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Let $I_C[F]=(2\pi {)}^{-1}{\int }_C{e}^{ikx-(k^2-i\alpha k)t}F(k)dk$. The drift-diffusion half-line global relation and the half-line drift null contour identity give the formula in the secondary title. Its real-space kernel is the difference of two drifted Gaussian heat kernels, and its boundary-forcing kernel is $x(2\sqrt{\pi }{t}^{3/2}{)}^{-1}\exp [-(x+\alpha t{)}^2/(4t)]$. This kernel is an approximate identity in time as $x\downarrow 0$, which verifies the Dirichlet boundary condition.

Set $\omega (k)=k^2-i\alpha k$ and $\nu (k)=i\alpha -k$, so $\omega (\nu )=\omega (k)$. For boundary traces $h_j(s)={\partial }_x^{j}q(0,s)$, defineThe Half-range Fourier transform of the equation is obtained by two integrations by parts:Solving this scalar equation gives the drift-diffusion half-line global relationIn particular the reflected relation is valid on $L$, since $\mathrm{Im}\nu (k)\le 0$ there.

In the region between the real axis and $L$, $\mathrm{Re}\omega \ge 0$. Thus a boundary contribution $e^{-\omega t}{G}_j={\int }_0^t{e}^{-\omega (t-s)}{h}_j(s)ds$ admits an upward contour deformation from the real axis to $L$. Large-contour estimates follow from the spatial exponential and the time integral; at the upper time endpoint one can integrate in time once or use a small contour displacement. The initial-data transform is retained on the real axis. Hence, if $I_C[F]=(2\pi {)}^{-1}{\int }_C{e}^{ikx-\omega (k)t}F(k)dk$, ordinary inversion first givesFor the Dirichlet boundary condition, take $h_0=g_0$. Subtract the null integral of the reflected time-dependent transform, using the inversion from part (a) with $c=-1$. The unknown derivative transform cancels because $i\nu +\alpha =-ik$. This derives the Dirichlet spectral formula for half-line constant drift:Only initial and prescribed boundary data occur in this expression.

For the Neumann boundary condition, take $h_1=g_1$. The reflected global relation reads $e^{\omega t}\hat{q}(\nu ,t)=Q(\nu )-G_1+ik{G}_0$. Solve for $G_0$ and substitute it in the preliminary representation. The leftover term is a null integral with multiplier $(ik+\alpha )/(ik)$, analytic above $L$; its pole at zero is below the contour. Thus the Neumann spectral formula for half-line constant drift isThere is no residue from the origin, since this elimination never moves the contour across it. The given derivative is the inward-coordinate derivative $q_x$, not the outward derivative $-q_x$. The initial compatibility for this case is $q_0^{\prime }(0)=g_1(0)$.