Andrei Martínez-Finkelshtein

Quantum information entropies and orthogonal polynomials

This is a survey of the present knowledge on the analytical determination of the Shannon information entropies for simple quantum systems: single-particle systems in central potentials. Emphasis is made on D-dimensional harmonic oscillator and Coulombian potentials in both position and momentum spaces. First of all, these quantities are explicitly shown to be controlled by the entropic integrals of some classical orthogonal polynomials (Hermite, Laguerre and Gegenbauer). Then, the connection of these integrals with more common mathematical objects, such as the logarithmic potential, energy and L p -norms of orthogonal polynomials, is brie:y described. Third, its asymptotic behaviour is discussed for both general and varying weights. The explicit computation of these integrals is carried out for the Chebyshev and Gegenbauer polynomials, which have a bounded orthogonality interval, as well as for Hermite polynomials to illustrate the di;culties encountered when the interval is unbounded. These results have allowed us to <nd the position and momentum entropies of the ground and excited states of the physical systems mentioned above. c 2001 Elsevier Science B.V. All rights reserved.

Information-theoretic measures for Morse and Pöschl–Teller potentials

The spreading of the quantum-mechanical probability cloud for the ground state of the Morse and modified Pöschl–Teller potentials, which controls the chemical and physical properties of some molecular systems, is studied in position and momentum space by means of global (Shannons information entropy, variance) and local (Fishers information) information-theoretic measures. We establish a general relation between variance and Fishers information, proving that, in the case of a real-valued and symmetric wavefunction, the well-known Cramer–Rao and Heisenberg uncertainty inequalities are equivalent. Finally, we discuss the asymptotics of all three information measures, showing that the ground state of these potentials saturates all the uncertainty relations in an appropriate limit of the parameter.

Strong asymptotics for Jacobi polynomials with varying nonstandard parameters

AbstractStrong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomialsPnαnβn are studied, assuming that1

Asymptotic Properties of Heine-Stieltjes and Van Vleck Polynomials

\mathop {\lim }\limits_{n \to \infty } \frac{{\alpha _n }}{n} = A, \mathop {\lim }\limits_{n \to \infty } \frac{{\beta _n }}{n} = B,

Information entropy of Gegenbauer polynomials

withA andB satisfyingA>−1,B>−1,A+B<−1. The asymptotic analysis is based on the non-Hermitian orthogonality of these polynomials and uses the Deift/Zhou steepest descent analysis for matrix Riemann-Hilbert problems. As a corollary, asymptotic zero behavior is derived. We show that in a generic case, the zeros distribute on the set of critical trajectories Γ of a certain quadratic differential according to the equilibrium measure on Γ in an external field. However, when either αnβn or αn+βn are geometrically close to ℤ, part of the zeros accumulate along a different trajectory of the same quadratic differential.

Analytic aspects of Sobolev orthogonal polynomials revisited

Abstract We study the asymptotic zero distribution of Laguerre L n ( α n ) and generalized Bessel B n ( α n ) polynomials with the parameter α n varying in such a way that the limit of α n / n exists. Our approach is based on a non-hermitian orthogonality satisfied by these sequences of polynomials. In the cases that remain open we formulate the corresponding conjectures.

General linearization formulae for products of continuous hypergeometric-type polynomials

We study the asymptotic behavior of the zeros of polynomial solutions of a class of generalized Lam? differential equations, when their coefficients satisfy certain asymptotic conditions. The limit distribution is described by an equilibrium measure in the presence of an external field, generated by charges at the singular points of the equation. Moreover, a case of non-positive charges is considered, which leads to an equilibrium with a non-convex external field.

Asymptotic properties of Sobolev orthogonal polynomials

Strong asymptotics of polynomials orthogonal on the unit circle with respect to a weight of the form