Upper bounds on Shannon and Renyi entropies for central 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 derive upper bounds on these quantities in terms of the second-order moment < r(2)> for general central potentials. This improves previous results of this type. The proof uses the Renyi maximization procedure with a covariance constraint due to Costa et al. [in Proceedings of the Fourth International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR), edited by A. Rangarajan, M. A. T. Figueiredo, and J. Zerubia (Springer-Verlag, Lisbon, 2003), [Lect. Notes Comput. Sci. 52, 211 (2003).]] The contributions to these bounds coming from the radial and angular parts of the physical wave functions are taken into account. Finally, the application to the d-dimensional (d >= 3) hydrogenic and oscillator-like systems is provided. (C) 2011 American Institute of Physics. [doi:10.1063/1.3549585]