Dynamic response of one-dimensional interacting fermions

We evaluate the dynamic structure factor $S(q,\omega )$ of interacting one-dimensional spinless fermions with a nonlinear dispersion relation. The combined effect of the nonlinear dispersion and of the interactions leads to new universal features of $S(q,\omega )$. The sharp peak $S\propto q\delta(\omega-uq)$, characteristic for the Tomonaga-Luttinger model, broadens up; $S(q,\omega )$ for a fixed $q$ becomes finite at arbitrarily large $\omega$. The main spectral weight, however, is confined to a narrow frequency interval of the width $\delta \omega \sim q^2\mathrm{/}m$. At the boundaries of this interval the structure factor exhibits power-law singularities with exponents depending on the interaction strength and on the wave number $q$.