The conductivity of carriers in a quasi-one-dimensional electron system over liquid helium is measured in the temperature interval0.5−1.9 Kin confining electric fields up to 4 kV/cm at a frequency of 1.1 MHz. Quasi-one-dimensional channels were created by using an optical diffraction grating covered with a thin helium layer. The carrier conductivity decreases exponentially with temperatureT,and the activation energy is of the order of a few degrees, thus pointing towards localization of electrons in a quasi-one-dimensional electron system. As the thickness of the helium layer covering the grating is increased, a departure from a mono exponential dependence is observed atT<0.8 K,which indicates that quantum effects begin to play an active role in electron mobility at these temperatures. An analysis of the obtained results leads to the assumption that under localization conditions, quasi-one-dimensional electron systems may contain two branches of the optical mode of plasma oscillations, viz., a high-frequency branch associated with electron oscillations in potential wells, and a low-frequency branch associated with the oscillations of the electron-dimple complex with a large effective mass. ©1997 American Institute of Physics.