arXiv subjects

Tobias Weich

Publications and source records attributed to Tobias Weich.

At least 19 recordsLinked to original sources

arXiv2026

The spectrum of Anosov representations

Given a $P$-Anosov representation into a noncompact semisimple real algebraic group $G$, where $P < G$ is a parabolic subgroup, we construct a natural resonance spectrum for the classical dynamics associated with the representation. This spectrum is a complex analytic hypersurface in $(\mathfrak{a}_P^*)_{\mathbb{C}}$, the complexified dual of the Lie algebra of the split component of the associated Levi group $L < P$. We reinterpret several objects from the theory of Anosov representations within this spectral framework and investigate, in higher rank, questions that are classically related to Pollicott-Ruelle theory in the rank-one setting. In particular, the ``leading resonance''--which is now a hypersurface--is identified with the critical hypersurface of the representation and, as in the rank-one case, the resonant states lead to continuous families of invariant measures. We prove that the multivariate zeta functions and Poincaré series associated with Anosov representations admit a meromorphic extension to $(\mathfrak{a}_P^*)_{\mathbb{C}}$. We also establish an asymptotic expansion in inverse powers of time for the correlation function of the diagonal flow under a Diophantine condition on the representation. Most of our results concerning Anosov representations are obtained as a byproduct of a general theory of Axiom A actions of type $(k,1)$, where $k+1:=\dim \mathfrak{a}_P$, that we introduce in the article.

math.RT↗

arXiv2026

Resonances on geometrically finite graphs

In analogy with the spectral theory of geometrically finite hyperbolic manifolds, we initiate the study of resonances on geometrically finite (q+1)-regular graphs of groups. We prove the meromorphic continuation of the resolvent of the adjacency operator on such spaces and give a geometric characterization of the resonant states. In contrast to the hyperbolic surfaces setting, geometrically finite graphs have only finitely many resonances, and these resonances may be computed explicitly, yet they exhibit many of the same qualitative phenomena as in the hyperbolic manifolds setting. Particularly interesting examples arise from algebraic curves over finite fields.

math.SP↗

arXiv2026

Spectral theory for transfer operators on compact quotients of Euclidean buildings

In this paper we generalize the geodesic flow on (finite) homogeneous graphs to a multiparameter flow on compact quotients of Euclidean buildings. Then we study the joint spectra of the associated transfer operators acting on suitable Lipschitz spaces. The main result says that outside an arbitrarily small neighborhood of zero in the set of spectral parameters the Taylor spectrum of the commuting family of transfer operators is contained in the joint point spectrum.

math.DS↗

arXiv2025

Polyhedral bounds on the joint spectrum and temperedness of locally symmetric spaces

Given a real semisimple connected Lie group $G$ and a discrete subgroup $Γ< G$ we prove a precise connection between growth rates of the group $Γ$, 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 $L^2(Γ\backslash G)$ in terms of Quint's growth indicator function. As an application of our sharp polyhedral bounds we prove temperedness of $L^2(Γ\backslash G)$ for all Borel Anosov subgroups $Γ$ in higher rank Lie groups $G$ not locally isomorphic to $\mathfrak{sl}_3(\mathbb{K}),\mathbb{K}=\R,\C,\mathbb H,$ or $\mathfrak{e}_{6(-26)}$.

math.RT↗

arXiv2024

Wave Front Sets of Nilpotent Lie Group Representations

Let $G$ be a nilpotent, connected, simply connected Lie group with Lie algebra $\mathfrak g$, and $π$ a unitary representation of $G$. The goal is to prove that the wave front set of $π$ coincides with the asymptotic cone of the orbital support of $π$, i.e. $\mathrm{WF}(π)=\mathrm{AC}(\bigcup_{σ\in \mathrm{supp}(π)}\mathcal O_σ)$, where $\mathcal O_σ\subset i\mathfrak g^\ast$ is the coadjoint orbit associated to the irreducible unitary representation $σ\in \hat{G}$ by Kirillov.

math.RT↗

arXiv2023

Decomposing large unitaries into multimode devices of arbitrary size

Decomposing complex unitary evolution into a series of constituent components is a cornerstone of practical quantum information processing. While the decompostion of an $n\times n$ unitary into a series of $2\times2$ subunitaries is well established (i.e. beamsplitters and phase shifters in linear optics), we show how this decomposition can be generalised into a series of $m\times m$ multimode devices, where $m>2$. If the cost associated with building each $m\times m$ multimode device is less than constructing with $\frac{m(m-1)}{2}$ individual $2\times 2$ devices, we show that the decomposition of large unitaries into $m\times m$ submatrices is is more resource efficient and exhibits a higher tolerance to errors, than its $2\times 2$ counterpart. This allows larger-scale unitaries to be constructed with lower errors, which is necessary for various tasks, not least Boson sampling, the quantum Fourier transform and quantum simulations.

quant-ph↗

arXiv2023

Invariant Ruelle Distributions on Convex-Cocompact Hyperbolic Surfaces -- A Numerical Algorithm via Weighted Zeta Functions

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.

math.DS↗

arXiv2023

SRB measures for Anosov actions

Given a general Anosov $\mathbb{R}^κ$ action on a closed manifold, we study properties of certain invariant measures that have recently been introduced in \cite{BGHW20} using the theory of Ruelle-Taylor resonances. We show that these measures share many properties of Sinai-Ruelle-Bowen measures for general Anosov flows such as smooth disintegrations along the unstable foliation, positive Lebesgue measure basins of attraction and a Bowen formula in terms of periodic orbits. Finally we show that if the action in the positive Weyl chamber is transitive, the measure is unique and has full support.

math.DS↗

arXiv2023

Spectral correspondences for finite graphs without dead ends

We compare the spectral properties of two kinds of linear operators characterizing the (classical) geodesic flow and its quantization on connected locally finite graphs without dead ends. The first kind are transfer operators acting on vector spaces associated with the set of non backtracking paths in the graphs. The second kind of operators are averaging operators acting on vector spaces associated with the space of vertices of the graph. The choice of vector spaces reflects regularity properties. Our main results are correspondences between classical and quantum spectral objects as well as some automatic regularity properties for eigenfunctions of transfer operators.

math.SP↗

arXiv2023

Absence of principal eigenvalues for higher rank locally symmetric spaces

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 $Γ\backslash G/K$ of higher rank. We derive dynamical assumptions on the $Γ$-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.

math.SP↗

arXiv2023

Temperedness of locally symmetric spaces: The product case

Let $X=X_1\times X_2$ be a product of two rank one symmetric spaces of non-compact type and $Γ$ a torsion-free discrete subgroup in $G_1\times G_2$. We show that the spectrum of $Γ\backslash X$ is related to the asymptotic growth of $Γ$ in the two direction defined by the two factors. We obtain that $L^2(Γ\backslash G)$ is tempered for large class of $Γ$.

math.SP↗

arXiv2023

Ruelle-Taylor resonances of Anosov actions

Combining microlocal methods and a cohomological theory developped by J. Taylor, we define for $\mathbb{R}^κ$-Anosov actions a notion of joint Ruelle resonance spectrum. We prove that these Ruelle-Taylor resonances fit into a Fredholm theory, are intrinsic and form a discrete subset of $\mathbb{C}^κ$, with $λ=0$ being always a leading resonance. The joint resonant states at $0$ give rise to some new measures of SRB type and the mixing properties of these measures are related to the existence of purely imaginary resonances. The spectral theory developed in this article applies in particular to the case of Weyl chamber flows and provides a new way to study such flows.

math.DS↗

arXiv2022

Poisson Transforms for Trees of Bounded Degree

We introduce a parameterized family of Poisson transforms on trees of bounded degree, construct explicit inverses for generic parameters, and characterize moderate growth of Laplace eigenfunctions by Hölder regularity of their boundary values.

math.SP↗

arXiv2022

Resonances and weighted zeta functions for obstacle scattering via smooth models

We 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 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.

math.DS↗

arXiv2022

Semiclassical Formulae For Wigner Distributions

In this paper we give an overview over some aspects of the modern mathematical theory of Ruelle resonances for chaotic, i.e. uniformly hyperbolic, dynamical systems and their implications in physics. First we recall recent developments in the mathematical theory of resonances, in particular how invariant Ruelle distributions arise as residues of weighted zeta functions. Then we derive a correspondence between weighted and semiclassical zeta functions in the setting of negatively curved surfaces. Combining this with results of Hilgert, Guillarmou and Weich yields a high frequency interpretation of invariant Ruelle distributions as quantum mechanical matrix coefficients in constant negative curvature. We finish by presenting numerical calculations of phase space distributions in the more physical setting of 3-disk scattering systems.

math-ph↗

arXiv2021

Meromorphic Continuation of Weighted Zeta Functions on Open Hyperbolic Systems

In this article we prove meromorphic continuation of weighted zeta functions in the framework of open hyperbolic systems by using the meromorphically continued restricted resolvent of Dyatlov and Guillarmou (2016). We obtain a residue formula proving equality between residues of weighted zetas and invariant Ruelle distributions. We combine this equality with results of Guillarmou, Hilgert and Weich (2021) in order to relate the residues to Patterson-Sullivan distributions. Finally we provide proof-of-principle results concerning the numerical calculation of invariant Ruelle distributions for 3-disc scattering systems.

math.DS↗

arXiv2021

Higher rank quantum-classical correspondence

For a compact Riemannian locally symmetric space $Γ\backslash G/K$ of arbitrary rank we determine the location of certain Ruelle-Taylor resonances for the Weyl chamber action. We provide a Weyl-lower bound on an appropriate counting function for the Ruelle-Taylor resonances and establish a spectral gap which is uniform in $Γ$ if $G/K$ is irreducible of higher rank. This is achieved by proving a quantum-classical correspondence, i.e. a 1:1-correspondence between horocyclically invariant Ruelle-Taylor resonant states and joint eigenfunctions of the algebra of invariant differential operators on $G/K$.

math.DS↗

arXiv2020

Spectral Asymptotics for Kinetic Brownian Motion on Hyperbolic Surfaces

The kinetic Brownian motion on the sphere bundle of a Riemannian manifold $M$ is a stochastic process that models a random perturbation of the geodesic flow. If $M$ is a orientable compact constant negatively curved surface, we show that in the limit of infinitely large perturbation the $L^2$-spectrum of the infinitesimal generator of a time rescaled version of the process converges to the Laplace spectrum of the base manifold. In addition, we give explicit error estimates for the convergence to equilibrium. The proofs are based on noncommutative harmonic analysis of $SL_2(\mathbb{R})$.

math.SP↗