growth of energy of eigenfunctions on hyperbolic surface

2012-11-21T13:07:36Z

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

Answer by Igor Rivin for growth of energy of eigenfunctions on hyperbolic surface

2012-11-21T14:32:34Z

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.

growth of energy of eigenfunctions on hyperbolic surface - MathOverflow

growth of energy of eigenfunctions on hyperbolic surface

Answer by Igor Rivin for growth of energy of eigenfunctions on hyperbolic surface