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

This expository article, written for the proceedings of the Journ\'ees EDP (Roscoff, June 2017), presents recent work joint with Jean Bourgain [arXiv:1612.09040] and Long Jin [arXiv:1705.05019]. We in particular show that eigenfunctions of the Laplacian on hyperbolic surfaces are bounded from below in $L^2$ norm on each nonempty open set, by a constant depending on the set but not on the eigenvalue.

We generalise the notion of the Pseudo-Laplacian on a hyperbolic Riemann surface with one cusp, that was studied by Lax and Phillips and Colin de Verdi\`ere, by considering a boundary condition of Robin type for the constant term instead of the classical Dirichlet condition. The resulting family of Pseudo-Laplacians is a holomorphic family of unbounded operators in the sense of Kato. By use of holomorphic perturbation theory, we study the eigenvalues and eigenvectors of this ...

The eigenvalues of the Laplace-Beltrami operator and the integrals of products of eigenfunctions and holomorphic $s$-differentials satisfy certain consistency conditions on closed hyperbolic surfaces. These consistency conditions can be derived by using spectral decompositions to write quadruple overlap integrals in terms of triple overlap integrals in different ways. We show how to efficiently construct these consistency conditions and use them to derive upper bounds on eige...

For a geometrically finite hyperbolic surface of infinite volume we write down the spectral decomposition for the Laplacian on 1-forms, generalize the Kudla and Millson's construction of hyperbolic Eisenstein series and other related results.

We develop a unified approach to the construction of the hyperbolic and elliptic Eisenstein series on a finite volume hyperbolic Riemann surface. Specifically, we derive expressions for the hyperbolic and elliptic Eisenstein series as integral transforms of the kernel of a wave operator. Established results in the literature relate the wave kernel to the heat kernel, which admits explicit construction from various points of view. Therefore, we obtain a sequence of integral tr...

For a hyperbolic surface embedded eigenvalues of the Laplace operator are unstable and tend to become resonances. A sufficient dissolving condition was identified by Phillips-Sarnak and is elegantly expressed in Fermi's Golden Rule. We prove formulas for higher approximations and obtain necessary and sufficient conditions for dissolving a cusp form with eigenfunction $u_j$ into a resonance. In the framework of perturbations in character varieties, we relate the result to the ...

We study the Green function gr_\Gamma\ for the Laplace operator on the quotient of the hyperbolic plane by a cofinite Fuchsian group \Gamma. We use a limiting procedure, starting from the resolvent kernel, and lattice point estimates for the action of \Gamma\ on the hyperbolic plane to prove an "approximate spectral representation" for gr_\Gamma. Combining this with bounds on Maa{\ss} forms and Eisenstein series for \Gamma, we prove explicit bounds on gr_\Gamma.

We present a quantum ergodicity theorem for fixed spectral window and sequences of compact hyperbolic surfaces converging to the hyperbolic plane in the sense of Benjamini and Schramm. This addresses a question posed by Colin de Verdi\`{e}re. Our theorem is inspired by results for eigenfunctions on large regular graphs by Anantharaman and the first-named author. It applies in particular to eigenfunctions on compact arithmetic surfaces in the level aspect, which connects it to...

By a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with generating set given by choosing a generator for each cyclic factor. In this article we study the spectral theory of the combinatorial Laplacian for sequences of discrete tori when the orders of the cyclic factors tend to infinity at comparable rates. First we show that the sequence of heat kernels corresponding to the degenerating family converges, after re-scaling, to the...

We consider the Ruelle zeta function $R(s)$ of a genus $g$ hyperbolic Riemann surface with $n$ punctures and $v$ ramification points. $R(s)$ is equal to $Z(s)/Z(s+1)$, where $Z(s)$ is the Selberg zeta function. The main result of this work is the leading behavior of $R(s)$ at $s=0$. If $n_0$ is the order of the determinant of the scattering matrix $\varphi(s)$ at $s=0$, we find that \begin{align*} \lim_{s\rightarrow 0}\frac{R(s)}{s^{2g-2+n-n_0}}=(-1)^{\frac{A}{2}+1}(2\pi)^{2g...