  Upon increasing the electron density in a quantum wire, the one-dimensional
electron system undergoes a transition to a quasi-one-dimensional state. In the
absence of interactions between electrons, this corresponds to filling up the
second subband of transverse quantization, and there are two gapless excitation
modes above the transition. On the other hand, strongly interacting
one-dimensional electrons form a Wigner crystal, and the transition corresponds
to it splitting into two chains (zigzag crystal). The two chains are locked, so
their relative motion is gapped, and only one gapless mode remains. We study
the evolution of the system as the interaction strength changes, and show that
only one gapless mode exists near the transition at any interaction strength.
