Mathematics – Spectral Theory
Scientific paper
2006-01-31
Ann. Henri Poincare 8 (2007), no. 2, 361--426.
Mathematics
Spectral Theory
54 pages, no figures. Added some references
Scientific paper
10.1007/s00023-006-0311-7
We relate two types of phase space distributions associated to eigenfunctions $\phi_{ir_j}$ of the Laplacian on a compact hyperbolic surface $X_{\Gamma}$: (1) Wigner distributions $\int_{S^*\X} a dW_{ir_j}=< Op(a)\phi_{ir_j}, \phi_{ir_j}>_{L^2(\X)}$, which arise in quantum chaos. They are invariant under the wave group. (2) Patterson-Sullivan distributions $PS_{ir_j}$, which are the residues of the dynamical zeta-functions $\lcal(s; a): = \sum_\gamma \frac{e^{-sL_\gamma}}{1-e^{-L_\gamma}} \int_{\gamma_0} a$ (where the sum runs over closed geodesics) at the poles $s = {1/2} + ir_j$. They are invariant under the geodesic flow. We prove that these distributions (when suitably normalized) are asymptotically equal as $r_j \to \infty$. We also give exact relations between them. This correspondence gives a new relation between classical and quantum dynamics on a hyperbolic surface, and consequently a formulation of quantum ergodicity in terms of classical ergodic theory.
Anantharaman Nalini
Zelditch Steve
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