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given the PDE Eigenvalue problem $ y^{2}( \partial _{x}^{2}f(x,y) +\partial _{y}^{2}f(x,y))= E_{n}f(x,y) $ (1)

if we are on the poincare disc so i impose the conditions

$ x'=x+1$ invariance

$ y' = \frac{y}{|cz+d|^{2}} $ invariance

then i can get the solution $ f(x,y)=y^{1/2+ik} \sum_{g}|cz+d|^{-1-2ik}+c.c $

here c.c complex conjugation and $ k^{2}+1/4=E $ with E the Eigenvalues of the equiaton (1)

once i get the solution how can be 'k' obtained from the boundary conditions ??

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    $k$ is presumably the weight? Usually it enters the PDE and the invariance.2012-03-21
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    'k' is what physics call the WAVE NUMBER whose square is the energy of the particle in classical and quantum mechanics :)2012-03-21
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    You should perhaps look at number theory stuff here. The things you are looking at are so called real analytic Eisenstein series and $k$ is the weight. I am always confuesed, why certain subfields adopt theories of other subfields, without sticking to the original notation...2012-03-21

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