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Foto: Universität Paderborn

Prof. Dr. Tobias Weich

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Prof. Dr. Tobias Weich

Spektral Analysis

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2022

Absence of principal eigenvalues for higher rank locally symmetric spaces

T. Weich, L.L. Wolf, in: arXiv:2205.03167, 2022

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.


Semiclassical Formulae For Wigner Distributions

P. Schütte, S. Barkhofen, T. Weich, 2022

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.


2021

Spectral Asymptotics for Kinetic Brownian Motion on Surfaces of Constant Curvature

M. Kolb, T. Weich, L.L. Wolf, Annales Henri Poincaré (2021), 23(4), pp. 1283-1296

<jats:title>Abstract</jats:title><jats:p>The kinetic Brownian motion on the sphere bundle of a Riemannian manifold <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbb {M}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> is a stochastic process that models a random perturbation of the geodesic flow. If <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbb {M}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> is an orientable compact constantly curved surface, we show that in the limit of infinitely large perturbation the <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>-spectrum of the infinitesimal generator of a time-rescaled version of the process converges to the Laplace spectrum of the base manifold.</jats:p>


Higher rank quantum-classical correspondence

J. Hilgert, T. Weich, L.L. Wolf, in: arXiv:2103.05667, 2021

For a compact Riemannian locally symmetric space $\Gamma\backslash G/K$ 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 $\Gamma$ 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 $G/K$.


Meromorphic Continuation of Weighted Zeta Functions on Open Hyperbolic Systems

P. Schütte, T. Weich, S. Barkhofen, 2021

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.


Resonances and weighted zeta functions for obstacle scattering via smooth models

P. Schütte, T. Weich, B. Delarue, 2021

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.


High frequency limits for invariant Ruelle densities

C. Guillarmou, J. Hilgert, T. Weich, Annales Henri Lebesgue (2021), 4, pp. 81-119


2020

Pollicott-Ruelle Resonant States and Betti Numbers

B. Küster, T. Weich, Communications in Mathematical Physics (2020), 378(2), pp. 917-941

<jats:title>Abstract</jats:title><jats:p>Given a closed orientable hyperbolic manifold of dimension <jats:inline-formula><jats:alternatives><jats:tex-math>$$\ne 3$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>≠</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> we prove that the multiplicity of the Pollicott-Ruelle resonance of the geodesic flow on perpendicular one-forms at zero agrees with the first Betti number of the manifold. Additionally, we prove that this equality is stable under small perturbations of the Riemannian metric and simultaneous small perturbations of the geodesic vector field within the class of contact vector fields. For more general perturbations we get bounds on the multiplicity of the resonance zero on all one-forms in terms of the first and zeroth Betti numbers. Furthermore, we identify for hyperbolic manifolds further resonance spaces whose multiplicities are given by higher Betti numbers. </jats:p>


2019

Spectral Asymptotics for Kinetic Brownian Motion on Hyperbolic Surfaces

M. Kolb, T. Weich, L.L. Wolf, in: arXiv:1909.06183, 2019

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 $SL_2(\mathbb{R})$.


Improved fractal Weyl bounds for hyperbolic manifolds. With an appendix by David Borthwick, Semyon Dyatlov and Tobias Weich

S. Dyatlov, D. Borthwick, T. Weich, Journal of the European Mathematical Society (2019), 21(6), pp. 1595-1639


2017

Global Normal Form and Asymptotic Spectral Gap for Open Partially Expanding Maps

F. Faure, T. Weich, Communications in Mathematical Physics (2017), 356(3), pp. 755-822


Classical and quantum resonances for hyperbolic surfaces

C. Guillarmou, J. Hilgert, T. Weich, Mathematische Annalen (2017), 370(3-4), pp. 1231-1275


Wave front sets of reductive Lie group representations III

B. Harris, T. Weich, Advances in Mathematics (2017), 313, pp. 176-236


Exkursinhalte in der fachmathematischen Lehramtsausbildung: Wie man das Wesen und die Rolle der Mathematik vermittelt.

T. Weich, M. Hoffmann, die hochschullehre (2017), 3


2016

Symmetry reduction of holomorphic iterated function schemes and factorization of Selberg zeta functions

D. Borthwick, T. Weich, Journal of Spectral Theory (2016), 6(2), pp. 267-329


On the Support of Pollicott–Ruelle Resonanant States for Anosov Flows

T. Weich, Annales Henri Poincaré (2016), 18(1), pp. 37-52


2015

Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps

J.F. ARNOLDI, F. FAURE, T. Weich, Ergodic Theory and Dynamical Systems (2015), 37(1), pp. 1-58

<jats:p>We consider a simple model of an open partially expanding map. Its trapped set <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0143385715000346_inline1" /><jats:tex-math>${\mathcal{K}}$</jats:tex-math></jats:alternatives></jats:inline-formula> in phase space is a fractal set. We first show that there is a well-defined discrete spectrum of Ruelle resonances which describes the asymptotic of correlation functions for large time and which is parametrized by the Fourier component <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0143385715000346_inline2" /><jats:tex-math>$\unicode[STIX]{x1D708}$</jats:tex-math></jats:alternatives></jats:inline-formula> in the neutral direction of the dynamics. We introduce a specific hypothesis on the dynamics that we call ‘minimal captivity’. This hypothesis is stable under perturbations and means that the dynamics is univalued in a neighborhood of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0143385715000346_inline3" /><jats:tex-math>${\mathcal{K}}$</jats:tex-math></jats:alternatives></jats:inline-formula>. Under this hypothesis we show the existence of an asymptotic spectral gap and a fractal Weyl law for the upper bound of density of Ruelle resonances in the semiclassical limit <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0143385715000346_inline4" /><jats:tex-math>$\unicode[STIX]{x1D708}\rightarrow \infty$</jats:tex-math></jats:alternatives></jats:inline-formula>. Some numerical computations with the truncated Gauss map and Bowen–Series maps illustrate these results.</jats:p>


Resonance Chains and Geometric Limits on Schottky Surfaces

T. Weich, Communications in Mathematical Physics (2015), 337(2), pp. 727-765


2014

Equivariant spectral asymptotics for<i>h</i>-pseudodifferential operators

T. Weich, Journal of Mathematical Physics (2014), 55(10), 101501


Resonance chains in open systems, generalized zeta functions and clustering of the length spectrum

S. Barkhofen, F. Faure, T. Weich, Nonlinearity (2014), 27(8), pp. 1829-1858


Formation and interaction of resonance chains in the open three-disk system

T. Weich, S. Barkhofen, U. Kuhl, C. Poli, H. Schomerus, New Journal of Physics (2014), 16(3), 033029


2012

Weyl asymptotics: From closed to open systems

A. Potzuweit, T. Weich, S. Barkhofen, U. Kuhl, H. Stöckmann, M. Zworski, Physical Review E (2012), 86(6), 066205


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