Rigorous bounds for Renyi entropies of spherically symmetric potentials

The Renyi and Shannon entropies are information-theoretic measures which have enabled to formulate the position-momentum uncertainty principle in a much more adequate and stringent way than the (variance-based) Heisenberg-like relation. Moreover, they are closely related to various energetic density-functionals of quantum systems. Here we find sharp upper bounds to these quantities in terms of the second order moment < r(2)> for general spherically symmetric potentials, which substantially improve previous results of this type, by means of the Reny maximization procedure with a covariance constraint due to Costa, Hero and Vignat [1]. The contributions to these bounds coming from the radial and angular parts of the physical wave functions are explicitly given.