@article{51204,
abstract = {{Given a real semisimple connected Lie group $G$ and a discrete torsion-free
subgroup $\Gamma < G$ we prove a precise connection between growth rates of the
group $\Gamma$, polyhedral bounds on the joint spectrum of the ring of
invariant differential operators, and the decay of matrix coefficients. In
particular, this allows us to completely characterize temperedness of
$L^2(\Gamma\backslash G)$ in this general setting.}},
author = {{Lutsko, Christopher and Weich, Tobias and Wolf, Lasse Lennart}},
journal = {{Duke Math. Journal }},
title = {{{Polyhedral bounds on the joint spectrum and temperedness of locally symmetric spaces}}},
volume = {{(to appear)}},
year = {{2026}},
}
@article{64569,
abstract = {{Abstract
We show how the Fourier transform for distributional sections of vector bundles over symmetric spaces of non‐compact type can be used for questions of solvability of systems of invariant differential equations in analogy to Hörmander's proof of the Ehrenpreis–Malgrange theorem. We get complete solvability for the hyperbolic plane and partial results for products and the hyperbolic 3‐space .}},
author = {{Olbrich, Martin and Palmirotta, Guendalina}},
issn = {{0025-584X}},
journal = {{Mathematische Nachrichten}},
number = {{2}},
pages = {{456--479}},
publisher = {{Wiley}},
title = {{{Solvability of invariant systems of differential equations on H2$\mathbb {H}^2$ and beyond}}},
doi = {{10.1002/mana.70100}},
volume = {{299}},
year = {{2026}},
}
@article{57580,
abstract = {{We investigate dispersive and Strichartz estimates for the Schrödinger equation involving the fractional Laplacian in real hyperbolic spaces and their discrete analogues, homogeneous trees. Due to the Knapp phenomenon, the Strichartz estimates on Euclidean spaces for the fractional Laplacian exhibit loss of derivatives. A similar phenomenon appears on real hyperbolic spaces. However, such a loss disappears on homogeneous trees, due to the triviality of the estimates for small times.}},
author = {{Palmirotta, Guendalina and Sire, Yannick and Anker, Jean-Philippe}},
journal = {{Journal of Differential Equations}},
keywords = {{Schrödinger equation, Fractional Laplacian, Dispersive estimates, Strichartz estimates, Real hyperbolic spaces, Homogeneous trees}},
publisher = {{Elsevier}},
title = {{{The Schrödinger equation with fractional Laplacian on hyperbolic spaces and homogeneous trees}}},
doi = {{10.1016/j.jde.2025.114065}},
year = {{2026}},
}
@unpublished{65232,
abstract = {{On finite regular graphs, we construct Patterson-Sullivan distributions associated with eigenfunctions of the discrete Laplace operator via their boundary values on the phase space. These distributions are closely related to Wigner distributions defined via a pseudo-differential calculus on graphs, which appear naturally in the study of quantum chaos. Using a pairing formula, we prove that Patterson-Sullivan distributions are also related to invariant Ruelle distributions arising from the transfer operator of the geodesic flow on the shift space. Both relationships provide discrete analogues of results for compact hyperbolic surfaces obtained by Anantharaman-Zelditch and by Guillarmou-Hilgert-Weich.}},
author = {{Arends, Christian and Palmirotta, Guendalina}},
booktitle = {{arXiv:2603.09779}},
pages = {{38}},
title = {{{Patterson-Sullivan distributions of finite regular graphs}}},
year = {{2026}},
}
@article{32099,
author = {{Weich, Tobias and Budde, Julia}},
journal = {{Journal of Functional Analysis}},
number = {{1}},
title = {{{Wave Front Sets of Nilpotent Lie Group Representations}}},
doi = {{ https://doi.org/10.1016/j.jfa.2024.110684}},
volume = {{288}},
year = {{2025}},
}
@article{53414,
abstract = {{By constructing a non-empty domain of discontinuity in a suitable homogeneous
space, we prove that every torsion-free projective Anosov subgroup is the
monodromy group of a locally homogeneous contact Axiom A dynamical system with
a unique basic hyperbolic set on which the flow is conjugate to the refraction
flow of Sambarino. Under the assumption of irreducibility, we utilize the work
of Stoyanov to establish spectral estimates for the associated complex Ruelle
transfer operators, and by way of corollary: exponential mixing, exponentially
decaying error term in the prime orbit theorem, and a spectral gap for the
Ruelle zeta function. With no irreducibility assumption, results of
Dyatlov-Guillarmou imply the global meromorphic continuation of zeta functions
with smooth weights, as well as the existence of a discrete spectrum of
Ruelle-Pollicott resonances and (co)-resonant states. We apply our results to
space-like geodesic flows for the convex cocompact pseudo-Riemannian manifolds
of Danciger-Gu\'eritaud-Kassel, and the Benoist-Hilbert geodesic flow for
strictly convex real projective manifolds.}},
author = {{Delarue, Benjamin and Monclair, Daniel and Sanders, Andrew}},
journal = {{Geometric and Functional Analysis (GAFA)}},
pages = {{673–735}},
title = {{{Locally homogeneous Axiom A flows I: projective Anosov subgroups and exponential mixing}}},
doi = {{10.1007/s00039-025-00712-2}},
volume = {{35}},
year = {{2025}},
}
@article{53412,
abstract = {{Let $M$ be a symplectic manifold carrying a Hamiltonian $S^1$-action with
momentum map $J:M \rightarrow \mathbb{R}$ and consider the corresponding
symplectic quotient $\mathcal{M}_0:=J^{-1}(0)/S^1$. We extend Sjamaar's complex
of differential forms on $\mathcal{M}_0$, whose cohomology is isomorphic to the
singular cohomology $H(\mathcal{M}_0;\mathbb{R})$ of $\mathcal{M}_0$ with real
coefficients, to a complex of differential forms on $\mathcal{M}_0$ associated
with a partial desingularization $\widetilde{\mathcal{M}}_0$, which we call
resolution differential forms. The cohomology of that complex turns out to be
isomorphic to the de Rham cohomology $H(\widetilde{ \mathcal{M}}_0)$ of
$\widetilde{\mathcal{M}}_0$. Based on this, we derive a long exact sequence
involving both $H(\mathcal{M}_0;\mathbb{R})$ and $H(\widetilde{
\mathcal{M}}_0)$ and give conditions for its splitting. We then define a Kirwan
map $\mathcal{K}:H_{S^1}(M) \rightarrow H(\widetilde{\mathcal{M}}_0)$ from the
equivariant cohomology $H_{S^1}(M)$ of $M$ to $H(\widetilde{\mathcal{M}}_0)$
and show that its image contains the image of $H(\mathcal{M}_0;\mathbb{R})$ in
$H(\widetilde{\mathcal{M}}_0)$ under the natural inclusion. Combining both
results in the case that all fixed point components of $M$ have vanishing odd
cohomology we obtain a surjection $\check \kappa:H^\textrm{ev}_{S^1}(M)
\rightarrow H^\textrm{ev}(\mathcal{M}_0;\mathbb{R})$ in even degrees, while
already simple examples show that a similar surjection in odd degrees does not
exist in general. As an interesting class of examples we study abelian polygon
spaces.}},
author = {{Delarue, Benjamin and Ramacher, Pablo and Schmitt, Maximilian}},
journal = {{Transformation Groups}},
title = {{{Singular cohomology of symplectic quotients by circle actions and Kirwan surjectivity}}},
doi = {{10.1007/s00031-025-09924-0}},
year = {{2025}},
}
@article{53413,
abstract = {{For negatively curved symmetric spaces it is known that the poles of the
scattering matrices defined via the standard intertwining operators for the
spherical principal representations of the isometry group are either given as
poles of the intertwining operators or as quantum resonances, i.e. poles of the
meromorphically continued resolvents of the Laplace-Beltrami operator. We
extend this result to classical locally symmetric spaces of negative curvature
with convex-cocompact fundamental group using results of Bunke and Olbrich. The
method of proof forces us to exclude the spectral parameters corresponding to
singular Poisson transforms.}},
author = {{Delarue, Benjamin and Hilgert, Joachim}},
issn = {{0949-5932}},
journal = {{Journal of Lie Theory}},
number = {{(4)}},
pages = {{787----804}},
title = {{{Quantum resonances and scattering poles of classical rank one locally symmetric spaces}}},
volume = {{35}},
year = {{2025}},
}
@article{51207,
abstract = {{Let $X=X_1\times X_2$ be a product of two rank one symmetric spaces of
non-compact type and $\Gamma$ a torsion-free discrete subgroup in $G_1\times
G_2$. We show that the spectrum of $\Gamma \backslash X$ is related to the
asymptotic growth of $\Gamma$ in the two direction defined by the two factors.
We obtain that $L^2(\Gamma \backslash G)$ is tempered for large class of
$\Gamma$.}},
author = {{Weich, Tobias and Wolf, Lasse Lennart}},
journal = {{Geom Dedicata}},
title = {{{Temperedness of locally symmetric spaces: The product case}}},
doi = {{https://doi.org/10.1007/s10711-024-00904-4}},
volume = {{218}},
year = {{2024}},
}
@book{55193,
author = {{Hoffmann, Max and Hilgert, Joachim and Weich, Tobias}},
isbn = {{9783662673560}},
publisher = {{Springer Berlin Heidelberg}},
title = {{{Ebene euklidische Geometrie. Algebraisierung, Axiomatisierung und Schnittstellen zur Schulmathematik}}},
doi = {{10.1007/978-3-662-67357-7}},
year = {{2024}},
}
@article{32101,
author = {{Weich, Tobias and Guedes Bonthonneau, Yannick and Guillarmou, Colin and Hilgert, Joachim}},
journal = {{J. Europ. Math. Soc.}},
number = {{8}},
pages = {{3085–3147}},
title = {{{Ruelle-Taylor resonances of Anosov actions}}},
doi = {{https://doi.org/10.4171/JEMS/1428}},
volume = {{27}},
year = {{2024}},
}
@unpublished{57582,
abstract = {{We prove that the Patterson-Sullivan and Wigner distributions on the unit
sphere bundle of a convex-cocompact hyperbolic surface are asymptotically
identical. This generalizes results in the compact case by
Anantharaman-Zelditch and Hansen-Hilgert-Schr\"oder.}},
author = {{Delarue, Benjamin and Palmirotta, Guendalina}},
booktitle = {{arXiv:2411.19782}},
title = {{{Patterson-Sullivan and Wigner distributions of convex-cocompact hyperbolic surfaces}}},
year = {{2024}},
}
@article{32097,
author = {{Weich, Tobias and Guedes Bonthonneau, Yannick and Guillarmou, Colin}},
journal = {{Journal of Differential Geometry}},
pages = {{959--1026}},
title = {{{SRB Measures of Anosov Actions}}},
doi = {{ DOI: 10.4310/jdg/1729092452}},
volume = {{128}},
year = {{2024}},
}
@unpublished{58873,
abstract = {{We prove that the Patterson-Sullivan and Wigner distributions on the unit
sphere bundle of a convex-cocompact hyperbolic surface are asymptotically
identical. This generalizes results in the compact case by
Anantharaman-Zelditch and Hansen-Hilgert-Schr\"oder.}},
author = {{Delarue, Benjamin and Palmirotta, Guendalina}},
booktitle = {{arXiv:2411.19782}},
title = {{{Patterson-Sullivan and Wigner distributions of convex-cocompact hyperbolic surfaces}}},
year = {{2024}},
}
@article{31189,
abstract = {{Given a geometrically finite hyperbolic surface of infinite volume it is a
classical result of Patterson that the positive Laplace-Beltrami operator has
no $L^2$-eigenvalues $\geq 1/4$. In this article we prove a generalization of
this result for the joint $L^2$-eigenvalues of the algebra of commuting
differential operators on Riemannian locally symmetric spaces $\Gamma\backslash
G/K$ of higher rank. We derive dynamical assumptions on the $\Gamma$-action on
the geodesic and the Satake compactifications which imply the absence of the
corresponding principal eigenvalues. A large class of examples fulfilling these
assumptions are the non-compact quotients by Anosov subgroups.}},
author = {{Weich, Tobias and Wolf, Lasse Lennart}},
journal = {{Communications in Mathematical Physics}},
title = {{{Absence of principal eigenvalues for higher rank locally symmetric spaces}}},
doi = {{https://doi.org/10.1007/s00220-023-04819-1}},
volume = {{403}},
year = {{2023}},
}
@unpublished{51206,
abstract = {{We present a numerical algorithm for the computation of invariant Ruelle
distributions on convex co-compact hyperbolic surfaces. This is achieved by
exploiting the connection between invariant Ruelle distributions and residues
of meromorphically continued weighted zeta functions established by the authors
together with Barkhofen (2021). To make this applicable for numerics we express
the weighted zeta as the logarithmic derivative of a suitable parameter
dependent Fredholm determinant similar to Borthwick (2014). As an additional
difficulty our transfer operator has to include a contracting direction which
we account for with techniques developed by Rugh (1992). We achieve a further
improvement in convergence speed for our algorithm in the case of surfaces with
additional symmetries by proving and applying a symmetry reduction of weighted
zeta functions.}},
author = {{Schütte, Philipp and Weich, Tobias}},
booktitle = {{arXiv:2308.13463}},
title = {{{Invariant Ruelle Distributions on Convex-Cocompact Hyperbolic Surfaces -- A Numerical Algorithm via Weighted Zeta Functions}}},
year = {{2023}},
}
@article{31210,
abstract = {{In this paper we complete the program of relating the Laplace spectrum for
rank one compact locally symmetric spaces with the first band Ruelle-Pollicott
resonances of the geodesic flow on its sphere bundle. This program was started
by Flaminio and Forni for hyperbolic surfaces, continued by Dyatlov, Faure and
Guillarmou for real hyperbolic spaces and by Guillarmou, Hilgert and Weich for
general rank one spaces. Except for the case of hyperbolic surfaces a countable
set of exceptional spectral parameters always left untreated since the
corresponding Poisson transforms are neither injective nor surjective. We use
vector valued Poisson transforms to treat also the exceptional spectral
parameters. For surfaces the exceptional spectral parameters lead to discrete
series representations of $\mathrm{SL}(2,\mathbb R)$. In higher dimensions the
situation is more complicated, but can be described completely.}},
author = {{Arends, Christian and Hilgert, Joachim}},
issn = {{2270-518X}},
journal = {{Journal de l’École polytechnique — Mathématiques}},
keywords = {{Ruelle resonances, Poisson transforms, locally symmetric spaces, principal series representations}},
pages = {{335--403}},
title = {{{Spectral correspondences for rank one locally symmetric spaces: the case of exceptional parameters}}},
doi = {{10.5802/jep.220}},
volume = {{10}},
year = {{2023}},
}
@unpublished{53404,
abstract = {{In this short note we observe, on locally symmetric spaces of higher rank, a
connection between the growth indicator function introduced by Quint and the
modified critical exponent of the Poincar\'e series equipped with the
polyhedral distance. As a consequence, we provide a different characterization
of the bottom of the $L^2$-spectrum of the Laplace-Beltrami operator in terms
of the growth indicator function. Moreover, we explore the relationship between
these three objects and the temperedness.}},
author = {{Wolf, Lasse L. and Zhang, Hong-Wei}},
booktitle = {{arXiv:2311.11770}},
title = {{{$L^2$-spectrum, growth indicator function and critical exponent on locally symmetric spaces}}},
year = {{2023}},
}
@article{53410,
abstract = {{AbstractWe consider a geodesic billiard system consisting of a complete Riemannian manifold and an obstacle submanifold with boundary at which the trajectories of the geodesic flow experience specular reflections. We show that if the geodesic billiard system is hyperbolic on its trapped set and the latter is compact and non-grazing, the techniques for open hyperbolic systems developed by Dyatlov and Guillarmou (Ann Henri Poincaré 17(11):3089–3146, 2016) can be applied to a smooth model for the discontinuous flow defined by the non-grazing billiard trajectories. This allows us to obtain a meromorphic resolvent for the generator of the billiard flow. As an application we prove a meromorphic continuation of weighted zeta functions together with explicit residue formulae. In particular, our results apply to scattering by convex obstacles in the Euclidean plane.}},
author = {{Delarue, Benjamin and Schütte, Philipp and Weich, Tobias}},
issn = {{1424-0637}},
journal = {{Annales Henri Poincaré}},
keywords = {{Mathematical Physics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics}},
number = {{2}},
pages = {{1607--1656}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Resonances and Weighted Zeta Functions for Obstacle Scattering via Smooth Models}}},
doi = {{10.1007/s00023-023-01379-x}},
volume = {{25}},
year = {{2023}},
}
@unpublished{53411,
abstract = {{We compute a Riemann-Roch formula for the invariant Riemann-Roch number of a
quantizable Hamiltonian $S^1$-manifold $(M,\omega,\mathcal{J})$ in terms of the
geometry of its symplectic quotient, allowing $0$ to be a singular value of the
moment map $\mathcal{J}:M\to\mathbb{R}$. The formula involves a new explicit
local invariant of the singularities. Our approach relies on a complete
singular stationary phase expansion of the associated Witten integral.}},
author = {{Delarue, Benjamin and Ioos, Louis and Ramacher, Pablo}},
booktitle = {{arXiv:2302.09894}},
title = {{{A Riemann-Roch formula for singular reductions by circle actions}}},
year = {{2023}},
}