  We consider the resolvent on asymptotically Euclidean warped product
manifolds in an appropriate 0-Gevrey class, with trapped sets consisting of
only finitely many components. We prove that the high-frequency resolvent is
either bounded by $C_\epsilon |\lambda|^\epsilon$ for any $\epsilon>0$, or
blows up faster than any polynomial (at least along a subsequence). A stronger
result holds if the manifold is analytic. The method of proof is to exploit the
warped product structure to separate variables, obtaining a one-dimensional
semiclassical Schr\"odinger operator. We then classify the microlocal resolvent
behaviour associated to every possible type of critical value of the potential,
and translate this into the associated resolvent estimates. Weakly stable
trapping admits highly concentrated quasimodes and fast growth of the
resolvent. Conversely, using a delicate inhomogeneous blowup procedure loosely
based on the classical positive commutator argument, we show that any weakly
unstable trapping forces at least some spreading of quasimodes.
  As a first application, we conclude that either there is a resonance free
region of size $| \Im \lambda | \leq C_\epsilon | \Re \lambda |^{-1-\epsilon}$
for any $\epsilon>0$, or there is a sequence of resonances converging to the
real axis faster than any polynomial. Again, a stronger result holds if the
manifold is analytic. As a second application, we prove a spreading result for
weak quasimodes in partially rectangular billiards.

Analysis of PDEs;Spectral Theory