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

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  • $\begingroup$ Interesting question. Where did you get the question from? $\endgroup$ Dec 3 '17 at 5:56
  • $\begingroup$ @JackyChong. That's the Schrödinger equation in 2 space dimensions ($x$ and $y$). Here $z$ is the time dimension. $\endgroup$
    – md2perpe
    Dec 3 '17 at 7:22
  • $\begingroup$ @md2perpe Thank you. That much I already know. But what I don't know is whether the solution is just the regular Schrodinger kernel. $\endgroup$ Dec 3 '17 at 7:31
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Here you can find the solution of the 2D diffusion equation. Just replace $t$ with $z$ and $k$ with $i/a$ to get "your" $G$.

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Using the observation made by md2perpe, I believe we have

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

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

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

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