Equidistribution of Eisenstein Series for Convex Co-compact Hyperbolic Manifolds Colin Guillarmou and Fr´ed´eric Naud
The quantum ergodic theorem, due to Schnirelman [4], Colin de Verdi`ere [1] and Zelditch [5], says that on any compact Riemannian manifold X whose geodesic flow is ergodic, one can find a full density sequence j ! C1 of eigenvalues of the Laplacian X such that the corresponding normalized eigenfunctions j are equidistributed i.e. for all f 2 L2 .X /, we have Z lim
j !C1 X
Z f .z/j
2 j .z/j d v.z/ D
f .z/d v.z/; X
where d v is the normalized volume measure. For non-compact manifolds, there can be continuous spectrum and the quantum ergodic theorem does not really make sense in general. However, for hyperbolic surfaces of finite volume and in particular arithmetic cases, Zelditch [6], Luo-Sarnak [3] prove a related statement involving the generalized eigenfunctions (also known as Eisenstein series). Let us recall their results. Let X D nH2 be a finite area surface where is a non co-compact cofinite Fuchsian group. The non compact ends of X are cusps related to fixed points cj in @H2 of parabolic elements in . The spectrum of the Laplacian X has a discrete part which corresponds to L2 .X /-eigenfunctions and may be infinite and the absolutely continuous part Œ1=4; C1/ which is parametrized (t 2 R) by the finite set of Eisenstein series EX .1=2 C i tI z; j / related to each cusp cj . The Eisenstein series EX .1=2 C i tI z; j / are smooth non-L2 .X / eigenfunctions
C. Guillarmou () DMA, U.M.R. 8553 CNRS, Ecole Normale Sup´erieure, 45 rue d’Ulm, F 75230 Paris cedex 05, France e-mail:
[email protected] F. Naud Laboratoire d’Analyse non lin´eaire et G´eom´etrie, Universit´e d’Avignon, 33 rue Louis Pasteur 84000 Avignon, France e-mail:
[email protected] D. Grieser et al. (eds.), Microlocal Methods in Mathematical Physics and Global Analysis, Trends in Mathematics, DOI 10.1007/978-3-0348-0466-0 21, © Springer Basel 2013
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X EX .1=2 C i tI z; j / D .1=4 C t 2 /EX .1=2 C i tI z; j /: For all t 2 R, define the density t by Z a.z/dt .z/ WD X
XZ j
a.z/jEX .1=2 C i tI z; j /j2 d v.z/;
X
where a 2 C01 .X /. In the case with only finitely many eigenvalues then Zelditch’s equidistribution result is as follows: for a 2 C01 .X /, 1 s.T /
Z
T T
Z ˇZ ˇ ˇ ˇ adt @t s.t/ a d vˇdt ! 0 as T ! 1 ˇ X
X
where s.t/ is the scattering phase appearing as a sort of regularization of Eisenstein series due to the fact that the Weyl law involves the continuous spectrum. On the other hand, for the modular surface X D PSL2 .Z/nH2 , Luo and Sarnak [3] showed that as t ! C1, Z Z 48 log.t/ adt D ad v C o.log.t//; X X which is a much stronger statement obtained via sharp estimates on certain L-functions. We report here some recent result of [2], where we studied the case of infinite volume hyperbolic manifolds without cusps, more precisely convex co-compacts quotients X D nHnC1 of the hyperbolic space. A discrete group of orientation preserving isometries of HnC1 is said to be convex co-compact if it admits a polygonal, finite sided fundamental domain whose closure does not intersect the limit set of . The limit set ƒ and the set of discontinuity are defined by ƒ WD :o \ S n ;
WD S n n ƒ ;
where o 2 HnC1 is any point in HnC1 . The quotient space X D nHnC1 has ‘funnel type’ ends and is the interior of a compact manifold with boundary X WD n.HnC1 [ /, the action of on .HnC1 [ / being free and totally discontinuous. By a result of Patterson and Sullivan, the Hausdorff dimension of ƒ ı WD dimHaus .ƒ / is also the exponent of convergence of the Poincar´e series, i.e. for all m; m0 2 HnC1 and s > 0, X 0 e sd. m;m / < 1 ” s > ı ; (1) 2 0
where d.m; m / denotes the hyperbolic distance.
Equidistribution of Eisenstein Series for Convex Co-compact Hyperbolic Manifolds
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In that case the spectrum of X consists of the absolutely continuous spectrum Œn2 =4; C1/ and a (possibly empty) finite set of eigenvalues in .0; n2 =4/. The Eisenstein functions are defined using the ball model of HnC1 to be the automorphic functions of m 2 HnC1 given by EX .sI m; / D
X 1 j mj2 s ; 4j m j2 2
2 ;
which are absolutely convergent for Re.s/ > ı and extend meromorphically to s 2 C. The Eisenstein series are non-L2.X / eigenfunctions of the Laplacian with eigenvalue s.n s/ on Re.s/ D n=2. We show the following Theorem 1. Let X D nHnC1 be a convex co-compact quotient with ı < n=2. Let a 2 C01 .X / and let EX .sI ; / be an Eisenstein series as above with a given point 2 @X at infinity. Then we have as t ! C1, Z ˇ2 ˇ n ˇ ˇ a.m/ˇEX . C i tI m; /ˇ d v.m/ D a.m/EX .nI m; /d v.m/ C O.t 2ı n /: 2 X X
Z
The limit measure on X is given by the harmonic density EX .nI m; / whose boundary limit is the Dirac mass at 2 @X . A microlocal extension of this theorem is also proved. We first need to introduce some adequate notations. Fix any 2 @X . Let L defined by L WD [ 2 L S X; where L are stable Lagrangian submanifolds of the unit cotangent bundle S X : the Lagrangian manifold L is defined to be the projection on nS HnC1 of f.m; .m// 2 S HnC1 I m 2 HnC1 g; where .m/ is the unit (co)vector tangent to the geodesic starting at m and pointing toward 2 S n . The set L “fibers” over X , and the fiber over a point m 2 X corresponds to the closure of the set of directions v 2 S X such that the geodesic starting at m with directions v converges to 2 @X as t ! C1. Since the closure of the orbit : satisfies : ƒ , L contains the forward trapped set TC WD f.m; / 2 S X W gt .m; / remains bounded as t ! C1g; where gt W S X ! S X is the geodesic flow. The Hausdorff dimension of L is n C ı C 1 and satisfies n C 1 < ı C n C 1 < 2n C 1 if is non elementary. Our phase-space statement is the following Theorem 2. Let A be a compactly supported 0-th order pseudodifferential operator with principal symbol a 2 C01 .X; T X /, then as t ! C1
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D AEX .
E n n C i tI ; /; EX . C i tI ; / D 2 2 L2 .X /
Z S X
a d C O.t min.1;n2ı / /
where is a gt -invariant measure supported on the fractal subset L S X . Notice that the fractal behaviour of the semi-classical limit can only be observed at the microlocal level. By averaging over the boundary with respect to the volume measure induced by S n on , we obtain as t ! C1 Z @X
Z
Z ˇ2 ˇ n ˇ ˇ n a.m/ˇEX . C i tI m; /ˇ d v.m/d D vol.S / a.m/d v.m/ C O.t 2ı n / 2 X X (2)
and Z D Z E n n d D a d C O.t min.n2ı ;1/ / AEX . C it I ; /; EX . C it I ; / 2 2 L2 .X / @X S X where denotes the Liouville measure. This is the perfect analog of the previously known results for the modular surface (actually with a remainder in our case).
References 1. Y. Colin de Verdi`ere, Ergodicit´e et fonctions propres du Laplacien, Comm. Math. Phys., 102 (1985), 497–502. 2. C. Guillarmou, F. Naud, Equidistribution of Eisenstein series on convex co-compact hyperbolic manifolds, arXiv:1107.2655 3. W. Luo, P. Sarnak, Quantum ergodicity of eigenfunctions on PSL2 .Z/=H2 , Publications Math´ematiques de l’IHES, 81 (1995), 207–237. 4. A. I. Schnirelman, Ergodic properties of eigenfunctions, Usp. Math. Nauk., 29 (1974), 181–182. 5. S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987), 919–941. 6. S. Zelditch, Mean Lindel¨of hypothesis and Equidistribution of Cusps forms and Eisenstein series, Journal of Functional Analysis 97 (1991), 1–49.
