TY - JOUR

T1 - Patterson-Sullivan distributions and quantum ergodicity

AU - Anantharaman, Nalini

AU - Zelditch, Steve

N1 - Funding Information:
Research partially supported by NSF grant #DMS-0302518 and NSF Focussed Research Grant # FRG 0354386.
Funding Information:
This work was begun while the first author was visiting Johns Hopkins University as part of the NSF focussed research grant # FRG 0354386. Much of it was written at the Time at Work program of the Institut Henri Poincaré in June, 2005.

PY - 2007/4

Y1 - 2007/4

N2 - This article gives relations between two types of phase space distributions associated to eigenfunctions φirj of the Laplacian on a compact hyperbolic surface X Γ: Wigner distributions ∫S*XΓ a dWirj = Op(a)φirj,φ 〈 irj L2Γ, which arise in quantum chaos. They are invariant under the wave group. Patterson-Sullivan distributions PSirj, which are the residues of the dynamical zeta-functions Z(s; a) := ∑γ -sL_γ1-e-L_γ∫γ 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 → ∞ . 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.

AB - This article gives relations between two types of phase space distributions associated to eigenfunctions φirj of the Laplacian on a compact hyperbolic surface X Γ: Wigner distributions ∫S*XΓ a dWirj = Op(a)φirj,φ 〈 irj L2Γ, which arise in quantum chaos. They are invariant under the wave group. Patterson-Sullivan distributions PSirj, which are the residues of the dynamical zeta-functions Z(s; a) := ∑γ -sL_γ1-e-L_γ∫γ 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 → ∞ . 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.

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U2 - 10.1007/s00023-006-0311-7

DO - 10.1007/s00023-006-0311-7

M3 - Article

AN - SCOPUS:34247601057

VL - 8

SP - 361

EP - 426

JO - Annales Henri Poincare

JF - Annales Henri Poincare

SN - 1424-0637

IS - 2

ER -