  We apply the random phase approximation (RPA) and its extension called
renormalized RPA to the quantum anharmonic oscillator with an O(2) symmetry. We
first obtain the equation for the RPA frequencies in the standard and in the
renormalized RPA approximations using the equation of motion method. In the
case where the ground state has a broken symmetry, we check the existence of a
zero frequency in the standard and in the renormalized RPA approximations. Then
we use a time-dependent approach where the standard RPA frequencies are
obtained as small oscillations around the static solution in the time-dependent
Hartree-Bogoliubov equation. We draw a parallel between the two approaches.
