  In this paper, we continue our study [16] on the long time dynamics of radial
solutions to defocusing energy critical wave equation with a trapping radial
potential in 3 + 1 dimensions. For generic radial potentials (in the
topological sense) there are only finitely many steady states which might be
either stable or unstable. We first observe that there can be stable excited
states (i.e., a steady state which is not the ground state) if the potential is
large and attractive, although all small excited states are unstable. We prove
that the set of initial data for which solutions scatter to any one unstable
excited state forms a finite co-dimensional connected $C^1$ manifold in energy
space. This amounts to the construction of the global path-connected, and
unique, center-stable manifold associated with, but not necessarily close to,
any unstable steady state. In particular, the set of data for which solutions
scatter to unstable states has empty interior in the energy space, and generic
radial solutions scatter to one of the stable steady states. Our main tools are
(i) near any given finite energy radial initial data $(u_0,u_1)$ for which the
solution $u(t)$ scatters to some unstable steady state $\phi$ we construct a
$C^1$ manifold containing $(u_0,u_1)$ with the property that any solution
starting on the manifold scatters to $\phi$; moreover, any solution remaining
near the manifold for all positive times lies on the manifold (ii) an exterior
energy inequality from [9,10,16]. The latter is used to obtain a result in the
spirit of the one-pass theorem [22], albeit with completely different
techniques
