Dirichlet gauge transform for constant drift ($w=e^{ax}u$)

For the advection-diffusion equation $u_t=u_{xx}+2a{u}_x$, the multiplication $w=e^{ax}u$ gives $w_t=w_{xx}-a^{2}w$. A Dirichlet boundary condition remains a prescribed value, multiplied by $e^{ax}$ at the endpoint. This real exponential conjugation removes the first derivative without changing a bounded spatial interval.

After the Dirichlet gauge transform for constant drift, put $q=\sqrt{p+a^2}$. The Dirichlet Green function of $p-{\partial }_x^2+a^2$ isThe 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.