  The asymptotic analysis of a linear high-field Wigner-BGK equation is
developped by a modified Chapman-Enskog procedure. By an expansion of the
unknown Wigner function in powers of the Knudsen number $\epsilon$, evolution
equations are derived for the terms of zeroth and first order in $\epsilon$. In
particular, it is obtained a quantum drift-diffusion equation for the position
density, which is corrected by field-dependent terms of order $\epsilon$.
Well-posedness and regularity of the approximate problems are established, and
it is proved that the difference between exact and asymptotic solutions is of
order $\epsilon ^2$, uniformly in time and for arbitrary initial data.
