  The proximity of antiferromagnetic order in high-temperature superconducting
materials is considered a possible clue to the electronic excitations which
form superconducting pairs. Here we study the transverse and longitudinal spin
excitation spectrum in a one-band model in the pure spin density wave (SDW)
state and in the coexistence state of SDW and the superconductivity. We start
from a Stoner insulator and study the evolution of the spectrum with doping,
including distinct situations with only hole pockets, with only electron
pockets and with pockets of both types. In addition to the usual spin-wave
modes, in the partially gapped cases we find significant weight of low-energy
particle-hole excitations. We discuss the implications of our findings for
neutron scattering experiments and for theories of Cooper-pairing in the
metallic SDW state.
