ASYMPTOTIC BEHAVIOR OF THE<i>L<sup>p</sup></i>-NORMS AND THE ENTROPY FOR GENERAL ORTHOGONAL POLYNOMIALS

Aptekarev, A. I.; Buyarov, V S; Degeza, I S

doi:10.1070/sm1995v082n02abeh003571

Cited by 59 publications

(130 citation statements)

References 6 publications

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“…These integrals are called ''entropies of the orthogonal polynomials p n (x),'' and they are closely related to the L p -norms, whose study is of independent interest in the theory of general orthogonal and extremal polynomials. 21 Asymptotic formulas for E n in the n→ϱ limit have been obtained in the case when p n (x) are general orthogonal polynomials on a finite interval, 22 or Freud orthogonal polynomials ͓w(x)ϭexp(Ϫ͉x͉ m ), mϾ0͔ on the whole real axis. 21,23,24 However, the analytical value of these entropies is only known for Chebyshev polynomials of the first and second kinds, in an exact form, and for Gegenbauer polynomials in an approximate way.…”

Section: Introductionmentioning

confidence: 99%

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

Sánchez-Ruiz¹

1997

Journal of Mathematical Physics

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The 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) ϭϪ͐ Ϫϱdx at the zeros of the Hermite polynomial H n (x). Here, a closed analytical expression for V n (t) is obtained, which in turn yields an exact analytical expression for the entropies when the exact location of the zeros of H n (x) is known. An inequality for the values of V n (t) at the zeros of H n (x) is conjectured, which leads to a new, nonvariational, upper bound for the entropies. Finally, the exact formula for V n (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 (nӷ1) limit of this alternative expression for the entropies is discussed, and the conjectured upper bound for the entropies is proved to be asymptotically valid.

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Section: Introductionmentioning

confidence: 99%

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

Sánchez-Ruiz¹

1997

Journal of Mathematical Physics

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“…However, to prove (or disprove) that the previous formula holds indeed in the general case is a very difficult problem, whose solution would require the use of a higher-order WKB approximation and a stronger version of the key theorem used in section 2 (lemma 2.1 in [4]), and, up to now, we have not been able to obtain any result of this kind. The generalization of the asymptotic formulae obtained in this paper to the D-dimensional case is straightforward for systems whose Hamiltonian is completely separable in Cartesian coordinates [5],…”

Section: Conclusion and Open Problemsmentioning

confidence: 79%

“…Accordingly, there has been a growing interest in the calculation of S Q for physically interesting quantum states. However, the exact calculation of S Q is a very difficult mathematical problem, even for simple systems as the harmonic oscillator and hydrogen atom [2], which has attracted interest to its approximate calculation, specially for very excited or Rydberg stationary states [3,4].…”

Section: S = − P (X) Ln P (X) DX = − Ln P (X)mentioning

confidence: 99%

“…A rigorous proof of this statement can be achieved by making the change of variable u(x) = θ and applying, with g(θ) = cos 2q θ, the following theorem (lemma 2.1 in [4]),…”

Section: Generalization Of the Asymptotic Formulae To Rényi Entropiesmentioning

confidence: 99%

“…In recent years, the Rényi and Tsallis entropies corresponding to the quantum probability densities of position and momentum have also been applied to the mathematical formulation of the uncertainty principle [14], and their approximate calculation for Rydberg states of physically interesting systems has been the subject of active research [4]. The aims of this paper are (i) to find the generalization of equations (2), (10) and (11) to the whole set of Rényi entropies, and (ii) to check these generalized asymptotic formulae in the simplest particular case of the infinite potential well, where, as we shall see, this can be done in a fully analytical way.…”

Section: S = − P (X) Ln P (X) DX = − Ln P (X)mentioning

confidence: 99%

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Asymptotic formulae for the quantum Rényi entropies of position: application to the infinite well

Sánchez-Ruiz

1999

J. Phys. A: Math. Gen.

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Abstract. General asymptotic formulae are derived by means of the WKB approximation for the continuous and discrete Rényi entropies of position of one-dimensional quantum systems in energy eigenstates, in terms of the corresponding entropies for a microcanonical ensemble of analogous classical systems. These results are checked in the simplest particular case of the infinite potential well, where the asymptotic formula for continuous entropies holds as an exact identity. For the discrete entropies, analytical expressions are obtained from which the asymptotic formulae given for the limiting cases of large and small measurement resolution can both be verified.

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Information theory of D-dimensional hydrogenic systems: Application to circular and Rydberg states

Dehesa

López-Rosa

Martínez–Finkelshtein

et al. 2009

Int. J. Quantum Chem.

151

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ABSTRACT:The analytic information theory of quantum systems includes the exact determination of their spatial extension or multidimensional spreading in both position and momentum spaces by means of the familiar variance and its generalization, the power and logarithmic moments, and, more appropriately, the Shannon entropy and the Fisher information. These complementary uncertainty measures have a global or local character, respectively, because they are power-like (variance, moments), logarithmic (Shannon) and gradient (Fisher) functionals of the corresponding probability distribution. Here we explicitly discuss all these spreading measures (and their associated uncertainty relations) in both position and momentum for the main prototype in D-dimensional physics, the hydrogenic system, directly in terms of the dimensionality and the hyperquantum numbers which characterize the involved states. Then, we analyze in detail such measures for s-states, circular states (i.e., single-electron states of highest angular momenta allowed within an electronic manifold characterized by a given principal hyperquantum number), and Rydberg states (i.e., states with large radial hyperquantum numbers n).

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ASYMPTOTIC BEHAVIOR OF THEL^p-NORMS AND THE ENTROPY FOR GENERAL ORTHOGONAL POLYNOMIALS

Cited by 59 publications

References 6 publications

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

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

Asymptotic formulae for the quantum Rényi entropies of position: application to the infinite well

Information theory of D-dimensional hydrogenic systems: Application to circular and Rydberg states

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ASYMPTOTIC BEHAVIOR OF THELp-NORMS AND THE ENTROPY FOR GENERAL ORTHOGONAL POLYNOMIALS

Cited by 59 publications

References 6 publications

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

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

Asymptotic formulae for the quantum Rényi entropies of position: application to the infinite well

Information theory of D-dimensional hydrogenic systems: Application to circular and Rydberg states

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ASYMPTOTIC BEHAVIOR OF THEL^p-NORMS AND THE ENTROPY FOR GENERAL ORTHOGONAL POLYNOMIALS