Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces Küster, Benjamin Weich, Tobias General Mathematics <jats:title>Abstract</jats:title> <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> Oxford University Press (OUP) 2019 info:eu-repo/semantics/article doc-type:article text http://purl.org/coar/resource_type/c_6501 https://ris.uni-paderborn.de/record/53416 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> eng info:eu-repo/semantics/altIdentifier/doi/10.1093/imrn/rnz068 info:eu-repo/semantics/altIdentifier/issn/1073-7928 info:eu-repo/semantics/altIdentifier/issn/1687-0247 info:eu-repo/semantics/closedAccess