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The so-called Darboux III oscillator is an exactly solvable $N$-dimensionalnonlinear oscillator defined on a radially symmetric space with non-constantnegative curvature. This oscillator can be interpreted as a smooth(super)integrable deformation of the usual $N$-dimensional harmonic oscillatorin terms of a non-negative parameter $\lambda$ which is directly related to thecurvature of the underlying space. In this paper, a detailed study of theShannon information entropy for the quantum version of the Darboux IIIoscillator is presented, and the interplay between entropy and curvature isanalysed. In particular, analytical results for the Shannon entropy in theposition space can be found in the $N$-dimensional case, and the known resultsfor the quantum states of the $N$-dimensional harmonic oscillator are recoveredin the limit of vanishing curvature $\lambda \to 0$. However, the Fouriertransform of the Darboux III wave functions cannot be computed in exact form,thus preventing the analytical study of the information entropy in momentumspace. Nevertheless, we have computed the latter numerically both in the oneand three-dimensional cases and we have found that by increasing the absolutevalue of the negative curvature (through a larger $\lambda$ parameter) theinformation entropy in position space increases, while in momentum space itbecomes smaller. This result is indeed consistent with the spreading propertiesof the wave functions of this quantum nonlinear oscillator, which areexplicitly shown. The sum of the entropies in position and momentum spaces hasbeen also analysed in terms of the curvature: for all excited states such totalentropy decreases with $\lambda$, but for the ground state the total entropy isminimised when $\lambda$ vanishes, and the corresponding uncertainty relationis always fulfilled.

Shannon information entropy for a quantum nonlinear oscillator on a space of non-constant curvature