Quantum potentials with q-Gaussian ground states

We determine families of spherically symmetrical D-dimensional quantum potential functions V(r) having ground-state wavefunctions that exhibit, either in configuration space or in momentum space, the form of an isotropic q-Gaussian. These wavefunctions admit a maximum-entropy description in terms of Sq power-law entropies. We show that the potentials with a ground state of the q-Gaussian form in momentum space admit the Coulomb potential −1/r as a particular instance. Furthermore, all these potentials behave asymptotically as the Coulomb potential for large r for all values of the parameter q such that 0<q<1.