Shannon-like integrals of hypergeometric orthogonal polynomials with large parameters and applications to high-dimensional harmonic and hydrogenic systems

Abstract

In this talk we determine the asymptotics of various logarithmic-type integral functionals of hypergeometric orthogonal polynomials (Laguerre, Gegenbauer) when their parameter α -> ∞. Then, we apply the corresponding results to find the physical Shannon entropies for all the stationary states of harmonic and hydrogenic systems with a very high dimensionality D. Briefly, it is found that these entropies have the same rate of growth, O (D log D), when D -> ∞ 1 for both types of quantum systems.