LECTURE 24-25: QUANTUM ERGODICITY 1. Classical dynamics on cotangent bundle T ∗M {Hamiltonian flow on cotangent bundle. Recall from Lecture 2 that associated to any Hamiltonian function H(x; ξ) 2 C1(R2n), there is a Hamiltonian vector field X @H @ @H @ ΞH = − @ξk @xk @xk @ξk k which \dominates" the classical behavior of the system. More precisely, if we denote by γ(t) = γx0,ξ0 (t) the integral curve of ΞH starting at the point γx0,ξ0 (0) = (x0; ξ0), then the time-evolution of the system is given by the Hamiltonian flow tΞH 2n 2n ρt = e : R ! R ; (x; ξ) 7! ρt(x; ξ) := γx,ξ(t): In particular, we have the conservation of energy ∗ ρt H = H −1 (which implies that each energy surface H (E) is preserved under the flow ρt), and the evolution equation of classical observable d ρ∗a = fH; ag; dt t where {·; ·} is the Poisson bracket so that for any f; g 2 C1(R2n), n X @f @g @f @g 1 2n ff; gg = − 2 C (R ): @ξk @xk @xk @ξk k=1 Now suppose M be a smooth manifold, and T ∗M its cotangent bundle. For any smooth function p 2 C1(T ∗M), we can also define its Hamiltonian vector field 1 via X @p @ @p @ (1) Ξp = − : @ξk @xk @xk @ξk k Of course one has to check that Ξp is well-defined, namely it is independent of the choice of coordinate charts. Recall from Lecture 18 that under the coordinate change @x T y = y(x) on the base manifold, the cotangent variables are related by η = ( @y ) ξ. So we have @ @x @ ( ) = ( )T ( ) @y @y @x 1 We will give a more intrinsic definition of Ξp via the symplectic structure later. 1 2 LECTURE 24-25: QUANTUM ERGODICITY and @ @η @ @x @ ( ) = ( )T ( ) = ( )( ): @ξ @ξ @η @y @η It follows @p @ @p @x @x @ @p @ ( )T ( ) = ( )T ( )−1( )( ) = ( )T ( ) @x @ξ @y @y @y @η @y @η and similarly @p @ @p @ ( )T ( ) = ( )T ( ): @ξ @x @η @y n So the vector field Ξp is well-defined. By repeating the theory for R , we can define the Hamiltonian flow associated to p on T ∗M via tΞp ∗ ∗ ρt = e : T M ! T M; (x; ξ) 7! ρt(x; ξ) := γx,ξ(t); where γx,ξ(t) is the integral curve of Ξp starting at the point γx,ξ(0) = (x; ξ). Again we have the conservation of energy ∗ ρt p = p −1 (which implies that each energy surface p (E) is preserved under the flow ρt), and the evolution equation of classical observable d ρ∗a = fp; ag; dt t where {·; ·} is the Poisson bracket on T ∗M so that for any f; g 2 C1(T ∗M), n X @f @g @f @g 1 ∗ ff; gg = − = Xf (g) 2 C (T M): @ξk @xk @xk @ξk k=1 Remark. Let (M; g) be a smooth Riemannian manifold, and let 1 1 X (2) p(x; ξ) = kξk2 = gijξ ξ 2 g 2 i j be half of the Riemannian norm square on the cotangent bundle. Then the integral curves of Ξp (i.e. the trajectories of the Hamiltonian flow of p), when projected to M, are geodesics of the Riemannian manifold (M; g). Conversely, every parametrized geodesic arises in this way. (For a proof, c.f. my Riemannian geometry notes.) Note that as a consequence of the conservation law of energy, the cosphere bundle ∗ ∗ S M := f(x; ξ) 2 T M j kξkg = 1g ∗ is invariant under the flow ρt. As a consequence, we get an induced flow ρt : S M ! S∗M. This flow is usually called the geodesic flow on S∗M. LECTURE 24-25: QUANTUM ERGODICITY 3 −1 {Dynamical system on p (c) generated by the flow ρt. Similar computations also show that the n-form dx1 ^ dξ1 ^ · · · ^ dxn ^ dξn is a well-defined volume form on T ∗M, which is usually known as the Liouville volume form or the symplectic volume form. The induced measure dxdξ on T ∗M is called the Liouville measure. Now suppose a < b and assume that on the set a ≤ p(x; ξ) ≤ b, jdpj ≥ c0 > 0. So in particular for each c 2 [a; b], the level set −1 Σc := p (c) is a smooth (2n−1)-dimensional hyper-surface in T ∗M. Moreover, for each c 2 [a; b], there is an induced Liouville measure on Σc defined via the formula Z Z b Z fdxdξ = fdµcdc: −1 p ([a;b]) a Σc In other words, dµc is the measure associated to the induced volume form on the orientable hypersurface Σc. For example, if p(x; ξ) = jξj, then each Σc is a cosphere ∗ ∗ bundle of different radius in T M, and the induced Liouville measure on Σ1 = S M is nothing but dxdSn−1 (ξ). Recall that the Hamiltonian flow of the Hamiltonian function p(x; ξ) preserves any level set Σc. We will prove that the Liouville volume form and thus the Liouville measure is invariant under the Hamiltonian flow ρt. As a consequence, the induced Liouville measure µc on Σc is invariant under the flow ρt generated by p. {Classical ergodicity. A very important class of dynamical systems, known as measure preserving flow, is a triple (X; µ, ρt), where (X; µ) is a measure space with µ(X) < +1, and ρt : X ! X is a measure preserving flow on X, namely • For any t 2 R, ρt :(X; µ) ! (X; µ) is measure-preserving. • ρt is a flow: ρt+s = ρt ◦ ρs. Among all classes of dynamical systems, two extremal cases are widely studied: the integrable case and the ergodic case. Roughly speaking, an integrable dynamical system is a system with maximal conserved quantities and thus is very \regular", while ergodic system is very \chaotic". Here is a precise definition of ergodicity: Definition 1.1. We say a measure-preserving flow ρt :(X; µ) ! (X; µ) is ergodic if any ρt-invariant measurable subset of X either has measure 0 or has full measure. Example. For any compact Riemannian manifold with negative sectional curvature, the geodesic flow is ergodic. (This was first proved by Hopf for n = 2, and by Anosov and Sinai for higher dimensions. Example. The geodesic flow on Sn is NOT ergodic. (It is integrable.) 4 LECTURE 24-25: QUANTUM ERGODICITY The following theorem is a classical result in the theory of dynamical systems, which claims that for an ergodic system, the \time-average" of any L1-function equals to its \space-average": Theorem 1.2 (Birkhorff). Suppose ρt is an ergodic flow on (X; µ), then for any f 2 L1(X; µ), 1 Z T 1 Z lim f(ρt(x))dt ! f(y)dµ T !1 T 0 µ(X) X for a.e. x 2 X. In other words, the flows of ergodic systems are equidistributed in the phase space, which is in contrast to the fact that classical completely integrable systems generally have periodic orbits in phase space. Birkhoff ergodicity theorem is a very strong theorem. What we will need is the following weaker ergodicity theorem: 2 Theorem 1.3 (L -mean ergodic theorem). Suppose ρt is an ergodic flow on (X; µ), then for any f 2 L2(X; µ), 2 Z 1 Z T 1 Z lim f(ρt(x))dt − f(y)dµy dµx = 0: T !1 X T 0 m(X) X (For a proof, c.f. Zworski, page 367-368.) Notations for the time-average 1 Z T hfiT (x) := f(ρt(x))dt: T 0 and the space-average 1 Z hfiX := f(y)dµy: µ(X) X Then we can rewrite the mean ergodic theorem as Z 2 lim (hfiT (x) − hfiX ) dµx = 0 T !1 X while the Birkhoff ergodicity theorem claims that hfiT (x) ! hfiX as T ! 1 for a.e. x 2 X. 2. Quantum ergodicity In this section we will always assume (1)( M; g) is a compact Riemannian manifold, (2) p 2 Sm(T ∗M) is an (almost) elliptic classical symbol, where m > 0. (3) a < b, and on the set a ≤ p(x; ξ) ≤ b, jdpj ≥ c0 > 0. LECTURE 24-25: QUANTUM ERGODICITY 5 Note that by the condition (3), each c 2 [a; b] is a regular value and thus each energy −1 level set Σc = p (c) is a smooth compact manifold on which we will always endow the induced Liouville measure µc. {Quantum ergodicity. In the classical side, we have the Hamiltonian flow on each energy level set p−1(c) (which is compact since p is proper, where we assume that c is a regular value of p). In the quantum part, we have the eigenvalues/eigenfunctions of P , P'j = λj'j; 1 2 2 where 'j 2 C (M) and f'jg form an L -orthonormal basis of L (M), and Spec(P ): λ1 ≤ λ2 ≤ · · · ≤ λn ≤ · · · ! 1: and people would like to understand the relation between the dynamical behavior of the classical Hamiltonian flow and the quantum eigenvalue/eigenfunction data. The main theorem in this section is the quantum ergodicity theorem which describes the behavior of eigenfunctions when the corresponding classical system is ergodic. Theorem 2.1 (Schnirelman-Zelditch-Colin de Verdiere Quantum Ergodicity Theo- rem, Version 1). Suppose the Hamiltonian flow of p is ergodic on (Σc; µc) for each c 2 [a; b]. Then there exists a family of subsets Λ(~) ⊂ Spec(P ) \ [a; b] which has density 1 in the sense #Λ( ) (3) lim ~ = 1; ~!0 #(Spec(P ) \ [a; b]) such that for any semiclassical pseudodifferential operator A 2 Ψ0(M) whose symbol σ(A) satisfy the condition that the quantity 1 Z (4) σ(A)dµc Vol(Σc) Σc is independent of c 2 [a; b], we have, as ~ ! 0, 1 Z (5) hA'j;'ji ! −1 σ(A)dxdξ Vol(p ([a; b])) a≤p≤b for λj 2 Λ(~).