  We compute asymptotics of eigenvalues approaching the bottom of the
continuous spectrum, and associated resonances, for Schr\"odinger operators in
dimension two. We distinguish persistent eigenvalues, which have associated
resonances, from disappearing ones, which do not. We illustrate the
significance of this distinction by computing corresponding scattering phase
asymptotics and numerical Breit--Wigner peaks. We prove all of our results for
circular wells, and extend some of them to more general problems using recent
resolvent techniques.
