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 ??