R\'enyi and Tsallis information entropies for the Darboux III quantum nonlinear oscillator

The Darboux III oscillator is an exactly solvable $N$-dimensional nonlinear oscillator defined on a radially symmetric space with non-constant negative curvature. Its one-dimensional version can be seen as a position dependent mass system whose mass function $\mu = (1 + \lambda x^2)$ depends on the nonlinearity parameter $\lambda$, such that in the limit $\lambda \to 0$ the harmonic oscillator is recovered. In this paper, a detailed study of the entropic moments and of the R\'enyi and Tsallis information entropies for the quantum version of the one-dimensional Darboux III oscillator is presented. In particular, analytical expressions for the aforementioned quantities in position space are obtained. Since the Fourier transform of the Darboux III wave functions does not admit a closed form expression, a numerical analysis of these quantities has been performed. Throughout the paper the interplay between the entropy parameter $\alpha$ and the nonlinearity parameter $\lambda$ is analysed, and known results for the Shannon entropy of the Darboux III and for the R\'enyi and Tsallis entropies of the harmonic oscillator are recovered in the limits $\alpha \to 1$ and $\lambda \to 0$, respectively. Finally, motivated by the strong non-linear effects arising when large values of $\lambda$ and/or highly excited states are considered, an approximation to the probability density function valid in those regimes is presented. From it, an analytical approximation to the probability density in momentum space can be obtained, and some of the previously observed effects arising from the interplay between $\alpha$ and $\lambda$ can be explained.