Quantum-classical correspondence on associated vector bundles over locally symmetric spaces

摘要: For a compact Riemannian locally symmetric space $\mathcal M$ of rank one and an associated vector bundle $\mathbf V_\tau$ over the unit cosphere $S^\ast\mathcal M$, we give precise description those classical (Pollicott-Ruelle) resonant states on that vanish under covariant derivatives in Anosov-unstable directions chaotic geodesic flow M$. In particular, show they are isomorphically mapped by natural pushforwards into generalized common eigenspaces algebra invariant differential operators $D(G,\sigma)$ compatible bundles W_\sigma$ As consequence this description, obtain exact band structure Pollicott-Ruelle spectrum. Further, some mild assumptions representations $\tau$ $\sigma$ defining W_\sigma$, very explicit eigenspaces. This allows us to relate resonances quantum eigenvalues Laplacian suitable Hilbert sections W_\sigma$. Our methods proof based representation theory Lie theory.