  We evaluate the p-wave $K^-N$ amplitudes from the chiral Lagrangians and from
there construct the p-wave part of the $K^-$ nucleus optical potential plus a
small s-wave part induced from the elementary p-wave amplitude and the nuclear
Fermi motion. Simultaneously, the momentum and energy dependence of the s-wave
optical potential, previously developed, are taken into account and shown to
generate a small p-wave correction to the optical potential. All the
corrections considered are small compared to the leading s-wave potential, and
lead to changes in the shifts and widths which are smaller than the
experimental errors.
  A thorough study of the threshold region and low densities is conducted,
revealing mathematical problems for which a physical solution is given.
