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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? –  Thomas Andrews Jul 24 '12 at 19:29
    
er sorry about that $ z=x+iy $ and $ i= \sqrt -1 $ –  Jose Garcia Jul 24 '12 at 19:34
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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? –  anon Jul 24 '12 at 19:42

1 Answer 1

Modulo some garbling or imprecisions, this is approximately correct, but perhaps the imprecisions are sufficient to require attention before proceeding further.

First, as commented, the sum used to try to define $\Psi$ is not what one wants, even for $1/2+ik$ replaces by $s\in\mathbb C$ with $\Re(s)\gg 1$ for convergence purposes. That is, even with the latter constraint, the sum should be over $\Gamma_\infty\backslash SL_2(\mathbb Z)$, where $\Gamma_\infty=\pmatrix{*&*\\0&*}$ in $SL_2(\mathbb Z)$. And that extra leading summand is a repeat of the identity element summand in the sum, so should be dropped.

Edit: also, the summands will look like $y^s/|cz+d|^{2s}$. In the expression given in the question, there is a mismatch of exponents, in the sense that the exponent of $y$ in the numerator should not be identical to the exponent of $|cz+d|$ in the denominator, but half.

With these adjustments, and with $z=x+iy$ in the complex upper half-plane, $\Psi_s(x,y)$ is (one normalization of) an Eisenstein series, and admits a meromorphic continuation to $\Re(s)=1/2$, indeed.

The other expression $G_s(z)$ above is $\zeta(s)$ times $y^{-s/2}\Psi_{s/2}$.

Yes, $\Psi_s$ is an $s(s-1)$-eigenfunction of the indicated $SL_2(\mathbb R)$-invariant differential operator.

There are many sources for this, both on-line and off-line, going back many decades. Googling "Eisenstein series" should work. Also, on my "intro to modular forms" course website, at http://www.math.umn.edu/~garrett/m/mfms/ and other course-notes pages there are various fairly accessible discussions of such things. But there are many other sources out there, too.

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