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 Andos 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.
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