Entropy of orthogonal polynomials with Freud weights and information entropies of the harmonic oscillator potential

The information entropy of the harmonic oscillator potential V(x)=1/2λx 2 in both position and momentum spaces can be expressed in terms of the so‐called ‘‘entropy of Hermite polynomials,’’ i.e., the quantity S n (H):= −∫−∞ +∞ H 2 n (x)log H 2 n (x) e −x 2 dx. These polynomials are instances of the polynomials orthogonal with respect to the Freud weights w(x)=exp(−‖x‖ m ), m≳0. Here, a very precise and general result of the entropy of Freud polynomials recently established by Aptekarev et al. [J. Math. Phys. 35, 4423–4428 (1994)], specialized to the Hermite kernel (case m=2), leads to an important refined asymptotic expression for the information entropies of very excited states (i.e., for large n) in both position and momentum spaces, to be denoted by S ρ and S γ, respectively. Briefly, it is shown that, for large values of n, S ρ+1/2logλ≂log(π√2n/e)+o(1) and S γ−1/2log λ≂log(π√2n/e)+o(1), so that S ρ+S γ≂log(2π2 n/e 2)+o(1) in agreement with the generalized indetermination relation of Byalinicki‐Birula and Mycielski [Commun. Math. Phys. 44, 129–132 (1975)]. Finally, the rate of convergence of these two information entropies is numerically analyzed. In addition, using a Rakhmanov result, we describe a totally new proof of the leading term of the entropy of Freud polynomials which, naturally, is just a weak version of the aforementioned general result.