We compute asymptotics of  eigenvalues approaching the bottom of the continuous spectrum, and associated resonances, for Schrödinger operators in dimension two for which the potential depends on a parameter.  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.  While we concentrate on the case of the circular well for 
illustrative and computational purposes, we also prove some of our results for more general potentials, using recent results on low-energy resonance expansions.

This talk is based on joint work with Kiril Datchev and Colton Griffin,  and is part of a larger project with K. Datchev.