete Schroedinger equations whose potential functions differ by a quantity that can be considered small in a suitable sense as the index n \rightarrow \infty. With simple assumptions on the growth rate of the solutions of the original system, we show that the perturbed system has a fundamental set of solutions with the same behavior at infinity, employing a variation-of-constants scheme to produce a convergent iteration for the solutions of the second equation in terms of those of the original one. We use the relations between the solution sets to derive exponential dichotomy of solutions and elucidate the structure of transfer matrices. Later, we present a sharp discrete analogue of the Liouville-Green (WKB) transformation, making it possible to derive exponential behavior at infinity of a single difference equation, by explicitly constructing a comparison equation to which our perturbation results apply. In addition, we point out an exact relationship connecting the diagonal part of the Green matrix to the asymptotic behavior of solutions. With both of these tools it is possible to identify an Agmon metric, in terms of which, in some situations, any decreasing solution must decrease exponentially.This talk is based on joint work with Evans Harrell.

The term "BMV Conjecture" was introduced in 2004 by Lieb and Seiringer for a conjecture introduced in 1975 by Bessis, Moussa and Villani, and they also introduced a new form for it : all coefficients of the polynomial Tr(A+xB)^k are non negative as soon as the hermitian matrices A and B are positive definite. A recent proof of the conjecture has been given recently by Herbert Stahl. The question occurs in various domains: complex analysis, combinatorics, operator algebras and statistical mechanics.

Chromatic derivatives are special, numerically robust linear differential operators which provide a unification framework for a broad class of orthogonal polynomials with a broad class of special functions. They are used to define chromatic expansions which generalize the Neumann series of Bessel functions. Such expansions are motivated by signal processing; they grew out of a design of a switch mode power amplifier. Chromatic expansions provide local signal representation complementary to the global signal representation given by the Shannon sampling expansion. Unlike the Taylor expansion which they are intended to replace, they share all the properties of the Shannon expansion which are crucial for signal processing. Besides being a promising new tool for signal processing, chromatic derivatives and expansions have intriguing mathematical properties connecting in a novel way orthogonal polynomials with some familiar concepts and theorems of harmonic analysis. For example, they introduce novel spaces of almost periodic functions which naturally correspond to a broad class of families of orthogonal polynomials containing most classical families. We also present a conjecture which generalizes the Paley Wiener Theorem and which relates the growth rate of entire functions with the asymptotic behavior of the recursion coefficients of a corresponding family of orthogonal polynomials.

Normal 0 false false false EN-US X-NONE X-NONE MicrosoftInternetExplorer4 By the classical Weierstrass theorem, any function continuous on a compact set can be uniformly approximated by algebraic polynomials. In this talk we shall discuss possible extensions of this basic result of analysis to approximation by homogeneous algebraic polynomials on central symmetric convex bodies. We shall also consider a related question of approximating convex bodies by convex algebraic level surfaces. It has been known for some time time that any convex body can be approximated arbirarily well by convex algebraic level surfaces. We shall present in this talk some new results specifying rate of convergence.