Quantization of the conductance of ballistic point contacts beyond the adiabatic approximation

We review the adiabatic (slowly varying confining walls) approximation to the quantized conductance introduced by Glazman et al. We show that all the corrections to this approximation are exponentially small in the smoothness parameter of the constriction. A condition for accurate quantization is given, based on the result that the reflections due to the sudden widening of the constriction (existing in real devices) are highly suppressed if a small adiabatic widening and/or a potential barrier precede the sudden widening. An interesting collimation effect associated with the adiabatic picture is mentioned.