Consider the torus $\mathbb T^2:=\mathbb C/(\mathbb Z+i \mathbb Z)$ and the operator $$ T = (2D_{\bar z}-\lambda)^{-1} $$ on the torus with periodic boundary conditions. This one is well-defined for $\lambda \notin 2\pi ( \mathbb Z+i \mathbb Z)$ where $2D_{\bar z}= (-i \partial_{x}+\partial_y).$

My question is if there is anything known about the Green's function $K$ satisfying $$ Tf(x) = \int_{\mathbb T^2} K(x,y) f(y) \ dy $$ for $f$ a periodic function.

Addendum: The case $\lambda=0$ is actually rather simple and can be easily deduced from the paper Class of hypocomplex structures on the two dimensional torus. But is probably already well-known in this much simpler case. Then, in terms of the Jacobi $\Theta$ function $$ T f(x) = \int_{\mathbb T^2} M(x,y) f(y) \ dy $$ where $$ M(x,y) = \frac{\Theta'(y-x+\frac{1+i}{2})}{\Theta(y-x+\frac{1+i}{2})}. $$

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    $\begingroup$ Hi! Details on the general case for higher dimensional surfaces can be found in Lang's book "Introduction to Arakelov theory", done by Robert Coleman. In your special case, the Green function does have a closed form and it can be expressed using Dedekind eta functions. This is well known, I think. $\endgroup$ Nov 17 at 3:55
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    $\begingroup$ @Bombyxmori interesting, so where could I find this closed form? $\endgroup$
    – Guido
    Nov 17 at 4:06
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    $\begingroup$ @Bombyxmori that is to say, I did not see that formula in the book you mentioned, but maybe I overlooked sth? $\endgroup$
    – Guido
    Nov 18 at 15:38
  • $\begingroup$ The diagonal_approximation to $M(x,y)$, for small $|x-y|$, is given on page 1073 of Dimers and famillies of Cauchy-Riemann operators. $\endgroup$ Nov 20 at 7:32
  • $\begingroup$ is the conclusion from this question that only the diagonal approximation to $M(x,y)$ is known? $\endgroup$ Nov 25 at 7:54

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