Pining effect in Pierles doped systems with deviation from half-filling of energy band

A study is made of the effect of deviation from half-filling of the energy band ($\mu \ne 0$) on the Fröhlich collective mode in onedimensional impurity systems. A low impurity concentration is considered, and the infinite series of impurity scattering is taken into account self-consistently in the determination of the collective mode Green's function. The conductivity $\sigma (\omega)$ is found in terms of this Green's function, and an analytic expression is obtained for $\sigma (\omega)$ at $\omega \sim \omega _T$ ($\omega _T$ is the pinning frequency). It is shown that for the ratio $\operatorname {Re}\frac {\sigma (\omega)}{\sigma _{\max}}$ a universal formula arises. It differs from the results of Kurihara in the expression for $\omega _T$, which contains an essential dependence on $\mu$ in the incommensurate state of the charge density wave. It is also shown that the width of the peak in the dependence $\sigma (\omega)$ and its position increase with increasing $\mu$.