What is the Green's function of $\partial_x^2 + \partial_y^2 + ia\partial_z $?

Question

What is the solution to $G(\mathbf x)$ to $$(\partial_x^2 + \partial_y^2 + ia\partial_z)G(\mathbf x)=\delta (\mathbf x)$$ ?

Answer 1

Here you can find the solution of the 2D diffusion equation. Just replace $t$ with $z$ and $k$ with $i/a$ to get "your" $G$.

Answer 2

Using the observation made by md2perpe, I believe we have

$$\Big(\partial_z-\frac{i}{a}\nabla_{xy}\Big)(ia G(\mathbf x))= \delta(\mathbf x)$$

$$ia G(\mathbf x) =\Theta(z)\left(\frac{a}{4\pi i z}\right)\mathrm e^{ia \rho^2/(4z)}$$

$$ G(\mathbf x) = -\frac{\Theta(z)}{4\pi z}e^{ia \rho^2/(4z)} $$