  Quantum point contacts exhibit mysterious conductance anomalies in addition
to well known conductance plateaus at multiples of 2e^2/h. These 0.7 and
zero-bias anomalies have been intensively studied, but their microscopic origin
in terms of many-body effects is still highly debated. Here we use the charged
tip of a scanning gate microscope to tune in situ the electrostatic potential
of the point contact. While sweeping the tip distance, we observe repetitive
splittings of the zero-bias anomaly, correlated with simultaneous appearances
of the 0.7 anomaly. We interpret this behaviour in terms of alternating
equilibrium and non-equilibrium Kondo screenings of different spin states
localized in the channel. These alternating Kondo effects point towards the
presence of a Wigner crystal containing several charges with different
parities. Indeed, simulations show that the electron density in the channel is
low enough to reach one-dimensional Wigner crystallization over a size
controlled by the tip position.
