growth of energy of eigenfunctions on hyperbolic surface - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T02:32:51Z http://mathoverflow.net/feeds/question/114056 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114056/growth-of-energy-of-eigenfunctions-on-hyperbolic-surface growth of energy of eigenfunctions on hyperbolic surface Paul 2012-11-21T13:07:36Z 2013-01-30T19:22:00Z <p>I am looking to the behavior of eigenfunctions associated to small eigenvalues on a degenerating hyperbolic surface. </p> <p>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.</p> <p>My idea was to study $\phi_n$ in the collar using the fact this region is isometric to</p> <p>$\mathbb{H} / {z\mapsto e^l z}$. Then in polar coordinate $$\phi_n(r,\theta)=\sum_n a_n(\theta) r^{\frac{2\pi i n}{l}}$$<br> where $$a_n" +\left( \frac{\lambda_n}{sin^2(\theta)} -\left(\frac{2\pi n}{l} \right)^2\right)a_n=0$$</p> <p>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 $1/4$.</p> <p>So I am looking for any new ideas or references on that questions. Thanks in advance, Paul</p> http://mathoverflow.net/questions/114056/growth-of-energy-of-eigenfunctions-on-hyperbolic-surface/114061#114061 Answer by Igor Rivin for growth of energy of eigenfunctions on hyperbolic surface Igor Rivin 2012-11-21T14:32:34Z 2012-11-21T14:32:34Z <p>You might want to look at Chris Judge's papers, particularly:</p> <p>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. </p>