{"abstract":[{"text":"Abstract\r\n 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.","lang":"eng"}],"date_created":"2022-05-17T12:00:36Z","keyword":["General Mathematics"],"issue":"11","publication_status":"published","page":"8225-8296","doi":"10.1093/imrn/rnz068","author":[{"full_name":"Küster, Benjamin","last_name":"Küster","first_name":"Benjamin"},{"last_name":"Weich","full_name":"Weich, Tobias","first_name":"Tobias"}],"publisher":"Oxford University Press (OUP)","year":"2021","intvolume":" 2021","status":"public","citation":{"bibtex":"@article{Küster_Weich_2021, title={Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces}, volume={2021}, DOI={10.1093/imrn/rnz068}, number={11}, journal={International Mathematics Research Notices}, publisher={Oxford University Press (OUP)}, author={Küster, Benjamin and Weich, Tobias}, year={2021}, pages={8225–8296} }","chicago":"Küster, Benjamin, and Tobias Weich. “Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces.” *International Mathematics Research Notices* 2021, no. 11 (2021): 8225–96. https://doi.org/10.1093/imrn/rnz068.","short":"B. Küster, T. Weich, International Mathematics Research Notices 2021 (2021) 8225–8296.","mla":"Küster, Benjamin, and Tobias Weich. “Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces.” *International Mathematics Research Notices*, vol. 2021, no. 11, Oxford University Press (OUP), 2021, pp. 8225–96, doi:10.1093/imrn/rnz068.","ama":"Küster B, Weich T. Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces. *International Mathematics Research Notices*. 2021;2021(11):8225-8296. doi:10.1093/imrn/rnz068","ieee":"B. Küster and T. Weich, “Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces,” *International Mathematics Research Notices*, vol. 2021, no. 11, pp. 8225–8296, 2021, doi: 10.1093/imrn/rnz068.","apa":"Küster, B., & Weich, T. (2021). Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces. *International Mathematics Research Notices*, *2021*(11), 8225–8296. https://doi.org/10.1093/imrn/rnz068"},"external_id":{"arxiv":["1710.04625"]},"user_id":"49178","volume":2021,"publication_identifier":{"issn":["1073-7928","1687-0247"]},"department":[{"_id":"10"},{"_id":"623"},{"_id":"548"}],"_id":"31261","title":"Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces","language":[{"iso":"eng"}],"publication":"International Mathematics Research Notices","type":"journal_article","date_updated":"2022-05-25T06:42:01Z"}