  We give examples of semiclassical Schr\"odinger operators with exponentially
large cutoff resolvent norms, even when the supports of the cutoff and
potential are very far apart. The examples are radial, which allows us to
analyze the resolvent kernel in detail using ordinary differential equation
techniques. In particular, we identify a threshold spatial radius where the
resolvent behavior changes. We apply these results to wave equations with
radial wavespeed, identifying a corresponding threshold radius at which wave
decay properties change.
