= Resolvent kernel for Dirichlet advection-diffusion on an interval
{title2=$e^{-ax}R_q(x,y)e^{ay}$}
After the <Dirichlet gauge transform for constant drift>, put $q=\sqrt{p+a^2}$. The <Dirichlet Green function> of $p-\partial_x^2+a^2$ is
$$
R_q(x,y)=\frac{\sinh(q\min(x,y))\sinh(q[L-\max(x,y)])}{q\sinh(qL)}.
$$
The original unweighted resolvent kernel is $e^{-ax}R_q(x,y)e^{ay}$. Its poles are $p=-a^2-(n\pi/L)^2$. The kernel is even in $q$, so the apparent square-root branch point is removable.
Back to article page
