  We study the Complex Absorbing Potential (CAP) Method in computing quantum
resonances of width $c(h) = O(h^N)$, $N\gg1$. We show that up to
$h^{-M}\sqrt{c(h)} +\Oh$ error, $M\gg1$, resonances are perturbed eigenvalues
of the CAP Hamiltonian $P(h)-iW$, and vice versa, where $W$ is the CAP with
non-negative real part supported outside the trapping region. In some cases,
the error terms are exponentially small.
