Friedel oscillations in a gas of interacting one-dimensional fermionic atoms confined in a harmonic trap

Using an asymptotic phase representation of the particle density operator ρ̂(z) in the one-dimensional harmonic trap, the part δ ρ̂_F(z) which describes the Friedel oscillations is extracted. The expectation value 〈δ ρ̂_F(z)〉 with respect to the interacting ground state requires the calculation of the mean square average of a properly defined phase operator. This calculation is performed analytically for the Tomonaga-Luttinger model with harmonic confinement. It is found that the envelope of the Friedel oscillations at zero temperature decays with the boundary exponent ν = (K + 1)/2 away from the classical boundaries. This value differs from that known for open boundary conditions or strong pinning impurities. The soft boundary in the present case thus modifies the decay of Friedel oscillations. The case of two components is also discussed.