= Drift-diffusion half-line global relation
{title2=$\omega(k)=k^2-i\alpha k$}
For $q_t=q_{xx}+\alpha q_x$ on $x>0$, write $Q(k)=\int_0^\infty e^{-ikx}q(x,0)dx$ and $G_j(k,t)=\int_0^t e^{\omega(k)s}\partial_x^jq(0,s)ds$. Twice integrating by parts gives $e^{\omega(k)t}\widehat q(k,t)=Q(k)-G_1(k,t)-(ik+\alpha)G_0(k,t)$ in the lower transform half-plane. The <dispersion relation> is invariant under $k\mapsto i\alpha-k$. This <global relation> eliminates one unknown boundary trace after the symmetry maps the inversion contour into the transform's analytic domain.
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