  We consider a system composed of a trapped atom and a trapped ion. The ion
charge induces in the atom an electric dipole moment, which attracts it with an
r^{-4} dependence at large distances. In the regime considered here, the
characteristic range of the atom-ion interaction is comparable or larger than
the characteristic size of the trapping potential, which excludes the
application of the contact pseudopotential. The short-range part of the
interaction is described in the framework of quantum-defect theory, by
introducing some short-range parameters, which can be related to the s-wave
scattering length. When the separation between traps is changed we observe
trap-induced shape resonances between molecular bound states and vibrational
states of the external trapping potential. Our analysis is extended to
quasi-one-dimensional geometries, when the scattering exhibit
confinement-induced resonances, similar to the ones studied before for
short-range interactions. For quasi-one-dimensional systems we investigate the
effects of coupling between the center of mass and relative motion, which
occurs for different trapping frequencies of atom and ion traps. Finally, we
show how the two types of resonances can be employed for quantum state control
and spectroscopy of atom-ion molecules.
