Logarithmic potential of Hermite polynomials and information entropies of the harmonic oscillator eigenstates

13 pages, 1 figure.-- PACS nrs.: 03.65.Ge, 02.10.Nj, 02.10.Sp.MR#: MR1471913 (99c:81031)Zbl#: Zbl 0891.33007The problem of calculating the information entropy in both position and momentum spaces for the nth stationary state of the one-dimensional quantum harmonic oscillator reduces to the evaluation of the logarithmic potential V_n(t)=-\int e\sp {-x\sp 2}H_n\sp 2(x)\log -t x at the zeros of the Hermite polynomial Hn(x). Here, a closed analytical expression for Vn(t) is obtained, which in turn yields an exact analytical expression for the entropies when the exact location of the zeros of Hn(x) is known. An inequality for the values of Vn(t) at the zeros of Hn(x) is conjectured, which leads to a new, nonvariational, upper bound for the entropies. Finally, the exact formula for Vn(t) is written in an alternative way, which allows the entropies to be expressed in terms of the even-order spectral moments of the Hermite polynomials. The asymptotic (n>>1) limit of this alternative expression for the entropies is discussed, and the conjectured upper bound for the entropies is proved to be asymptotically validThe author gratefully acknowledges the financial support from the Fundació Aula (Barcelona, Spain).Publicad

Asymptotics for multiple Meixner polynomials

. We use an algebraic function formulation for the solution of the equilibrium problem with constraint to describe their zero distribution. Moreover, analyzing the limiting behavior of the coefficients of the recurrence relations for Multiple Meixner polynomials we obtain the main term of their asymptotics. (C) 2013 Elsevier Inc. All rights reserved.The research of J. Arvesú was partially supported by the research grant MTM2012-36732-C03-01 of the Ministerio de Educación y Ciencia of Spain and grants CCG07-UC3M/ESP-3339 and CCG08-UC3M/ESP-4516 from Comunidad Autónoma de Madrid

Asymptotics for Hermite-Pade rational approximants for two analytic functions with separated pairs of branch points (case of genus 0)

We investigate the asymptotic behavior for type II Hermite-Pade approximation to two functions, where each function has two branch points and the pairs of branch points are separated. We give a classification of the cases such that the limiting counting measures for the poles of the Hermite-Pade approximants are described by an algebraic function of order 3 and genus 0. This situation gives rise to a vector-potential equilibrium problem for three measures and the poles of the common denominator are asymptotically distributed like one of these measures. We also work out the strong asymptotics for the corresponding Hermite-Pade approximants by using a 3x3 Riemann-Hilbert problem that characterizes this Hermite-Pade approximation problem.Comment: 102 pages, 31 figure

A scalar Riemann boundary value problem approach to orthogonal polynomials on the circle

8 pages, no figures.-- MSC2000 codes: 33C47; 42C05.MR#: MR2252097 (2007k:33010)Zbl#: Zbl 1130.42025A scalar Riemann boundary value problem defining orthogonal polynomials on the unit circle and the corresponding functions of the second kind is obtained. The Riemann problem is used for the asymptotic analysis of the polynomials orthogonal with respect to an analytical real-valued weight on the circle.The research was supported by INTAS Research Network NeCCA 03-51-6637. The first author was also supported by the Grants RFBR 05-01-00522, NSh-1551.2003.1 and by the program N1 DMS, RAS. The second authorwas supported by Ministerio de Ciencia y Tecnología under Grant number MTM2005-01320. The third author was supported by Ministerio de Ciencia y Tecnología under Grant number BFM2003-06335-C03-02.Publicad

Strong asymptotics for the Pollaczek multiple orthogonal polynomials

The asymptotic properties of multiple orthogonal polynomials with respect to two Pollaczek weights with different parameters are considered. This set of weights is a Nikishin system generated by two measures with unbounded supports; moreover, the second measure is discrete. During the last years, multiple orthogonal polynomials with respect to Nikishin systems of this type have found wide applications in the theory of random matrices. Strong asymptotic formulas for the polynomials under consideration are obtained by means of the matrix Riemann-Hilbert method.The first author acknowledges the support of the Rus sian Science Foundation, project no. 14 -21- 00025. The second author’s research was supported by MTM2012 -36732 -C03 -01 MEC. The third author’s research was supported by MTM2011-28952-C02-01 ERDF, Exc. Grant no. P11- FQM -7276, Res. Group FQM -229, and CEIMAR Universidad de Almeria