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Let $\Delta$ be the Laplacian (a positive operator) on $H^2$ the hyperbolic plane. My question is what is the expression of the eigenfunction $\Delta f= f/4$? (say in the ploar coordiante)

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    $\begingroup$ You should be more precise: given the eigenvalue you ask about, I guess you are considering the Laplacian acting on $L^2$ functions. Also, you could note that you are interested in eigenfunctions up to isometries, but that could be considered implicitly obvious. $\endgroup$
    – Benoît Kloeckner
    Commented Oct 23, 2014 at 19:41

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There are many such functions, indeed an infinite-dimensional space of them. For instance, the function $f:x+iy\mapsto y^s$ for $s\in\mathbb C$ satisfies $\Delta f=s(s-1)f$, so you can choose $s$ appropriately. Next, once you found one eigenfunction $f$, then $f\circ\gamma$ is a new one, if $\gamma$ lies in the group of biholomorphic maps of $\mathbb H$.

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