Abstract
A comparative study of one-dimensional quantum structures which allow analytic expressions for the position and momentum R\'{e}nyi R(α) and Tsallis T(α) entropies, focuses on extracting the most characteristic physical features of these one-parameter functionals. Consideration of the harmonic oscillator reconfirms a special status of the Gaussian distribution: at any parameter α it converts into the equality both R\'{e}nyi and Tsallis uncertainty relations removing for the latter an additional requirement 1/2≤α≤1 that is a necessary condition for all other geometries. It is shown that the lowest limit of the semi infinite range of the dimensionless parameter α where \emph{momentum} components exist strongly depends on the \emph{position} potential and/or boundary condition for the \emph{position} wave function. Asymptotic limits reveal that in either space the entropies R(α) and T(α) approach their Shannon counterpart, α=1, along different paths. Similarities and differences between the two entropies and their uncertainty relations are exemplified. Some unsolved problems are pointed at too.