  The transport properties of an interacting one-dimensional quantum dot
capacitively coupled to an atomic force microscope probe are investigated. The
dot is described within a Luttinger liquid framework which captures both
Friedel and Wigner oscillations. In the linear regime, we demonstrate that both
the conductance peak position and height oscillate as the tip is scanned along
the dot. A pronounced beating pattern in the conductance maximum is observed,
connected to the oscillations of the electron density. Signatures of the
effects induced by a Wigner molecule are clearly identified and their stability
against the strength of Coulomb interactions are analyzed. While the
oscillations of the peak position due to Wigner get enhanced at strong
interactions, the peak height modulations are suppressed as interactions grow.
Oscillations due to Friedel, on the other hand, are robust against interaction.
