A one-dimensional system of a particle interacting with a dynamic potential well is studied as a primitive analog, with regard to methods and results, for more realistic three-dimensional models. A $\ensuremath{\delta}$-function approximation (linear in the deformation) to the interaction leads to an unnormalizable wave function and negatively infinite eigenenergies for levels of the ground-state nucleonic band when a certain parameter of the system exceeds unity. Results of numerical computations of eigenenergies by the following approximation methods are given and compared: (1) adiabatic Born-Oppenheimer, (2) static Rainwater, (3) no-recoil, single-exciton Tamm-Dancoff, (4) energy-shell, ${t}_{m}$-exciton TammDancoff. Of these, the first is based on the "true" interaction, the last three on the \ensuremath{\delta} function. The no-recoil, unlike the energy-shell, procedure is capable of yielding a reasonable approximation to eigenenergies of the true interaction for the ground-state band in spite of the divergence. Eigenenergies of a higher band computed by method (4) as functions of coupling strength and of ${t}_{m}$ for ${t}_{m}\ensuremath{\le}4$ converge rapidly but are less depressed by the interaction than those computed by (1).Scattering of the particle by the well is similarly investigated by (1) distorted-wave Born, (2) adiabatic, and (3) energy-shell Tamm-Dancoff approximations. From numerical computations, resonances through quasi-bound intermediate states are found to dominate the energy dependence of cross sections except at very weak coupling, so that the Born and adiabatic approximations are unsatisfactory. The single-level Breit-Wigner formula is shown to be unsuitable for describing these resonances when inelastic channels are open. The single-particle level structure of the potential well modulates the resonances, and single-particle antiresonances suppress the effect of intermediate states of lower particle energy, influencing convergence with respect to the number of wall excitons included, ${t}_{m}$. Qualitative inferences are made concerning three-dimensional models involving collective nuclear motion.