  By definition, the exterior asymptotic energy of a solution to a wave
equation on $\mathbb{R}^{1+N}$ is the sum of the limits as $t\to \pm\infty$ of
the energy in the the exterior $\{|x|>|t|\}$ of the wave cone. In our previous
work (JEMS 2012, arXiv:1003.0625), we have proved that the exterior asymptotic
energy of a solution of the linear wave equation in odd space dimension $N$ is
bounded from below by the conserved energy of the solution.
  In this article, we study the analogous problem for the linear wave equation
with a potential \begin{equation}
  \label{abstractLW} \tag{*} \partial_t^2u+L_Wu=0,\quad L_W:=-\Delta
-\frac{N+2}{N-2}W^{\frac{4}{N-2}} \end{equation} obtained by linearizing the
energy critical wave equation at the ground-state solution $W$, still in odd
space dimension. This equation admits nonzero solutions of the form $A+tB$,
where $L_WA=L_WB=0$ with vanishing asymptotic exterior energy. We prove that
the exterior energy of a solution of \eqref{abstractLW} is bounded from below
by the energy of the projection of the initial data on the orthogonal
complement of the space of initial data corresponding to these solutions. This
will be used in a subsequent paper to prove soliton resolution for the
energy-critical wave equation with radial data in all odd space dimensions.
  We also prove analogous results for the linearization of the energy-critical
wave equation around a Lorentz transform of $W$, and give applications to the
dynamics of the nonlinear equation close to the ground state in space
dimensions $3$ and $5$.
