LECTURE 24-25: QUANTUM

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∞(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 = {H, a}, dt t where {·, ·} is the Poisson bracket so that for any f, g ∈ C∞(R2n), n   X ∂f ∂g ∂f ∂g ∞ 2n {f, g} = − ∈ 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 ∈ C∞(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 = {p, a}, dt t where {·, ·} is the Poisson bracket on T ∗M so that for any f, g ∈ C∞(T ∗M), n   X ∂f ∂g ∂f ∂g ∞ ∗ {f, g} = − = Xf (g) ∈ 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 of the Riemannian manifold (M, g). Conversely, every parametrized 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. LECTURE 24-25: QUANTUM ERGODICITY 3

−1 ¶ on p (c) generated by the flow ρt.

Similar computations also show that the n-form dx1 ∧ dξ1 ∧ · · · ∧ dxn ∧ dξn is a well-defined 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, |dp| ≥ c0 > 0. So in particular for each c ∈ [a, b], the level set −1 Σc := p (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 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) = |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 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 ∈ L1(X, µ), 1 Z T 1 Z lim f(ρt(x))dt → f(y)dµ T →∞ T 0 µ(X) X 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: 2 Theorem 1.3 (L -mean ergodic theorem). Suppose ρt is an ergodic flow on (X, µ), then for any f ∈ L2(X, µ), 2 Z  1 Z T 1 Z  lim f(ρt(x))dt − f(y)dµy dµx = 0. T →∞ 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 →∞ X while the Birkhoff ergodicity theorem claims that

hfiT (x) → hfiX as T → ∞ for a.e. x ∈ X.

2. Quantum ergodicity In this we will always assume (1)( M, g) is a compact Riemannian manifold, (2) p ∈ Sm(T ∗M) is an (almost) elliptic classical symbol, where m > 0. (3) a < b, and on the set a ≤ p(x, ξ) ≤ b, |dp| ≥ c0 > 0. LECTURE 24-25: QUANTUM ERGODICITY 5

Note that by the condition (3), each c ∈ [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 ϕj = λjϕj, ∞ 2 2 where ϕj ∈ C (M) and {ϕj} form an L -orthonormal basis of L (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 Theo- rem, 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 #Λ( ) (3) lim ~ = 1, ~→0 #(Spec(P ) ∩ [a, b]) such that for any semiclassical pseudodifferential operator A ∈ 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 ∈ [a, b], we have, as ~ → 0, 1 Z (5) hAϕj, ϕji → −1 σ(A)dxdξ Vol(p ([a, b])) a≤p≤b 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 2 • eigenvalues are λj,~ = ~ λj, 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 6 LECTURE 24-25: QUANTUM ERGODICITY

• A ∈ Ψ0(M) and satisfies the condition (4): 1 Z 1 Z σ(A)dµc = f(x)dx. Vol(Σc) Σc Vol(M) M 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 #{j ≤ N} lim k = 1, N→∞ N such that for each f ∈ C∞(M), Z Z 2 1 (6) |ϕjk | fdx → f(x)dx. M Vol(M) M

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 2 Vol(U) |ϕjk | dx → . 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 man- ifold with negative sectional curvature. Then as k → ∞, Z Z 2 1 |ϕk| fdx → f(x)dx. M Vol(M) M

2The limit can’t be too bad. See the theorem at the end of this lecture 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 1 Z α := hσ(A)iΣc = σ(A)dµc Vol(Σc) Σc is independent of c ∈ [a, b]. Then as ~ → 0, ZZ 2 n X 1 (7) (2π~) hAϕj, ϕji − −1 σ(A)dxdξ → 0. Vol(p ([a, b])) a≤p≤b a≤λj ≤b

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 ZZ n X (2π~) hBϕj, ϕji → σ(B)dxdξ. a≤p≤b a≤λj ≤b

2 (3) The L -mean ergodic theorem (Theorem 1.3 above): Suppose ρt is an ergodic flow on (X, µ), then for any f ∈ L2(X, µ), Z 2 lim (hfiT − hfiX ) dµx = 0. T →∞ X Proof of Theorem 2.4. By our assumption on σ(A), we have 1 Z −1 σ(A)dxdξ = hσ(A)iΣc = α. Vol(p ([a, b])) p−1([a,b]) ∞ We choose a cut-off χ ∈ C0 (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

n X 2 (8) (2π~) |hBϕj, ϕji| → 0. a≤λj ≤b 8 LECTURE 24-25: QUANTUM ERGODICITY

The idea is to consider “the time-average of B”, 1 Z T hBiT := U(−t)BU(t)idt. T 0 For any t ∈ R, we have

hU(−t)BU(t)ϕj, ϕji = hBϕj, ϕji. Thus hBϕj, ϕji = hhBiT ϕj, ϕji. It follows from Cauchy-Schwarz inequality that 2 2 ∗ |hBϕj, ϕji| ≤ khBiT ϕjk = hBiT hBiT ϕj, ϕj . By weak Egorov theorem,

khBiT − hBeiT kL(L2) = OT (~), where OT (~) means O(~) with constants depending on T , and 1 Z T hBei = Be(t)dt, T 0 and Be(t) ∈ Ψ0(M) with ∗ σ(Be(t)) = ρt σ(B). It follows Z T 1 ∗ σ(hBeiT ) = ρt σ(B)dt = hσ(B)iT . T 0 and thus ∗ 2 2 2 σ(hBeiT hBeiT ) = |σ(hBeiT )| = |hσ(A)iT − α| = |hσ(A)iT − hσ(A)iΣc | . So according to the generalized Weyl law, for each fixed T > 0, n X 2 n X ∗ lim sup(2π~) |hBϕj, ϕji| ≤ lim sup(2π~) hBiT hBiT ϕj, ϕj ~→0 ~→0 a<λj

¶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 N 1 X lim ak = 0, N→∞ N k=1

then there exists a density one subset {akj } of {ai} such that

lim aj = 0. k→∞ k 2 It is easy to see that for the case P = ~ ∆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, ϕji → 0 as ~ → 0. Step 1. Construct Λ(~) for a fixed B. As we have seen in the proof of Theorem 2.4, n X 2 ε(~) := (2π~) |hBϕj, ϕji| → 0. a≤λj ≤b So if we define 2 1/2 Γ(~) := {λj ∈ [a, b]: |hBϕj, ϕji| ≥ ε(~) }, then we must have n 1/2 (2π~) #Γ(~) ≤ ε(~) . Let’s denote Λ(~) := (Spec(P ) ∩ [a, b]) \ Γ(~). Then by construction, for λj ∈ Λ(~) we have 1/4 |hBϕj, ϕji| < ε(~) → 0. Moreover, according to Weyl’s law, as ~ → 0, #Λ( ) #Γ( ) (2π )n#Γ( ) ~ = 1 − ~ = 1 − ~ ~ → 1. #Spec(P ) ∩ [a, b] #Spec(P ) ∩ [a, b] Vol(p−1([a, b])) + o(1) 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(~) ∩ Λ0(~). 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 #Λ ( ) 1 k ~ ≥ 1 − , ∀0 < < (k). #Spec(P ) ∩ [a, b] k ~ ~ Moreover, we can choose ~(k) so that they decrease to 0: ~(k) & 0 as k → ∞. Now we define Λ∞(~) := Λk(~), ~(k + 1) ≤ ~ < ~(k). Then we have #Λ ( ) 1 ∞ ~ ≥ 1 − #Spec(P ) ∩ [a, b] k for all k and all 0 < ~ < ~(k), and thus #Λ ( ) ∞ ~ → 1 #Spec(P ) ∩ [a, b]

as ~ → 0. So Λ∞(~) is a density 1 subset.

Moreover, for each k, since Λ∞(~) ⊂ Λk(~) for ~ < ~(k), for λj = λj,~ ∈ Λ∞(~) we still have |hBkϕj, ϕji| → 0 as ~ → 0. This completes Step 2. Step 3. Construct Λ(~) that works for all B simultaneously.

The idea is to choose a “dense sequence” Bk. Suppose we can find a sequence B so that for any B ∈ Ψ−∞(M) with R σ(B)dµ = 0 for all c ∈ [a, b] and for any k Σc c ε > 0, there exists k and ~0 such that for all 0 < ~ < ~0, we have

kBk − BkL(L2) < ε,

then for λj = λj,~ ∈ Λ∞(~) (constructed in Step 2),

lim suphBϕj, ϕji < ε and lim infhBϕj, ϕji < ε →0 ~→0 ~ and the conclusion follows.

It remains to construct such a dense sequence Bk. According to Corollary 2.2 in Lecture 15 (see also the remark at the end of Lecture 11), 1/2 kBk − BkL(L2) ≤ kσ(Bk) − σ(B)kL∞(T ∗M) + O(~ ). So it is enough to find a sequence {b } ⊂ S−∞(T ∗M) with R b dµ = 0, such that k σc k c for any b ∈ S−∞(T ∗M) with R bdµ = 0 and any ε > 0, there exists k such that σc c kb − bkkL∞(T ∗M) < ε. This is possible, since LECTURE 24-25: QUANTUM ERGODICITY 11

• The space of continuous functions that vanishes at infinity, ∗ ∗ c C0(T M) = {f ∈ C(T M) | ∀ε > 0, ∃ compact K s.t. |f(x)| < ε on K }, is separable (i.e. has a countable dense subset). (This can be proven via the Stone-Weierstrass theorem for locally compact Hausdorff spaces.) ∗ ∞ • The space C0(T M) is a metric space with respect to the L -metric. • Any subspace of a separable metric space is separable. (In general a subspace of a separable space may be non-separable). −∞ ∗ ∗ 0 ∗ • S (T M) ⊂ C0(T M). (We can’t use S (T M) in the proof since the space of bounded continuous functions is not separable.) This completes the proof. 

3. Semiclassical defect measure ¶Quantum ergodicity from measure point of view. Of course in the conclusion of Corollary 2.2, we may replace f ∈ C∞(M) by f ∈ C(M). In other words, we can restate the quantum ergodicity as If the geodesic flow of (M, g) is ergodic, then there exists a density 1 subset {jk} ⊂ N such that the sequence of probability measure 2 |ϕjk | dx on M converges weakly to the uniform probability measure: 1 |ϕ (x)|2dx * dx. jk Vol(M) Similarly, one can explain the conclusion of Theorem 2.1 using the language of measure: For most eigenfunctions, if we regard the expected value of the quantum observable A to a system in the state ϕj,~ (c.f. Lecture 2),

hAϕj,~, ϕj,~i, as an integral of the symbol σ(A) with respect to a measure on the classical phase space3, then the measure converges weakly to the uniform Liouville measure 1 dxdξ vol(p−1([a, b])) on p−1([a, b]) (or the uniform Liouville measure 1 dµ on Σ ). This explains the vol(Σc) c c following description of quantum ergodicity in Wikipedia: [Wiki: Quantum ergodicity states, roughly, that in the high-energy limit, the probability distributions associated to energy eigenstates of a quantized ergodic Hamiltonian tend to a uniform distribution in the classical phase space.]

3We will make this sentence more clear in the next page. 12 LECTURE 24-25: QUANTUM ERGODICITY

¶Semiclassical defect measure. 2 Let {u~} be a family of L -normalized functions. Definition 3.1. A semiclassical defect measure (also known as semiclassical mea-

sure in some literature) associated with {u~} is a nonnegative Radon measure µ such that for some subsequence hj → 0, we have Z lim haW u , u i → a(x, ξ)dµ j→∞ b ~j ~j R2n ∞ 2n for any symbol a ∈ C0 (R ). The existence of a defect measure is guaranteed by the standard diagonal trick:

Proposition 3.2. For any {u~}, there exists at least one semiclassical defect mea- sure dµ.

∞ 2n Proof. We first choose a countable dense subset {ak} in the set C0 (R ). By bound- W (1) edness of ba (Corollary 2.2 in Lecture 15), we can pick ~j → 0 such that W ha1 u~, u~i (1) → α1. b ~=~j (2) (1) Similarly we can pick a subsequence ~j of ~j such that W ha2 u~, u~i (2) → α2 b ~=~j (j) and so on. Now we define ~j := ~j . Then ha W u , u i → α bk ~ ~ ~=~j k for any k.

Now consider a map Φ which maps ak to αk. Note that by Corollary 2.2 in Lecture 15, we have |αk| ≤ kakkL∞ . So Φ is bounded on the dense subset αk, it also keep any possible “linear relation” on the set {ak}, and thus can be extended to a ∞ 2n bounded linear map on Φ : C0 (R ) → R with

|Φ(a)| ≤ kakL∞ . By Riesz representation theorem, there exists a unique (complex-valued) Radon measure µ on R2n such that Z Φ(a) = adµ.

It remains to prove that dµ is nonnegative, namely Φ(a) ≥ 0 for a ≥ 0. For this ∞ 2n purpose we need a more precise formula for Φ(a). In fact, for any a ∈ C0 (R ), we ∞ may approximate a by a sequence akj . If kak − akL < ε, then W W lim sup ha u~, u~i − hak u~, u~i = ≤ ε j→∞ b b ~ ~j LECTURE 24-25: QUANTUM ERGODICITY 13 which implies W Φ(a) = lim ha u , u i = . j→∞ b ~ ~ ~ ~j Thus according to the sharp G˚ardinginequality, for any a ≥ 0, Φ(a) = lim haW u , u i ≥ 0. b ~ ~ ~=~j ~j →0 So dµ is a positive measure and the proof is finished. 

Recall from Lecture 21 that the wavefront set WF(u~) of {u~} (which describes the “concentration” of {u~} in the phase space) is defined via its complement: ∞ 2n (x0, ξ0) 6∈ WF(u~) if for any b ∈ C0 (R ) with support sufficiently close to (x0, ξ), W 2 we have kb u~kL2 = O(~ ). This implies W lim hb u , u i = = 0 j→∞ ~ ~ ~ ~j for any subsequence ~j → 0. As a result, for any semiclassical measure µ associated with {u }, we must have ~ Z b(x, ξ)dµ = 0. R2n Since this holds for all b with b(x0, ξ0) = 1 with support sufficiently close to (x0, ξ), we must have µ(x0, ξ0) = 0. In other words,

Proposition 3.3. Any semiclassical defect measure µ associated with {u~} is sup- ported in WF(u~). Remark. The conception of defect measure as well as the propositions just proved can be easily extended (with small modifications) to T ∗M, where M is any compact Riemannian manifold.

¶Semiclassical defect measure associated to eigenfunctions.

Now let P be a semiclassical pseudodifferential operator as in §2, and let u~ be 2 L -normalized eigenfunctions of P associated with eigenvalue λ~:

P u~ = λ~u~. We have

Theorem 3.4. Let {u~} be a sequence of eigenfunctions so that λ~ → E0 as ~ → 0. Then any semiclassical defect measure µ associated with {u~} is a probability measure −1 supported on p (E0) which is invariant under the flow ρt generated by p.

Proof. According to Theorem 2.5 in Lecture 21, we have4 −1 suppµ ⊂ WF(u~) ⊂ p (E0).

4 Here it is enough to assume P~u~ = λ~u~ + o(1). 14 LECTURE 24-25: QUANTUM ERGODICITY

R 2n Since | adµ| = |Φ(a)| ≤ kakL∞ , we have µ(R ) ≤ 1. Conversely, according to ∞ Theorem 1.3 in Lecture 21, for any a ∈ C0 such that a ≤ 1 and a ≡ 1 in a −1 W ∞ neighborhood of p (E0), we have ba u~ = u~ + O(~ ) and thus Z 2n W µ( ) ≥ adµ = lim ha u , u i = ≥ 1. R j→∞ b ~ ~ ~ ~j R2n

To prove the invariance of µ under the Hamiltonian flow ρt associated with p, it is enough to prove ∗ (9) Φ(a) = Φ(ρt a) ∞ 2n W for any a ∈ C0 (R ) and any t ∈ R. In fact, if we denote A = ba , then W [P,A] = ~{\p, a} + O( 2). i ~ But

h[P,A]u~, u~i = hAu~, P u~i − hP u~, Au~i = hAu~, λ~u~i − hλ~u~, Au~i = 0. W 5 It follows h{\p, a} u~, u~i = O(~) and thus Z W {p, a}dµ = lim h{\p, a} u , u i = 0. j→∞ ~ ~ R2n Now the equation (9) follows, since d d Z Z d Z Φ(ρ∗a) = ρ∗adµ = ρ∗adµ = {p, a}dµ = 0. dt t dt t dt t  For the geodesic flow on the cosphere bundle of compact Riemannian manifold- s, here are some invariant probability measures which are possible candidates of semiclassical defect measures of the Laplacian eigenfunctions:

• The Liouville measure dµLiouvlle, • The dirac delta measure on closed geodesics, • Combinations of the above... The QUE conjecture claims that for negatively curved manifolds, the only semiclas- sical defect measure is the Liouville measure. Some recent works: • Lindenstrauss 2006, Soundararajan 2010: the conjecture is true for modular surface SL2(Z) \ H. • Anantharaman 2008, Anantharaman-Nonnenmacher 2007: the KS-entropy of the semiclassical measure has a positive lower bound (and in particular rules out the delta measures concentrated on closed geodesics). • Dyatlov-Jin 2018, Dyatlov-Jin-Nonnenmacher 2019: For negatively curved surfaces, every semiclassical measure has full support.

5 Here it is enough to assume P~u~ = λ~u~ + o(~).