  We have investigated the intersubband transitions of surface state electrons
(SSE) on liquid $^3$He induced by microwave radiation at temperatures from 1.1
K down to 0.01 K. Above 0.4 K, the transition linewidth is proportional to the
density of $^3$He vapor atoms. This proportionality is explained well by Ando's
theory, in which the linewidth is determined by the electron - vapor atom
scattering. However, the linewidth is larger than the calculation by a factor
of 2.1. This discrepancy strongly suggests that the theory underestimates the
electron - vapor atom scattering rate. At lower temperatures, the absorption
spectrum splits into several peaks. The multiple peak structure is partly
attributed to the spatial inhomogeneity of the static holding electric field
perpendicular to the electron sheet.
