  Spin dynamics of a metallic ferromagnetic film imbedded between normal metal
layers is studied using the spin-pumping theory of Tserkovnyak et al. [Phys.
Rev. Lett. 88, 117601 (2002)]. The scattering matrix for this structure is
obtained using a spin-dependent potential with quantum well in the
ferromagnetic region. Owing to multiple reflections in the well, the excess
Gilbert damping and the gyromagnetic ratio exhibit quantum oscillations as a
function of the thickness of the ferromagnetic film. The wavelength of the
oscillations is given by the depth of the quantum well. For iron film imbedded
between gold layers, the amplitude of the oscillations of the Gilbert damping
is in an order of magnitude agreement with the damping observed by Urban et al.
[Phys. Rev. Lett. 87, 217204 (2001)]. The results are compared with the linear
response theory of Mills [Phys. Rev. B 68, 0144419 (2003)].
