A Pseudogap in the Single-Particle Density of States of a   Tomonaga-Luttinger Liquid
  We study a single-particle density of states (DOS) in a model of Tomonaga-Luttinger liquid with small (large) values of the Luttinger parameter $K_\rc < 0.17$, ($K_\rc > 5.82$), in the charge sector. We explain that such values of $K_\rc$ may be achieved by electron-phonon interactions without generating the spin gap. We suggest that electron-phonon interactions can be partially incorporated into the Tomonaga-Luttinger liquid scheme by introducing frequency dependent $K_\rc(\omega)$. We demonstrate that when the low-frequency asymptotic value $K_\rc(0) < 0.17$ or $K_\rc(0) > 5.82$, the single-particle DOS has a pseudogap behavior in frequency where $\rho(\omega)/\omega$ vanishes at small frequencies.   The DOS exhibits a peak the position of which scales as $[K(0) + K^{-1}(0)]\omega_0/2$ where $\omega_0$ is the characteristic phonon frequency. 
Emiliano Papa, Alexei M. Tsvelik