Related papers: Selberg trace formula in hyperbolic band theory
For cofinite Kleinian groups, with finite-dimensional unitary representations, we derive the Selberg trace formula. As an application we define the corresponding Selberg zeta-function and compute its divisor, thus generalizing results of…
An exact trace formula for the perturbation of the Laplacian by a Dirac delta potential on a compact hyperbolic Riemann surface is derived. The formula can be considered an analogue of the Selberg trace formula. The difference of perturbed…
For cofinite Kleinian groups, with finite-dimensional unitary representations, we derive the Selberg trace formula. As an application we define the corresponding Selberg zeta-function and compute its divisor, thus generalizing results of…
We study geometric and spectral properties of typical hyperbolic surfaces of high genus, excluding a set of small measure for the Weil-Petersson probability measure. We first prove Benjamini-Schramm convergence to the hyperbolic plane H as…
We present a rigorous scheme that makes it possible to compute eigenvalues of the Laplace operator on hyperbolic surfaces within a given precision. The method is based on an adaptation of the method of particular solutions to the case of…
For weighted Bergman spaces on the unit disk, we give trace formulas of semicommutators of Toeplitz operators with $\mathscr{C}^2(\overline{\mathbb{D}})$ symbols. We generalize this formula to weighted Bergman spaces on the unit ball in…
These lecture notes provide a basic introduction to Selberg's trace formula. We discuss the simplest possible case: the spectrum of the Laplacian on a compact Riemannian surface of constant negative curvature. (To appear in Springer LNP.)
In the 80s, Zagier and Jacquet-Zagier tried to derive the Selberg trace formula by applying the Rankin-Selberg method to the automorphic kernel function. Their derivation was incomplete due to a puzzle of the computation of a residue. We…
We study a variety of problems in the spectral theory of automorphic forms using entirely analytic techniques such as Selberg trace formula, asymptotics of Whittaker functions and behavior of heat kernels. Error terms for Weyl's law and an…
The goal of the course was a review of results mainly due to M. Olbrich and the first author. We consider a discrete cocompact subgroup $\Gamma$ of a semisimple Lie group $G$. We relate the group cohomology of $\Gamma$ with coefficients in…
We constrain the low-energy spectra of Laplace operators on closed hyperbolic manifolds and orbifolds in three dimensions, including the standard Laplace-Beltrami operator on functions and the Laplacian on powers of the cotangent bundle.…
We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on $ {\mathbb H}^{n+1}$: in strips parallel to the imaginary axis the zeta function is bounded by $ \exp (C |s|^\delta) $ where $ \delta $…
A certain generalization of the Selberg trace formula was proved by the first named author in 1999. In this generalization instead of considering the integral of $K(z,z)$ (where $K(z,w)$ is an automorphic kernel function) over the…
Hyperbolic lattices underlie a new form of quantum matter with potential applications to quantum computing and simulation and which, to date, have been engineered artificially. A corresponding hyperbolic band theory has emerged, extending…
We give a new, simple proof of the trace formula for Hecke operators on modular forms for finite index subgroups of the modular group. The proof uses algebraic properties of certain universal Hecke operators acting on period polynomials of…
The trace formula is a versatile tool for computing sums of spectral data across families of automorphic forms. Using specialized test functions, one can treat small families with refined spectral properties. This has proven fruitful in…
For $\Gamma={\hbox{PSL}_2( {\mathbb Z})}$ the hyperbolic circle problem aims to estimate the number of elements of the orbit $\Gamma z$ inside the hyperbolic disc centered at $z$ with radius $\cosh^{-1}(X/2)$. We show that, by averaging…
We study Selberg zeta functions $Z(s,\sigma)$ associated to locally homogeneous vector bundles over the unit-sphere bundle of a complete odd-dimensional hyperbolic manifold of finite volume. We assume a certain condition on the fundamental…
The Eichler-Selberg trace formula expresses the trace of Hecke operators on spaces of cusp forms as weighted sums of Hurwitz-Kronecker class numbers. We extend this formula to a natural class of relations for traces of singular moduli,…
We stabilize the full Arthur-Selberg trace formula for the metaplectic covering of symplectic groups over a number field. This provides a decomposition of the invariant trace formula for metaplectic groups, which encodes information about…