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Denote the function $$ \Psi (x,y)= y^{1/2+ik}+ \sum_{g\in SL(2,\Bbb Z)} \frac{y^{1/2+ik}} {|c_{g}z+d_{g}|^{1/2+ik}}\tag{1}$$

My question is if I can write the wave function in terms of the Eisenstein series

$$ G_{s}(z)= \sum_{(c,d)\in\Bbb Z\setminus(0,0)}|cz+d|^{-s}$$ as the solution

$$\Psi (x,y)= y^{1/2+ik}+ y^{1/2+ik}G_{1/2+ik}(z)$$

Equation $(1)$ is the solution to the Laplace equation $ y^{2}( \partial _{x}^{2}+ \partial _{y}^{2})+(1-s)s=0$ imposing boundary conditions on $SL(2,\Bbb Z)$. [I assume $s=1/2+ik$ and $z=x+iy$, $-$anon]

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    What is $z$ in the first function?2012-07-24
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    er sorry about that $ z=x+iy $ and $ i= \sqrt -1 $2012-07-24
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    For every $a,b,c,d$ with $ad-bc=1$, we have $$\begin{pmatrix}a+nc&b+nd\\c&d\end{pmatrix}\in SL(2,\Bbb Z)$$ for each $n\in\Bbb Z$, so I don't see how the sum in $(1)$ is well-defined (each term appears an infinite number of times). Also note that $g\in SL(2,\Bbb Z)\implies \gcd(c_g,d_g)=1$ via Bezout's identity, so not every pair $\in\Bbb Z^2\setminus(0,0)$ corresponds to a $g\in SL(2,\Bbb Z)$. Is it supposed to be a Haar integral instead? Can you provide some sort of reference?2012-07-24

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