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Asymptotics of \(L_p\)-norms of Hermite polynomials and Rényi entropy of Rydberg oscillator states. (English) Zbl 1318.94027
Arvesú, Jorge (ed.) et al., Recent advances in orthogonal polynomials, special functions, and their applications. 11th international symposium, Universidad Carlos III de Madrid, Leganés, Spain, August 29 – September 2, 2011. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-6896-6/pbk). Contemporary Mathematics 578, 19-29 (2012).
Summary: The asymptotics of the weighted \(L_{p}\)-norms of Hermite polynomials, which describes the Rényi entropy of order \(p\) of the associated quantum oscillator probability density, is determined for \(n\to\infty\) and \(p>0\). Then, it is applied to the calculation of the Rényi entropy of the quantum-mechanical probability density of the highly-excited (Rydberg) states of the isotropic oscillator.
For the entire collection see [Zbl 1250.00015].

MSC:
94A17 Measures of information, entropy
11B37 Recurrences
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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