  In this work we study the increasing resolution of linear inverse scattering
problems at a large fixed frequency. We consider the problem of recovering the
density of a Herglotz wave function, and the linearized inverse scattering
problem for a potential. It is shown that the number of features that can be
stably recovered (stable region) becomes larger as the frequency increases,
whereas one has strong instability for the rest of the features (unstable
region). To show this rigorously, we prove that the singular values of the
forward operator stay roughly constant in the stable region and decay
exponentially in the unstable region. The arguments are based on structural
properties of the problems and they involve the Courant min-max principle for
singular values, quantitative Agmon-H\"ormander estimates, and a Schwartz
kernel computation based on the coarea formula.
