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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 $Spropto qdelta(omega-uq)$, 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 $deltaomegasim q^2/m$. 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 $q$.
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The density distribution of the one-dimensional Hubbard model in a harmonic trapping potential is investigated in order to study the effect of the confining trap. Strong superimposed oscillations are always present on top of a uniform density cloud, which show universal scaling behavior as a function of increasing interactions. An analytical formula is proposed on the basis of bosonization, which describes the density oscillations for all interaction strengths. The wavelength of the dominant oscillation changes with interaction, which indicates the crossover to a spin-incoherent regime. Using the Bethe ansatz the shape of the uniform fermion cloud is analyzed in detail, which can be described by a universal scaling form.
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