 Drift-diffusion half-line global relation

For $q_t=q_{xx}+\alpha q_x$ on $x>0$, write $Q(k)={\int }_0^{\infty }{e}^{-ikx}q(x,0)dx$ and $G_j(k,t)={\int }_0^t{e}^{\omega (k)s}{\partial }_x^{j}q(0,s)ds$. Twice integrating by parts gives $e^{\omega (k)t}\hat{q}(k,t)=Q(k)-G_1(k,t)-(ik+\alpha )G_0(k,t)$ 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.