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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. –  plusepsilon.de Mar 21 '12 at 17:47
    
'k' is what physics call the WAVE NUMBER whose square is the energy of the particle in classical and quantum mechanics :) –  Jose Garcia Mar 21 '12 at 18:22
    
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... –  plusepsilon.de Mar 21 '12 at 18:59

1 Answer 1

In fact, for all complex "k" (in your notation), that series (analytically continued, anyway, because it only converges for Re(1/2+ik)>1...) is an eigenfunction for the hyperbolic Laplacian. These eigenfunctions are never square-integrable on the quotient $SL(2,\mathbb Z)\backslash H$, however. Nevertheless, integrals (often called "wave packets") over $k$ of these Eisenstein series are square integrable.

Apart from the periodicity requirements, there really are no boundary conditions here. We have a continuum of these "Eisenstein series".

In fact, the bulk of the space of square-integrable $SL(2,\mathbb Z)$-periodic functions on the upper half-plane is made of up "cuspforms", which are genuine square-integrable eigenfunctions for this Laplace-Beltrami operator. It is very difficult to give explicit numerical examples of these cuspforms, despite the fact (proven by Selberg in the 1950s, and thereafter) that they satisfy "Weyl's Law" about asymptotics of eigenfunctions.

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