LECTURE24-25:QUANTUMERGODICITY1.CLASSICAL DYNAMICS ON COTANGENTBUNDLET*MHamiltonian flow on cotangent bundle.Recall from Lecture 2 that associated to any Hamiltonian function H(r,)Co(R2n), there is a Hamiltonian vector field(OHH a三H=3EOCkOCkOEkkwhich"dominates"the classical behavior of the system.More precisely,if we denoteby (t) = ro. (t) the integral curve of EH starting at the point ro.o(O) = (ro, So),then the time-evolution of the system is given by the Hamiltonian flowPt = et=H : R2n → R2n, (r, S) → p(r,s) := e(t)Inparticular,we have the conservation of energyPH=H(which implies that each energy surface H-1(E) is preserved under the flow pt), andthe evolution equation of classical observableddtpia = (H,a),where (., J is the Poisson bracket so that for any f,g E C(R2n),(af ogaf og>E C(R2n).(f,g] =aOrkOrok=1Now suppose M be a smooth manifold, and T*M its cotangent bundle.For anysmooth function p E Co(T*M), we can also define its Hamiltonian vector field 1 viaopaop (1)三pakOCkOrkOEkOf course one has to check that E, is well-defined,namely it is independent of thechoice of coordinate charts.Recallfrom Lecture 18 that under the coordinate changey = y(r) on the base manifold, the cotangent variables are related by n = (o)Te.So wehaveo9aauarQulWe will give a more intrinsic definition of Ep via the symplectic structure later
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) ∈ C ∞(R 2n ), there is a Hamiltonian vector field ΞH = X k ∂H ∂ξk ∂ ∂xk − ∂H ∂xk ∂ ∂ξk 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 ρt = e tΞH : R 2n → R 2n , (x, ξ) 7→ ρt(x, ξ) := γx,ξ(t). In particular, we have the conservation of energy ρ ∗ t H = H (which implies that each energy surface H−1 (E) is preserved under the flow ρt), and the evolution equation of classical observable d dtρ ∗ t a = {H, a}, where {·, ·} is the Poisson bracket so that for any f, g ∈ C ∞(R 2n ), {f, g} = Xn k=1 ∂f ∂ξk ∂g ∂xk − ∂f ∂xk ∂g ∂ξk ∈ C ∞(R 2n ). Now suppose M be a smooth manifold, and T ∗M its cotangent bundle. For any smooth function p ∈ C ∞(T ∗M), we can also define its Hamiltonian vector field 1 via (1) Ξp = X k ∂p ∂ξk ∂ ∂xk − ∂p ∂xk ∂ ∂ξk . 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 y = y(x) on the base manifold, the cotangent variables are related by η = ( ∂x ∂y ) T ξ. So we have ( ∂ ∂y ) = (∂x ∂y ) T ( ∂ ∂x) 1We will give a more intrinsic definition of Ξp via the symplectic structure later. 1
2LECTURE24-25:QUANTUMERGODICITYandarduItfollowsOPand similarly()()=()()aaraSo the vector field E, is well-defined. By repeating the theory for Rn, we can definethe Hamiltonian flow associated to p on T*M viaPt =et=p : T*M -→ T*M, (r,E) → pt(r,E) := re(t),where r.(t) is the integral curve of Ep starting at the point .s(O)= (r,s).Againwehavetheconservation of energyptp=p(which implies that each energy surface p-i(E) is preserved under the fow pt), andthe evolution equation of classical observableddipia = (p,a),where (., I is the Poisson bracket on T*M so that for any f, g E Co(T*M),(afogafog(f,g) =/= Xr(g) E C~(T*M).COEOrkOKOEKk=1Remark. Let (M,g) be a smooth Riemannian manifold, and letp(r,)=Il,=,g"t,(2)be half of the Riemannian norm square on the cotangent bundle. Then the integralcurves of E, (i.e. the trajectories of the Hamiltonian flow of p), when projected to M,are geodesics of the Riemannian manifold (M, g). Conversely, every parametrizedgeodesic arises in this way. (For a proof, c.f.my Riemannian geometry notes.)Notethat as a consequence of the conservation law of energy, the cosphere bundleS*M := ((r, $) E T*M I IlSllg = 1)is invariant under theflowpt.As a consequence,wegetan induced flow pt:S*M-→S*M. This flow is usually called the geodesic flow on S*M
2 LECTURE 24-25: QUANTUM ERGODICITY and ( ∂ ∂ξ ) = (∂η ∂ξ ) T ( ∂ ∂η ) = (∂x ∂y )( ∂ ∂η ). It follows ( ∂p ∂x) T ( ∂ ∂ξ ) = (∂p ∂y ) T ( ∂x ∂y ) −1 ( ∂x ∂y )( ∂ ∂η ) = (∂p ∂y ) T ( ∂ ∂η ) and similarly ( ∂p ∂ξ ) T ( ∂ ∂x) = (∂p ∂η ) T ( ∂ ∂y ). So the vector field Ξp is well-defined. By repeating the theory for R n , we can define the Hamiltonian flow associated to p on T ∗M via ρt = e tΞp : 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 ρ ∗ t p = p (which implies that each energy surface p −1 (E) is preserved under the flow ρt), and the evolution equation of classical observable d dtρ ∗ t a = {p, a}, where {·, ·} is the Poisson bracket on T ∗M so that for any f, g ∈ C ∞(T ∗M), {f, g} = Xn k=1 ∂f ∂ξk ∂g ∂xk − ∂f ∂xk ∂g ∂ξk = Xf (g) ∈ C ∞(T ∗M). Remark. Let (M, g) be a smooth Riemannian manifold, and let (2) p(x, ξ) = 1 2 kξk 2 g = 1 2 Xg ij ξiξj 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 := {(x, ξ) ∈ T ∗M | kξkg = 1} 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
3LECTURE24-25:QUANTUMERGODICITY[Dynamical system on p-1(c) generated by the flow pt.Similar computations also show that the n-form dri Adei A.. Adrn A den is awell-defined volume form on T*M, which is usually known as the Liouville volumeform or the symplectic volume form. The induced measure drde on T*M is calledthe Liouville measure.Now suppose a 0.So in particular for each c E [a, b], the level setZ := p-1(c)is a smooth (2n-1)-dimensional hyper-surface in T*M. Moreover, for each c e Ja, b],thereisan induced Liouvillemeasureon Ecdefined viatheformula fdrde =fdpedc(o.blIn other words, dμe is themeasure associated to the induced volume form on theorientable hypersurface Ee.For example, if p(r,)=[sl, then each is a cospherebundle of different radius in T*M, and the induced Liouville measure on Zi = S*Misnothing butdrdgn-i($).Recall that the Hamiltonian ffow of the Hamiltonian function p(c,)preservesany level set Ze.We will prove that the Liouville volumeform and thus the Liouvillemeasure is invariant under the Hamiltonian flow pt.As a consequence, the inducedLiouville measure μe on Z。is invariant under the fow pt generated by p.Classical ergodicity.A very important class of dynamical systems, known as measure preservingflow, is a triple (X, μ, pt), where (X, μ) is a measure space with μ(X) < +oo, andPt : X →X is a measure preserving flow on X, namely. For any t e R, pt : (X,μ) → (X,μ) is measure-preservingPtisaflow:Pt+s=Ptops.Among all classes of dynamical systems, two extremal cases are widely studied: theintegrable case and the ergodic case.Roughly speaking,an integrable dynamicalsystem 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 pt : (X, μ) → (X, μ) is ergodic ifany Pt-invariant measurable subset of X either has measure O or has full measure.Erample. 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 Anosovand Sinai for higher dimensions.Erample. The geodesic flow on Sn is NOT ergodic. (It is integrable.)
LECTURE 24-25: QUANTUM ERGODICITY 3 ¶Dynamical system on p −1 (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 0. So in particular for each c ∈ [a, b], the level set Σc := p −1 (c) is a smooth (2n−1)-dimensional hyper-surface in T ∗M. Moreover, for each c ∈ [a, b], there is an induced Liouville measure on Σc defined via the formula Z p−1([a,b]) f dxdξ = Z b a Z Σc f dµcdc. In other words, dµc is the measure associated to the induced volume form on the orientable hypersurface Σc. For example, if p(x, ξ) = |ξ|, 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) < +∞, and ρt : X → X is a measure preserving flow on X, namely • For any t ∈ 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 S n is NOT ergodic. (It is integrable.)
4LECTURE24-25:QUANTUMERGODICITYThe following theorem is a classical result in the theory of dynamical systems,which claims that for an ergodic system, the "time-average" of any Li-functionequals to its “space-average":Theorem1.2(Birkhorff).Supposeptisan ergodic flowon (X,μ),then foranyf E L'(X,μ),1limf(pt(r)dt →f(y)dpV+00Tμ(X)for a.e. r EX.In other words,the flows of ergodic systems are equidistributed in the phasespace, which is in contrast to the fact that classical completely integrable systemsgenerally have periodic orbits in phase space.Birkhoff ergodicity theorem is a very strong theorem. What we will need is thefollowing weaker ergodicity theorem:Theorem 1.3 (L?-mean ergodic theorem). Suppose pt is an ergodic flow on (X, μ),then for any f E L?(X,μ),m(ro)a-/f(g)dpy)dμr=01(Fora proof,c.f.Zworski,page367-368.)Notations for the time-average(f)r(r) :=f(pt(r))dtand the space-average(f)x :=f(y)dμyμ(X)Then we canrewrite themean ergodic theoremaslim(0.(3) a 0
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 L 1 -function equals to its “space-average”: Theorem 1.2 (Birkhorff). Suppose ρt is an ergodic flow on (X, µ), then for any f ∈ L 1 (X, µ), lim T→∞ 1 T Z T 0 f(ρt(x))dt → 1 µ(X) Z X f(y)dµ for a.e. x ∈ 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: Theorem 1.3 (L 2 -mean ergodic theorem). Suppose ρt is an ergodic flow on (X, µ), then for any f ∈ L 2 (X, µ), lim T→∞ Z X 1 T Z T 0 f(ρt(x))dt − 1 m(X) Z X f(y)dµy 2 dµx = 0. (For a proof, c.f. Zworski, page 367-368.) Notations for the time-average hfiT (x) := 1 T Z T 0 f(ρt(x))dt. and the space-average hfiX := 1 µ(X) Z X f(y)dµy. Then we can rewrite the mean ergodic theorem as lim T→∞ Z X (hfiT (x) − hfiX) 2 dµx = 0 while the Birkhoff ergodicity theorem claims that hfiT (x) → hfiX as T → ∞ for a.e. x ∈ X. 2. Quantum ergodicity In this section we will always assume (1) (M, g) is a compact Riemannian manifold, (2) p ∈ S m(T ∗M) is an (almost) elliptic classical symbol, where m > 0. (3) a 0
5LECTURE24-25:QUANTUMERGODICITYNote that by the condition (3), each c e[a,b] is a regular value and thus each energylevel set。=p-1(c)isa smoothcompactmanifold on whichwewill always endowthe induced Liouville measureμe.Quantum ergodicityIn 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,Ppj=Ajpj,where E C(M) and () form an L?-orthonormal basis of L?(M), andSpec(P) :.≤00.and people would like to understand the relation between the dynamical behaviorof the classical Hamiltonian flow and the quantum eigenvalue/eigenfunction data.The main theorem in this section is the quantum ergodicity theorem whichdescribes the behavior of eigenfunctions when the corresponding classical system isergodic.Theorem 2.1 (Schnirelman-Zelditch-Colin de Verdiere Quantum Ergodicity Theo-rem, Version 1). Suppose the Hamiltonian flow of p is ergodic on (Ze, μe) for eachcE[a,b]. : Then there erists a family of subsets A(h) C Spec(P)n[a,b] which hasdensity 1 in the sense#A(h)(3)lim胃0 升(Spe(P)n[a,可) = 1,such that for any semiclassical pseudodifferential operator A e o(M) whose symbolo(A)satisfythe conditionthatthequantity1(4)α(A)dμcVol() Jis independent ofce Ja,bl,we have,as ho1(5)o(A)drdE(Apj, Pi) →ol(p-1([a, b) J), E A(h).Before we go to the proof, let's state an important corollary. Let's take P-?,and take A= M, to be the “multiplication by f" map, where f e C(M). Then. eigenvalues are Aj,h = h?>, where >, are the standard Laplacian eigenvaluesof (M,g): the eigenfunctions Pj are the standard Laplacian eigenfunctions which areindependent of h. each Zis a cosphere bundle
LECTURE 24-25: QUANTUM ERGODICITY 5 Note that by the condition (3), each c ∈ [a, b] is a regular value and thus each energy level set Σc = p −1 (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 ϕj = λjϕj , where ϕj ∈ C ∞(M) and {ϕj} form an L 2 -orthonormal basis of L 2 (M), and Spec(P) : λ1 ≤ λ2 ≤ · · · ≤ λn ≤ · · · → ∞. 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 Theorem, Version 1). Suppose the Hamiltonian flow of p is ergodic on (Σc, µc) for each c ∈ [a, b]. . Then there exists a family of subsets Λ(~) ⊂ Spec(P) ∩ [a, b] which has density 1 in the sense (3) lim ~→0 #Λ(~) #(Spec(P) ∩ [a, b]) = 1, such that for any semiclassical pseudodifferential operator A ∈ Ψ0 (M) whose symbol σ(A) satisfy the condition that the quantity (4) 1 Vol(Σc) Z Σc σ(A)dµc is independent of c ∈ [a, b], we have, as ~ → 0, (5) hAϕj , ϕj i → 1 Vol(p −1 ([a, b])) Z a≤p≤b σ(A)dxdξ for λj ∈ Λ(~). Before we go to the proof, let’s state an important corollary. Let’s take P = ~ 2∆, and take A = Mf to be the “multiplication by f” map, where f ∈ C ∞(M). Then • eigenvalues are λj,~ = ~ 2λj , where λj are the standard Laplacian eigenvalues of (M, g) • the eigenfunctions ϕj are the standard Laplacian eigenfunctions which are independent of ~ • each Σc is a cosphere bundle
6LECTURE24-25:QUANTUMERGODICITY.A E o(M)and satisfies the condition (4):o(A)de= Vol(M),f(r)drVol(2)In this case, by taking [a,b] = [o,b] we getCorollary 2.2 (Quantum ergodicity for Laplacian eigenfunctions). Suppose (M, g)is a compact Riemannianmanifold with ergodic geodesic filow.Then there eristsasequence jk-oo which has density1 in the sense#(k≤N)=1limNN-→such that for each f e Co(M)(6)lpi2fda-f(r)drVol(MIn particular, if we take U C M to be an open subset, and f a sequence ofsmooth functions that approximates the characteristic function xu of U, then wewill getVol(U)piPda-Vol(M)In other words, for most eigenfunctions, the “mass"will tends to be equidistributedon M.In the statement of quantum ergodicity theorems above, one can't rule outthe possibility of the existence of quantum scar, namely a density O subsequencealong which the integral (6) concentrate on a closed geodesic (or the correspondingintegral (5) will concentrate on a periodic orbit of the geodesic flow)?. For the caseof negatively curved manifolds (in which case it is known that the geodesic flowsatisfy a stronger ergodicity condition), it was conjectured by Z. Rudnick and P.Sarnakin1994thatthereis no quantumscar.This isknownasQuantum UniqueErgodicity Conjecture, and is currently one of the most famous open problems inspectral geometry:Conjecture 2.3 (QUE conjecture). Suppose (M,g) is a compact Riemannian man-ifold with negative sectional curvature. Then as k → oo,f(r)drIpxP fde→ Vol(M)2The limit can't be too bad. See the theorem at the end of this lecture
6 LECTURE 24-25: QUANTUM ERGODICITY • A ∈ Ψ0 (M) and satisfies the condition (4): 1 Vol(Σc) Z Σc σ(A)dµc = 1 Vol(M) Z M f(x)dx. In this case, by taking [a, b] = [0, b] we get Corollary 2.2 (Quantum ergodicity for Laplacian eigenfunctions). Suppose (M, g) is a compact Riemannian manifold with ergodic geodesic flow. Then there exists a sequence jk → ∞ which has density 1 in the sense lim N→∞ #{jk ≤ N} N = 1, such that for each f ∈ C ∞(M), (6) Z M |ϕjk | 2 f dx → 1 Vol(M) Z M f(x)dx. In particular, if we take U ⊂ M to be an open subset, and f a sequence of smooth functions that approximates the characteristic function χU of U, then we will get Z U |ϕjk | 2 dx → Vol(U) Vol(M) . In other words, for most eigenfunctions, the “mass” will tends to be equidistributed on M. In the statement of quantum ergodicity theorems above, one can’t rule out the possibility of the existence of quantum scar, namely a density 0 subsequence along which the integral (6) concentrate on a closed geodesic (or the corresponding integral (5) will concentrate on a periodic orbit of the geodesic flow)2 . For the case of negatively curved manifolds (in which case it is known that the geodesic flow satisfy a stronger ergodicity condition), it was conjectured by Z. Rudnick and P. Sarnak in 1994 that there is no quantum scar. This is known as Quantum Unique Ergodicity Conjecture, and is currently one of the most famous open problems in spectral geometry: Conjecture 2.3 (QUE conjecture). Suppose (M, g) is a compact Riemannian manifold with negative sectional curvature. Then as k → ∞, Z M |ϕk| 2 f dx → 1 Vol(M) Z M f(x)dx. 2The limit can’t be too bad. See the theorem at the end of this lecture
LECTURE24-25:QUANTUMERGODICITY7Proof of Quantum ergodicity theorem (version 2).To prove the quantum ergodicity theorem, namely, Theorem 2.1, we first provethe followingTheorem 2.4 (Quantum ergodicity theorem, Version 2). Suppose the Hamiltonianflow of p(r,s) is ergodic in p-i([a,b]), and assume A e o(M) is a pseudodifferentialoperator whose symbol o(A) satisfy the condition that the quantity1o(A)dμeQ := (o(A)≥。= Vol(2) J2.is independent of ce Ja,bl.Then ash-→O,(2元h)n /(A)drds(7)<Ai, i) -Vol(p-(a,b)→0a≤a<bThere are three main ingredients in the proof:(1) Weak Egorov theorem (PSet 4): Let U(t) = e-itP/n. Then for any a ES-(T*M), we haveIIU(-t)Op(a)U(t) - Op(a 0 pt)lc(L2) = O(h).(2) Generalized Weyl's law (Lecture 23): Suppose B E o(M) and a <b. Thenash→0wehave(2h)n Z(Bj,i) →o(B)drde.<<a<Aj<b(3) The L?-mean ergodic theorem (Theorem 1.3 above): Suppose p is an ergodicflow on (X,μ), then for any f e L?(X,μ),lim( (<F)T -《f)x)’dμz = 0.Proof of Theorem 2.4.By our assumption on o(A),we have1(A)drdE = (α(A))z = αVol(p-1([a, b]))-1([a,b])We choose a cut-off x E Co(R) so that x = 1 on [a, b]. LetB = (A - αId)x(P).Then B e -(M) (so that we can apply weak Egorov theorem), and in view ofthefactBpi=Api-apiforA[a,b],wearereduced toprove(2h)n KB, )P→0.(8)a≤a≤b
LECTURE 24-25: QUANTUM ERGODICITY 7 ¶Proof of Quantum ergodicity theorem (version 2). To prove the quantum ergodicity theorem, namely, Theorem 2.1, we first prove the following Theorem 2.4 (Quantum ergodicity theorem, Version 2). Suppose the Hamiltonian flow of p(x, ξ) is ergodic in p −1 ([a, b]), and assume A ∈ Ψ0 (M) is a pseudodifferential operator whose symbol σ(A) satisfy the condition that the quantity α := hσ(A)iΣc = 1 Vol(Σc) Z Σc σ(A)dµc is independent of c ∈ [a, b]. Then as ~ → 0, (7) (2π~) n X a≤λj≤b hAϕj , ϕj i − 1 Vol(p −1 ([a, b])) Z Z a≤p≤b σ(A)dxdξ 2 → 0. There are three main ingredients in the proof: (1) Weak Egorov theorem (PSet 4): Let U(t) = e −itP/~ . Then for any a ∈ S −∞(T ∗M), we have kU(−t)Op(a)U(t) − Op(a ◦ ρt)kL(L2) = O(~). (2) Generalized Weyl’s law (Lecture 23): Suppose B ∈ Ψ0 (M) and a < b. Then as ~ → 0 we have (2π~) n X a≤λj≤b hBϕj , ϕj i → Z Z a≤p≤b σ(B)dxdξ. (3) The L 2 -mean ergodic theorem (Theorem 1.3 above): Suppose ρt is an ergodic flow on (X, µ), then for any f ∈ L 2 (X, µ), lim T→∞ Z X (hfiT − hfiX) 2 dµx = 0. Proof of Theorem 2.4. By our assumption on σ(A), we have 1 Vol(p −1 ([a, b])) Z p−1([a,b]) σ(A)dxdξ = hσ(A)iΣc = α. We choose a cut-off χ ∈ C ∞ 0 (R) so that χ ≡ 1 on [a, b]. Let B = (A − αId)χ(P). Then B ∈ Ψ−∞(M) (so that we can apply weak Egorov theorem), and in view of the fact Bϕj = Aϕj − αϕj for λj ∈ [a, b], we are reduced to prove (8) (2π~) n X a≤λj≤b |hBϕj , ϕj i|2 → 0
8LECTURE24-25:QUANTUMERGODICITYThe idea is to consider "the time-average of B"(B)T :=U(-t)BU(t))dtTFor any t e R, we have(U(-t)BU(t)pj,i) = (Bj, Pi),Thus(Bpj,pi)=《(B)Ti,pi)It follows from Cauchy-Schwarz inequality thatKB,i)P≤II(B)ll2=((B)(B)Tj,))By weak Egorov theorem,II(B)T - (B)llc(L2) = Or(),where Or(h) means O(h) with constants depending on T, and(B)=B(t)dt,and B(t) e (M) witho(B(t)) = pio(B).It followso(B)r) =pto(B)dt = (α(B))Tand thusα(B)(B)T) = [α(B)r)2 = (o(A)T - α/2 = K(A)T - (α(A))ze/2.So according to thegeneralized Weyl law, for each fixed T > o.limsup(2h)n <Bj,;)2≤limsup(2h)n 《B)(B)j,i)h→0h→0<Ai<b<lim sup(2元h)n 《《B)(B)TPj,Pi) +Or(h)h-0a<A<bo((B)(B)T)drde1 ([a,b])Ko(A)T - (o(A))z,’dμedcFinally we use the L?-mean ergodic theorem to conclude (8), which completes theproof.口Remark.Roughly speaking, in theproof we are trying to prove that the operator(B)r is “"small", or equivalently, the difference between the “time-average operator"(A)r and the “space-average operator" (A)x := (o(A)x - Id is small
8 LECTURE 24-25: QUANTUM ERGODICITY The idea is to consider “the time-average of B”, hBiT := 1 T Z T 0 U(−t)BU(t)idt. For any t ∈ R, we have hU(−t)BU(t)ϕj , ϕj i = hBϕj , ϕj i. Thus hBϕj , ϕj i = hhBiTϕj , ϕj i. It follows from Cauchy-Schwarz inequality that |hBϕj , ϕj i|2 ≤ khBiTϕjk 2 = hBi ∗ T hBiTϕj , ϕj . By weak Egorov theorem, khBiT − hBeiT kL(L2) = OT (~), where OT (~) means O(~) with constants depending on T, and hBei = 1 T Z T 0 Be(t)dt, and Be(t) ∈ Ψ0 (M) with σ(Be(t)) = ρ ∗ t σ(B). It follows σ(hBeiT ) = 1 T Z T 0 ρ ∗ t σ(B)dt = hσ(B)iT . and thus σ(hBei ∗ T hBeiT ) = |σ(hBeiT )| 2 = |hσ(A)iT − α| 2 = |hσ(A)iT − hσ(A)iΣc | 2 . So according to the generalized Weyl law, for each fixed T > 0, lim sup ~→0 (2π~) n X a<λj<b |hBϕj , ϕj i|2 ≤ lim sup ~→0 (2π~) n X a<λj<b hBi ∗ T hBiTϕj , ϕj ≤ lim sup ~→0 (2π~) n X a<λj<b hBei ∗ T hBeiTϕj , ϕj + OT (~) = Z Z p−1([a,b]) σ(hBei ∗ T hBeiT )dxdξ = Z b a Z Σc |hσ(A)iT − hσ(A)iΣc | 2 dµcdc. Finally we use the L 2 -mean ergodic theorem to conclude (8), which completes the proof. Remark. Roughly speaking, in the proof we are trying to prove that the operator hBiT is “small”, or equivalently, the difference between the “time-average operator” hAiT and the “space-average operator” hAiX := hσ(A)iX · Id is small.
9LECTURE24-25:QUANTUMERGODICITYProof of Quantum ergodicity theorem (version 1).NowwearereadytoprovetheSchnirelman-Zelditch-ColindeVerdiereQuantumErgodicity Theorem.To get Theorem 2.1from Theorem 2.4.it is helpful to noticethe following beautiful fact from mathematical analysis:Lemma 2.5. Suppose a; are non-negative real numbers andN0limk-1then there erists a density one subset [ak,] of [ai] such thatlim aj = 0.It is easy to see that for the case P = ?△。 and Ja,b] = [0, ], Theorem 2.1 isadirect consequencefrom Theorem 2.4 together with Lemma 2.5.Unfortunatelyin the general semiclassical setting, we can't directly apply Lemma 2.5 since thesummands in (7)are different fordifferenth.So what we need isavariantofLemma2.5adapted to oursemiclassical setting.Proof of Theorem 2.1As intheproof of Theorem2.4,isenough to constructdensity1 subsets A(h)such that for >j = ^j,h E A(h) and for all B = (A -Qld)x(P) -(M),(Bpj,pi) →0as h→0.Step 1.Construct A(h) for a fixed BAs we have seen in the proof of Theorem 2.4,e() := (2h)n KBj,)2→0.a≤Aj≤bSo if we defineI(h) := [, E [a,b] : (Bj,p;)P ≥e(h)1/2]thenwemusthave(2元h)"#(h)≤e()1/2Let's denoteA(h) := (Spec(P) n [a,b) /T(h)Then by construction, for ^,EA(h)we haveKBj,Pi)/ <e(h)/4 →0.Moreover, according to Weyl's law, as h → 0,#Λ(h)#r(h)(2元h)n#(h)?1.#Spec(P)n[a, b]#Spec(P) n[a,b]Vol(p-1([a, b])) + o(1)
LECTURE 24-25: QUANTUM ERGODICITY 9 ¶Proof of Quantum ergodicity theorem (version 1). Now we are ready to prove the Schnirelman-Zelditch-Colin de Verdiere Quantum Ergodicity Theorem. To get Theorem 2.1 from Theorem 2.4, it is helpful to notice the following beautiful fact from mathematical analysis: Lemma 2.5. Suppose ai are non-negative real numbers and lim N→∞ 1 N X N k=1 ak = 0, then there exists a density one subset {akj } of {ai} such that lim k→∞ ajk = 0. It is easy to see that for the case P = ~ 2∆g and [a, b] = [0, λ], Theorem 2.1 is a direct consequence from Theorem 2.4 together with Lemma 2.5. Unfortunately in the general semiclassical setting, we can’t directly apply Lemma 2.5 since the summands in (7) are different for different ~. So what we need is a variant of Lemma 2.5 adapted to our semiclassical setting. Proof of Theorem 2.1. As in the proof of Theorem 2.4, is enough to construct density 1 subsets Λ(~) such that for λj = λj,~ ∈ Λ(~) and for all B = (A − αId)χ(P) ∈ Ψ−∞(M), hBϕj , ϕj i → 0 as ~ → 0. Step 1. Construct Λ(~) for a fixed B. As we have seen in the proof of Theorem 2.4, ε(~) := (2π~) n X a≤λj≤b |hBϕj , ϕj i|2 → 0. So if we define Γ(~) := {λj ∈ [a, b] : |hBϕj , ϕj i|2 ≥ ε(~) 1/2 }, then we must have (2π~) n#Γ(~) ≤ ε(~) 1/2 . Let’s denote Λ(~) := (Spec(P) ∩ [a, b]) \ Γ(~). Then by construction, for λj ∈ Λ(~) we have |hBϕj , ϕj i| < ε(~) 1/4 → 0. Moreover, according to Weyl’s law, as ~ → 0, #Λ(~) #Spec(P) ∩ [a, b] = 1 − #Γ(~) #Spec(P) ∩ [a, b] = 1 − (2π~) n#Γ(~) Vol(p −1 ([a, b])) + o(1) → 1
10LECTURE24-25:QUANTUMERGODICITYStep 2.Construct A(h) that works for a sequence B simultaneously.This can be done by using the standard diagonal trick.For each k, we haveconstructed in Step 1 density1 subsets A,(h) CSpec(P)n[a,b) so that the theoremholds for Bk and Ar(h).Observation:If A(h) and A'(h)aredensity 1 subsets, so is A(h)nA(h)So if we replace A(h) by Ni O small enough so that#A(h)1V0 0, there exists k and ho such that for all 0 0, there exists k such thatIb - ballL(T*M) < . This is possible, since
10 LECTURE 24-25: QUANTUM ERGODICITY Step 2. Construct Λ(~) that works for a sequence Bk simultaneously. This can be done by using the standard diagonal trick. For each k, we have constructed in Step 1 density 1 subsets Λk(~) ⊂ Spec(P)∩[a, b] so that the theorem holds for Bk and Λk(~). Observation: If Λ(~) and Λ0 (~) are density 1 subsets, so is Λ(~) ∩ Λ 0 (~). So if we replace Λk(~) by ∩1≤j≤kΛj (~), then we get density 1 subsets Λk(~) ⊂ Spec(P) ∩ [a, b] with Λk+1(~) ⊂ Λk(~), so that the theorem holds for Ak and Λk(~). Now for each k ∈ N, we take ~(k) > 0 small enough so that #Λk(~) #Spec(P) ∩ [a, b] ≥ 1 − 1 k , ∀0 0, there exists k and ~0 such that for all 0 0, there exists k such that kb − bkkL∞(T ∗M) < ε. This is possible, since