Change of variable in integral of product of exponential and gaussian functions

I have the integral

$\int_{-\infty}^{\infty}dx \int_{-\infty}^{\infty}dy e^{-\xi \vert x-y\vert}e^{-x^2}e^{-y^2}$

where $\xi$ is a constant. I would like to transform by some change of variables in the form

$\int_{-\infty}^{\infty}dx F(x) \int_{-\infty}^{\infty}dy G(y)$

the problem is that due to absolute value in the integral one must take in account where x is greater or less than y,

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 Recognitions: Homework Help First observe that $e^{-\xi \vert x-y\vert}e^{-x^2}e^{-y^2}=e^{-\xi \vert x-y\vert}e^{-(x-y)^2/2}e^{-(x+y)^2/2}$ Then you can either change variables such as u=(x+y)/sqrt(2) v=(x-y)/sqrt(2) or break into two regions xy
 Hi ! the clolsed form of the integral involves a special function (erf). Attached Thumbnails

Change of variable in integral of product of exponential and gaussian functions

Nice trick! Thank you so much!