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 or in momentum space, the form of an isotropic $q$-Gaussian. These wavefunctions admit a maximum entropy description in terms of $S_q$ 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 $0<q<1.$