  We develop a realistic and analytically tractable model to describe the spin
current which arises in a quantum point contact (QPC) with spin-orbit
interaction (SOI) upon a small voltage is applied. In the model, the QPC is
considered as a saddle point of two-dimensional potential landscape. The SOI
acts within a finite region and is absent deep in the reservoirs. The SOI
strength is not supposed to be strong. It is shown that the spin polarization
appears in the third order of the perturbation theory as a result of definite
combinations of electron transitions. They include two intersubband transitions
to nearest subbands and one intrasubband transition. The spin current is
proportional to the cube of the SOI strength and strongly depends on geometric
parameters of the saddle point. The spin is polarized in the plane of the QPC
and directed normally to the electron current if the SOI is of Rashba type. As
a function of the saddle-point potential (i.e., the height of the QPC barrier),
the spin conductance and especially the spin polarization have characteristic
features (specifically, peaks) correlated with the charge conductance
quantization steps. The peak shape depends on the length of the region where
the SOI acts. In QPCs with sharp potential landscape, this picture is distorted
by interference processes.
