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I am looking to the behavior of eigenfunctions associated to small eigenvalues on a degenerating hyperbolic surface.

Let $(\Sigma_n, h_n)$ a sequence of compact surfaces with area equal to $1$ and curvature equal to $-1$. Let $\phi_n$ an eigenfunction associated to $\lambda_n$ which goes to $0$, such that $\Vert \phi_n\Vert_2=1$. Hence we easily prove that $\Vert \nabla \phi_n \Vert_2 =\sqrt{\lambda_n}$. But I guess that the energy in smaler in the thick part, I guess we have $\Vert \nabla {\phi_n}_{\vert K} \Vert_2 =O(\lambda_n)$ on every $K\subset \Sigma_n$ where the injectivity radius in bounded from below.

My idea was to study $\phi_n$ in the collar using the fact this region is isometric to

$\mathbb{H} / \{z\mapsto e^l z\} $. Then in polar coordinate $$\phi_n(r,\theta)=\sum_n a_n(\theta) r^{\frac{2\pi i n}{l}}$$
where $$a_n" +\left( \frac{\lambda_n}{sin^2(\theta)} -\left(\frac{2\pi n}{l} \right)^2\right)a_n=0$$

whose solutions are given by Legendre functions and then I try to get some estimates on the growth of the energy. Unfortunately I didn't succeed and I didn't find references about this precise subject. Since I am not a specialist of this field, I tryed to read classical references such as Buser book or the paper of Wolpert 'Spectral limit for hyperbolic surfaces' which studies notably the growth of the $L^2$ norm for eigenfunction associated to eigenvalue bigger than $1/4$.

So I am looking for any new ideas or references on that questions. Thanks in advance, Paul

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You might want to look at Chris Judge's papers, particularly:

Judge, Christopher M.(1-IN) Tracking eigenvalues to the frontier of moduli space. I. Convergence and spectral accumulation. (English summary) J. Funct. Anal. 184 (2001), no. 2, 273–290.

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Thank you for the reference. The proof is clearer than in Wolpert but it "just" prove that $\phi_n$ converge to some $\phi_*$(in my case a constant). I would like to know the rate of convergence on the thick part. – Paul Nov 22 '12 at 13:41
You might want to just write to Chris... – Igor Rivin Nov 22 '12 at 17:57

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