[{"type":"journal_article","status":"public","user_id":"49178","department":[{"_id":"10"},{"_id":"548"},{"_id":"91"}],"_id":"31190","citation":{"ieee":"J. Hilgert, T. Weich, and L. L. Wolf, “Higher rank quantum-classical correspondence,” <i>Analysis &#38; PDE</i>, vol. 16, no. 10, pp. 2241–2265, 2023, doi: <a href=\"https://doi.org/10.2140/apde.2023.16.2241\">https://doi.org/10.2140/apde.2023.16.2241</a>.","chicago":"Hilgert, Joachim, Tobias Weich, and Lasse Lennart Wolf. “Higher Rank Quantum-Classical Correspondence.” <i>Analysis &#38; PDE</i> 16, no. 10 (2023): 2241–2265. <a href=\"https://doi.org/10.2140/apde.2023.16.2241\">https://doi.org/10.2140/apde.2023.16.2241</a>.","ama":"Hilgert J, Weich T, Wolf LL. Higher rank quantum-classical correspondence. <i>Analysis &#38; PDE</i>. 2023;16(10):2241–2265. doi:<a href=\"https://doi.org/10.2140/apde.2023.16.2241\">https://doi.org/10.2140/apde.2023.16.2241</a>","apa":"Hilgert, J., Weich, T., &#38; Wolf, L. L. (2023). Higher rank quantum-classical correspondence. <i>Analysis &#38; PDE</i>, <i>16</i>(10), 2241–2265. <a href=\"https://doi.org/10.2140/apde.2023.16.2241\">https://doi.org/10.2140/apde.2023.16.2241</a>","bibtex":"@article{Hilgert_Weich_Wolf_2023, title={Higher rank quantum-classical correspondence}, volume={16}, DOI={<a href=\"https://doi.org/10.2140/apde.2023.16.2241\">https://doi.org/10.2140/apde.2023.16.2241</a>}, number={10}, journal={Analysis &#38; PDE}, publisher={MSP}, author={Hilgert, Joachim and Weich, Tobias and Wolf, Lasse Lennart}, year={2023}, pages={2241–2265} }","short":"J. Hilgert, T. Weich, L.L. Wolf, Analysis &#38; PDE 16 (2023) 2241–2265.","mla":"Hilgert, Joachim, et al. “Higher Rank Quantum-Classical Correspondence.” <i>Analysis &#38; PDE</i>, vol. 16, no. 10, MSP, 2023, pp. 2241–2265, doi:<a href=\"https://doi.org/10.2140/apde.2023.16.2241\">https://doi.org/10.2140/apde.2023.16.2241</a>."},"intvolume":"        16","page":"2241–2265","author":[{"first_name":"Joachim","id":"220","full_name":"Hilgert, Joachim","last_name":"Hilgert"},{"first_name":"Tobias","full_name":"Weich, Tobias","id":"49178","orcid":"0000-0002-9648-6919","last_name":"Weich"},{"first_name":"Lasse Lennart","last_name":"Wolf","orcid":"0000-0001-8893-2045","id":"45027","full_name":"Wolf, Lasse Lennart"}],"volume":16,"date_updated":"2026-02-18T10:39:36Z","doi":"https://doi.org/10.2140/apde.2023.16.2241","publication":"Analysis & PDE","abstract":[{"lang":"eng","text":"For a compact Riemannian locally symmetric space $\\Gamma\\backslash G/K$ of\r\narbitrary rank we determine the location of certain Ruelle-Taylor resonances\r\nfor the Weyl chamber action. We provide a Weyl-lower bound on an appropriate\r\ncounting function for the Ruelle-Taylor resonances and establish a spectral gap\r\nwhich is uniform in $\\Gamma$ if $G/K$ is irreducible of higher rank. This is\r\nachieved by proving a quantum-classical correspondence, i.e. a\r\n1:1-correspondence between horocyclically invariant Ruelle-Taylor resonant\r\nstates and joint eigenfunctions of the algebra of invariant differential\r\noperators on $G/K$."}],"external_id":{"arxiv":["2103.05667"]},"language":[{"iso":"eng"}],"issue":"10","year":"2023","date_created":"2022-05-11T10:41:35Z","publisher":"MSP","title":"Higher rank quantum-classical correspondence"},{"intvolume":"       398","page":"655-678","citation":{"short":"P. Schütte, T. Weich, S. Barkhofen, Communications in Mathematical Physics 398 (2023) 655–678.","bibtex":"@article{Schütte_Weich_Barkhofen_2023, title={Meromorphic Continuation of Weighted Zeta Functions on Open Hyperbolic Systems}, volume={398}, DOI={<a href=\"https://doi.org/10.1007/s00220-022-04538-z\">https://doi.org/10.1007/s00220-022-04538-z</a>}, journal={Communications in Mathematical Physics}, author={Schütte, Philipp and Weich, Tobias and Barkhofen, Sonja}, year={2023}, pages={655–678} }","mla":"Schütte, Philipp, et al. “Meromorphic Continuation of Weighted Zeta Functions on Open Hyperbolic Systems.” <i>Communications in Mathematical Physics</i>, vol. 398, 2023, pp. 655–78, doi:<a href=\"https://doi.org/10.1007/s00220-022-04538-z\">https://doi.org/10.1007/s00220-022-04538-z</a>.","apa":"Schütte, P., Weich, T., &#38; Barkhofen, S. (2023). Meromorphic Continuation of Weighted Zeta Functions on Open Hyperbolic Systems. <i>Communications in Mathematical Physics</i>, <i>398</i>, 655–678. <a href=\"https://doi.org/10.1007/s00220-022-04538-z\">https://doi.org/10.1007/s00220-022-04538-z</a>","ama":"Schütte P, Weich T, Barkhofen S. Meromorphic Continuation of Weighted Zeta Functions on Open Hyperbolic Systems. <i>Communications in Mathematical Physics</i>. 2023;398:655-678. doi:<a href=\"https://doi.org/10.1007/s00220-022-04538-z\">https://doi.org/10.1007/s00220-022-04538-z</a>","chicago":"Schütte, Philipp, Tobias Weich, and Sonja Barkhofen. “Meromorphic Continuation of Weighted Zeta Functions on Open Hyperbolic Systems.” <i>Communications in Mathematical Physics</i> 398 (2023): 655–78. <a href=\"https://doi.org/10.1007/s00220-022-04538-z\">https://doi.org/10.1007/s00220-022-04538-z</a>.","ieee":"P. Schütte, T. Weich, and S. Barkhofen, “Meromorphic Continuation of Weighted Zeta Functions on Open Hyperbolic Systems,” <i>Communications in Mathematical Physics</i>, vol. 398, pp. 655–678, 2023, doi: <a href=\"https://doi.org/10.1007/s00220-022-04538-z\">https://doi.org/10.1007/s00220-022-04538-z</a>."},"year":"2023","volume":398,"date_created":"2022-05-04T12:27:46Z","author":[{"first_name":"Philipp","last_name":"Schütte","id":"50168","full_name":"Schütte, Philipp"},{"first_name":"Tobias","full_name":"Weich, Tobias","id":"49178","last_name":"Weich","orcid":"0000-0002-9648-6919"},{"last_name":"Barkhofen","full_name":"Barkhofen, Sonja","id":"48188","first_name":"Sonja"}],"date_updated":"2026-02-18T10:41:07Z","doi":"https://doi.org/10.1007/s00220-022-04538-z","title":"Meromorphic Continuation of Weighted Zeta Functions on Open Hyperbolic Systems","publication":"Communications in Mathematical Physics","type":"journal_article","status":"public","abstract":[{"lang":"eng","text":"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."}],"department":[{"_id":"10"},{"_id":"548"},{"_id":"623"},{"_id":"15"}],"user_id":"49178","external_id":{"arxiv":["2112.05791"]},"_id":"31059","language":[{"iso":"eng"}]},{"_id":"31982","user_id":"70575","department":[{"_id":"548"}],"keyword":["General Mathematics"],"language":[{"iso":"eng"}],"type":"journal_article","publication":"Inventiones mathematicae","abstract":[{"text":"<jats:title>Abstract</jats:title><jats:p>We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Sigma $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mi>Σ</mml:mi>\r\n                </mml:math></jats:alternatives></jats:inline-formula> with Betti number <jats:inline-formula><jats:alternatives><jats:tex-math>$$b_1$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:msub>\r\n                    <mml:mi>b</mml:mi>\r\n                    <mml:mn>1</mml:mn>\r\n                  </mml:msub>\r\n                </mml:math></jats:alternatives></jats:inline-formula>, the order of vanishing of the Ruelle zeta function at zero equals <jats:inline-formula><jats:alternatives><jats:tex-math>$$4-b_1$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mn>4</mml:mn>\r\n                    <mml:mo>-</mml:mo>\r\n                    <mml:msub>\r\n                      <mml:mi>b</mml:mi>\r\n                      <mml:mn>1</mml:mn>\r\n                    </mml:msub>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>, while in the hyperbolic case it is equal to <jats:inline-formula><jats:alternatives><jats:tex-math>$$4-2b_1$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mn>4</mml:mn>\r\n                    <mml:mo>-</mml:mo>\r\n                    <mml:mn>2</mml:mn>\r\n                    <mml:msub>\r\n                      <mml:mi>b</mml:mi>\r\n                      <mml:mn>1</mml:mn>\r\n                    </mml:msub>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula>. This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott–Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that the first variation is generically nonzero we introduce a new identity relating pushforwards of products of resonant and coresonant 2-forms on the sphere bundle <jats:inline-formula><jats:alternatives><jats:tex-math>$$S\\Sigma $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mrow>\r\n                    <mml:mi>S</mml:mi>\r\n                    <mml:mi>Σ</mml:mi>\r\n                  </mml:mrow>\r\n                </mml:math></jats:alternatives></jats:inline-formula> with harmonic 1-forms on <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Sigma $$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mi>Σ</mml:mi>\r\n                </mml:math></jats:alternatives></jats:inline-formula>.</jats:p>","lang":"eng"}],"status":"public","date_updated":"2022-06-21T11:55:15Z","publisher":"Springer Science and Business Media LLC","author":[{"first_name":"Mihajlo","last_name":"Cekić","full_name":"Cekić, Mihajlo"},{"first_name":"Benjamin","last_name":"Delarue","full_name":"Delarue, Benjamin","id":"70575"},{"first_name":"Semyon","last_name":"Dyatlov","full_name":"Dyatlov, Semyon"},{"first_name":"Gabriel P.","full_name":"Paternain, Gabriel P.","last_name":"Paternain"}],"date_created":"2022-06-20T08:24:17Z","volume":229,"title":"The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds","doi":"10.1007/s00222-022-01108-x","publication_status":"published","publication_identifier":{"issn":["0020-9910","1432-1297"]},"issue":"1","year":"2022","citation":{"ama":"Cekić M, Delarue B, Dyatlov S, Paternain GP. The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds. <i>Inventiones mathematicae</i>. 2022;229(1):303-394. doi:<a href=\"https://doi.org/10.1007/s00222-022-01108-x\">10.1007/s00222-022-01108-x</a>","ieee":"M. Cekić, B. Delarue, S. Dyatlov, and G. P. Paternain, “The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds,” <i>Inventiones mathematicae</i>, vol. 229, no. 1, pp. 303–394, 2022, doi: <a href=\"https://doi.org/10.1007/s00222-022-01108-x\">10.1007/s00222-022-01108-x</a>.","chicago":"Cekić, Mihajlo, Benjamin Delarue, Semyon Dyatlov, and Gabriel P. Paternain. “The Ruelle Zeta Function at Zero for Nearly Hyperbolic 3-Manifolds.” <i>Inventiones Mathematicae</i> 229, no. 1 (2022): 303–94. <a href=\"https://doi.org/10.1007/s00222-022-01108-x\">https://doi.org/10.1007/s00222-022-01108-x</a>.","apa":"Cekić, M., Delarue, B., Dyatlov, S., &#38; Paternain, G. P. (2022). The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds. <i>Inventiones Mathematicae</i>, <i>229</i>(1), 303–394. <a href=\"https://doi.org/10.1007/s00222-022-01108-x\">https://doi.org/10.1007/s00222-022-01108-x</a>","short":"M. Cekić, B. Delarue, S. Dyatlov, G.P. Paternain, Inventiones Mathematicae 229 (2022) 303–394.","bibtex":"@article{Cekić_Delarue_Dyatlov_Paternain_2022, title={The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds}, volume={229}, DOI={<a href=\"https://doi.org/10.1007/s00222-022-01108-x\">10.1007/s00222-022-01108-x</a>}, number={1}, journal={Inventiones mathematicae}, publisher={Springer Science and Business Media LLC}, author={Cekić, Mihajlo and Delarue, Benjamin and Dyatlov, Semyon and Paternain, Gabriel P.}, year={2022}, pages={303–394} }","mla":"Cekić, Mihajlo, et al. “The Ruelle Zeta Function at Zero for Nearly Hyperbolic 3-Manifolds.” <i>Inventiones Mathematicae</i>, vol. 229, no. 1, Springer Science and Business Media LLC, 2022, pp. 303–94, doi:<a href=\"https://doi.org/10.1007/s00222-022-01108-x\">10.1007/s00222-022-01108-x</a>."},"intvolume":"       229","page":"303-394"},{"year":"2022","issue":"3","title":"Ruelle–Pollicott resonances for manifolds with hyperbolic cusps","date_created":"2023-01-05T16:23:34Z","publisher":"European Mathematical Society - EMS - Publishing House GmbH","publication":"Journal of the European Mathematical Society","language":[{"iso":"eng"}],"keyword":["Applied Mathematics","General Mathematics"],"intvolume":"        24","page":"851-923","citation":{"mla":"Guedes Bonthonneau, Yannick, and Tobias Weich. “Ruelle–Pollicott Resonances for Manifolds with Hyperbolic Cusps.” <i>Journal of the European Mathematical Society</i>, vol. 24, no. 3, European Mathematical Society - EMS - Publishing House GmbH, 2022, pp. 851–923, doi:<a href=\"https://doi.org/10.4171/jems/1103\">10.4171/jems/1103</a>.","bibtex":"@article{Guedes Bonthonneau_Weich_2022, title={Ruelle–Pollicott resonances for manifolds with hyperbolic cusps}, volume={24}, DOI={<a href=\"https://doi.org/10.4171/jems/1103\">10.4171/jems/1103</a>}, number={3}, journal={Journal of the European Mathematical Society}, publisher={European Mathematical Society - EMS - Publishing House GmbH}, author={Guedes Bonthonneau, Yannick and Weich, Tobias}, year={2022}, pages={851–923} }","short":"Y. Guedes Bonthonneau, T. Weich, Journal of the European Mathematical Society 24 (2022) 851–923.","apa":"Guedes Bonthonneau, Y., &#38; Weich, T. (2022). Ruelle–Pollicott resonances for manifolds with hyperbolic cusps. <i>Journal of the European Mathematical Society</i>, <i>24</i>(3), 851–923. <a href=\"https://doi.org/10.4171/jems/1103\">https://doi.org/10.4171/jems/1103</a>","ieee":"Y. Guedes Bonthonneau and T. Weich, “Ruelle–Pollicott resonances for manifolds with hyperbolic cusps,” <i>Journal of the European Mathematical Society</i>, vol. 24, no. 3, pp. 851–923, 2022, doi: <a href=\"https://doi.org/10.4171/jems/1103\">10.4171/jems/1103</a>.","chicago":"Guedes Bonthonneau, Yannick, and Tobias Weich. “Ruelle–Pollicott Resonances for Manifolds with Hyperbolic Cusps.” <i>Journal of the European Mathematical Society</i> 24, no. 3 (2022): 851–923. <a href=\"https://doi.org/10.4171/jems/1103\">https://doi.org/10.4171/jems/1103</a>.","ama":"Guedes Bonthonneau Y, Weich T. Ruelle–Pollicott resonances for manifolds with hyperbolic cusps. <i>Journal of the European Mathematical Society</i>. 2022;24(3):851-923. doi:<a href=\"https://doi.org/10.4171/jems/1103\">10.4171/jems/1103</a>"},"publication_identifier":{"issn":["1435-9855"]},"publication_status":"published","doi":"10.4171/jems/1103","volume":24,"author":[{"first_name":"Yannick","last_name":"Guedes Bonthonneau","full_name":"Guedes Bonthonneau, Yannick"},{"first_name":"Tobias","full_name":"Weich, Tobias","id":"49178","last_name":"Weich","orcid":"0000-0002-9648-6919"}],"date_updated":"2023-01-06T08:47:35Z","status":"public","type":"journal_article","department":[{"_id":"10"},{"_id":"623"},{"_id":"548"}],"user_id":"49178","_id":"35306"},{"title":"Semiclassical formulae For Wigner distributions","publisher":"IOP Publishing Ltd","date_created":"2022-05-04T12:23:11Z","year":"2022","issue":"24","language":[{"iso":"eng"}],"external_id":{"arxiv":["2201.04892"]},"abstract":[{"lang":"eng","text":"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."}],"publication":"Journal of Physics A: Mathematical and Theoretical","doi":"10.1088/1751-8121/ac6d2b","date_updated":"2024-02-06T20:40:45Z","volume":55,"author":[{"full_name":"Barkhofen, Sonja","id":"48188","last_name":"Barkhofen","first_name":"Sonja"},{"id":"50168","full_name":"Schütte, Philipp","last_name":"Schütte","first_name":"Philipp"},{"last_name":"Weich","orcid":"0000-0002-9648-6919","id":"49178","full_name":"Weich, Tobias","first_name":"Tobias"}],"intvolume":"        55","citation":{"apa":"Barkhofen, S., Schütte, P., &#38; Weich, T. (2022). Semiclassical formulae For Wigner distributions. <i>Journal of Physics A: Mathematical and Theoretical</i>, <i>55</i>(24), Article 244007. <a href=\"https://doi.org/10.1088/1751-8121/ac6d2b\">https://doi.org/10.1088/1751-8121/ac6d2b</a>","short":"S. Barkhofen, P. Schütte, T. Weich, Journal of Physics A: Mathematical and Theoretical 55 (2022).","mla":"Barkhofen, Sonja, et al. “Semiclassical Formulae For Wigner Distributions.” <i>Journal of Physics A: Mathematical and Theoretical</i>, vol. 55, no. 24, 244007, IOP Publishing Ltd, 2022, doi:<a href=\"https://doi.org/10.1088/1751-8121/ac6d2b\">10.1088/1751-8121/ac6d2b</a>.","bibtex":"@article{Barkhofen_Schütte_Weich_2022, title={Semiclassical formulae For Wigner distributions}, volume={55}, DOI={<a href=\"https://doi.org/10.1088/1751-8121/ac6d2b\">10.1088/1751-8121/ac6d2b</a>}, number={24244007}, journal={Journal of Physics A: Mathematical and Theoretical}, publisher={IOP Publishing Ltd}, author={Barkhofen, Sonja and Schütte, Philipp and Weich, Tobias}, year={2022} }","ama":"Barkhofen S, Schütte P, Weich T. Semiclassical formulae For Wigner distributions. <i>Journal of Physics A: Mathematical and Theoretical</i>. 2022;55(24). doi:<a href=\"https://doi.org/10.1088/1751-8121/ac6d2b\">10.1088/1751-8121/ac6d2b</a>","chicago":"Barkhofen, Sonja, Philipp Schütte, and Tobias Weich. “Semiclassical Formulae For Wigner Distributions.” <i>Journal of Physics A: Mathematical and Theoretical</i> 55, no. 24 (2022). <a href=\"https://doi.org/10.1088/1751-8121/ac6d2b\">https://doi.org/10.1088/1751-8121/ac6d2b</a>.","ieee":"S. Barkhofen, P. Schütte, and T. Weich, “Semiclassical formulae For Wigner distributions,” <i>Journal of Physics A: Mathematical and Theoretical</i>, vol. 55, no. 24, Art. no. 244007, 2022, doi: <a href=\"https://doi.org/10.1088/1751-8121/ac6d2b\">10.1088/1751-8121/ac6d2b</a>."},"article_number":"244007","article_type":"review","_id":"31057","department":[{"_id":"623"},{"_id":"548"},{"_id":"10"}],"user_id":"49178","status":"public","type":"journal_article"},{"user_id":"49063","department":[{"_id":"10"},{"_id":"623"},{"_id":"548"},{"_id":"91"}],"_id":"35322","language":[{"iso":"eng"}],"keyword":["Geometry and Topology","Mathematical Physics","Statistical and Nonlinear Physics"],"type":"journal_article","publication":"Journal of Spectral Theory","status":"public","date_created":"2023-01-06T08:49:06Z","author":[{"full_name":"Bux, Kai-Uwe","last_name":"Bux","first_name":"Kai-Uwe"},{"first_name":"Joachim","last_name":"Hilgert","id":"220","full_name":"Hilgert, Joachim"},{"first_name":"Tobias","last_name":"Weich","orcid":"0000-0002-9648-6919","full_name":"Weich, Tobias","id":"49178"}],"volume":12,"publisher":"European Mathematical Society - EMS - Publishing House GmbH","date_updated":"2024-02-19T06:28:12Z","doi":"10.4171/jst/414","title":"Poisson transforms for trees of bounded degree","issue":"2","publication_status":"published","publication_identifier":{"issn":["1664-039X"]},"citation":{"mla":"Bux, Kai-Uwe, et al. “Poisson Transforms for Trees of Bounded Degree.” <i>Journal of Spectral Theory</i>, vol. 12, no. 2, European Mathematical Society - EMS - Publishing House GmbH, 2022, pp. 659–81, doi:<a href=\"https://doi.org/10.4171/jst/414\">10.4171/jst/414</a>.","bibtex":"@article{Bux_Hilgert_Weich_2022, title={Poisson transforms for trees of bounded degree}, volume={12}, DOI={<a href=\"https://doi.org/10.4171/jst/414\">10.4171/jst/414</a>}, number={2}, journal={Journal of Spectral Theory}, publisher={European Mathematical Society - EMS - Publishing House GmbH}, author={Bux, Kai-Uwe and Hilgert, Joachim and Weich, Tobias}, year={2022}, pages={659–681} }","short":"K.-U. Bux, J. Hilgert, T. Weich, Journal of Spectral Theory 12 (2022) 659–681.","apa":"Bux, K.-U., Hilgert, J., &#38; Weich, T. (2022). Poisson transforms for trees of bounded degree. <i>Journal of Spectral Theory</i>, <i>12</i>(2), 659–681. <a href=\"https://doi.org/10.4171/jst/414\">https://doi.org/10.4171/jst/414</a>","chicago":"Bux, Kai-Uwe, Joachim Hilgert, and Tobias Weich. “Poisson Transforms for Trees of Bounded Degree.” <i>Journal of Spectral Theory</i> 12, no. 2 (2022): 659–81. <a href=\"https://doi.org/10.4171/jst/414\">https://doi.org/10.4171/jst/414</a>.","ieee":"K.-U. Bux, J. Hilgert, and T. Weich, “Poisson transforms for trees of bounded degree,” <i>Journal of Spectral Theory</i>, vol. 12, no. 2, pp. 659–681, 2022, doi: <a href=\"https://doi.org/10.4171/jst/414\">10.4171/jst/414</a>.","ama":"Bux K-U, Hilgert J, Weich T. Poisson transforms for trees of bounded degree. <i>Journal of Spectral Theory</i>. 2022;12(2):659-681. doi:<a href=\"https://doi.org/10.4171/jst/414\">10.4171/jst/414</a>"},"intvolume":"        12","page":"659-681","year":"2022"},{"issue":"1","publication_identifier":{"issn":["0232-704X","1572-9060"]},"publication_status":"published","intvolume":"        63","citation":{"mla":"Olbrich, Martin, and Guendalina Palmirotta. “Delorme’s Intertwining Conditions for Sections of Homogeneous Vector Bundles on Two- and Three-Dimensional Hyperbolic Spaces.” <i>Annals of Global Analysis and Geometry</i>, vol. 63, no. 1, 9, Springer Science and Business Media LLC, 2022, doi:<a href=\"https://doi.org/10.1007/s10455-022-09882-w\">10.1007/s10455-022-09882-w</a>.","short":"M. Olbrich, G. Palmirotta, Annals of Global Analysis and Geometry 63 (2022).","bibtex":"@article{Olbrich_Palmirotta_2022, title={Delorme’s intertwining conditions for sections of homogeneous vector bundles on two- and three-dimensional hyperbolic spaces}, volume={63}, DOI={<a href=\"https://doi.org/10.1007/s10455-022-09882-w\">10.1007/s10455-022-09882-w</a>}, number={19}, journal={Annals of Global Analysis and Geometry}, publisher={Springer Science and Business Media LLC}, author={Olbrich, Martin and Palmirotta, Guendalina}, year={2022} }","apa":"Olbrich, M., &#38; Palmirotta, G. (2022). Delorme’s intertwining conditions for sections of homogeneous vector bundles on two- and three-dimensional hyperbolic spaces. <i>Annals of Global Analysis and Geometry</i>, <i>63</i>(1), Article 9. <a href=\"https://doi.org/10.1007/s10455-022-09882-w\">https://doi.org/10.1007/s10455-022-09882-w</a>","ama":"Olbrich M, Palmirotta G. Delorme’s intertwining conditions for sections of homogeneous vector bundles on two- and three-dimensional hyperbolic spaces. <i>Annals of Global Analysis and Geometry</i>. 2022;63(1). doi:<a href=\"https://doi.org/10.1007/s10455-022-09882-w\">10.1007/s10455-022-09882-w</a>","ieee":"M. Olbrich and G. Palmirotta, “Delorme’s intertwining conditions for sections of homogeneous vector bundles on two- and three-dimensional hyperbolic spaces,” <i>Annals of Global Analysis and Geometry</i>, vol. 63, no. 1, Art. no. 9, 2022, doi: <a href=\"https://doi.org/10.1007/s10455-022-09882-w\">10.1007/s10455-022-09882-w</a>.","chicago":"Olbrich, Martin, and Guendalina Palmirotta. “Delorme’s Intertwining Conditions for Sections of Homogeneous Vector Bundles on Two- and Three-Dimensional Hyperbolic Spaces.” <i>Annals of Global Analysis and Geometry</i> 63, no. 1 (2022). <a href=\"https://doi.org/10.1007/s10455-022-09882-w\">https://doi.org/10.1007/s10455-022-09882-w</a>."},"year":"2022","volume":63,"date_created":"2026-02-20T20:02:50Z","author":[{"last_name":"Olbrich","full_name":"Olbrich, Martin","first_name":"Martin"},{"last_name":"Palmirotta","full_name":"Palmirotta, Guendalina","id":"109467","first_name":"Guendalina"}],"date_updated":"2026-02-20T20:03:38Z","publisher":"Springer Science and Business Media LLC","doi":"10.1007/s10455-022-09882-w","title":"Delorme’s intertwining conditions for sections of homogeneous vector bundles on two- and three-dimensional hyperbolic spaces","publication":"Annals of Global Analysis and Geometry","type":"journal_article","status":"public","department":[{"_id":"10"},{"_id":"548"}],"user_id":"109467","_id":"64570","language":[{"iso":"eng"}],"extern":"1","article_number":"9"},{"year":"2022","issue":"2","title":"A topological Paley-Wiener-Schwartz Theorem for sections of homogeneous vector bundles on $G/K$","publisher":"Heldermann Verlag","date_created":"2026-02-20T20:04:49Z","abstract":[{"lang":"eng","text":"We study the Fourier transform for compactly supported distributional sections of complex homogeneous vector bundles on symmetric spaces of non-compact type $X = G/K$. We prove a characterisation of their range. In fact, from Delorme's Paley-Wiener theorem for compactly supported smooth functions on a real reductive group of Harish-Chandra class, we deduce topological Paley-Wiener and Paley-Wiener-Schwartz theorems for sections."}],"publication":"Journal of Lie theory","language":[{"iso":"eng"}],"external_id":{"arxiv":["2202.06905"]},"citation":{"ama":"Olbrich M, Palmirotta G. A topological Paley-Wiener-Schwartz Theorem for sections of homogeneous vector bundles on $G/K$. <i>Journal of Lie theory</i>. 2022;34(2):53--384.","chicago":"Olbrich, Martin, and Guendalina Palmirotta. “A Topological Paley-Wiener-Schwartz Theorem for Sections of Homogeneous Vector Bundles on $G/K$.” <i>Journal of Lie Theory</i> 34, no. 2 (2022): 53--384.","ieee":"M. Olbrich and G. Palmirotta, “A topological Paley-Wiener-Schwartz Theorem for sections of homogeneous vector bundles on $G/K$,” <i>Journal of Lie theory</i>, vol. 34, no. 2, pp. 53--384, 2022.","apa":"Olbrich, M., &#38; Palmirotta, G. (2022). A topological Paley-Wiener-Schwartz Theorem for sections of homogeneous vector bundles on $G/K$. <i>Journal of Lie Theory</i>, <i>34</i>(2), 53--384.","bibtex":"@article{Olbrich_Palmirotta_2022, title={A topological Paley-Wiener-Schwartz Theorem for sections of homogeneous vector bundles on $G/K$}, volume={34}, number={2}, journal={Journal of Lie theory}, publisher={Heldermann Verlag}, author={Olbrich, Martin and Palmirotta, Guendalina}, year={2022}, pages={53--384} }","short":"M. Olbrich, G. Palmirotta, Journal of Lie Theory 34 (2022) 53--384.","mla":"Olbrich, Martin, and Guendalina Palmirotta. “A Topological Paley-Wiener-Schwartz Theorem for Sections of Homogeneous Vector Bundles on $G/K$.” <i>Journal of Lie Theory</i>, vol. 34, no. 2, Heldermann Verlag, 2022, pp. 53--384."},"intvolume":"        34","page":"53--384","publication_status":"published","date_updated":"2026-02-20T20:07:31Z","author":[{"full_name":"Olbrich, Martin","last_name":"Olbrich","first_name":"Martin"},{"last_name":"Palmirotta","full_name":"Palmirotta, Guendalina","id":"109467","first_name":"Guendalina"}],"volume":34,"status":"public","type":"journal_article","extern":"1","_id":"64571","user_id":"109467","department":[{"_id":"10"},{"_id":"548"}]},{"issue":"6","publication_identifier":{"unknown":["1540-2347","1527-5256"]},"publication_status":"published","page":"1281 - 1337","intvolume":"        19","citation":{"ama":"Delarue B, Ramacher P. Asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions. <i>Journal of Symplectic Geometry</i>. 2021;19(6):1281-1337. doi:<a href=\"https://doi.org/10.4310/JSG.2021.v19.n6.a1\">10.4310/JSG.2021.v19.n6.a1</a>","ieee":"B. Delarue and P. Ramacher, “Asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions,” <i>Journal of Symplectic Geometry</i>, vol. 19, no. 6, pp. 1281–1337, 2021, doi: <a href=\"https://doi.org/10.4310/JSG.2021.v19.n6.a1\">10.4310/JSG.2021.v19.n6.a1</a>.","chicago":"Delarue, Benjamin, and Pablo Ramacher. “Asymptotic Expansion of Generalized Witten Integrals for Hamiltonian Circle Actions.” <i>Journal of Symplectic Geometry</i> 19, no. 6 (2021): 1281–1337. <a href=\"https://doi.org/10.4310/JSG.2021.v19.n6.a1\">https://doi.org/10.4310/JSG.2021.v19.n6.a1</a>.","bibtex":"@article{Delarue_Ramacher_2021, title={Asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions}, volume={19}, DOI={<a href=\"https://doi.org/10.4310/JSG.2021.v19.n6.a1\">10.4310/JSG.2021.v19.n6.a1</a>}, number={6}, journal={Journal of Symplectic Geometry}, author={Delarue, Benjamin and Ramacher, Pablo}, year={2021}, pages={1281–1337} }","short":"B. Delarue, P. Ramacher, Journal of Symplectic Geometry 19 (2021) 1281–1337.","mla":"Delarue, Benjamin, and Pablo Ramacher. “Asymptotic Expansion of Generalized Witten Integrals for Hamiltonian Circle Actions.” <i>Journal of Symplectic Geometry</i>, vol. 19, no. 6, 2021, pp. 1281–337, doi:<a href=\"https://doi.org/10.4310/JSG.2021.v19.n6.a1\">10.4310/JSG.2021.v19.n6.a1</a>.","apa":"Delarue, B., &#38; Ramacher, P. (2021). Asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions. <i>Journal of Symplectic Geometry</i>, <i>19</i>(6), 1281–1337. <a href=\"https://doi.org/10.4310/JSG.2021.v19.n6.a1\">https://doi.org/10.4310/JSG.2021.v19.n6.a1</a>"},"year":"2021","volume":19,"date_created":"2022-06-20T08:46:56Z","author":[{"last_name":"Delarue","id":"70575","full_name":"Delarue, Benjamin","first_name":"Benjamin"},{"first_name":"Pablo","last_name":"Ramacher","full_name":"Ramacher, Pablo"}],"date_updated":"2022-06-21T11:54:50Z","doi":"10.4310/JSG.2021.v19.n6.a1","title":"Asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions","publication":"Journal of Symplectic Geometry","type":"journal_article","status":"public","department":[{"_id":"548"}],"user_id":"70575","_id":"32016","language":[{"iso":"eng"}],"article_type":"original"},{"citation":{"ama":"Schütte P, Weich T, Delarue B. Resonances and weighted zeta functions for obstacle scattering via smooth models. Published online 2021.","chicago":"Schütte, Philipp, Tobias Weich, and Benjamin Delarue. “Resonances and Weighted Zeta Functions for Obstacle Scattering via Smooth Models,” 2021.","ieee":"P. Schütte, T. Weich, and B. Delarue, “Resonances and weighted zeta functions for obstacle scattering via smooth models.” 2021.","bibtex":"@article{Schütte_Weich_Delarue_2021, title={Resonances and weighted zeta functions for obstacle scattering via smooth models}, author={Schütte, Philipp and Weich, Tobias and Delarue, Benjamin}, year={2021} }","short":"P. Schütte, T. Weich, B. Delarue, (2021).","mla":"Schütte, Philipp, et al. <i>Resonances and Weighted Zeta Functions for Obstacle Scattering via Smooth Models</i>. 2021.","apa":"Schütte, P., Weich, T., &#38; Delarue, B. (2021). <i>Resonances and weighted zeta functions for obstacle scattering via smooth models</i>."},"year":"2021","date_created":"2022-05-04T12:25:58Z","author":[{"first_name":"Philipp","full_name":"Schütte, Philipp","id":"50168","last_name":"Schütte"},{"first_name":"Tobias","id":"49178","full_name":"Weich, Tobias","last_name":"Weich","orcid":"0000-0002-9648-6919"},{"first_name":"Benjamin","full_name":"Delarue, Benjamin","last_name":"Delarue"}],"date_updated":"2022-05-17T12:05:52Z","title":"Resonances and weighted zeta functions for obstacle scattering via smooth models","type":"preprint","status":"public","abstract":[{"lang":"eng","text":"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."}],"department":[{"_id":"10"},{"_id":"548"}],"user_id":"50168","_id":"31058","external_id":{"arxiv":["2109.05907"]},"language":[{"iso":"eng"}]},{"publication_identifier":{"issn":["1073-7928","1687-0247"]},"publication_status":"published","page":"8225-8296","intvolume":"      2021","citation":{"mla":"Küster, Benjamin, and Tobias Weich. “Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces.” <i>International Mathematics Research Notices</i>, vol. 2021, no. 11, Oxford University Press (OUP), 2021, pp. 8225–96, doi:<a href=\"https://doi.org/10.1093/imrn/rnz068\">10.1093/imrn/rnz068</a>.","bibtex":"@article{Küster_Weich_2021, title={Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces}, volume={2021}, DOI={<a href=\"https://doi.org/10.1093/imrn/rnz068\">10.1093/imrn/rnz068</a>}, number={11}, journal={International Mathematics Research Notices}, publisher={Oxford University Press (OUP)}, author={Küster, Benjamin and Weich, Tobias}, year={2021}, pages={8225–8296} }","short":"B. Küster, T. Weich, International Mathematics Research Notices 2021 (2021) 8225–8296.","apa":"Küster, B., &#38; Weich, T. (2021). Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces. <i>International Mathematics Research Notices</i>, <i>2021</i>(11), 8225–8296. <a href=\"https://doi.org/10.1093/imrn/rnz068\">https://doi.org/10.1093/imrn/rnz068</a>","ama":"Küster B, Weich T. Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces. <i>International Mathematics Research Notices</i>. 2021;2021(11):8225-8296. doi:<a href=\"https://doi.org/10.1093/imrn/rnz068\">10.1093/imrn/rnz068</a>","chicago":"Küster, Benjamin, and Tobias Weich. “Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces.” <i>International Mathematics Research Notices</i> 2021, no. 11 (2021): 8225–96. <a href=\"https://doi.org/10.1093/imrn/rnz068\">https://doi.org/10.1093/imrn/rnz068</a>.","ieee":"B. Küster and T. Weich, “Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces,” <i>International Mathematics Research Notices</i>, vol. 2021, no. 11, pp. 8225–8296, 2021, doi: <a href=\"https://doi.org/10.1093/imrn/rnz068\">10.1093/imrn/rnz068</a>."},"volume":2021,"author":[{"full_name":"Küster, Benjamin","last_name":"Küster","first_name":"Benjamin"},{"first_name":"Tobias","last_name":"Weich","full_name":"Weich, Tobias"}],"date_updated":"2022-05-25T06:42:01Z","doi":"10.1093/imrn/rnz068","type":"journal_article","status":"public","department":[{"_id":"10"},{"_id":"623"},{"_id":"548"}],"user_id":"49178","_id":"31261","issue":"11","year":"2021","date_created":"2022-05-17T12:00:36Z","publisher":"Oxford University Press (OUP)","title":"Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces","publication":"International Mathematics Research Notices","abstract":[{"lang":"eng","text":"<jats:title>Abstract</jats:title>\r\n               <jats:p>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.</jats:p>"}],"external_id":{"arxiv":["1710.04625"]},"language":[{"iso":"eng"}],"keyword":["General Mathematics"]},{"page":"81-119","intvolume":"         4","citation":{"short":"C. Guillarmou, J. Hilgert, T. Weich, Annales Henri Lebesgue 4 (2021) 81–119.","mla":"Guillarmou, Colin, et al. “High Frequency Limits for Invariant Ruelle Densities.” <i>Annales Henri Lebesgue</i>, vol. 4, Cellule MathDoc/CEDRAM, 2021, pp. 81–119, doi:<a href=\"https://doi.org/10.5802/ahl.67\">10.5802/ahl.67</a>.","bibtex":"@article{Guillarmou_Hilgert_Weich_2021, title={High frequency limits for invariant Ruelle densities}, volume={4}, DOI={<a href=\"https://doi.org/10.5802/ahl.67\">10.5802/ahl.67</a>}, journal={Annales Henri Lebesgue}, publisher={Cellule MathDoc/CEDRAM}, author={Guillarmou, Colin and Hilgert, Joachim and Weich, Tobias}, year={2021}, pages={81–119} }","apa":"Guillarmou, C., Hilgert, J., &#38; Weich, T. (2021). High frequency limits for invariant Ruelle densities. <i>Annales Henri Lebesgue</i>, <i>4</i>, 81–119. <a href=\"https://doi.org/10.5802/ahl.67\">https://doi.org/10.5802/ahl.67</a>","chicago":"Guillarmou, Colin, Joachim Hilgert, and Tobias Weich. “High Frequency Limits for Invariant Ruelle Densities.” <i>Annales Henri Lebesgue</i> 4 (2021): 81–119. <a href=\"https://doi.org/10.5802/ahl.67\">https://doi.org/10.5802/ahl.67</a>.","ieee":"C. Guillarmou, J. Hilgert, and T. Weich, “High frequency limits for invariant Ruelle densities,” <i>Annales Henri Lebesgue</i>, vol. 4, pp. 81–119, 2021, doi: <a href=\"https://doi.org/10.5802/ahl.67\">10.5802/ahl.67</a>.","ama":"Guillarmou C, Hilgert J, Weich T. High frequency limits for invariant Ruelle densities. <i>Annales Henri Lebesgue</i>. 2021;4:81-119. doi:<a href=\"https://doi.org/10.5802/ahl.67\">10.5802/ahl.67</a>"},"publication_identifier":{"issn":["2644-9463"]},"publication_status":"published","doi":"10.5802/ahl.67","volume":4,"author":[{"last_name":"Guillarmou","full_name":"Guillarmou, Colin","first_name":"Colin"},{"full_name":"Hilgert, Joachim","id":"220","last_name":"Hilgert","first_name":"Joachim"},{"first_name":"Tobias","full_name":"Weich, Tobias","id":"49178","orcid":"0000-0002-9648-6919","last_name":"Weich"}],"date_updated":"2024-02-19T06:27:43Z","status":"public","type":"journal_article","department":[{"_id":"10"},{"_id":"623"},{"_id":"548"},{"_id":"91"}],"user_id":"49063","_id":"31263","year":"2021","title":"High frequency limits for invariant Ruelle densities","date_created":"2022-05-17T12:05:17Z","publisher":"Cellule MathDoc/CEDRAM","publication":"Annales Henri Lebesgue","language":[{"iso":"eng"}],"external_id":{"arxiv":["1803.06717"]}},{"publication":"Annales Henri Poincaré","language":[{"iso":"eng"}],"keyword":["Mathematical Physics","Nuclear and High Energy Physics","Statistical and Nonlinear Physics"],"year":"2021","issue":"11","title":"Spectral Theory of the Frame Flow on Hyperbolic 3-Manifolds","date_created":"2022-06-20T08:37:52Z","publisher":"Springer Science and Business Media LLC","status":"public","type":"journal_article","department":[{"_id":"548"}],"user_id":"70575","_id":"32006","page":"3565-3617","intvolume":"        22","citation":{"apa":"Guillarmou, C., &#38; Küster, B. (2021). Spectral Theory of the Frame Flow on Hyperbolic 3-Manifolds. <i>Annales Henri Poincaré</i>, <i>22</i>(11), 3565–3617. <a href=\"https://doi.org/10.1007/s00023-021-01068-7\">https://doi.org/10.1007/s00023-021-01068-7</a>","short":"C. Guillarmou, B. Küster, Annales Henri Poincaré 22 (2021) 3565–3617.","bibtex":"@article{Guillarmou_Küster_2021, title={Spectral Theory of the Frame Flow on Hyperbolic 3-Manifolds}, volume={22}, DOI={<a href=\"https://doi.org/10.1007/s00023-021-01068-7\">10.1007/s00023-021-01068-7</a>}, number={11}, journal={Annales Henri Poincaré}, publisher={Springer Science and Business Media LLC}, author={Guillarmou, Colin and Küster, Benjamin}, year={2021}, pages={3565–3617} }","mla":"Guillarmou, Colin, and Benjamin Küster. “Spectral Theory of the Frame Flow on Hyperbolic 3-Manifolds.” <i>Annales Henri Poincaré</i>, vol. 22, no. 11, Springer Science and Business Media LLC, 2021, pp. 3565–617, doi:<a href=\"https://doi.org/10.1007/s00023-021-01068-7\">10.1007/s00023-021-01068-7</a>.","ieee":"C. Guillarmou and B. Küster, “Spectral Theory of the Frame Flow on Hyperbolic 3-Manifolds,” <i>Annales Henri Poincaré</i>, vol. 22, no. 11, pp. 3565–3617, 2021, doi: <a href=\"https://doi.org/10.1007/s00023-021-01068-7\">10.1007/s00023-021-01068-7</a>.","chicago":"Guillarmou, Colin, and Benjamin Küster. “Spectral Theory of the Frame Flow on Hyperbolic 3-Manifolds.” <i>Annales Henri Poincaré</i> 22, no. 11 (2021): 3565–3617. <a href=\"https://doi.org/10.1007/s00023-021-01068-7\">https://doi.org/10.1007/s00023-021-01068-7</a>.","ama":"Guillarmou C, Küster B. Spectral Theory of the Frame Flow on Hyperbolic 3-Manifolds. <i>Annales Henri Poincaré</i>. 2021;22(11):3565-3617. doi:<a href=\"https://doi.org/10.1007/s00023-021-01068-7\">10.1007/s00023-021-01068-7</a>"},"publication_identifier":{"issn":["1424-0637","1424-0661"]},"publication_status":"published","doi":"10.1007/s00023-021-01068-7","volume":22,"author":[{"first_name":"Colin","full_name":"Guillarmou, Colin","last_name":"Guillarmou"},{"first_name":"Benjamin","full_name":"Küster, Benjamin","last_name":"Küster"}],"date_updated":"2024-04-11T12:39:23Z"},{"doi":"10.1007/s00220-020-03793-2","volume":378,"author":[{"first_name":"Benjamin","full_name":"Küster, Benjamin","last_name":"Küster"},{"full_name":"Weich, Tobias","id":"49178","last_name":"Weich","orcid":"0000-0002-9648-6919","first_name":"Tobias"}],"date_updated":"2022-05-19T10:13:48Z","intvolume":"       378","page":"917-941","citation":{"bibtex":"@article{Küster_Weich_2020, title={Pollicott-Ruelle Resonant States and Betti Numbers}, volume={378}, DOI={<a href=\"https://doi.org/10.1007/s00220-020-03793-2\">10.1007/s00220-020-03793-2</a>}, number={2}, journal={Communications in Mathematical Physics}, publisher={Springer Science and Business Media LLC}, author={Küster, Benjamin and Weich, Tobias}, year={2020}, pages={917–941} }","mla":"Küster, Benjamin, and Tobias Weich. “Pollicott-Ruelle Resonant States and Betti Numbers.” <i>Communications in Mathematical Physics</i>, vol. 378, no. 2, Springer Science and Business Media LLC, 2020, pp. 917–41, doi:<a href=\"https://doi.org/10.1007/s00220-020-03793-2\">10.1007/s00220-020-03793-2</a>.","short":"B. Küster, T. Weich, Communications in Mathematical Physics 378 (2020) 917–941.","apa":"Küster, B., &#38; Weich, T. (2020). Pollicott-Ruelle Resonant States and Betti Numbers. <i>Communications in Mathematical Physics</i>, <i>378</i>(2), 917–941. <a href=\"https://doi.org/10.1007/s00220-020-03793-2\">https://doi.org/10.1007/s00220-020-03793-2</a>","ama":"Küster B, Weich T. Pollicott-Ruelle Resonant States and Betti Numbers. <i>Communications in Mathematical Physics</i>. 2020;378(2):917-941. doi:<a href=\"https://doi.org/10.1007/s00220-020-03793-2\">10.1007/s00220-020-03793-2</a>","ieee":"B. Küster and T. Weich, “Pollicott-Ruelle Resonant States and Betti Numbers,” <i>Communications in Mathematical Physics</i>, vol. 378, no. 2, pp. 917–941, 2020, doi: <a href=\"https://doi.org/10.1007/s00220-020-03793-2\">10.1007/s00220-020-03793-2</a>.","chicago":"Küster, Benjamin, and Tobias Weich. “Pollicott-Ruelle Resonant States and Betti Numbers.” <i>Communications in Mathematical Physics</i> 378, no. 2 (2020): 917–41. <a href=\"https://doi.org/10.1007/s00220-020-03793-2\">https://doi.org/10.1007/s00220-020-03793-2</a>."},"publication_identifier":{"issn":["0010-3616","1432-0916"]},"publication_status":"published","department":[{"_id":"10"},{"_id":"623"},{"_id":"548"}],"user_id":"49178","_id":"31264","status":"public","type":"journal_article","title":"Pollicott-Ruelle Resonant States and Betti Numbers","date_created":"2022-05-17T12:06:06Z","publisher":"Springer Science and Business Media LLC","year":"2020","issue":"2","language":[{"iso":"eng"}],"keyword":["Mathematical Physics","Statistical and Nonlinear Physics"],"abstract":[{"text":"<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\">\r\n                  <mml:mrow>\r\n                    <mml:mo>≠</mml:mo>\r\n                    <mml:mn>3</mml:mn>\r\n                  </mml:mrow>\r\n                </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.\r\n</jats:p>","lang":"eng"}],"publication":"Communications in Mathematical Physics"},{"issue":"2","year":"2020","publisher":"Springer Science and Business Media LLC","date_created":"2024-04-11T12:33:03Z","title":"Pollicott-Ruelle Resonant States and Betti Numbers","publication":"Communications in Mathematical Physics","abstract":[{"lang":"eng","text":"<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\">\r\n                  <mml:mrow>\r\n                    <mml:mo>≠</mml:mo>\r\n                    <mml:mn>3</mml:mn>\r\n                  </mml:mrow>\r\n                </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.\r\n</jats:p>"}],"keyword":["Mathematical Physics","Statistical and Nonlinear Physics"],"language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"issn":["0010-3616","1432-0916"]},"citation":{"bibtex":"@article{Küster_Weich_2020, title={Pollicott-Ruelle Resonant States and Betti Numbers}, volume={378}, DOI={<a href=\"https://doi.org/10.1007/s00220-020-03793-2\">10.1007/s00220-020-03793-2</a>}, number={2}, journal={Communications in Mathematical Physics}, publisher={Springer Science and Business Media LLC}, author={Küster, Benjamin and Weich, Tobias}, year={2020}, pages={917–941} }","mla":"Küster, Benjamin, and Tobias Weich. “Pollicott-Ruelle Resonant States and Betti Numbers.” <i>Communications in Mathematical Physics</i>, vol. 378, no. 2, Springer Science and Business Media LLC, 2020, pp. 917–41, doi:<a href=\"https://doi.org/10.1007/s00220-020-03793-2\">10.1007/s00220-020-03793-2</a>.","short":"B. Küster, T. Weich, Communications in Mathematical Physics 378 (2020) 917–941.","apa":"Küster, B., &#38; Weich, T. (2020). Pollicott-Ruelle Resonant States and Betti Numbers. <i>Communications in Mathematical Physics</i>, <i>378</i>(2), 917–941. <a href=\"https://doi.org/10.1007/s00220-020-03793-2\">https://doi.org/10.1007/s00220-020-03793-2</a>","ama":"Küster B, Weich T. Pollicott-Ruelle Resonant States and Betti Numbers. <i>Communications in Mathematical Physics</i>. 2020;378(2):917-941. doi:<a href=\"https://doi.org/10.1007/s00220-020-03793-2\">10.1007/s00220-020-03793-2</a>","ieee":"B. Küster and T. Weich, “Pollicott-Ruelle Resonant States and Betti Numbers,” <i>Communications in Mathematical Physics</i>, vol. 378, no. 2, pp. 917–941, 2020, doi: <a href=\"https://doi.org/10.1007/s00220-020-03793-2\">10.1007/s00220-020-03793-2</a>.","chicago":"Küster, Benjamin, and Tobias Weich. “Pollicott-Ruelle Resonant States and Betti Numbers.” <i>Communications in Mathematical Physics</i> 378, no. 2 (2020): 917–41. <a href=\"https://doi.org/10.1007/s00220-020-03793-2\">https://doi.org/10.1007/s00220-020-03793-2</a>."},"intvolume":"       378","page":"917-941","date_updated":"2024-04-11T12:36:53Z","author":[{"last_name":"Küster","full_name":"Küster, Benjamin","first_name":"Benjamin"},{"first_name":"Tobias","id":"49178","full_name":"Weich, Tobias","orcid":"0000-0002-9648-6919","last_name":"Weich"}],"volume":378,"doi":"10.1007/s00220-020-03793-2","type":"journal_article","status":"public","_id":"53415","user_id":"70575","department":[{"_id":"548"}]},{"external_id":{"arxiv":["1512.00836"]},"language":[{"iso":"eng"}],"keyword":["Applied Mathematics","General Mathematics"],"publication":"Journal of the European Mathematical Society","date_created":"2022-05-17T12:06:41Z","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","issue":"6","year":"2019","user_id":"49178","department":[{"_id":"10"},{"_id":"623"},{"_id":"548"}],"_id":"31265","type":"journal_article","status":"public","author":[{"first_name":"Semyon","full_name":"Dyatlov, Semyon","last_name":"Dyatlov"},{"first_name":"David","full_name":"Borthwick, David","last_name":"Borthwick"},{"first_name":"Tobias","last_name":"Weich","orcid":"0000-0002-9648-6919","full_name":"Weich, Tobias","id":"49178"}],"volume":21,"date_updated":"2022-05-19T10:12:59Z","doi":"10.4171/jems/867","publication_status":"published","publication_identifier":{"issn":["1435-9855"]},"citation":{"apa":"Dyatlov, S., Borthwick, D., &#38; Weich, T. (2019). Improved fractal Weyl bounds for hyperbolic manifolds. With an appendix by David Borthwick, Semyon Dyatlov and Tobias Weich. <i>Journal of the European Mathematical Society</i>, <i>21</i>(6), 1595–1639. <a href=\"https://doi.org/10.4171/jems/867\">https://doi.org/10.4171/jems/867</a>","mla":"Dyatlov, Semyon, et al. “Improved Fractal Weyl Bounds for Hyperbolic Manifolds. With an Appendix by David Borthwick, Semyon Dyatlov and Tobias Weich.” <i>Journal of the European Mathematical Society</i>, vol. 21, no. 6, European Mathematical Society - EMS - Publishing House GmbH, 2019, pp. 1595–639, doi:<a href=\"https://doi.org/10.4171/jems/867\">10.4171/jems/867</a>.","short":"S. Dyatlov, D. Borthwick, T. Weich, Journal of the European Mathematical Society 21 (2019) 1595–1639.","bibtex":"@article{Dyatlov_Borthwick_Weich_2019, title={Improved fractal Weyl bounds for hyperbolic manifolds. With an appendix by David Borthwick, Semyon Dyatlov and Tobias Weich}, volume={21}, DOI={<a href=\"https://doi.org/10.4171/jems/867\">10.4171/jems/867</a>}, number={6}, journal={Journal of the European Mathematical Society}, publisher={European Mathematical Society - EMS - Publishing House GmbH}, author={Dyatlov, Semyon and Borthwick, David and Weich, Tobias}, year={2019}, pages={1595–1639} }","ama":"Dyatlov S, Borthwick D, Weich T. Improved fractal Weyl bounds for hyperbolic manifolds. With an appendix by David Borthwick, Semyon Dyatlov and Tobias Weich. <i>Journal of the European Mathematical Society</i>. 2019;21(6):1595-1639. doi:<a href=\"https://doi.org/10.4171/jems/867\">10.4171/jems/867</a>","ieee":"S. Dyatlov, D. Borthwick, and T. Weich, “Improved fractal Weyl bounds for hyperbolic manifolds. With an appendix by David Borthwick, Semyon Dyatlov and Tobias Weich,” <i>Journal of the European Mathematical Society</i>, vol. 21, no. 6, pp. 1595–1639, 2019, doi: <a href=\"https://doi.org/10.4171/jems/867\">10.4171/jems/867</a>.","chicago":"Dyatlov, Semyon, David Borthwick, and Tobias Weich. “Improved Fractal Weyl Bounds for Hyperbolic Manifolds. With an Appendix by David Borthwick, Semyon Dyatlov and Tobias Weich.” <i>Journal of the European Mathematical Society</i> 21, no. 6 (2019): 1595–1639. <a href=\"https://doi.org/10.4171/jems/867\">https://doi.org/10.4171/jems/867</a>."},"page":"1595-1639","intvolume":"        21"},{"abstract":[{"lang":"eng","text":"The kinetic Brownian motion on the sphere bundle of a Riemannian manifold $M$\r\nis a stochastic process that models a random perturbation of the geodesic flow.\r\nIf $M$ is a orientable compact constant negatively curved surface, we show that\r\nin the limit of infinitely large perturbation the $L^2$-spectrum of the\r\ninfinitesimal generator of a time rescaled version of the process converges to\r\nthe Laplace spectrum of the base manifold. In addition, we give explicit error\r\nestimates for the convergence to equilibrium. The proofs are based on\r\nnoncommutative harmonic analysis of $SL_2(\\mathbb{R})$."}],"status":"public","type":"preprint","publication":"arXiv:1909.06183","language":[{"iso":"eng"}],"_id":"31191","external_id":{"arxiv":["1909.06183"]},"user_id":"45027","department":[{"_id":"548"}],"year":"2019","citation":{"bibtex":"@article{Kolb_Weich_Wolf_2019, title={Spectral Asymptotics for Kinetic Brownian Motion on Hyperbolic Surfaces}, journal={arXiv:1909.06183}, author={Kolb, Martin and Weich, Tobias and Wolf, Lasse Lennart}, year={2019} }","short":"M. Kolb, T. Weich, L.L. Wolf, ArXiv:1909.06183 (2019).","mla":"Kolb, Martin, et al. “Spectral Asymptotics for Kinetic Brownian Motion on Hyperbolic Surfaces.” <i>ArXiv:1909.06183</i>, 2019.","apa":"Kolb, M., Weich, T., &#38; Wolf, L. L. (2019). Spectral Asymptotics for Kinetic Brownian Motion on Hyperbolic Surfaces. In <i>arXiv:1909.06183</i>.","ieee":"M. Kolb, T. Weich, and L. L. Wolf, “Spectral Asymptotics for Kinetic Brownian Motion on Hyperbolic Surfaces,” <i>arXiv:1909.06183</i>. 2019.","chicago":"Kolb, Martin, Tobias Weich, and Lasse Lennart Wolf. “Spectral Asymptotics for Kinetic Brownian Motion on Hyperbolic Surfaces.” <i>ArXiv:1909.06183</i>, 2019.","ama":"Kolb M, Weich T, Wolf LL. Spectral Asymptotics for Kinetic Brownian Motion on Hyperbolic Surfaces. <i>arXiv:190906183</i>. Published online 2019."},"title":"Spectral Asymptotics for Kinetic Brownian Motion on Hyperbolic Surfaces","date_updated":"2022-05-24T13:06:58Z","author":[{"first_name":"Martin","full_name":"Kolb, Martin","last_name":"Kolb"},{"orcid":"0000-0002-9648-6919","last_name":"Weich","full_name":"Weich, Tobias","id":"49178","first_name":"Tobias"},{"first_name":"Lasse Lennart","last_name":"Wolf","full_name":"Wolf, Lasse Lennart","id":"45027"}],"date_created":"2022-05-11T10:42:11Z"},{"type":"mastersthesis","status":"public","user_id":"49063","department":[{"_id":"548"}],"_id":"31302","language":[{"iso":"eng"}],"citation":{"short":"P. Schütte, Numerically Investigating Residues of Weighted Zeta Functions on Schottky Surfaces, 2019.","mla":"Schütte, Philipp. <i>Numerically Investigating Residues of Weighted Zeta Functions on Schottky Surfaces</i>. 2019.","bibtex":"@book{Schütte_2019, title={Numerically Investigating Residues of Weighted Zeta Functions on Schottky Surfaces}, author={Schütte, Philipp}, year={2019} }","apa":"Schütte, P. (2019). <i>Numerically Investigating Residues of Weighted Zeta Functions on Schottky Surfaces</i>.","chicago":"Schütte, Philipp. <i>Numerically Investigating Residues of Weighted Zeta Functions on Schottky Surfaces</i>, 2019.","ieee":"P. Schütte, <i>Numerically Investigating Residues of Weighted Zeta Functions on Schottky Surfaces</i>. 2019.","ama":"Schütte P. <i>Numerically Investigating Residues of Weighted Zeta Functions on Schottky Surfaces</i>.; 2019."},"year":"2019","author":[{"last_name":"Schütte","full_name":"Schütte, Philipp","id":"50168","first_name":"Philipp"}],"date_created":"2022-05-17T13:41:53Z","supervisor":[{"orcid":"0000-0002-9648-6919","last_name":"Weich","full_name":"Weich, Tobias","id":"49178","first_name":"Tobias"},{"first_name":"Joachim","full_name":"Hilgert, Joachim","id":"220","last_name":"Hilgert"}],"date_updated":"2024-02-19T06:21:23Z","title":"Numerically Investigating Residues of Weighted Zeta Functions on Schottky Surfaces"},{"publisher":"Oxford University Press (OUP)","date_updated":"2024-04-11T12:36:33Z","date_created":"2024-04-11T12:33:46Z","author":[{"first_name":"Benjamin","last_name":"Küster","full_name":"Küster, Benjamin"},{"first_name":"Tobias","id":"49178","full_name":"Weich, Tobias","orcid":"0000-0002-9648-6919","last_name":"Weich"}],"volume":2021,"title":"Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces","doi":"10.1093/imrn/rnz068","publication_status":"published","publication_identifier":{"issn":["1073-7928","1687-0247"]},"issue":"11","year":"2019","citation":{"short":"B. Küster, T. Weich, International Mathematics Research Notices 2021 (2019) 8225–8296.","mla":"Küster, Benjamin, and Tobias Weich. “Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces.” <i>International Mathematics Research Notices</i>, vol. 2021, no. 11, Oxford University Press (OUP), 2019, pp. 8225–96, doi:<a href=\"https://doi.org/10.1093/imrn/rnz068\">10.1093/imrn/rnz068</a>.","bibtex":"@article{Küster_Weich_2019, title={Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces}, volume={2021}, DOI={<a href=\"https://doi.org/10.1093/imrn/rnz068\">10.1093/imrn/rnz068</a>}, number={11}, journal={International Mathematics Research Notices}, publisher={Oxford University Press (OUP)}, author={Küster, Benjamin and Weich, Tobias}, year={2019}, pages={8225–8296} }","apa":"Küster, B., &#38; Weich, T. (2019). Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces. <i>International Mathematics Research Notices</i>, <i>2021</i>(11), 8225–8296. <a href=\"https://doi.org/10.1093/imrn/rnz068\">https://doi.org/10.1093/imrn/rnz068</a>","chicago":"Küster, Benjamin, and Tobias Weich. “Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces.” <i>International Mathematics Research Notices</i> 2021, no. 11 (2019): 8225–96. <a href=\"https://doi.org/10.1093/imrn/rnz068\">https://doi.org/10.1093/imrn/rnz068</a>.","ieee":"B. Küster and T. Weich, “Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces,” <i>International Mathematics Research Notices</i>, vol. 2021, no. 11, pp. 8225–8296, 2019, doi: <a href=\"https://doi.org/10.1093/imrn/rnz068\">10.1093/imrn/rnz068</a>.","ama":"Küster B, Weich T. Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces. <i>International Mathematics Research Notices</i>. 2019;2021(11):8225-8296. doi:<a href=\"https://doi.org/10.1093/imrn/rnz068\">10.1093/imrn/rnz068</a>"},"intvolume":"      2021","page":"8225-8296","_id":"53416","user_id":"70575","department":[{"_id":"548"}],"keyword":["General Mathematics"],"language":[{"iso":"eng"}],"type":"journal_article","publication":"International Mathematics Research Notices","abstract":[{"lang":"eng","text":"<jats:title>Abstract</jats:title>\r\n               <jats:p>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.</jats:p>"}],"status":"public"},{"language":[{"iso":"eng"}],"_id":"31301","department":[{"_id":"548"},{"_id":"288"}],"user_id":"50168","status":"public","type":"bachelorsthesis","title":"Identifying and Realizing Symmetries in Quantum Walks - Symmetry Classes and Quantum Walks","date_updated":"2022-05-17T13:42:20Z","author":[{"first_name":"Philipp","last_name":"Schütte","id":"50168","full_name":"Schütte, Philipp"}],"date_created":"2022-05-17T13:40:30Z","supervisor":[{"first_name":"Tobias","full_name":"Weich, Tobias","id":"49178","orcid":"0000-0002-9648-6919","last_name":"Weich"},{"full_name":"Silberhorn, Christine","id":"26263","last_name":"Silberhorn","first_name":"Christine"}],"year":"2017","citation":{"ieee":"P. Schütte, <i>Identifying and Realizing Symmetries in Quantum Walks - Symmetry Classes and Quantum Walks</i>. 2017.","chicago":"Schütte, Philipp. <i>Identifying and Realizing Symmetries in Quantum Walks - Symmetry Classes and Quantum Walks</i>, 2017.","ama":"Schütte P. <i>Identifying and Realizing Symmetries in Quantum Walks - Symmetry Classes and Quantum Walks</i>.; 2017.","apa":"Schütte, P. (2017). <i>Identifying and Realizing Symmetries in Quantum Walks - Symmetry Classes and Quantum Walks</i>.","bibtex":"@book{Schütte_2017, title={Identifying and Realizing Symmetries in Quantum Walks - Symmetry Classes and Quantum Walks}, author={Schütte, Philipp}, year={2017} }","short":"P. Schütte, Identifying and Realizing Symmetries in Quantum Walks - Symmetry Classes and Quantum Walks, 2017.","mla":"Schütte, Philipp. <i>Identifying and Realizing Symmetries in Quantum Walks - Symmetry Classes and Quantum Walks</i>. 2017."}}]