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
