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)



What is the expression of first eigen function of Laplacian on Hyperbolic plane?
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 $\mathbb H$.


