of (if the former is well de ned) on a compact negatively curved manifold of ... A general theme of semi-classical analysis is to nd relations between the as- .... concentrate on a closed geodesic, or more generally whether their microlocal lifts ..... this result have been established for non-compact hyperbolic surfaces of nite ...
CLASSICAL LIMITS OF EIGENFUNCTIONS FOR SOME COMPLETELY INTEGRABLE SYSTEMS DMITRY JAKOBSON AND STEVE ZELDITCH Abstract. We give an overview of some old results on weak* limits of eigenfunctions and prove some new ones. We rst show that on = ( n ) every probability measure on which is invariant under the geodesic ow and time reversal is a weak* limit of a sequence of Wigner measures corresponding to eigenfunctions of . We next show that joint eigenfunctions of and a single Hecke operator on n cannot scar on a single closed geodesic. We nally use the estimates of [Z3] on the rate of quantum ergodicity to prove that adding a DO of order ? + 2 doesn't change the level spacings distribution of (if the former is well de ned) on a compact negatively curved manifold of dimension . In dimension two this shows that the level spacings distributions of quantizations of certain Hamiltonians do not depend on the quantization. M
S
S
; can
M
S
n
n
1. Introduction A general theme of semi-classical analysis is to nd relations between the asymptotic properties of the eigenfunctions of a quantum system and the dynamics of the classical limit system. In this paper, the quantum system will consist of the p wave group Ut = eit of a compact Riemannian manifold (M; g), whose eigenfunctions Ut j = eitj j represent standing waves, and the classical limit system will consist of the geodesic ow Gt on the unit sphere bundle S M. Our purpose is to describe some results on the asymptotic behavior of matrix elements (A j ; j ) of observables A 2 o (M) relative to eigenfunctions of certain quantum completely integrable systems. In part we will be reviewing known results, but our primary goal is to present some new results on asymptotics of matrix elements, especially in the case of the standard spheres (S n ; can): Before describing the new results, let us mention a few of the issues of current interest in the semiclassical analysis of matrix elements (for more details, see x1). The basic problem is to determine the possible classical limits of the diagonal and o -diagonal matrix elements (A i ; j ): This problem is raised in many standard texts of quantum mechanics, such as the classic Landau-Lifshitz text [L-L], but the discussions are non-rigorous and often contain implicit assumptions on the behavior of the classical limit systems (such as its complete integrability). One reason to be interested in the diagonal matrix elements (A j ; j ) is that they are the eigenvalues of the quantum time average 1 Z T U AU dt hAi := w ? Tlim !1 2T ?T t t Date : December 9, 1997. 1991 Mathematics Subject Classi cation. 81Q50, 58C40, 58G15. The rst author was supported by the NSF postdoctoral fellowship. 1
2
DMITRY JAKOBSON AND STEVE ZELDITCH
of the observable A (at least when the spectrum Sp() is simple). The limit points of f(A j ; j )g therefore ll out the essential spectrum Spess (hAi): Since the high eigenvalue limit is a semiclassical limit, one may hope or suspect that Spess(hAi) can be described in terms of the classical limit. Were hAi a nice observable (i.e. pseudodi erential operator) then Spess (hAi) would be the essential range of hai, the time-averaged symbol. It is an open question to decide how generally this relation actually holds. Intuitively, the limits of the matrix elements describe the concentration and oscillation properties of the eigenfunctions. In the case of completely integrable systems, one imagines that subsequences of the eigenfunctions correspond to certain invariant tori for the geodesic ow (speci cally, the quantizable ones), and concentrate on them in the classical limit. Although there are rigorous results in this direction (cf. x2), it is quite an open problem to establish such results for general completely integrable systems. The diculty is that the tori actually correspond to quasi-modes rather than actual modes (eigenfunctions), and the relation between quasi-modes and modes is notoriously unclear (see [Ar]). For KAM systems, in which a positive measure of the tori break up, essentially nothing rigorous is known. In the case of ergodic systems, it is known that the only point of density in Sp(hAi) is the constant hai (the space average of the symbol); but there may exist other limit points. This is often viewed as the `scarring problem': can sequences of eigenfunctions of quantum ergodic systems singularly concentrate on closed geodesics? The answer is known to be no in the case of some arithmetic hyperbolic manifolds [R-S], but is generally not known. There are also many computer studies of eigenfunctions, devoted to unraveling the patterns of critical points and nodal surfaces. They suggest that eigenfunctions of quantum chaotic systems might have local gaussian limit distributions, a statement going far beyond the analysis of matrix elements (which involve only the rst two moments of the distribution). Another much-studied topic in the physics literature is the comparison of matrix elements (A i ; j ) of chaotic systems with those of various kinds of random matrix ensembles. Let us now state the speci c problems and results of this article more precisely. To study matrix elements, we consider the distributions j 2 D0 (S M) de ned for a 2 C 1 (S M) by: Z adj := hOp(a) j ; j i S M Here, Op is a choice of quantization from symbols to pseudodi erential operators; the j 's of course depend on the choice of Op, but the semi-classical limit results will not. The general problem on any manifold (M; g) is to determine the set Q = w ?limfj g MI of weak* limit points of the sequence fj g. It is well-known, and easy to prove, that Q MI where MI is the convex set of probability measures invariant under Gt . When the eigenfunctions j are real, then the functionals j and their limits are also invariant under the time-reversal involution (x; ) = (x; ?). But frequently we are interested in complex eigenfunctions such as exponentials. Generally speaking, it is very dicult to determine the set Q. There are essentially no general tools available besides trace formulae and a number of symmetry and convexity principles. But our rst result gives a complete solution of this problem on the standard n-spheres (M; g) = (S n ; can) (the sphere with the standard metric).
CLASSICAL LIMITS OF EIGENFUNCTIONS
3
Theorem 1.1. Suppose (M; g) = (S n ; can). Then Q = MI .
Our second topic concerns the singular concentration of eigenfunctions on closed geodesics. As mentioned above, one of the well-known problems of quantum chaos is whether eigenfunctions jk of quantizations of chaotic systems can singularly concentrate on a closed geodesic, or more generally whether their microlocal lifts jk can tend to any limit measure besides Liouville measure. A related problem is whether eigenfunctions of quantum chaotic systems can `scar' on a closed geodesic. As became clear from discussions during the IMA meeting, these two types of concentration are not the same. In this article, we will only be considering the singular concentration of eigenfunctions. Our goal here is to develop the proof of Rudnick-Sarnak that joint eigenfunctions of the Laplacian and of Hecke operators on certain arithmetic quotients cannot concentrate on nite unions of closed geodesics. Speci cally, we lift their argument from the con guration space to phase space and adapt it to simple Hecke operators on the spheres. Our results here are only the simplest of the kinds that could be envisioned in this context. To state the result, we will need to recall the de nition of a Hecke operator on a Riemannian manifold (M; g): it is is de ned to be a self-adjoint operator on L2 (M; dvol) of the form N 1 X f(Cj x) + f(Cj?1 x) (1) TC f(x) = 2N j =1 ~ ~g) and where where the Cj 's are isometries of the universal Riemannian cover (M; TC is assumed to commute with the deck transformation group ? of M~ ! M and hence to preserve C 1(M). The classical examples are the Hecke operators associated to discrete arithmetic subgroups of SL2 (R) [Shi]. Since the Cj 's are isometries, [; TC ] = 0 and hence there exists an orthonormal basis of joint eigenfunctions j = j j TC j = j (C) j In the notation and terminology above, we are interested in the weak* limits of the linear functionals j (A) = (A j ; j ) corresponding to these eigenfunctions on the algebra o of bounded pseudodi erential operators. As mentioned above, from the fact that j is a Laplace eigenfunction it follows immediately p that j is it invariant under the automorphism t(A) = Ut AUt , where Ut = e is the wave group, and hence any weak limit of the j 's is an invariant probability measure for the geodesic ow Gt. On the other hand the Hecke operator does not de ne an automorphism of o . In fact it does not even de ne an endomorphism since TC ATC is not usually a pseudodi erential operator. Hence we have the following questions ([CV4], [R-S], [Z4]): Question 1.2. What invariance property of the classical limit measures of fj g follows from the fact that the j 's are Hecke eigenfunctions? Question 1.3. Which invariant probability measures for Gt have this additional Hecke-invariance property? Can a periodic orbit measure arise as the classical limit of Hecke eigenfunctions?
We will give a rather complete answer to these questions in the case of the standard spheres. In particular, we will prove:
4
DMITRY JAKOBSON AND STEVE ZELDITCH
Theorem 1.4. Suppose TC is a Hecke operator on S n de ned by (1) such that
(i) Any closed geodesic on S n is xed by at most two isometries Cj ; Cj?1 in (1); and by at most two words of length two in the free group F generated by the symbols Cj . (ii) Let be a closed geodesic xed by some word W 2 F of length at most two, and assume that W is not a power of another element in F . Then the only words in F of length at most four xing are those which reduce to the powers of W . Then there is no sequence 'j of joint TC ? -eigenfunctions such that the corresponding j -s converge to where is the delta-measure on a single closed geodesic.
Our nal result is a small observation related to a very dicult problem, namely that of determining the level spacings distribution of a Laplacian. It is de ned as follows: Suppose (M; g) is a compact Riemannian manifold of dimension n, and denote the eigenvalues of its Laplacian by 0 = 0 < 1 : : :. The growth rate of the spectrum is given by Weyl's law: (2) N() = #fj g Cnvol(M) n2 where Cn is a constant depending only on the dimension. To eliminate the constants, let us rescale the metric so that Cnvol(M) = 1. To make the consecutive eigenvalue spacings j +1 ? j equal to 1 on naverage, we also renormalize ! . By the Weyl law, the spectrum of = 2 has growth rate n N () = #fj2 g (3) so its eigenvalues do have unit mean level spacing. To detect statistical regularities in the spacings, one then forms the local level spacings distribution on [0; 1] by N ([a; b]) := #fj < N :Nxj 2 [a; b]g (4) for any 0 a b < 1. The problem is then to determine whether there exists a unique weak limit N ! as N ! 1 and if so, to compute it. The physicists conjecture (and sometimes claim to have proved) that there exists a unique weak limit in case the classical limit dynamics is chaotic, namely the GOE level spacings distribution GOE : However, from the mathematical standpoint there is no proof that even a uniquely de ned limit measure exists. Moreover, it is invisible why the limit measure should be so universal and in particular depend only the principal symbol of . Indeed, addition of a perturbation term ! + V with V a pseudodi erential operator of order ?n+2 will move the eigenvalues by amounts at or above the mean level spacing. We may assume the mean value of V with respect to Liouville measure d! on S M equals zero since addition of a constant will not change the level spacings. Our observation is that in the borderline case a perturbation V 2 ?n+2 will not change the level spacings distribution if it exists. Theorem 1.5. Suppose the Laplacian of a compact Riemannian n-manifold has a well-de ned level spacings distribution . Then if the curvature is negative, is also the level spacings distribution of + V where V is any pseudodi erential operator of order ?n + 2.
CLASSICAL LIMITS OF EIGENFUNCTIONS
5
The authors would like to thank Yves Colin de Verdiere, Dennis Hejhal, Andrei Komech, Denis Kosygin, Zeev Rudnick, Peter Sarnak, John Toth and Maciej Zworski for helpful comments. The authors would also like to thank the organizers of the conference for their hospitality. 2. Background The matrix element (A j ; j ) represents the expected value of the observable A in the energy state j . The o -diagonal elements (A i ; j ) represent amplitudes for making transitions between energy states. The problem of determining their semiclassical asymptotics can be found in many classical books on quantum mechanics such as the classic text of Landau-Lifshits [L-L, x48, 51]. In this text, the system is assumed to have one degree of freedom, and the discussion is implicitly based on the fact that the system to be completely integrable. 2.1. Completely integrable systems: quantum torus action. A geodesic
ow on a compact n-dimensional compact M is completely integrable if there exist n integrals of motion in involution, i.e. n functions (p1 ; : : :pn) on T M (homogep neous of degree one) commuting with each other (and with the generator j j2 P of Gt) and formingPan elliptic system (i.e. i pi(x; )2 > 0 on T M nf0g; it is often the case that i p2i = j j2). There are stronger and weaker versions of complete integrability; the strongest is called Hamiltonian torus action. It assumes the existence of (globally de ned) action-angle variables on T M. The examples include at tori and convex surfaces of revolution. For such systems the level sets of (p1; : : :pn )-s are Lagrangian tori, and one can de ne the global action variables (q1; : : :qn) generating 2-periodic Hamiltonian ows on these tori. A quantization of a completely integrable system ([CV2], [Z2]) is a choice of n commuting DO-s (P1; : : : ; Pn) of order 1 whose principal symbols are pj -s. For systems with Hamiltoniantorus action one can de ne action operators (Q1; : : : ; Qn) with Qj = qj and exp(2iQj ) = cj Id. The joint spectrum of Qj -s is C \ Zn + where C is a conic set in Rn and 2 Zn =4 is a Maslov index. The joint spectrum is simple (perhaps with nitely many exceptions). The preimages under q = (q1 ; : : :qn ) of the points in the spectrum are Lagrangean tori satisfying the quantization conditions of Bohr-Sommerfeld-Maslov. In the completely integrable case one can associate sequences of approximate WKB eigenfunctions (quasi-modes) to the tori satisfying the quantization condition ([CV1]). For quantum torus action systems it has been shown (cf. [G-S], [CV2] and [Z2]) that quasimodes correspond to actual eigenfunctions. Accordingly, given a ray l 2 C and a sequence of points fj g in the spectrum approaching l1 there is an associated sequence of eigenfunctions 'j = 'j such that dj -s converge to the Lebesgue measure on the torus q?1(l). 2.2. General completely integrable systems; special cases. It is generally believed that the eigenfunctions of a completely integrable system split up into subsequences concentrating on invariant tori for the system. However, to the authors' knowledge, this has only been proved for the quantum torus action systems, and then only for the special basis of joint eigenfunctions of the action operators. This 1 The points cannot lie on a ray pointing in an irrational direction; in that case one can take a sequence of lattice points corresponding to Diophantine approximants of the slope.
6
DMITRY JAKOBSON AND STEVE ZELDITCH
is not generally appreciated because the distinction between modes (eigenfunctions) and quasi-modes is often blurred or lost in the physics literature. For a general completely integrable system, there do not exist global actionangle charts (see [D]). Rather, one needs to cover the phase space with a nite number of charts. Moreover, the moment map (p1 ; : : :; pn) is generally many-toone. On the quantum level, the joint spectrum of (P1 ; : : :; Pn) is multiple, and one cannot associate a unique eigenfunction to each torus satisfying the BohrSommerfeld-Maslov quantization conditions. In fact, the correspondence between actual eigenfunctions and the lattice of BSM tori has not been worked out in very many examples. For some results in the case of separable systems, see J. Toth ([T1]) for geodesic ow on an ellipsoid and motion of a rigid body in vacuum; and Bleher, Kosygin, Minasov, Sinai and others ([Bl], [BKS], [K-M-S]) for geodesic ow on Liouville surfaces.2 2.3. Spheres. Our results in this paper show that on (S n ; can) the set Q of weak* limit points of the sequence j -s is the largest possible. However, the second author showed in [Z5] that for a random choice of an orthonormal basis of eigenfunctions (in the sense made precise in [Z5]), Theorem 2.4 holds, as in the ergodic case. Moreover, J. VanderKam showed in [Va] that a random orthonormal basis on (S 2 ; can) is uniquely ergodic, that is there are no exceptional subsequences in the statement of Theorem 2.4. 2.4. Flat tori. We next discuss the classical limits on at tori. Let Tn = Rn=(2 L) and let ' = eih;i be an eigenfunction on Tn (here 2 L ), and let dn be the distribution on S Tn corresponding to 'n . Then it is easy to see that dn -s converge to the delta-measure on the invariant torus Tx i for Gt consisting of unit vectors pointing in the direction =j j. The projection of any T onto the base Tn is an isometry, so delta-measure on any T projects to the Lebesgue measure on Tn. In [J1] the rst author studied the case L = Zn, the standard integer lattice. The multiplicity of the eigenvalue 2 N is then equal to the number of ways of representing as a sum of n squares p (or, equivalently, to the number of lattice points on the sphere S of radius centered at the origin), and it is well known that multiplicities can become arbitrarily large for n 2. Since an eigenfunction ' satisfying ' + ' = 0 can be arbitrary linear combinations of ' -s for j j2 = , unbounded multiplicities may give rise to more complicated limits than deltameasure on T -s, and in particular their projections may be more complicated than the Lebesgue measure. Most of the results in [J1] deal with those projections, which are nothing but weak* limits of all possible sequences of j'j j2-s for various eigenfunctions 'j of with eigenvalues j ! 1. We rst state the result of Bourgain (cf. [J1]): Theorem 2.1. On Tn , weak* limits of j'j j2-s are absolutely continuous with respect to the Lebesgue measure for all n. Accordingly, we will speak of possible densities (with respect to the Lebesgue measure) of weak* limits. It is easy to see that the number of frequencies in j'j j2 can be arbitrarily large for n 2 (those are just chords connecting lattice points 2 There are several examples in the literature(cf. [C-P] and [T2]) of eigenfunctionsof completely integrable separable systems concentrating on unstable closed geodesics.
CLASSICAL LIMITS OF EIGENFUNCTIONS
7
on S ); accordingly, the following result (proved by the rst author in [J1]) seems rather surprising: Theorem 2.2. The density of any weak* limit of j'j j2-s on T2 is a trigonometric
polynomial all of whose non-zero frequencies lie on the union of (at most) two circles centered at the origin.
The proof follows from the niteness of the number of solutions of a system of two Pell equations. It is also shown that the densities of weak* limits on T3 have absolutely convergent Fourier series, while on T4 they are in L2 (T4 ). The Lp properties of the densities of weak* limits on Tn for n 5 turn out to be related to possible generalizations (or lack thereof) to higher dimensions (n ? 2 and n ? 3) of the result of Zygmund ([Zyg]) on the uniform bound on jj'jj4=jj'jj2 for the eigenfunctions ' on T2 . In the present paper we wish to prove a result complementing Theorem 2.1. Namely, we show that
Proposition 2.3. The Lebesgue measure on Tn is absolutely continuous with respect to any weak* limit of j'j j2-s on Tn for all n. Proof. The statement follows easily from [Con, Lemma 2]. Having studied possible projections of limits of dj -s on Tn , one wishes to determine the limits d! which project to a given measure d = f(x) dx with X f(x) = 1 + c eih ;xi Writing d! as
06= 2Zn
X
2Zn
g () eih ;xi dxd
(where is the variable on S n?1 and where g () projects to c ), one can show easily that for 6= 0 the support of g lies on the (n ? 2)-dimensional sphere S( ) of the unit vectors on S n?1 orthogonal to (guaranteeing the invariance of d! under Gt). There are no such restrictions on the support of g0 . The more detailed information about g is given by the asymptotic distribution of lattice points on certain \parallels" Sj on Sj := Sj -s (namely, the locus of the endpoints of chords on Sj that are translates of ); the information about g0 is given by the distribution of lattice points on the whole Sj0 := Sj . If the lattice points on Sj become equidistributed then g can be any nonnegative measure on S( ) with certain upper bounds on its mass (depending on c ). However, studying the asymptotic distribution of lattice points on Sj for all -s is a dicult problem which the authors cannot solve. 2.5. Ergodic systems. We next turn to the discussion of weak* limits of j -s for manifolds with ergodic geodesic ows. The following result is due to Shnirelman ([Shn1], [Shn2]), the second author ([Z6]) and Colin de Verdiere ([CV3]): Theorem 2.4. Let M be a compact manifold with ergodic geodesic ow. Then for any orthonormal basis 'j of eigenfunctions of on M there exists a subsequence 'jk of density one such that the weak* limit of jk -s is the Liouville measure on S M .
8
DMITRY JAKOBSON AND STEVE ZELDITCH
In other words, almost all eigenfunctions become equidistributed. Analogues of this result have been established for non-compact hyperbolic surfaces of nite area in [Z7]; there one also considers the Eisenstein series which comprise the continuous spectrum of the hyperbolic Laplacian on these surfaces. Theorem 2.4 follows from the estimate 1 X j(A' ; ' )j2 = 0 lim (5) j j !1 N() j
R
for every pseudo-di erential operator A of order 0 with S M A = 0. The rate of convergence in (5) was studied in [Z3]. Better bounds in the case of arithmetic hyperbolic surfaces were proved in [L-S] and [J3]. A natural question related to Theorem 2.4 is the existence of \exceptional" subsequences of 'j -s (on manifolds with ergodic geodesic ows) such that j -s do not converge to the Liouville measure. Such subsequences are sometimes called strong scars. Natural candidates for limits of such subsequences are -measures on unstable closed geodesics on manifolds with ergodic geodesic ow. Rudnick and Sarnak conjecture in [R-S] that such subsequences don't exist for compact manifolds of negative curvature (this conjecture is sometimes called quantum unique ergodicity). A theorem of Rudnick and Sarnak quoted below supports this conjecture for arithmetic hyperbolic surfaces.3 Also, quantum unique ergodicity has been established for Eisenstein series for arithmetic subgroups of SL2 (R) by Luo, Sarnak and the rst author ([L-S], [J2]). To end our discussion of weak* limits in the ergodic case we mention a converse result to Theorem 2.4 proved by Sunada and the second author (cf. [Z1]). They prove that if (5) holds (together with another condition on the \o -diagonal" terms (A'i ; 'j )) for eigenfunctions of on a compact manifold M then the geodesic ow on M is ergodic; the second condition holds for manifolds with ergodic geodesic
ows. We conclude this section by stating the only rigorous result (to our knowledge) giving answers to questions 1.2 and 1.3 for arithmetic hyperbolic surfaces. This result is due to Rudnick and Sarnak ([R-S]):
Theorem 2.5. Joint Hecke-Laplace eigenfunctions on arithmetic hyperbolic surfaces cannot singularly concentrate on a nite union of closed geodesics.
3. Proof of Theorem 1.1. Our purpose is to prove that, in the case (M; g) = (S n ; can), any invariant probability measure 2 MI arises as a weak* limit of a sequence of j 's. To do so, it will be convenient to express all the relevant objects, e.g. invariant measures, eigenfunctions, and limits of their microlocal lifts, as integrals over the space G(S n ; can) of unoriented geodesics of the standard metric. Let us rst recall the properties of this space. Since the geodesic ow of can is periodic it de nes a free S 1 -action on the unit sphere bundle S S n . The quotient space, G+ (S n ; can), is the symplectic manifold formed by the set of all oriented great circles of (S n ; can): Since each oriented great circle determines an oriented plane, G(S n ; can) can be identi ed with the Grassmann manifold Gr(2; n) of oriented planes thru the origin in Rn. In particular, when n=2, G(S 2 ; can) = S 2 : In 3
See [SQC] for an overview of results about arithmetic hyperbolic manifolds.
CLASSICAL LIMITS OF EIGENFUNCTIONS
9
general, SO(n + 1) acts on G(S n ; can) and the space of oriented geodesics is the symmetric space G(S n ; can) = SO(n + 1)=SO(2) SO(n ? 1): When the eigenfunctions are real, the j 's are invariant under the the canonical involution : (x; ) ! (x; ?) on T S n , which we will refer to as the `time reversal.' It is obvious that takes an oriented closed geodesic to the same closed geodesic with the opposite orientation. Hence acts on S S n and on G(S n ; can), and the quotient G(S n ; can)= = G? (S n ; can); the space of unoriented closed geodesics. However, we will be interested in complex eigenfunctions, so will work on the space of oriented closed geodesics. Corresponding to each oriented closed geodesic
we Rnow associate two objects: R rst, the periodic orbit measure with S M fd = fds, and second the sequence k of highest weight spherical harmonics associated to : Let us recall the de nition of the latter. The spectral decomposition of on (S n ; can) takes the form 1 X 2 n L (S ; can) = Hk ; jHk = k(n + k ? 1); dimHk = pn (k) k=0
where pn is a certain polynomial of degree n-1. The eigenspaces Hk are invariant and irreducible under the action of SO(n + 1). Henceforth we x one geodesic o and express any other geodesic in the form g o with g 2 SO(n + 1): We then put: k = g ko : Hence it suces to de ne ko : To do so we recall that the Lie algebra o(n+1) has the root space decomposition M o(n + 1) = t g
where t is a choice of Cartan subalgebra and where f g runs over the set of positive roots of o(n + 1). As is well-known, a presentation of o(n + 1) is then given by a (Chevalley) basis fH ; X + ; X ? g where each triple (for xed ) forms an sl2 : By a highest weight vector in Hk relative to this basis (i.e. choice of t) we mean a vector satisfying X + v = 0 for all : Such a vector is unique by the theorem of the highest weight [B-D]. In the case n = 2 the Cartan subalgebra is onedimensional and corresponds to a choice of an axis of revolution. The o(3) basis is then given by Lz ; L+ ; L? in the usual physics notation; and the highest weight vector in Hk is the spherical harmonic usually denoted by Ykk := Ck (x + iy)k : For n > 2, one can choose the maximal torus in SO(n + 1) and a basis e1 ; : : : ; e with = [(n + 1)=2] for its Lie algebra t such that e1 generates rotations in the x1x2plane. The highest weight vector is then ko = Ck (x1 +ix2)k , where o is the great circle x21 + x22 = 1; x3 = : : : = 0 ([Ta, (3.34)]). Since Hk is irreducible, every vector in it is cyclic. In particular, any vector v 2 Hk may be expressed as a linear combination of the elements k . That is, we may express each eigenfunction in the form Z Z
k = k d( ) = Tg ko d(g) n G(S ;can)
SO(n+1)
with d(g) a probability measure on SO(n+1) and with Tg the translation operator by g on L2 (S n ), i.e. Tg f(x) = f(g?1 x): In particular, matrix elements may be written in the form hA k ; ki = hT AT ko ; ko i
10
DMITRY JAKOBSON AND STEVE ZELDITCH
with
Z
T :=
SO(n+1)
Tg d(g):
As above, let us denote by k the linear functional hk ; ai := hOp(a) k ; ki where Op is a xed choice of quantization. We then make the standard observation that k (a) depends only on the diagonal part k Op(a)k of Op(a) where k : L2 (S n ) ! Hk is the orthogonal projection. As is also well-known, the direct sum of the diagonal parts is just the time average of Op(a), that is, 1 M 1 Z 2 U Op(a)U dt k Op(a)k =: Opave(a) := 2 t t 0 k=0 q
with Ut = exp(itD), D = + ( n?4 1 )2 ? n?4 1 : As may be easily veri ed, DjHk = k so that Ut is a periodic group and Opave (a) commutes with : It is also a standard observation (by the Egorov theorem) that the principal symbol of Opave(a) is the classical time average 1 Z 2 a Gtdt aave := 2 0
geodesic ow. Hence the weak limits of k depend only to consider ow-invariant symbols. Thus, we may assume henceforth that a 2 C 1(G(S n ; can)): With these identi cations, our aim is to construct a sequence of k k 's so that R k k (a) ! S S n ad for all ow-invariant a 2 C(S S n ), where 2 MI is a given invariant measure. Since corresponds to a measure on C(G(S n ; can)) by
where Gt denotes the on aave and it suces
Z
SS
ad = n
where a 2 C(S S n ), and where
R(a)( ) =
Z
G(S n ;can)
Z
R(a)( )d ( )
ads 2 C(G(S n ; can)) R
is the Radon transform, it is equivalent to ask that the limit be G(S n ;can) ad : That is, our aim is to prove that the set Q of weak* limit points, viewed as a class of measures on C(G(S n ; can)), equals the entire class MG(S n ;can) of regular Borel probability measures on this space. To this end, it suces to show that any discrete probability measure d =
N X j =1
cj j ;
cj > 0;
X
cj = 1
arises as a weak* limit point of some sequence fdkg: Indeed, it is well known (by the Krein-Milman theorem) that convex combinations of point masses are weak* dense in the compact convex set MG(S n ;can) (equipped with the weak* topology). Moreover, this space is Hausdor so that the set of weak* limit points of the class of the measures fdk g is necessarily closed in MG(S n ;can) . The theorem then follows from:
CLASSICAL LIMITS OF EIGENFUNCTIONS
11
Lemma 3.1. Let d be as above, and let d = PNj=1 pcj gj where gj o = j : Also, let k be the microlocal lift of k . Then k ! d: Proof: What we want to prove is that Z
ave o o hT Op (a)T k ; k i ! n ad; 8a 2 C(G(S n ; can)): G(S ;can)
To do so, we form the generating function () :=
1 X
k=0
hT Opave(a)T ko ; ko ieik :
In a well-known way (cf. [B-G]), the singularities of () will determine the asymptotics of the Fourier coecients. The main point is to prove that () is a Lagrangean distribution and to determine its principal symbol. Since it is also a Hardy function (i.e. has only positive frequencies), the Lagrangean property will automatically imply a complete asymptotic expansion as k ! 1 of hT Opave(a)T ko ; ko i with principal asymptote given by the principal symbol of . To show that is Lagrangean and to determine its symbol, we express it as the trace: () = Tr eiD o T Opave(a)T where D is (as above) the positive elliptic rst order pseudodi erential operator equaling k on Hk and where o =
1 X
k=0
ko ko
is the canonical Toeplitz projection associated to o . That this projection is a Toeplitz projector follows from a theorem due to V.Guillemin [G]. We now claim that the trace above is a clean composition of Fourier Integral and Toeplitz operators (see [B-G] for the terminology). To analyse the trace, we write out (with A = Opave(a)) X T AT = pcj ck Tgj ATgk j;k
and consider the individual traces Tr eiD o pcj ck Tgj ATgk : It is clear that Tg is an FIO associated to the graph of the canonical transformation g on T S n lifted from Tg on S n . Hence Tgj ATgk is an FIO associated to gj?1 gk : Also, eiD is associated to G , the geodesic ow, and o is the restriction to
o o : Hence the singularities of the trace occur at values of for which the xed point set Fix(G gj?1 gk j o ) is non-empty. It is clear that no xed points can occur unless gj?1gk xes o . Since we may assume with no loss of generality that gj o are distinct geodesics, gj?1gk will only x o when j = k: Moreover, when j = k, the only for which Fix(G gj?1 gk j o ) is non-empty is = 0: The xed point set is obviously clean, and so by the composition formula [B-G, Theorem 7.5] we have Tr eiD o pcj ck Tgj ATgk 2 I (S 1 ; T0 (S 1 ))
12
DMITRY JAKOBSON AND STEVE ZELDITCH R
with principal symbol jk ck o a Tgk ds: Summing over (j; k) we get that () 2 I (S 1 ; T0 (S 1 ));
=
It then follows by [B-G, Ch. 12] that
hk ; ai !
X
k
ck
Z
Z
G(S n ;can)
o
a Tgk ds =
Z
G(S n ;can)
ad:
ad;
completing the proof of the Lemma 3.1 and hence of the Theorem 1.1. 4. Scarring on a closed geodesic: Proof of Theorem 1.4 Let us begin by discussing joint eigenfunctions of Hecke-Laplace operators in the framework of states on algebras of pseudodi erential and Fourier integral operators. 4.1. Hecke operators. Hecke operators are nite Radon transforms associated to nite Riemannian covering diagrams ~ ~g) (M; . & (M; g) (M; g) where, we emphasize, both and are covers of M~ ! M: The diagram gives rise to an isometric correspondence C : M ! M; C(x) = ?1 fxg and to the associated Hecke operator TC := : Our rst observation is that TC is an FIO (Fourier Integral operator), associated to the homogeneous canonical relation C := N (gr(C)) T (M) T (M) with gr(C) M M the graph of the correspondence C and with N the conormal bundle. We may describe this canonical relations in a more useful way, freely identifying cotangent and tangent objects by the metric. First, the covers and extend (by their derivatives) to the tangent (or cotangent) bundles T M~ ! T M:; we continue to denote them by ; : We then have a map: ~ ! T (M) T (M) : T (M) whose image is easily seen to be a di eomorphism onto its image, equal to C . It commutes with the action of R+ on the cotangent bundles, so we will usually slice the action to get the map ~ ! SC : S (M) with SC the unit length elements in C :
CLASSICAL LIMITS OF EIGENFUNCTIONS
13
4.2. Algebras, states and classical limits. We now reformulate the questions raised in the introduction from a C* algebraic point of view as in [Z1]. This begins with the fact that on any compact manifold M the C* algebra o (M) of bounded pseudodi erential operators (in the norm topology) ts into the exact sequence 0 ! K ! o ! C(S M) ! 0 with K the compact operators. Since the linear functionals j are states on o and since any weak limit annihilates ?1(M) (the -1st order operators) and hence annihilates K, may be identi ed with a state on C(S M), i.e. with a probability measure on S M. Moreover, invariance of j under t implies that is invariant under the quotient automorphism, the geodesic ow. To take Hecke operator TC into account, we will extend the algebra o to the algebra _ A := I o (M M; C ) fC g
generated by FIO's associated to the canonical relations C . The elements of I o (M M; C ) may all be expressed in the form B = A ~ In particular, this space includes operators of the form TC A and with A 2 o (M): ATC with A 2 o (M): The wave group still de nes an automorphism t of this algebra and the functionals j still de ne invariant states. Since any weak limit of the j 's, as functionals on A, annihilates compact operators, it de nes a functional on the symbol algebra S corresponding to A. This symbol algebra may be described as follows: any polynomial in the above FIO's is associated to a union of the canonical relations C . Hence its symbol consists of a collection of homogeneous 1/2-densities on the associated C 's. Since they are graphs, each C carries a canonical volume 1/2-density, and hence we may identify the symbols on them with the scalar coecient of this half-density. Thus the symbol is a collection of functions C , one on each component C : If C has components (as it will below) we may regard symbols as a collection of functions on the components. Via the maps above, each symbol C may be lifted back to the space ~ Thus we can regard the symbol as a collection of functions C on S (M): ~ S (M). It follows that any weak limit of the j 's on A correspond to a collection of ~ From the invariance of j under measures C , one on each component of S (M). the automorphisms t (A) = Ut AUt , it follows that each g is invariant under the ~ Note that the limit measures C 's correspond to geodesic ow Gt of g~ on S (M). sparse subsequences of j 's: Indeed C (TC ) is the limit eigenvalue of TC along the subsequence. So regarding the j 's as functionals on the larger algebra splits up the eigenfunctions into much smaller subsequences than on o . Our problem is to pin down , the limit measure on S (M). The following proposition answers Question (1) in the introduction, i.e. describes the symmetry property of the limit measures. Proposition 4.1. Let f jk g be a subsequence of eigenfunctions for which the jk 's have a (unique) weak limit on A. Then, for any Hecke operator TC , C = C = C (1):
14
DMITRY JAKOBSON AND STEVE ZELDITCH
Proof: We have:
j (ATC ) = j (TC A) = j (TC )j (A): Passing to the limit 1 along a subsequence as jk ! 1, we get 1 (ATC ) = 1 (TC A) = 1 (TC )1 (A): Identifying 1 with a collection of measures C as above, and noting that the ~ is (A ) while that of TC A is (A ), we symbol of ATC , qua function on S (M) get the equation C ( A ) = C ( A ) = C (1)(A) which is equivalent to the stated formula. Remark 4.2. Had TC been an isometry, the `conjugation' C (A) = TC ATC would have been an automorphism, C (1) would have equaled 1, and the maps (resp. ) would have been id (resp. C ). Hence the proposition would have established the invariance of under translation by C . We now specialize to the case of Hecke operators on M = (S n ; can). The main simpli cation is that each term TCj acts on M so that all the covers discussed above are trivial. We therefore introduce the situation in this level of generality. 4.3. Hecke operators for trivial covers. We will assume in this section that M~ = M f?N; : : :; N g = M M M 2N times: We will assume moreover that M is equipped with a metric g and N isometries Cj : M M and de ne (x; j) = x; (x; j) = Cjsgn(j )x where Cjsgn(j ) = Cj if j > 0 and = Cj?1 if j < 0: The Hecke operator TC is then given by TC f(x) =
N X j =1
f(Cj x) + f(Cj?1 x):
As discussed in the previous section, the limit measure C associated to a weakly convergent sequence fjk g may be identi ed with a set of measures Ck on the 2N ~ It is easy to see what they are: Indeed, consider an operator components of S M. of the form A with A a pseudodi erential operator on M~ which is zero except on the component M fkg: Let a be its principal symbol, viewed as a function on S M. Then we have Ck (a) = mlim !1hATCk jm ; jm i: By Egorov, we also have Ck (a Ck ) = mlim !1hTCk A jm ; jm i: Hence the invariance principle may be summarized in the form: N X
k=1
Ck + Ck?1 =
N X
k=1
Ck TCk + Ck?1 TCk?1 = 1 (C):
(6)
It implies the following semi-invariance properties, which in part are phase space versions of inequalities in [R-S]
CLASSICAL LIMITS OF EIGENFUNCTIONS
15
Proposition 4.3. Let be a classical limit of the states fj g on o and let Ck be the classical limit measures described above, i.e. = limm!1 jm and Ck (a) = limm!1 hATCk jm ; jm i: Then: (a) Ck
