Renyi and Tsallis entropies: three analytic examples

A comparative study of 1D quantum structures which allows analytic expressions for the position and momentum Renyi R(alpha) and Tsallis T(alpha) entropies, focuses on extracting the most characteristic physical features of these one-parameter functionals. Consideration of the harmonic oscillator reconfirms the special status of the Gaussian distribution: at any parameter a it converts into the equality both Renyi and Tsallis uncertainty relations removing for the latter an additional requirement 1/2 <= alpha <= 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 a where momentum components exist strongly depends on the position potential and/or boundary condition for the position wave function. Asymptotic limits reveal that in either space the entropies R(alpha) and T(alpha) approach their Shannon counterpart, alpha = 1, along different paths. Similarities and differences between the two entropies and their uncertainty relations are exemplified. Some unsolved problems are also indicated.