On the resolvent of the Laplacian on functions for degenerating surfaces of finite geometry

In this paper we describe a simple method that allows for a fast direct computation of the scattering matrix for a surface with hyperbolic cusps from the Neumann-to-Dirichlet map on the compact manifold with boundary obtained by removing the cusps. We illustrate that even if the Neumann-to-Dirichlet map is obtained by a Finite Element Method (FEM) one can achieve good accuracy for the scattering matrix. We give various interesting examples of how this can be used to investiga...

We study the distribution of resonances for geometrically finite hyperbolic surfaces of infinite area by countting resonances numerically. The resonances are computed as zeros of the Selberg zeta function, using an algorithm for computation of the zeta function for Schottky groups. Our particular focus is on three aspects of the resonance distribution that have attracted attention recently: the fractal Weyl law, the spectral gap, and the concentration of decay rates.

We study the behavior of the spectrum of the Dirac operator on degenerating families of compact Riemannian surfaces, when the length $t$ of a simple closed geodesic shrinks to zero, under the hypothesis that the spin structure along the pinched geodesic is non-trivial. The difficulty of the problem stems from the non-compactness of the limit surface, which has finite area and two cusps. The main idea in this investigation is to construct an adapted pseudodifferential calculus...

We obtain an essential spectral gap for a convex co-compact hyperbolic surface $M=\Gamma\backslash\mathbb H^2$ which depends only on the dimension $\delta$ of the limit set. More precisely, we show that when $\delta>0$ there exists $\varepsilon_0=\varepsilon_0(\delta)>0$ such that the Selberg zeta function has only finitely many zeroes $s$ with $\Re s>\delta-\varepsilon_0$. The proof uses the fractal uncertainty principle approach developed by Dyatlov-Zahl [arXiv:1504.06589...

We consider a surface M with constant curvature cusp ends and its Eisenstein functions E_j(\lambda). These are the plane waves associated to the j-th cusp and the spectral parameter \lambda, (\Delta - 1/4 - \lambda^2)E_j = 0. We prove that as Re\lambda \to \infty and Im\lambda \to \nu > 0, E_j converges microlocally to a certain naturally defined measure decaying exponentially along the geodesic flow. In particular, for a sequence of \lambda's corresponding to scattering reso...

Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $\chi$ denote a finite dimensional unitary representation of the fundamental group of $M$. Let $\Delta$ denote the hyperbolic Laplacian which acts on smooth sections of the flat bundle over $M$ associated to $\chi$. From the spectral theory of $\Delta$, there are three distinct sequences of numbers: The first coming from the eigenvalues of $L^{2}$ eigenfunctions, t...

We prove that eigenfunctions of the Laplacian on a compact hyperbolic surface delocalise in terms of a geometric parameter dependent upon the number of short closed geodesics on the surface. In particular, we show that an $L^2$ normalised eigenfunction restricted to a measurable subset of the surface has squared $L^2$-norm $\varepsilon>0$, only if the set has a relatively large size -- exponential in the geometric parameter. For random surfaces with respect to the Weil-Peters...

We give some estimates for the asymptotic orders of degenerating Eisenstein series for some families of degenerating punctured Riemann surfaces, which is motivated by the question identifying $L_{2}$-cohomology of the Takhtajan-Zograf metric that is originally asked by To and Weng.

We apply topological methods to study the smallest non-zero number $\lambda_1$ in the spectrum of the Laplacian on finite area hyperbolic surfaces. For closed hyperbolic surfaces of genus two we show that the set $\{S \in {\mathcal{M}_2}: {\lambda_1}(S) > 1/4 \}$ is unbounded and disconnects the moduli space ${\mathcal{M}_2}$.

The relation between expander graphs and spectral gap for eigenvalues of the Laplacian on Riemannian manifolds is well-known from the work of Brooks, Burger, Sunada, and Bourgain--Gamburd--Sarnak. In this paper we prove an analogous result for resonances of convex compact hyperbolic surfaces. Given such a surface $X$, let $\delta$ be the dimension of its limit set, let $(X_n)_{n\in \mathbb{N}}$ be a family of finite coverings of $X$, and let $(\mathcal{G}_n)_{n\in \mathbb{N}}...