LECTUE NOTES ON QUANTUM CHAOS COUSE 18.158, MIT, APIL 2016, VESION 2 SEMYON DYATLOV Abstract. We give an overview of te quantum ergodicity result. 1. Quantum ergodicity in te pysical space 1.1. Concentration of eigenfunctions. First, let us consider te case wen M 2 is a bounded domain wit piecewise C boundary and we take te operator = 2 x 1 2 x 2. We study te Diriclet eigenvalues λ 0 < λ 1 λ 2... (wit multiplicities taken into account) and te corresponding L 2 normalized eigenfunctions u j H 1 0(M), so u j = λ 2 ju j, u j M = 0, u j L 2 (M) = 1. (1.1) We are interested in te following question regarding te ig energy limit: Question 1.1. How do u j concentrate as j? In general, u j become rapidly oscillating at ig energies, so we ave to study teir concentration in some roug sense. A natural way to do tat is to take te weak limits of te measures u j (x) 2 dx along subsequences: Definition 1.2. Let u jk be a subsequence of (u j ) and µ a probability measure on M. We say tat u jk µ weakly, if a(x) u jk (x) 2 dx a(x) dµ(x) for all a C0 (M) (1.2) We say tat u jk M M equidistributes in M, if it converges weakly to te volume measure: u jk dx Vol(M). (1.3) emarks. 1. By a standard density argument, once (1.2) olds for all a C 0 (M), it olds for all a C(M). 2. By a diagonal argument (see [Zw, Teorem 5.2]), tere always exists a subsequence of u j converging to some measure. 1
2 SEMYON DYATLOV As a basic example, consider te square M = [0, 1] 2. Te Diriclet eigenfunctions ave te form u jl (x 1, x 2 ) = 2 sin(jπx 1 ) sin(lπx 2 ), j, l N; λ jl = π j 2 + l 2. Exercise 1.3. Sow tat as j, u jj dx 1 dx 2 ; u 1j 2 sin 2 (πx 1 ) dx 1 dx 2, tat is te sequence u jj equidistributes in M but te sequence u 1j does not. More generally, we will consider te case wen (M, g) is a compact iemannian manifold wit piecewise smoot boundary and replace by te Laplace Beltrami operator g wic can be defined using te identity du, dv g d Vol g = ( g u)v d Vol g, u, v C0 (M). M M Te generalization of te measure (1.3) to tis case is given by te iemannian volume measure d Vol g Vol g (M). Exercise 1.4. Let M = S 2 be te two-dimensional spere embedded into 3. (a) Using te expression for Laplacian on 3 in sperical coordinates, sow tat eac omogeneous armonic polynomial v on 3 of degree m, te restriction u := v S 2 is an eigenfunction of S 2 wit eigenvalue m(m + 1). (In fact, wit a bit more work one can see tat all eigenfunctions of S 2 are obtained in tis way.) (b) Using te coordinates (x 1, x 2, x 3 ) in n, define for eac m N 0, v ± m = (x 1 ± ix 2 ) m, u ± m := c m v ± m S 2. were te constant c m is cosen so tat u ± m L 2 (S 2 ) = 1. Sow tat as m, u ± m converge weakly to a probability measure on S 2 wic is supported on te equator {x 2 1 + x 2 2 = 1, x 3 = 0}. We see tat te limit of u jk may depend on te coice of te sequence. It turns out tat te limits in fact also depend in an essential way on te dynamics of a natural flow on (M, g), and quantum caos studies in particular ow te dynamical properties of (M, g) influence te beavior of eigenstates.
LECTUE NOTES ON QUANTUM CHAOS 3 1.2. Caotic dynamics and ergodicity. For (M, g) a iemannian manifold, define te unit cotangent bundle Wen M is a domain in 2, we can write S M = {(x, ξ) T M : ξ g = 1}. S M = {(x, ξ) M 2 : ξ = 1} and parametrize tis space by (x 1, x 2, θ) were ξ = (cos θ, sin θ). (Te unit tangent and cotangent bundles can be identified wit eac oter using te metric g, and it will become apparent later wy it is muc more convenient for us to use te cotangent bundle ere.) We consider te geodesic billiard ball flow on M, ϕ t : S M S M, t. For M a domain in 2, every trajectory of ϕ t follows a straigt line wit velocity vector ξ until it its te boundary, wen it bounces off by te law of reflection. For (M, g) a iemannian manifold, straigt lines are replaced by geodesics induced by te metric g. If M as a boundary, ten te resulting map is not continuous and it is defined everywere except a measure zero set in S M, corresponding to trajectories tat eiter it non-smoot parts of te boundary or become tangent to te boundary. We will ignore tese issues in our note and send te reader to [ZeZw] for a detailed explanation of ow tey can be andled. A natural probability measure on S M is te Liouville measure, defined for a general iemannian manifold by dµ L = d Vol g(x)dµ S n 1(ξ), n = dim M, Vol g (M) Vol(S n 1 ) were µ S n 1 is te standard surface measure on te spere, transported to a measure on eac fiber of S M. For M a domain in 2, in coordinates (x 1, x 2, θ) we ave dµ L = dx 1dx 2 dθ 2π Vol(M). Te measure µ L is invariant under te flow: µ L (ϕ t (U)) = µ L (U), U S M, t. We now introduce te notion of ergodicity for te flow ϕ t, wic is a rater weak way of saying tat ϕ t is a caotic flow: Definition 1.5. We say tat ϕ t is ergodic wit respect to µ L, if for eac flow invariant set U S M; ϕ t (U) = U, t, we ave eiter µ L (U) = 0 or µ L (U) = 1.
4 SEMYON DYATLOV One important consequence of ergodicity is te following statement about ergodic averages a T := 1 T T 0 a ϕ t dt, T > 0, a L 1 (S M; µ L ). (1.4) Teorem 1 (L 2 ergodic teorem). Assume tat ϕ t is ergodic wit respect to µ L. Ten for eac a L 2 (S M; µ L ), a T a dµ L in L 2 (S M; µ L ). S M Proof. We will only sketc te proof, sending te reader to [Zw, Teorem 15.1] for an alternative proof, and we restrict ourselves to te case wen M as no boundary. Consider te vector field X on S M generating te flow, so tat ϕ t = exp(tx). Tis vector field gives rise to a first order differential operator, still denotes X. Since µ L is a ϕ t -invariant measure, we ave L X µ L = 0 and tus ix is an unbounded self-adjoint operator on L 2 (S M). Let de X be te spectral measure of ix, wic is an operator-valued measure on wic is constructed via te spectral teorem for unbounded self-adjoint operators. Ten for a L 2 (S M; µ L ), a ϕ t = exp(tx)a = e itλ de X (λ)a. Terefore, for T > 0 a T = ( 1 T T 0 ) e itλ dt de X (λ)a = e it λ 1 it λ de X(λ)a. Now te function eit λ 1 is bounded uniformly in T, λ, and it as te pointwise in λ it λ limit { e it λ 1 1, λ = 0; 1l {0} (λ) = as T. it λ 0, λ 0, Since integral over te spectral measure is a strongly continuous function of te interval, one can see from ere tat a T de X (λ)a in L 2 (S M, µ L ). (1.5) {0} Te rigt-and side is te ortogonal projection of a onto te space V 0 L 2 (S M, µ L ) of functions satisfying te equation Xf = 0. However, for eac suc f we ave f ϕ t = ϕ t and tus te sublevel sets {f c} are invariant under te flow (modulo a measure zero set wic can be removed). By ergodicity, V 0 must ten consist of
LECTUE NOTES ON QUANTUM CHAOS 5 constant functions. Ten te rigt-and side of (1.5) is te integral of a wit respect to µ L, finising te proof. Exercise 1.6. Sow tat neiter [0, 1] 2 nor S 2 ave ergodic ϕ t. (Hint: on S 2, te angular momentum wit respect to any axis gives a conserved quantity. Any sublevel set of tis function will be invariant under te flow.) Tere are many important examples of ergodic systems, including Sinai billiards; Bunimovic stadiums; iemannian manifolds (M, g) witout boundary wic ave negative sectional curvature, in particular closed negatively curved surfaces. 1.3. Statement of quantum ergodicity. Te following teorem (togeter wit its generalizations is Teorems 4, 8 below) is te main result to be proved in tis course: Teorem 2 (Quantum ergodicity in te pysical space). Assume tat ϕ t is ergodic wit respect to µ L. Ten tere exists a density 1 subsequence λ jk, tat is #{k λ jk } #{j λ j } suc tat u jk equidistributes in M: u jk 1 as, d Vol g Vol g (M). Tis teorem was stated by Snirelman [S] and proved by Zelditc [Ze] and Colin de Verdère [CdV]. Te case of te domains wit boundary was establised by Zelditc Zworski [ZeZw]. See [Zw, Teorem 15.5] for a detailed proof in te boundaryless case. (All of te results mentioned above prove te more general Teorems 4,8.) We see tat Teorem 2 uses information about te cotangent bundle on M to derive a statement on te manifold M itself. It turns out tat to prove it, we sould generalize te statement of equidistribution to T M, wic we call te pase space. 2. Pase space concentration and proof of quantum ergodicity We will encefort assume tat M as no boundary, referring te reader to [ZeZw] for te boundary case. 2.1. Semiclassical quantization. Assume tat a C 0 (T M). Semiclassical quantization associates to a, wic is called symbol or classical observable, an operator Op (a) : L 2 (M) L 2 (M)
6 SEMYON DYATLOV wic is called a semiclassical pseudodifferential operator or quantum observable. Tis procedure depends on a parameter > 0, called te semiclassical parameter, and we will be interested in te limit 0. Originally referred to (a dimensionless version of) Planck constant; in general it is te wavelengt at wic we want to study our eigenfunctions. We will not give a definition of Op (a) ere but will instead send te reader to [Zw, Capters 4 and 14], and will give some explanations regarding semiclassical quantization later in te course. We remark tat te procedure is independent of te coice of coordinates on M only modulo an O() remainder in te symbol, but te defined class of operators is geometrically invariant. In fact we may define Op (a) for a in a more general class S m (T M), m, given by te conditions a S m (T M) α x β ξ a(x, ξ) C αβ(1 + ξ ) m β. Te resulting operator acts on Sobolev spaces Op (a) : H s (M) H s m (M), s. We note tat if a(x, ξ) is a polynomial in ξ, a(x, ξ) = a γ (x)ξ γ, γ m a γ (x) C (M), ten Op is a differential operator; on n, te standard quantization procedure gives Op (a) = a γ (x)(d x ) γ, D x = 1 i x. (2.1) γ m In particular, if a(x, ξ) = a(x), ten we get a multiplication operator and on n, Op (a)u(x) = a(x)u(x), Op (ξ j ) = D xj = i x j. Also, if X is a vector field on M, ten we ave i X = Op (p X ) + O(), p X (x, ξ) = ξ, X(x). Tis explains wy our symbols are functions on te cotangent bundle rater tan te tangent bundle a vector field naturally gives a linear function on te fibers of te cotangent bundle. We list below some fundamental properties of te quantization operation. We leave te remainders ambiguous, but tey will ave appropriate mapping properties in Sobolev spaces.
LECTUE NOTES ON QUANTUM CHAOS 7 Teorem 3. For a S m (T M), b S k (T M), we ave Op (a) = Op (ā) + O(), (2.2) Op (a) Op (b) = Op (ab) + O(), (2.3) [Op (a), Op (b)] = i Op ({a, b}) + O( 2 ), (2.4) were {a, b} is te Poisson bracket, given in coordinates by {a, b} = j ( ξj a xj b xj a ξj b). Moreover, for a S 0 (T M) te operator norm of Op (a) on L 2 can be estimated as follows [Zw, Teorem 5.1]: for some constant C independent of a,, lim sup Op (a) L 2 L 2 C a L (T M). (2.5) 0 2.2. Quantum ergodicity in pase space. We now generalize Teorem 2 to a statement about quantum observables Op (a)u j, u j L 2, a S 0 (T M). For tat we need to pick te value of and it will be convenient to put j = 1 λ j. We ten define V j (a) := Op j (a)u j, u j L 2 (M), a S 0 (T M). Definition 2.1 (Weak limits in pase space). Let u jk be a subsequence of u j and µ be a measure on T M. We say tat u jk µ in te sense of semiclassical measures, if V jk (a) a dµ for all a S 0 (T M). T M We say tat u jk equidistribute in pase space if tey converge to te Liouville measure: u jk µ L. emark. Tere is always a subsequence converging to some measure, and all resulting measures are supported on te unit cospere bundle S M and invariant under te flow ϕ t see [Zw, Capter 5] and (2.8), (2.10) below. Teorem 4 (Quantum ergodicity in pase space). Assume ϕ t is ergodic wit respect to µ L. Ten tere exists a density 1 subsequence u jk suc tat u jk equidistribute in te pase space.
8 SEMYON DYATLOV Teorem 2 follows from ere by taking a to be a function of x, so tat V j (a(x)) = a(x) u j (x) 2 dx, and using te fact tat te pusforward of µ L to M is te volume measure: 1 a(x) dµ L = a(x) d Vol g. Vol g (M) S M In te rest of tis section, we prove Teorem 4, following several steps. M 2.3. Step 1: using te eigenfunction equation. We first rewrite te eigenfunction equation in te form were te symbol is cosen so tat g u j = λ 2 ju j M Op j (p)u j = 0 (2.6) p(x, ξ) = p 0 (x, ξ) + O(), p 0 (x, ξ) = ξ 2 g 1 2 P = Op (p) = 2 g 1. 2 Define te Hamiltonian vector field (2.7) H p0 = j ξj p xj xj p ξj, and note tat Define te flow H p0 a = {p 0, a}, ϕ t = exp(th p0 ), a C (T M). ten (explaining te coice of 1 2 in te definition of p 0) te restriction of ϕ t to S M is te geodesic flow. A key tool in te proof is te Scrödinger propagator ( U(t) = U(t; ) = exp itp ) : L 2 (M) L 2 (M). It quantizes te flow ϕ t as made precise by te following Teorem 5 (Egorov s Teorem). For a C 0 (T M), we ave U( t) Op (a)u(t) = Op (a ϕ t ) + O() L 2 (M) L 2 (M).
LECTUE NOTES ON QUANTUM CHAOS 9 Proof. We only sketc te proof, see [Zw, Teorem 15.2] for details. It is enoug to prove tat, denoting a t := a ϕ t, t ( U(t) Op (a t )U( t)) = O() L 2 (M) L 2 (M) Te left-and side is ( U(t) Op ( t a t ) i [ ]) P, Op (a t ) U( t) By (2.4), tis becomes U(t) Op ( t a t {p 0, a t })U( t) + O() and it remains to use tat t a t = {p 0, a t }. We ten ave te following Lemma 2.2. Assume tat a C0 (T M). Ten for any T > 0, ( ) V j (a) = V j a T + OT ( j ) were a T is defined in (1.4) and te constant in te remainder depends on T. Proof. We ave for eac t, U(t; j )u j = u j and tus V j (a) = Op j (a)u j, u j L 2 = U( t; j ) Op j (a)u(t; j )u j, u j L 2 = Op j (a ϕ t )u j, u j L 2 + O t ( j ) = V j (a ϕ t ) + O t ( j ) and it remains to average bot sides over t [0, T ]. (2.8) Tis statement uses te fact tat u j are eigenfunctions and features ergodic averages along te flow ϕ t. 2.4. Step 2: basic bounds. We record ere a few standard bounds on V j (a). First of all, by (2.5) we ave for some global constant C and eac a S 0 (T M), Moreover, if a vanises on S M, ten as follows immediately from lim sup V j (a) C a L (T M). (2.9) j lim V j(a) = 0 (2.10) j Lemma 2.3 (Elliptic bound). Assume a S 0 (T M) and a S M = 0. Ten as j, Op j (a)u j L 2 = O( j ).
10 SEMYON DYATLOV Proof. Since a vanises on S M, we may write a = bp 0 = bp + O(), were p, p 0 are defined in (2.7) and b S 2 (T M). Ten by (2.3), Op j (a) = Op j (b) Op j (p) + O( j ) L 2 L 2. Since Op j (p)u j = 0 by (2.6), te proof is finised. 2.5. Step 3: bounding averages over eigenfunctions. We know by Teorem 1 tat for large T, te average a T is close to te integral of a, but only in L 2 (T M). If we ad an L estimate instead, ten we could use (2.9) to control V j (a) for all j in te limit j. However, ergodic averages typically do not converge in L (tis can be seen for instance by considering a closed geodesic). Terefore we will ave to make te best out of te L 2 bound on a. It turns out tat it produces a bound on V j (a) on average in j see Lemma 2.4 below. Te key statement is te following teorem, wic we will try to prove later in te course (see 3.3): ( ) Teorem 6 (Local Weyl Law). Assume tat χ C0 (0, ) and a S 0 (T M). Ten as, ( λj χ j ) ( ) ( n V j (a) = χ ( ) ξ g a (x, 2π T M ξ ) ) dxdξ + O( 1 ). ξ g Taking a = 1 in Teorem 6, we in particular get ( λj ) ( ) n ( χ = χ ( ) ) ξ g dxdξ + O( 1 ). 2π j T M Approximating χ = 1l [0,1] by functions in C 0 ( (0, ) ), tis gives Teorem 7 (Weyl Law). We ave as, #{j λ j } = ω n (2π) Vol g(m) n + o( n ) n were ω n > 0 is te volume of te unit ball in n. Note also tat te integral on te rigt-and side in Teorem 6 is zero if a vanises on S M; tis is in line wit Lemma 2.3. For te proof of quantum ergodicity, we use te following corollary of Teorem 6: Lemma 2.4 (Variance bound). We ave for eac a S 0 (T M), as n V j (a) 2 C a 2 dµ L + O( 1 ) λ j [,2] S M were te constant C depends on M, but not on a or.
LECTUE NOTES ON QUANTUM CHAOS 11 ( ) Proof. Take nonnegative χ C0 (0, ) wit χ = 1 on [1, 2], ten it is enoug to estimate ( n λj ) χ Op j (a)u j 2 L = ( λj ) 2 n χ Op j (a) Op j (a)u j, u j L 2 j j and te rigt-and side is bounded by Teorem 6 using tat by (2.2) and (2.3) Op j (a) Op j (a) = Op j ( a 2 ) + O( j ) = Op j ( a 2 ) + O( 1 ). 2.6. Step 4: integrated quantum ergodicity. We can now prove te following integrated (or, strictly speaking, summed) form of Teorem 4: Teorem 8 (Integrated quantum ergodicity). Assume tat a S 0 (T M) and L a = a dµ L. (2.11) Ten as, n λ j [,2] S M V j (a) L a 2 0. Proof. By subtracting L a from a and using tat Op (1) is te identity operator, we reduce to te case L a = 0: a dµ L = 0. S M Moreover, by (2.10) we may assume tat a C 0 (T M). Take some T > 0. By Lemma 2.2 and ten Lemma 2.4, we ave ( ) n V j (a) 2 n Vj a T 2 + OT ( 1 ) λ j [,2] λ j [,2] C a T 2 L 2 (S M,µ L ) + O T ( 1 ) were te constant C is independent of T and. Taking te limit as, we ave lim sup n V j (a) 2 C a T 2 L 2 (S M,µ L ). λ j [,2] Te left-and side does not depend on T, and te rigt-and side converges to 0 by Teorem 1. 2.7. Step 5: end of te proof. It remains to derive Teorem 4 from Teorem 8, tat is to extract a density 1 sequence of eigenfunctions wic equidistributes in pase space. For tat we use Cebysev inequality and a diagonal argument on dyadic pieces of te spectrum. More precisely, for r N let N r := #{j λ j [2 r, 2 r+1 )},
12 SEMYON DYATLOV ten N r 2 nr as r by te Weyl law (Teorem 7). Take a sequence a s C 0 (T M), s = 1, 2,... wic is dense in C0 (T M) wit respect to te uniform norm. Put L s = a S M s dµ L and ( 1 ε l,r := max V j (a s ) L s ). 2 s l N r λ j [2 r,2 r+1 ) Ten ε l,r 0 as r for eac l by Teorem 8. We pick r(l) suc tat r(l+1) > r(l) and ε l,r < 100 l for r r(l). Define te disjoint collection of sets J l N as follows: j J l λ j [2 r(l), 2 r(l+1) ) and max s l V j(a s ) L s < 2 l. By Cebysev inequality, for r(l) r < r(l + 1), terefore # ( {j λ j [2 r, 2 r+1 )} \ J l ) lε l,r 2 2l N r 2 l N r, 1 #(J l ) # ( {j λ j [2 r(l), 2 r(l+1) )} 2 l. It follows from ere and te Weyl law tat te sequence j k, {j k } = l J l is a density one subsequence in N. On te oter and, we ave for eac s, V jk (a s ) a s dµ L as k. S M Using te bound (2.9) and te fact tat {a s } is dense in C0 we see tat V jk (a) a dµ L as k S M wit te uniform norm, for all a C 0 (T M). By (2.10) same is true for all a S 0 (T M), finising te proof. 3. Overview of semiclassical quantization We now briefly discuss ow to define te quantization procedure Op, sending te reader to [Zw] for details.
LECTUE NOTES ON QUANTUM CHAOS 13 3.1. Quantization on n. We consider te following symbol classes on T n = 2n, S m (T n ) C ( n ), m, defined as follows: a(x, ξ; ) S m (T n ) if for eac multiiindices α, β tere exists a constant C αβ suc tat for all x, ξ and small, α x β ξ a(x, ξ; ) C αβ(1 + ξ ) m β. Note tat for m N 0 tis class includes polynomials of order m in ξ wit coefficients bounded wit all derivatives in x. For a S m(t n ), we define te operator Op (a) on functions on n as follows: Op (a)f(x) = (2π) n e i x y,ξ a(x, ξ)f(y) dydξ. (3.1) 2n Te integral (3.1) does not always converge in te usual sense, so some explanations are in order. Assume first a is smoot and compactly supported, or more generally a lies in te Scwartz class S (T n ). If f S ( n ), ten integral in (3.1) converges absolutely and gives a Scwartz function. Wen a S m (T n ), we see using te semiclassical Fourier transform F f(ξ) = (2π) n/2 y,ξ f(y) dy tat e i n Op (a)f(x) = (2π) n/2 e i x,ξ a(x, ξ)f f(ξ) dξ (3.2) n and since F f(ξ) is Scwartz, te integral still converges; integrating by parts in ξ, we see tat it still gives a Scwartz function. In fact, for any a S m (T n ), one can define Op (a)f for f S ( n ), were S ( n ), te dual to S ( n ), is te space of tempered distributions. Tis can be seen eiter by duality or by treating (3.1) as an oscillatory integral, or by first considering te case of a S (T n ) and extending to general a by density. In eiter case, we obtain te quantization procedure on n, a S m (T n ) Op (a) : S ( n ) S ( n ), S ( n ) S ( n ). Moreover, rapidly decaying symbols produce smooting operators: a S (T n ) = Op (a) : S ( n ) S ( n ). Note tat te mapping properties above are for any fixed ; we make no statement about te uniformity of norms as 0 at tis point. Exercise 3.1. Using (3.2), sow tat wen a S m (T n ) is polynomial in ξ, te operator Op (a) is te differential operator defined in (2.1).
14 SEMYON DYATLOV In wat follows, we will often ignore wat appens as x, ξ, so our proofs would immediately work for Scwartz symbols a S (T n ) and wit more work can be extended to general symbols. 3.2. Basic properties of quantization and stationary pase. We now want to establis some properties of te quantization procedure Op. We start wit te product formula (2.3). Assume tat a, b S (T n ) uniformly in. We would like to write Op (a) Op (b) = Op (c), c(x, ξ; ) S (T n ), (3.3) and understand te asymptotics of c as 0. We first find a formula for c using te following statement, known as oscillatory testing; see [Zw, Teorem 4.19]: Lemma 3.2. Assume tat a S (T n ). Ten for eac fixed > 0, 1. We can recover te symbol a from te operator A = Op (a) as follows: a(x, ξ) = e i x,ξ A(e i,ξ ). (3.4) 2. If A : S ( n ) S ( n ) and te function a S (T n ) satisfies (3.4), ten A = Op (a). We now write out te symbol c from (3.3) as follows: c(x, ξ) = e i x,ξ Op (a) Op (b)(e i,ξ ) = e i x,ξ Op (a)(b(, ξ; )e i,ξ ) = (2π) n e i x y,η ξ a(x, η; )b(y, ξ; ) dydη. n (3.5) To understand te beavior of c as 0, we use te following (see [Zw, Teorem 3.16]) Teorem 9 (Metod of stationary pase). Assume U n is an open set and ϕ C (U; ) as only one critical point x 0 U, tat is ϕ 0 on U \ {x 0 }. Assume also tat x 0 is a nondegenerate critical point, tat is te Hessian 2 ϕ(x 0 ) gives a nondegenerate quadratic form. Denote by sgn( 2 ϕ(x 0 )) te signature of tis form (te number of positive eigenvalues minus te number of negative eigenvalues). Ten for eac a C0 (U; C), we ave as 0 e iϕ(x) a(x) dx (2π) n/2 e iϕ(x 0 ) j L j (a) x=x0 (3.6) U were eac L j is a ϕ-dependent linear differential operator of order 2j. In particular j=0 L 0 (a) x=x0 = e iπ 4 sgn 2 ϕ(x 0 ) det 2 ϕ(x 0 ) 1/2 a(x 0 ).
LECTUE NOTES ON QUANTUM CHAOS 15 Proof. We only sketc a proof in a special case known as quadratic stationary pase: n = 1, ϕ(x) = x2 2. By Fubini s Teorem and a linear cange of variables, one can pass from ere to te case wen ϕ is a nondegenerate quadratic form in iger dimensions. Te general case can ten be andled by te Morse Lemma, wic gives a cange of variables conjugating a general pase ϕ locally to a quadratic form. We compute in terms of te standard (nonsemiclassical) Fourier transform â(ξ), e ix2 iπ 2 a(x) dx = e 4 e iξ2 2 â(ξ) dξ. (3.7) 2π Tis follows from te more general statement true for any z C, e z 0, z 0: e zx2 1 2 a(x) dx = e ξ2 2z â(ξ) dξ. (3.8) 2πz Te statement (3.8) follows for z > 0 by direct calculation using te Fourier transform of te Gaussian and for all z by analytic continuation. Now, taking te Taylor expansion of e iξ2 /2 as 0 and using tat â S, we get e ix2 iπ 1 ( ) jâ(ξ) 2 a(x) dx e 4 iξ2 dξ 2π j! 2 finising te proof. j=0 e iπ 1 i ) j 4 2π 2j x a(0) j!( 2 In te case (3.5) we integrate over y, η, tus te dimension is 2n. Te pase is given by j=0 (y, η) x y, η ξ, and te only critical point is y = x, η = ξ. Te value of te pase at te critical point is equal to 0. Te expansion (3.6) can be computed explicitly from quadratic stationary pase and yields c(x, ξ) a(x, ξ)b(x, ξ) + i n ξj a(x, ξ) xj b(x, ξ) + O( 2 ), j=1 explaining (2.3), (2.4).
16 SEMYON DYATLOV 3.3. More on semiclassical quantization. On a manifold M, we define te quantization Op (a) by covering M wit a locally finite system of coordinate carts, splitting a into pieces using a partition of unity, quantizing it separately on eac cart using (3.1), and adding te pieces back togeter. However, if we take different carts or te partition of unity, te resulting operator will cange by an operator wit symbol in S m 1 (T M). Terefore, it is more convenient to consider te class of semiclassical pseudodifferential operators Ψ m (M) = {Op (a) a S m (T M)} wic is independent of te coice of quantization, and te principal symbol map σ : Ψ m (M) S m (T M)/S m 1 (T M), σ (Op (a)) = a + S m 1 (T M) wic is also independent of te quantization. We ave te sort exact sequence 0 Ψ m 1 (M) Ψ m (M) σ S m (T M)/S m 1 (T M) 0. Te symbolic calculus makes it possible to construct more pseudodifferential operators by calculating teir symbol term by term. For instance, we can find approximate inverses of operators wit nonvanising symbols: Proposition 3.3. Assume a S 0(T M), p S m(t M), and p 0 on supp a. Ten tere exists b S m (T M) suc tat Op (a) = Op (b) Op (p) + O( ). emark. Tis generalizes Lemma 2.3 in te following sense: if ( 2 g 1)u = 0, ten Op (a)u L 2 = O( ) u L 2 for all a S 0 (T M), supp a { ξ g = 1} =. Sketc of proof. We first take b 0 = a p S m (T M). Ten by (2.3) we ave for some r 1 S 1 (T M), Op (a) = Op (b 0 ) Op (p) + Op (r 1 ) + O( ). Moreover, one can arrange so tat supp r 1 {a 0}. Ten we repeat te procedure, putting b 1 = r 1 p S m 1 (T M). Arguing tis way we construct some symbols b j j S m j (T M) and it remains to take b suc tat b b j. j
LECTUE NOTES ON QUANTUM CHAOS 17 We can also take functions of pseudodifferential operators, wic we present in a special case. Namely we ave Teorem 10. Assume (M, g) is a compact iemannian manifold witout boundary, and put P = 2 g = Op (p) + O() Ψ 1 were p(x, ξ) = ξ 2 g. Ten for eac χ C0 () and all N, we ave χ(p ) Ψ N (M); σ (χ(p )) = χ(p). Sketc of proof. We write using te Fourier transform ˆχ, χ(p ) = 1 ˆχ(t)e itp dt. (3.9) 2π For bounded t, we ave as can be done solving te equation e itp = Op (p t ), p t = e itp + O(), (3.10) t Op (p t ) = ip Op (p t ) in symbolic calculus. Wen ˆχ is compactly supported, we get te desired formula. Oterwise we can write (3.10) up to t ε for some small ε, were te symbol p t will ave derivatives mildly growing in, and use te integral (3.9) wit te fact tat ˆχ is Scwartz. From Teorem 10 we can derive te following version of local Weyl law of Teorem 6 (te original version, wit depending on u j, can be proved using a rescaling and Lemma 2.3): Teorem 11. For χ C0 (), a S 0 (T M), and λ j, u j defined in (1.1), we ave as 0 χ( 2 λ j ) Op (a)u j, u j = (2π) n χ(p(x, ξ))a(x, ξ) dxdξ + O( 1 n ). j T M Proof. Putting P := 2 g, te left-and side is te trace tr ( χ(p ) Op (a) ). However, we know tat χ(p ) Op (a) Ψ N (M) for all N, so we write χ(p ) Op (a) = Op (b) + O( ) were b = χ(p)a + O() is rapidly decreasing in ξ. It remains to use te following trace formula for pseudodifferential operators: tr Op (b) = (2π) n b(x, ξ) dxdξ + O( 1 n ) T M wic reduces to te case of quantization on n and tere te trace can be computed by integrating te Scwartz kernel.
18 SEMYON DYATLOV 3.4. Anoter application of stationary pase: concentration of Lagrangian states. Assume tat U n is open and we are given a pase function ϕ C (U; ) and an amplitude b C 0 (U; C). We define te family of functions u C 0 (U; C), > 0, by u (x) = e iϕ(x)/ b(x). We would like to understand te limits as 0 of observables Op (a)u, u L 2, a C 0 (T n ), specifically to write for some measure µ, Op (a)u, u L 2 a dµ. T n (3.11) Tis can be done by applying te metod of stationary pase to Op (a)u (x) = (2π) n e i ( x y,ξ +ϕ(y)) a(x, ξ)b(y) dydξ. 2n Te pase function is te stationary point is given by (y, ξ) x y, ξ + ϕ(y), x = y, ξ = ϕ(x), and te value of te pase at te stationary point is equal to ϕ(x). Applying (3.6), we obtain Op (a)u (x) = e iϕ(x)/ a(x, ϕ(x))b(x) + O(). Terefore we ave te limit (3.11) wit µ given by a dµ = a(x, ϕ(x)) b(x) 2 dx. In particular, µ lives on T M U Λ ϕ = {(x, ϕ(x)) x U} wic is a Lagrangian submanifold of T n. Exercise 3.4. Find te semiclassical limits (in te sense of Definition 2.1) of te functions u 1j and u jj from Exercise 1.3. (Ignore te boundary issues by testing tese functions against operators supported strictly inside te square.)
LECTUE NOTES ON QUANTUM CHAOS 19 eferences [CdV] Yves Colin de Verdière, Ergodicité et fonctions propres du Laplacien, Comm. Mat. Pys. 102(1985), 497 502. [S] Alexander Snirelman, Ergodic properties of eigenfunctions, Usp. Mat. Nauk. 29(1974), 181 182. [Ze] Steve Zelditc, Uniform distribution of eigenfunctions on compact yperbolic surfaces, Duke Mat. J. 55(1987), 919 941. [ZeZw] Steve Zelditc and Maciej Zworski, Ergodicity of eigenfunctions for ergodic billiards, Comm. Mat. Pys. 175(1996), 673 682. [Zw] Maciej Zworski, Semiclassical analysis, Graduate Studies in Matematics 138, AMS, 2012.