Spatial entropy of central potentials and strong asymptotics of orthogonal polynomials

The Boltzmann–Shannon information entropy of quantum-mechanical systems in central potentials can be expressed in terms of the entropy Sn of the classical orthogonal polynomials. Here, an asymptotic formula for the entropy of general orthogonal polynomials on finite intervals is obtained. It is shown that this entropy is intimately related to the relative entropy I (ρ0,ρ) of the equilibrium measure ρ0(x) and the weight function ρ(x) of the polynomials. To do so, the theory of strong asymptotics of orthogonal polynomials on compact sets is used.