@article{31264, abstract = {{AbstractGiven a closed orientable hyperbolic manifold of dimension $$\ne 3$$ ≠ 3 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. }}, author = {{Küster, Benjamin and Weich, Tobias}}, issn = {{0010-3616}}, journal = {{Communications in Mathematical Physics}}, keywords = {{Mathematical Physics, Statistical and Nonlinear Physics}}, number = {{2}}, pages = {{917--941}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Pollicott-Ruelle Resonant States and Betti Numbers}}}, doi = {{10.1007/s00220-020-03793-2}}, volume = {{378}}, year = {{2020}}, } @article{53415, abstract = {{AbstractGiven a closed orientable hyperbolic manifold of dimension $$\ne 3$$ ≠ 3 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. }}, author = {{Küster, Benjamin and Weich, Tobias}}, issn = {{0010-3616}}, journal = {{Communications in Mathematical Physics}}, keywords = {{Mathematical Physics, Statistical and Nonlinear Physics}}, number = {{2}}, pages = {{917--941}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Pollicott-Ruelle Resonant States and Betti Numbers}}}, doi = {{10.1007/s00220-020-03793-2}}, volume = {{378}}, year = {{2020}}, } @article{31265, author = {{Dyatlov, Semyon and Borthwick, David and Weich, Tobias}}, issn = {{1435-9855}}, journal = {{Journal of the European Mathematical Society}}, keywords = {{Applied Mathematics, General Mathematics}}, number = {{6}}, pages = {{1595--1639}}, publisher = {{European Mathematical Society - EMS - Publishing House GmbH}}, title = {{{Improved fractal Weyl bounds for hyperbolic manifolds. With an appendix by David Borthwick, Semyon Dyatlov and Tobias Weich}}}, doi = {{10.4171/jems/867}}, volume = {{21}}, year = {{2019}}, } @unpublished{31191, abstract = {{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 $SL_2(\mathbb{R})$.}}, author = {{Kolb, Martin and Weich, Tobias and Wolf, Lasse Lennart}}, booktitle = {{arXiv:1909.06183}}, title = {{{Spectral Asymptotics for Kinetic Brownian Motion on Hyperbolic Surfaces}}}, year = {{2019}}, } @article{53416, abstract = {{Abstract For a compact Riemannian locally symmetric space $\mathcal M$ of rank 1 and an associated vector bundle $\mathbf V_{\tau }$ over the unit cosphere bundle $S^{\ast }\mathcal M$, we give a precise description of those classical (Pollicott–Ruelle) resonant states on $\mathbf V_{\tau }$ that vanish under covariant derivatives in the Anosov-unstable directions of the chaotic geodesic flow on $S^{\ast }\mathcal M$. In particular, we show that they are isomorphically mapped by natural pushforwards into generalized common eigenspaces of the algebra of invariant differential operators $D(G,\sigma )$ on compatible associated vector bundles $\mathbf W_{\sigma }$ over $\mathcal M$. As a consequence of this description, we obtain an exact band structure of the Pollicott–Ruelle spectrum. Further, under some mild assumptions on the representations $\tau$ and $\sigma$ defining the bundles $\mathbf V_{\tau }$ and $\mathbf W_{\sigma }$, we obtain a very explicit description of the generalized common eigenspaces. This allows us to relate classical Pollicott–Ruelle resonances to quantum eigenvalues of a Laplacian in a suitable Hilbert space of sections of $\mathbf W_{\sigma }$. Our methods of proof are based on representation theory and Lie theory.}}, author = {{Küster, Benjamin and Weich, Tobias}}, issn = {{1073-7928}}, journal = {{International Mathematics Research Notices}}, keywords = {{General Mathematics}}, number = {{11}}, pages = {{8225--8296}}, publisher = {{Oxford University Press (OUP)}}, title = {{{Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces}}}, doi = {{10.1093/imrn/rnz068}}, volume = {{2021}}, year = {{2019}}, } @article{51389, author = {{Hilgert, Joachim and Weich, Tobias and Guillarmou, C.}}, journal = {{Math. Ann.}}, pages = {{1231--1275}}, title = {{{Classical and quantum resonances for hyperbolic surfaces}}}, volume = {{370}}, year = {{2018}}, } @article{31268, author = {{Faure, Frédéric and Weich, Tobias}}, issn = {{0010-3616}}, journal = {{Communications in Mathematical Physics}}, keywords = {{Mathematical Physics, Statistical and Nonlinear Physics}}, number = {{3}}, pages = {{755--822}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Global Normal Form and Asymptotic Spectral Gap for Open Partially Expanding Maps}}}, doi = {{10.1007/s00220-017-3000-0}}, volume = {{356}}, year = {{2017}}, } @article{31272, author = {{Harris, Benjamin and Weich, Tobias}}, issn = {{0001-8708}}, journal = {{Advances in Mathematics}}, keywords = {{General Mathematics}}, pages = {{176--236}}, publisher = {{Elsevier BV}}, title = {{{Wave front sets of reductive Lie group representations III}}}, doi = {{10.1016/j.aim.2017.03.025}}, volume = {{313}}, year = {{2017}}, } @article{31267, author = {{Guillarmou, Colin and Hilgert, Joachim and Weich, Tobias}}, issn = {{0025-5831}}, journal = {{Mathematische Annalen}}, keywords = {{General Mathematics}}, number = {{3-4}}, pages = {{1231--1275}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Classical and quantum resonances for hyperbolic surfaces}}}, doi = {{10.1007/s00208-017-1576-5}}, volume = {{370}}, year = {{2017}}, } @article{31377, author = {{Weich, Tobias and Hoffmann, Max}}, journal = {{die hochschullehre}}, title = {{{Exkursinhalte in der fachmathematischen Lehramtsausbildung: Wie man das Wesen und die Rolle der Mathematik vermittelt.}}}, volume = {{3}}, year = {{2017}}, } @article{31274, author = {{Borthwick, David and Weich, Tobias}}, issn = {{1664-039X}}, journal = {{Journal of Spectral Theory}}, keywords = {{Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics}}, number = {{2}}, pages = {{267--329}}, publisher = {{European Mathematical Society - EMS - Publishing House GmbH}}, title = {{{Symmetry reduction of holomorphic iterated function schemes and factorization of Selberg zeta functions}}}, doi = {{10.4171/jst/125}}, volume = {{6}}, year = {{2016}}, } @article{31289, author = {{Weich, Tobias}}, issn = {{1424-0637}}, journal = {{Annales Henri Poincaré}}, keywords = {{Mathematical Physics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics}}, number = {{1}}, pages = {{37--52}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{On the Support of Pollicott–Ruelle Resonanant States for Anosov Flows}}}, doi = {{10.1007/s00023-016-0514-5}}, volume = {{18}}, year = {{2016}}, } @article{31291, abstract = {{We consider a simple model of an open partially expanding map. Its trapped set ${\mathcal{K}}$ 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 $\unicode[STIX]{x1D708}$ 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 ${\mathcal{K}}$. 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 $\unicode[STIX]{x1D708}\rightarrow \infty$. Some numerical computations with the truncated Gauss map and Bowen–Series maps illustrate these results.}}, author = {{ARNOLDI, JEAN FRANCOIS and FAURE, FRÉDÉRIC and Weich, Tobias}}, issn = {{0143-3857}}, journal = {{Ergodic Theory and Dynamical Systems}}, keywords = {{Applied Mathematics, General Mathematics}}, number = {{1}}, pages = {{1--58}}, publisher = {{Cambridge University Press (CUP)}}, title = {{{Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps}}}, doi = {{10.1017/etds.2015.34}}, volume = {{37}}, year = {{2015}}, } @article{31293, author = {{Weich, Tobias}}, issn = {{0010-3616}}, journal = {{Communications in Mathematical Physics}}, keywords = {{Mathematical Physics, Statistical and Nonlinear Physics}}, number = {{2}}, pages = {{727--765}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Resonance Chains and Geometric Limits on Schottky Surfaces}}}, doi = {{10.1007/s00220-015-2359-z}}, volume = {{337}}, year = {{2015}}, } @article{31294, author = {{Weich, Tobias}}, issn = {{0022-2488}}, journal = {{Journal of Mathematical Physics}}, keywords = {{Mathematical Physics, Statistical and Nonlinear Physics}}, number = {{10}}, publisher = {{AIP Publishing}}, title = {{{Equivariant spectral asymptotics forh-pseudodifferential operators}}}, doi = {{10.1063/1.4896698}}, volume = {{55}}, year = {{2014}}, } @article{31296, author = {{Barkhofen, Sonja and Faure, F and Weich, Tobias}}, issn = {{0951-7715}}, journal = {{Nonlinearity}}, keywords = {{Applied Mathematics, General Physics and Astronomy, Mathematical Physics, Statistical and Nonlinear Physics}}, number = {{8}}, pages = {{1829--1858}}, publisher = {{IOP Publishing}}, title = {{{Resonance chains in open systems, generalized zeta functions and clustering of the length spectrum}}}, doi = {{10.1088/0951-7715/27/8/1829}}, volume = {{27}}, year = {{2014}}, } @article{31297, author = {{Weich, Tobias and Barkhofen, Sonja and Kuhl, U and Poli, C and Schomerus, H}}, issn = {{1367-2630}}, journal = {{New Journal of Physics}}, keywords = {{General Physics and Astronomy}}, number = {{3}}, publisher = {{IOP Publishing}}, title = {{{Formation and interaction of resonance chains in the open three-disk system}}}, doi = {{10.1088/1367-2630/16/3/033029}}, volume = {{16}}, year = {{2014}}, } @article{31298, author = {{Barkhofen, Sonja and Weich, Tobias and Potzuweit, A. and Stöckmann, H.-J. and Kuhl, U. and Zworski, M.}}, issn = {{0031-9007}}, journal = {{Physical Review Letters}}, keywords = {{General Physics and Astronomy}}, number = {{16}}, publisher = {{American Physical Society (APS)}}, title = {{{Experimental Observation of the Spectral Gap in Microwave n-Disk Systems}}}, doi = {{10.1103/physrevlett.110.164102}}, volume = {{110}}, year = {{2013}}, } @article{31300, author = {{Potzuweit, A. and Weich, Tobias and Barkhofen, Sonja and Kuhl, U. and Stöckmann, H.-J. and Zworski, M.}}, issn = {{1539-3755}}, journal = {{Physical Review E}}, keywords = {{Industrial and Manufacturing Engineering, Metals and Alloys, Strategy and Management, Mechanical Engineering}}, number = {{6}}, publisher = {{American Physical Society (APS)}}, title = {{{Weyl asymptotics: From closed to open systems}}}, doi = {{10.1103/physreve.86.066205}}, volume = {{86}}, year = {{2012}}, }