LLONG TIME QUANTUM EVOLUTION OF OBSERVABLES ONCUSP SURFACES
YANNICK BONTHONNEAU
Abstract.
We build a semi-classical quantization procedure for finite volume man-ifolds with hyperbolic cusps, adapted to a geometrical class of symbols. We provean Egorov Lemma until Ehrenfest times on such manifolds. Then we give a versionof Quantum Unique Ergodicity for the Eisenstein series for values of s convergingslowly to the unitary axis. In this paper, we work with non-compact complete manifolds (
M, g ) of finite volumewith hyperbolic ends. Such manifolds are called cusp manifolds . They decompose intoa compact manifold with boundary and a finite number of cusp-ends Z , . . . Z m ; thatis, of the type: Z Λ a = [ a, + ∞ ) y × T dθ with the metric ds = dy + dθ y where dθ is the canonical flat metric on the d -dimensionnal torus T d = R d / Λ. TheLaplacian on compactly supported smooth functions on M is essentially self-adjoint,so it has a unique self-adjoint extension ∆ to L ( M ). Here, we take the analyst’s con-vention that ∆ is a non-positive operator. In [CdV83], Yves Colin de Verdi`ere provedthat for cusp surfaces , the resolvent of the Laplacian has a meromorphic continuationthrough the continuous spectrum. Another proof was given in any dimension witha more general definition of cusps by M¨uller in [M¨ul83]. It gives the following. Theoperator defined on L for (cid:60) s > d/ s / ∈ [ d/ , d ], R ( s ) = ( − ∆ − s ( d − s )) − has a meromorphic continuation to the whole complex plane, as an operator C ∞ c ( M ) → C ∞ ( M ). The poles are called resonances. Those on the vertical line (cid:60) s = d/ , d ] correspond to discrete L eigenvalues.However the others are associated to continuous spectrum. The way to prove this uses ameromorphic family of eigenfunctions, the so-called Eisenstein series { E i ( s ) } s ∈ C ,i =1 ...m .Those are smooth, not L , and satisfy − ∆ E i ( s ) = s ( d − s ) E i ( s ) , s ∈ C . The E ( s ) naturally replace the L eigenfunctions as spectral data for the continuousspectrum. Actually, the data can be alternatively considered to be the values of E for s on the unitary axis, the full family { E ( s ) } s ∈ C , or the poles and the residues at the a r X i v : . [ m a t h . SP ] N ov YANNICK BONTHONNEAU poles of the family — a.k.a the resonant states. Observe that it is not always easyto translate results between those formulations. We are trying to determine whetherproperties of L eigenfunctions for compact manifolds remain true for this spectraldata in the non-compact case; we also seek to know if new behavior arise.? QUEQE Liouville µ η d/ d/ η The figure gives a synthetic vision of what is known(including our result) on the semi-classical measures:arrows represent asymptotics of sequences of s ’s, andwe give the associated semi-classical measure next toit.In [Dya12], in the case of surfaces, Semyon Dyatlovproved that when |(cid:61) s | → ∞ and (cid:60) s → / η with η >
0, the microlocal measures associated to E i ( s )converge to an explicit measure µ i,η on S ∗ M — theunit cotangent sphere. It satisfies ( X − η ) µ i,η = 0where X is the generator of the geodesic flow. Thisresult does not rely upon dynamical properties of thegeodesic flow such as ergodicity. We recover a simi-lar result in any dimension. When the curvature ofthe manifold is (strictly) negative, we adopt Babillot’sargument [Bab02] which relies on the Local ProductStructure to prove (see lemma 2.11) that µ i,η (cid:42) η → L where L t is the normalized Liouville measure on tS ∗ M . In this paper, we build a semi-classical pseudo-differential calculus Ψ( M ) with symbols S ( M ) and a quantization Op h (sections 1 and 2.1). For s ∈ C not a resonance, let µ hi ( s ) be the distribution on T ∗ M (cid:104) µ hi,j ( s ) , σ (cid:105) := (cid:104) Op h ( σ ) E i ( s ) , E j ( s ) (cid:105) . Also consider (cid:104) µ h ( s ) , σ (cid:105) := (cid:88) i (cid:104) Op h ( σ ) E i ( s ) , E i ( s ) (cid:105) . We prove:
Theorem 1.
Assume M is a cusp manifold of negative curvature. Then there is apositive constant C such that the following holds. η × µ hi,j (cid:18) d η + ih (cid:19) −→ h → ,η → δ ij π L as long as (1) η > C log | log h || log h | . ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 3 If s is a pole of E , then the resonant state at s is a linear combination of the E i ’sat 1 − s . Hence, if we have a sequence of resonances s ( h ) such that 1 − s satisfies thehypothesis of theorem 1, the semi-classical measures associated to the correspondingsequence of resonant states converges to Liouville after a suitable rescaling. However,we have no information on that rescaling; it is related to the size of the residues of thescattering determinant — see section 2.2.1 for a definition — and that seems to be adifficult problem.However, apart from the case of arithmetic cusp surfaces, it is quite possible that theregion of the plane we are considering contains no resonances. What is more η → ν > E ( s ) on the unitary axis, in the case of hyperbolic surfaces.The set of poles of { E ( s ) } s ∈ C is encoded in what is called the Scattering phase , whichis a function S on R (see section 2.1.1). Theorem (Zelditch) . For any
T > , h (cid:90) T − T (cid:12)(cid:12)(cid:12)(cid:12) µ h (cid:18) d ith (cid:19) − π S (cid:48) (cid:18) th (cid:19) L t /h (cid:12)(cid:12)(cid:12)(cid:12) dt −→ h → (cid:60) s = d/ µ hi ( d/ i/h ) without averaging on the spectrum. It is an open problemhow close to the spectrum one can get without some averaging; it is also not easy tomake a reasonable and precise conjecture on the behaviour of those Wigner measureswhen s gets closer to the axis without being on it. This can be contrasted with theconvex cocompact case — replacing cusps by funnels — for which results have recentlyappeared — see [GN14] and [DG14].The only surface, as far as we know, for which such results have been obtained isthe modular surface . In 2000, Luo and Sarnak proved in [LS95] that(2) µ h (cid:18)
12 + ih (cid:19) ∼ π | log h | L . Recently [PRR11], Petridis, Raulf and Risager extended this to(3) µ h (cid:18)
12 + η + ih (cid:19) ∼ π | log h | L . as long as η | log h | →
0. However, the case of the modular surface, for which most ofthe spectrum is discrete, is very particular. There is no reason to expect that (2) and(3) should hold in general for other surfaces.
YANNICK BONTHONNEAU
The main tool that enables us to improve Dyatlov’s theorem is a long time Egorovlemma. Let p be the principal symbol of the Laplacian, that is, the metric p ( x, ξ ) = | ξ | x . Theorem 2.
Let σ ∈ S be supported in a set where p is bounded. There is a symbol σ t in an exotic class such that for | t | ≤ C | log h | e − ith ∆ Op( σ ) e ith ∆ = Op( σ t ) + O ( h ∞ ) . This holds as long as
C > remains smaller than an explicit constant C max . As in most situations, the proof of our Egorov lemma is relatively easy once thesettings, and in particular, the relevant properties of the quantization have been es-tablished. Let us explain; apart from some abstract nonsense, the analysis in Egorovlemma is contained in an estimate of the derivatives of the flow. In the compact case,the choice of how those derivatives are estimated is not important. However, in anon-compact case, one has to use norms consistent with the geometry of the problem,and those norms determine the class of symbols to use.One way to avoid those problems is to use a compactly supported quantization andoperators. This is adapted in cases where the whole interesting part of the underlyingdynamics takes place in a compact set (see for example [DG14]). However in our case,a crucial part of the dynamics happens in the cusps, so we wanted to allow for symbolssupported in the cusps, and we had to give a treatment for the ends.The cusp -calculus of Melrose sees the cusp as conformal to a Euclidean cylinder(see [MM98]), and the corresponding class of symbols has a nice behavior with respectto the euclidean dynamics on a half cylinder with boundary, considering the point atinfinity of a cusp as a circle boundary. Another description we found was in [Bou14].For symbols only depending on the vertical variables in the cusps, this is probablyeasier to manipulate. However it does not allow for flexible behavior on the θ variable.Let us observe however that [Bou14] deals with cusps that are much more general thanthe ones we consider.The only closely related description of pseudo-differential operators we found was[Zel86] by Zelditch. In this paper, he develops a quantization procedure for the hyper-bolic plane, using Fourier-Helgason transform on the unit disk. Then, he shows thatthose operators can be symmetrized to operate on compact hyperbolic surfaces. Insome sense, we use the class of symbols that are compatible with the metric, and thatclass is similar to that introduced by Zelditch. However, the quantization procedureis different: we use only euclidean Fourier transform to build operators specificallyon cusps — see (6). We prove composition stability without any proper support as-sumption. This is available in any dimension ≥
2, in a semi-classical formulation. Wealso state usual theorems on pseudors, including L bounds, and a trace formula. Wedetailed the proofs, with elementary tools, only referring to [Zwo12]. ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 5
This will be part of the author’s PhD thesis.
Acknowledgement . We thank Nalini Anantharaman and Colin Guillarmou fortheir fruitful and extensive advice. We thank Semyon Dyatlov for his suggestions. Wethank Barbara Schapira for her explanations on dynamical matters.1.
Quantizing in a full cusp
Symbols.
Let Z Λ be a full cusp. That is Z Λ = R + × R d / Λ , where Λ is some lattice in R d . The first coordinate, y is called the height ; the secondis denoted by θ , and we write x = ( y, θ ) = ( y, θ , . . . , θ d ). The cusp is endowed with acusp metric : ds = dy + dθ y , where dθ = dθ + · · · + dθ d is the canonical flat metric on R d . By rescaling, we seethat Z Λ and Z t Λ are isometric whenever t >
0, so we assume that Λ has covolume 1.In the first part of the article, we will write just Z for Z Λ because Λ will not change.The Riemannian volume in Z is d vol( x ) = dydθ d y d +1 . We refer to the space of square integrable functions with respect to this volume as L ( Z ). The Laplacian is ∆ = y ∆ eucl − ( d − y ∂∂y , where ∆ eucl is the Laplacian for the Euclidean cylinder. On the cotangent bundle T ∗ Z ,we let Y and J be the dual coordinates to ∂ y and ∂ θ , with ξ = Y dy + J dθ ( J is avector in R d ). We also write ξ = ( Y, J ) = (
Y, J , . . . , J d ). The usual Poisson Bracketon T ∗ Z writes { f, g } = ∂ Y f ∂ y g − ∂ y f ∂ Y g + ∇ J f. ∇ θ g −∇ θ f. ∇ J g with ∇ the usual flat connexion on R n .The riemmannian metric on Z gives an isomorphism between T Z and T ∗ Z , and T ∗ Z is thus endowed with the metric | ξ | x = y ( Y + | J | ) . In appendix A, we recall how to define the spaces C n ( Z ) of functions using covariantderivatives. This definition is intrinsic of the metric, however it is not very practicalfor computations. Let X y := y∂ y X θ i := y∂ θ i . YANNICK BONTHONNEAU If α = ( α , . . . , α k ) is a sequence with α i ∈ { y, θ , . . . , θ d } — we say a space-index oflength k — we denote X α . . . X α k by X α , and | α | = k is the length. Then we prove in(26) that (cid:107) . (cid:107) C n ( Z ) and (cid:88) | α |≤ n (cid:107) X α f (cid:107) L ∞ ( Z ) are equivalent norms.Since [ X y , X θ ] = X θ , the order in which the derivatives are taken has little importance. Definition 1.1.
We call coefficients the elements of C ∞ b ( Z ) = ∩ n ≥ C n ( Z ) , that is,smooth functions f on Z such that for any space-index α , (cid:107) X α f (cid:107) L ∞ ( Z ) < ∞ . Elements of C ∞ b,h ( Z ) := C ∞ ( R + → C ∞ b ( Z )) are called (semi-classical) coefficients. Both C ∞ b ( Z ) and C ∞ b,h ( Z ) have a natural ring structure. Lemma 1.2.
The module generated by X y and X θ ’s over the coefficients C ∞ b ( Z ) is aLie algebra.Proof. It suffices to consider the behavior of [ aX y , bX θ i ] with a , b in C ∞ b ( Z ) :[ aX y , bX θ i ] = ( ab + aX y b ) X θ i − ( bX θ i a ) X y (cid:3) If we consider hX y and hX θ i ’s as vector fields on Z with a parameter h ≥
0, wededuce that the module they generate over C ∞ b,h ( Z ) is also a Lie algebra. Hence itmakes sense to speak of its universal envelopping algebra V ( Z ). This is an algebra of semi-classical differential operators on Z . From now on, all differential operators weuse will be in V ( Z ).Inside of V ( Z ), we can consider the subalgebra generated by hX y and hX θ i ’s over C ; those are the constant coefficients differential operators. Proposition 1.3.
The elements of V ( Z ) can be decomposed as finite sums of the type (cid:88) i ≥ ,α h | α | + i a i,α ( h, x ) X α with a i,α ’s in C ∞ b,h ( Z ) . We define : P = − h . It is in V ( Z ) and in some sense, V ( Z ) has been taylored to satisfy this property.Actually, P is a constant coefficient operator:∆ = X y − d × X y + X θ + · · · + X θ d . Using the algebraic properties described above, one can prove:
ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 7
Proposition 1.4.
Let A ∈ V ( Z ) , and ( x, ξ ) ∈ T ∗ Z . Let φ be a smooth function on Z ,with φ ( x ) = 0 and dφ ( x ) = ξ . Then we let σ ( A )( x, ξ ) = lim h → A h ( e iφ/h )( x ) . This limit exists and does not depend on the choice of φ . It is called the principalsymbol of A . It is a polynomial in the ξ variable, smoothly depending on x and h .What is more, its monomials are of the form a ( x ) y k + l Y k J l where a ∈ C ∞ b ( Z ) .The mapping σ from V ( Z ) to functions on T ∗ Z is linear. What is more, if A, B are in V ( Z ) , σ ( AB ) = σ ( A ) σ ( B ) and σ (cid:18) h [ A, B ] (cid:19) = (cid:8) σ ( A ) , σ ( B ) (cid:9) σ (cid:18) hi X y (cid:19) = yY and σ (cid:18) hi X θ i (cid:19) = yJ i We deduce that the principal symbol of P is the metric(4) p = | ξ | x / . If we consider the set of functions S V on T ∗ Z obtained by taking principal symbolsof differential operators, we see that it is stable by the action of the vectors X y and X θ i ’s. In the ξ direction, we introduce X Y := 1 y ∂ Y X J i := 1 y ∂ J i . Then X Y and X J i also stabilize S V . Let us refine this description. S V is a gradedalgebra decomposing as S V = ∪ n ≥−∞ S n V , where S n V is the set of q ’s of degree lesserthan n . Then X y and X θ ’s map S n V to itself while X Y and X J ’s map S n V to S n − V . Alsoremark that whenever q ∈ S n V , q = O ( (cid:104) yξ (cid:105) n ) where (cid:104) x (cid:105) is the usual bracket √ x .The above motivates the introduction of the following class of symbols: Definition 1.5.
Let σ be a smooth function on T ∗ Z . We say that σ is a (hyperbolic)symbol on Z of order n ∈ R if for any finite sequence { α k } with α k ∈ { y, θ ...d , Y, J ...d } ,if ( α ) is the number of Y ’s and J ’s in the sequence, (5) q n,α := sup T ∗ X i | X α . . . X α k a ( x, ξ ) | ≤ C (cid:104) yξ (cid:105) n − ( α ) . YANNICK BONTHONNEAU
We let S ( Z ) be the set of symbols, S n ( Z ) be the set of symbols of order n , and S −∞ = ∩ n ∈ R S n ( Z ) . S is graded by the order, and S n V ⊂ S n ( Z ) X y , X θ i : S n ( Z ) → S n ( Z ) X Y , X J i : S n ( Z ) → S n − ( Z ) σ ∈ S m , µ ∈ S n ⇒ { σ, µ } ∈ S m + n − The family of semi-norms q n,α gives a structure of metric space to S n . We have not specified an h -dependency. Actually, we will need to let symbols dependon h in a slightly rough fashion, so we use the classes : Definition 1.6.
Let ≤ ρ < / . Consider complex functions σ of h > and ( x, ξ ) ∈ T ∗ Z such that for fixed h , σ h is in S n . If α is a sequence of indices, let | α | beits length. Assume that σ additionnally satisfy the family of estimates q n,α,ρ := sup h h ρ | α | q n,α ( σ h ) < ∞ , for α finite sequence of y, θ i , Y, J i . Then we say that σ is an exotic symbol of order ( n, ρ ) , and write σ ∈ S nρ ( Z ) . We alsodefine S ρ ( Z ) = ∪ S nρ ( Z ) and S −∞ ρ ( Z ) = ∩ S nρ ( Z ) . We have: S n V ⊂ S n ( Z ) h ρ X y , h ρ X θ i : S n ( Z ) → S n ( Z ) h ρ X Y , h ρ X J i : S n ( Z ) → S n − ( Z ) σ ∈ S m , µ ∈ S n ⇒ h ρ { σ, µ } ∈ S m + n − The semi-norms q n,α,ρ also give a structure of metric space to S nρ . The rest of section 1 is devoted to describing a quantization procedure for this algebraof symbols. In section 1.2, we give a definition, and prove that we obtain pseudo-differential operators in the usual sense. Then, in section 1.3, we give a stationaryphase lemma. We use it to prove usual properties of the quantization in section 1.4.1.2.
Quantizing symbols.
After we give a quantization procedure Op in 1.7 for σ ∈ S ρ , we first prove that the operators we obtain have pseudo-differential behaviourlocally. That is, if γ is a diffeomorphism from a relatively compact open set U in R n onto its image in Z , the pullback( γ ∗ Op( σ )) f ( x ) = [Op( σ )( f ◦ γ )] ( γ ( x )) . is a pseudo-differential operator in U . Then we prove (lemma 1.15) that Op is pseudo-local in the following sense : if φ and φ are two coefficients on Z not depending on ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 9 θ , with disjoint support, φ Op( σ ) φ = O H − n → H n ( h ∞ ) for every n ∈ N . The Sobolev spaces on cusps are defined in appendix A. To get to 1.15, we haveto prove that our operators have some
Sobolev regularity — see proposition 1.12.This is deduced, with a usual parametrix argument, of the crucial lemma 1.9 aboutregularity on L ( Z ). The regularity we prove in this section is certainly not optimal,and we will get better statements — see proposition 1.22 — once we obtain stabilityby composition.For convenience, we will use some expressions in the half-space H d +1 = R + × R d ,which is the universal cover of Z . If a is some function on Z (resp. T ∗ Z ), we identifyit with its unique lift to H d +1 (resp. T ∗ H d +1 ). We denote by Op wh the usual Weylquantization of a symbol on R d +1 :(6) Op wh ( η ) f = 1(2 πh ) d +1 (cid:90) R d +1 × R d +1 e i (cid:104) x − x (cid:48) ,ξ (cid:105) /h η (cid:18) x + x (cid:48) , ξ (cid:19) f ( x (cid:48) ) dx (cid:48) dξ. If η and f are only defined on the upper half space, this also makes sense (continuingall functions by 0 in the lower half space). Now, assume that η is actually a symbol oforder n , and f is a smooth compactly supported function on Z — that is a periodicfunction in θ in H d +1 , with compact support in y . Then, in each fiber of T ∗ Z , η is asymbol in the usual sense of R n , with uniform estimates as long as y stays in a compactset of R + ∗ . We deduce that the Fourier transform of η fiberwise is a distribution in S (cid:48) ,whose singular support is { } and with fast decay at infinity, with estimates uniformin x as long as y stays in a compact set. We deduce that Op wh ( η ) f is well defined.Actually, applying a finite number of X y ’s and X θ ’s, we can repeat the argument andobserve it is also a smooth function. A similar argument will be detailed in the proofof lemma 1.9. Now, we can give our basic definition : Definition 1.7.
Let σ ∈ S ( Z ) be some hyperbolic symbol on T ∗ Z . Then, for any f ∈ C ∞ c ( Z ) , we let Op h ( σ ) f ( y, θ ) = y d +12 Op wh ( σ ) (cid:18) y d +12 f (cid:19) . This is seen to be a Λ -periodic function in the θ direction. Op h ( σ ) defines an operator C ∞ c ( Z ) → C ∞ ( Z ) . When σ is a symbol, Op( σ ) denotes the family { Op h ( σ ) } h .We let Ψ nρ be the set of { Op( σ ) | σ ∈ S nρ } . The introduction of ( y/y (cid:48) ) ( d +1) / corresponds the conjugacy by the unitary map f → y ( d +1) / f from L ( dydθ ) to L ( Z ). The principal symbol.
In this section, we check that:
Lemma 1.8.
The definition of the semi-classical principal symbol given for elementsof V ( Z ) in 1.4 extends to operators Op( σ ) ’s, and σ (Op( σ )) = σ .Proof. Let χ be some C ∞ c ( Z ) function equal to 1 near x ∈ Z . Let ξ ∈ T ∗ x Z and φ some smooth function on Z such that φ ( x ) = 0 and dφ ( x ) = ξ . Let σ ∈ S ρ , take ˜ x alift of x in H d +1 and integrating over H d +1 × R d +1 Op h ( σ ) (cid:0) χe iφ/h (cid:1) ( x ) = 1(2 πh ) d +1 (cid:90) e i ( (cid:104) ˜ x − x (cid:48) ,ξ (cid:105) + φ ( x (cid:48) )) /h (cid:18) yy (cid:48) (cid:19) ( d +1) / σ (cid:18) ˜ x + x (cid:48) , ξ (cid:19) χ ( x (cid:48) ) dx (cid:48) dξ The proof here will be very similar to the case of R n , so we just insist on the differences.We let ˜ χ be some smooth compactly supported function equal to 1 around ˜ x ; we insert1 = ˜ χ + (1 − ˜ χ ) to break the integral into two parts (I) and (II).For the first term (I), observe that it is an integral over a fixed compact set in the x (cid:48) variable, with an integrand that has symbolic behavior in the ξ variable, and a verysimple phase function. Classical stationary phase results directly apply to give that :lim h → (I) = σ ( x, ξ ) . To estimate (II), we integrate by parts in ξ ; we just have to introduce suitable powersof ( y + y (cid:48) ) to obtain the new integrand C k h k − d − e i ( (cid:104) ˜ x − x (cid:48) ,ξ (cid:105) + φ ( x (cid:48) )) /h ( y + y (cid:48) ) k (1 − ˜ χ ( x (cid:48) )) | ˜ x − x (cid:48) | k (cid:18) yy (cid:48) (cid:19) ( d +1) / σ ∗ k (cid:18) ˜ x + x (cid:48) , ξ (cid:19) χ ( x (cid:48) ) , where σ ∗ k = ( X Y + X J ) k σ and C k = ( i/ k − d − π − d − . With k big enough, this is O ( h (1 − ρ )2 k − d − ) in L ( dx (cid:48) dξ ). (cid:3) Basic boundedness estimates.
Lemma 1.9.
For all (cid:15) > , the elements of Ψ − d − − (cid:15)ρ extend to bounded operators on L ( Z ) , with O ( h − ρd − ) norm as h → . In the subsequent developments, we will call
Schwarz Kernel of A : C ∞ c → C ∞ thedistribution defined by : Af ( x ) = (cid:90) Z K ( x, x (cid:48) ) f ( x (cid:48) ) dx (cid:48) where dx (cid:48) = dydθ is the Lebesgue measure in the half-cylinder. Let us state a modifiedSchur lemma — see page 82 in [Zwo12] for the original version. ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 11
Lemma 1.10.
Let A be an operator from C ∞ c ( Z ) to C ∞ ( Z ) , with Schwarz kernel K .Assume that for some τ ∈ R , (7) C ( A, τ ) := sup x (cid:90) x (cid:48) ∈ Z (cid:18) y (cid:48) y (cid:19) d +1+ τ | K ( x, x (cid:48) ) | dx (cid:48) × sup x (cid:48) (cid:90) x ∈ Z (cid:18) yy (cid:48) (cid:19) τ | K ( x, x (cid:48) ) | dx < ∞ . Then A can be extended to a bounded operator on L with (cid:107) A (cid:107) L → L ≤ C ( A, τ ) . Proof.
We follow the classical proof. All the integrals are over Z . if u ∈ C ∞ c ( Z ), | Au ( x ) | ≤ (cid:90) (cid:112) y (cid:48) d +1+ τ | K | (cid:112) y (cid:48)− d − − τ | K || u | dx (cid:48) ≤ (cid:107) y (cid:48) d +1+ τ | K |(cid:107) / L ( x (cid:48) ) (cid:20)(cid:90) y (cid:48)− τ | K ( x, x (cid:48) ) || u | y (cid:48)− d − dx (cid:48) (cid:21) / (cid:90) | Au ( x ) | y − d − dx ≤ (cid:34) sup x (cid:90) x (cid:48) (cid:18) y (cid:48) y (cid:19) d +1+ τ | K ( x, x (cid:48) ) | (cid:35) (cid:90) x,x (cid:48) (cid:18) yy (cid:48) (cid:19) τ | K ( x, x (cid:48) ) | | u | y (cid:48) d +1 ≤ C ( A, τ ) (cid:90) | u | dx (cid:48) y (cid:48) d +1 (cid:3) Now, we prove lemma 1.9
Proof.
Let σ ∈ S − d − − (cid:15)ρ with some (cid:15) >
0. Formula (6) actually defines Op w ( σ ) actingon the half plane. We let K wσ ( x, x (cid:48) ) be its kernel, and we let K σ be the kernel of Op h ( σ )— K σ depends on h . Then we have(8) K σ ( y, θ, y (cid:48) , θ (cid:48) ) = (cid:18) yy (cid:48) (cid:19) d +12 (cid:88) k ∈ Λ K wσ ( y, θ, y (cid:48) , θ (cid:48) + k ) . Plugging this identity in (7), we see that instead of integrating over the cusp Z , wecan integrate over the half space; this way, we prove that C (Op( σ ) , τ ) is less than (cid:34) sup x (cid:90) x (cid:48) ∈ H d +1 (cid:18) y (cid:48) y (cid:19) d +12 + (cid:15) | K wσ ( x, x (cid:48) ) | dx (cid:48) (cid:35) (cid:34) sup x (cid:48) (cid:90) x ∈ H d +1 (cid:18) yy (cid:48) (cid:19) d +12 + (cid:15) | K wσ ( x, x (cid:48) ) | dx (cid:35) . By linearity of Op, it suffices to consider real symbols, so we assume that σ is real.Then K wσ ( x, x (cid:48) ) = K wσ ( x (cid:48) , x ). By symmetry of the two terms in the above equation, itsuffices to prove that for some τ ∈ R , the first term is finite.Since σ is of order strictly less than − d −
1, it is integrable in the fibers, and we canestimate : | K wσ ( x, x (cid:48) ) | ≤ πh ) d +1 (cid:90) R d +1 dξ (cid:12)(cid:12)(cid:12)(cid:12) σ ( x + x (cid:48) , ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:15) h d +1 y + y (cid:48) ) d +1 q d +1+ (cid:15), ( σ ) . With no decay in θ , this is obviously not sufficient to prove boundedness. We observethe following fact :(9) θ − θ (cid:48) ih y + y (cid:48) K wσ = K wX J σ . Using (9) d + 1 times, and the definition of symbols, we get that :(10) | K wσ ( x, x (cid:48) ) | ≤ C h d +1 y + y (cid:48) ) d +1
11 + h ( d +1) ρ (cid:12)(cid:12)(cid:12) θ − θ (cid:48) h ( y + y (cid:48) ) (cid:12)(cid:12)(cid:12) d +1 . Now, we integrate (10) in the θ (cid:48) variable. Actually, we translate by θ , and we rescalewith µ = h ρ − ( θ (cid:48) − θ ) / ( y + y (cid:48) ) to get: (cid:90) | K ( y, θ, y (cid:48) , θ (cid:48) ) | dθ (cid:48) ≤ Ch − ρd − y + y (cid:48) (cid:90) R d d µ
11 + | µ | d +1 ≤ Ch ρd y + y (cid:48) . We just have to find τ such thatsup y> (cid:90) y (cid:48) > y + y (cid:48) (cid:18) y (cid:48) y (cid:19) d +12 + τ dy (cid:48) < ∞ . and then the norm of Op( σ ) on L will be O ( h − ρd − ) times some symbol norm. Chang-ing variables to u = y (cid:48) /y , the LHS issup y> (cid:90) ∞ u d +12 + τ u du = (cid:90) ∞ u d +12 + τ u du. For τ ∈ ] − ( d + 1) / − , − ( d + 1) / (cid:3) If we wanted an optimal statement in terms of regularity, we could remark here thatwe only use d + 1 symbol estimates (differentiating only in the J direction) to obtainthis result.1.2.3. Sobolev regularity.
Recall that all functionnal spaces are defined in appendix A.We need a parametrix lemma for the composition of a pseudor with a differentialoperator:
Lemma 1.11.
Let σ ∈ S nρ , Q , , be constant-coefficient elements of V ( Z ) , of order k , . Then there exists a symbol ˜ σ ∈ S n + k + k ρ such that Q Op( σ ) Q = Op(˜ σ ) , ˜ σ = σ × σ ( Q ) σ ( Q ) + O ( h − ρ Ψ n + k + k − ρ ) . Additionally, let Q , also be constant-coefficient differential operators with order k , , and N ∈ N , satisfying the ellipticity condition that σ ( Q ) σ ( Q ) does not vanish. ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 13
Then, there exists a symbol ˜ σ N of order n + k + k − k − k , such that : Q Op( σ ) Q = Q Op(˜ σ N ) Q + O ( h N Ψ − Nρ ) , (11) ˜ σ N = σ σ ( Q ) σ ( Q ) σ ( Q ) σ ( Q ) + O ( h − ρ Ψ n + k + k − k − k − ρ ) . Proof.
We start with the first part. Proceeding by induction, we see that it is enoughto prove the property when Q , are constants, or first order differential operators. Thecase of constants is straightforward. Now, let us assume Q = hX θ and Q = 1. Thekernel of Q Op h ( σ ) is1(2 πh ) d +1 (cid:90) R d +1 e i (cid:104) x − x (cid:48) ,ξ (cid:105) (cid:18) yy (cid:48) (cid:19) d +12 (cid:20) iyJ σ (cid:18) x + x (cid:48) , ξ (cid:19) + h y∂ θ σ (cid:18) x + x (cid:48) , ξ (cid:19)(cid:21) dξ Decomposing y = ( y + y (cid:48) ) / y − y (cid:48) ) /
2, integrating by part in the ξ variable whennecessary, we get= 1(2 πh ) d +1 (cid:90) R d +1 e i (cid:104) x − x (cid:48) ,ξ (cid:105) (cid:18) yy (cid:48) (cid:19) d +12 (cid:20) σ θ (cid:18) x + x (cid:48) , ξ (cid:19)(cid:21) dξ where σ θ = iyJ σ − h yJ X Y σ + h X θ σ + i h X θ X Y σ.σ θ = iyJ σ + O ( h − ρ S nρ ) . similarly, for Q = hX y , we get Q Op( σ ) = Op( σ y ) with σ y = iyY σ + h d + 1) σ − yY X Y σ ) + h X y σ + i h X Y X y σ. The case when Q = 1 and Q = hX θ , hX y will lead to similar computations, and thesame conclusion.Now, we prove the second part of the lemma. We look for a semi-classical expansionfor ˜ σ N , in the following form : ˜ σ N = (cid:80) ∞ h (1 − ρ ) k σ k with σ k ∈ S ( n + k + k − k − k ) − k .Injecting this formal development in (11), we find a linear system of equations on the σ k ’s. Actually, identifying powers of h , we see that this system is in lower-triangularform. The diagonal coefficients are all the same, equal to σ ( Q ) σ ( Q ) σ ( Q ) σ ( Q ) . The ellipticitycondition is sufficient to see that this system has a unique solution of the above form.Our formal series does not converge, so we truncate at order M for some integer M (cid:29)
1. The remainder is then O ( h ( M +1)(1 − ρ ) Ψ n + k + k − k − k − M − ρ ). This is O ( h N Ψ − Nρ )for M big enough; we take ˜ σ N to be this truncated series. (cid:3) Proposition 1.12.
For all (cid:15) > , the elements of Ψ n − d − − (cid:15) are bounded from H s to H s − n for all s, n real numbers, with norm O ( h −| s |−| n |− ρd − ) . Proof.
Proceeding by interpolation, we only need to prove this result for s, n evenintegers. Let σ ∈ S kρ with k < n − d −
1. Then by (28) (cid:107) Op h ( σ ) (cid:107) H s → H s − n = h −| s |−| n | (cid:107) ( P + 1) ( s − n ) / Op h ( σ )( P + 1) − s/ (cid:107) L → L . By lemma 1.11, there is a symbol ˜ σ N ∈ S k − nρ such that( P +1) ( s − n ) / Op h ( σ )( P +1) − s/ = Op h (˜ σ N )+( P +1) − ( s − n ) − / (cid:2) O ( h N Ψ − Nρ ) (cid:3) ( P +1) − s + / . Now, we only have to apply lemma 1.9 to ˜ σ N to conclude since ( P + 1) − k is boundedon L as soon as k ≥ (cid:3) Pseudo-locality statements.
Before going on to prove pseudo-locality, we needto define what we mean when we say that a family of operators is smoothing.
Definition 1.13.
We say that a family of operators { A h } h> on L ( Z ) is smoothing iffor every h > and n > , A h maps H − n to H n in a continuous fashion. Additionnaly,we require that the following semi-norms (cid:107) . (cid:107) n,n = sup h> (cid:107) . (cid:107) H − n → H n , n ∈ N . are finite. We refer to the space of smoothing operators as Ψ −∞ . The semi-norms givea topology to that space. We let Ψ ρ = Ψ −∞ ∪ n Ψ nρ .A family { A h } h> is said to be asymptotically smoothing if for every n > , there isa h n > such that for every < h < h n , A h is uniformly bounded from H − n to H n .This space is also endowed with semi-norms (cid:107) . (cid:107) n,n,k = sup Corollary 1.14. The composition of a negligible (resp. asymptotically negligible) op-erator with a pseudor is still a negligible (resp. asymptotically negligible) operator. Notation 1. We denote by S ( R , α ) the class of symbols on R of order α , meaning that η ∈ S ( R , α ) when for all k ≥ , there is a constant C k > with η ( k ) ( u ) ≤ C k (cid:104) u (cid:105) α − k . Let K be the kernel of some operator A on C ∞ c ( Z ) . Let η ∈ S ( R , α ) . Then define A η to be the operator with kernel K η ( x, x (cid:48) ) = K ( x, x (cid:48) ) η (cid:18) y (cid:48) y − yy (cid:48) (cid:19) . ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 15 Proposition 1.15. Let η ∈ S ( R , α ) vanish near with α ≤ . Let σ ∈ S ρ . Then Op( σ ) η is negligible.Proof. We first give bounds on L . Recall that K wσ is the kernel of the operator in (6).Similarly to (9), we have: y − y (cid:48) ih y + y (cid:48) K wσ = K wX Y σ . (12)from this, we deduce that for all N ∈ N ,Op( σ ) η = ( ih/ N Op(( X Y ) N σ ) η N , where η N (cid:18) t − t (cid:19) = (cid:18) t − t (cid:19) N η (cid:18) t − t (cid:19) , so that η N ∈ S ( R , α ). For N big enough, X NY σ ∈ S − d − ρ . From lemma 1.9, we thusdetermine that for all N ≥ (cid:107) Op h ( σ ) η (cid:107) L → L = O ( h N (1 − ρ ) ) , where the constant depends on (cid:107) η N (cid:107) ∞ — which is finite since α ≤ h , the H − N → H N norm is bounded by(13) (cid:107) ( P + 1) N Op h ( σ ) η ( P + 1) N (cid:107) L → L . Observe that composition with X θ commutes with the A → A η operation. Further, X y (cid:20) η ( yy (cid:48) − y (cid:48) y ) (cid:21) = (cid:20) yy (cid:48) + y (cid:48) y (cid:21) η (cid:48) ( yy (cid:48) − y (cid:48) y ) = η ∗ (cid:18) t − t (cid:19) with η ∗ ∈ S ( R , α ) . Combining this with (12), we deduce that if we expand both ( P + 1) N ’s in (13), wewill get a finite sum of Op( σ ∗ ) η ∗ , with σ ∗ in S ρ , and η ∗ in S ( R , α ) still vanishing near0. We can apply the first part of our proof to conclude. (cid:3) Remark 1. Actually, if we take η ( u ) = ˜ η ( u/h ρ (cid:48) ) , and go through the above proof,we see that it works as long as ρ (cid:48) < − ρ . We deduce that the kernel of Op h ( σ ) isessentially supported at distance h − ρ of { y = y (cid:48) } . Stationary Phase. Now that we proved that off-diagonal terms in the kernel ofour pseudors give rise to negligible operators, it is legitimate to cutoff the kernels andkeep only the part supported near the diagonal. While proving composition formulae,or when changing quantizations, this will produce in the equations expressions of thetype σ ( x , ξ ) × σ ( x , ξ ) × χ ( x , x ) where χ ( x , x ) is a function of y /y supported near 1. This motivates the introductionof Definition 1.16. For (cid:15) > , let Ω k,(cid:15) be the subset of ( T ∗ Z ) k +1 : Ω k,(cid:15) = (cid:26) ( x , ξ ; x , ξ , . . . , x k , ξ k ) ∈ ( T ∗ Z ) k +1 | ∀ i, (cid:15) ≤ (cid:12)(cid:12)(cid:12)(cid:12) y i y (cid:12)(cid:12)(cid:12)(cid:12) ≤ /(cid:15) (cid:27) . Let σ be some smooth function ( T ∗ Z ) k +1 × [0 , h [ , supported in some Ω k,(cid:15) . We say that σ is a ( k, ρ ) -symbol if it is a symbol w.r.t the weights {(cid:104) y ξ (cid:105) β . . . (cid:104) y ξ k (cid:105) β k | ( β , . . . , β k ) ∈ R k +1 } and the vector fields X x,i = y ∂ x i and X ξ,i = 1 /y ∂ ξ i , losing a constant h − ρ whendifferentiating. By this we mean that there is β ∈ R k +1 such that whenever α is a finitesequence of indices α j ∈ { ( x, i ) , ( ξ, i ) | i = 0 . . . k } , if n i is the number of ( ξ, i ) in thesequence, (14) | X α σ | ≤ C α (cid:104) y ξ (cid:105) β − n . . . (cid:104) y ξ k (cid:105) β k − n k for some constant C α > .In particular, a (0 , ρ ) -symbol is just a symbol in S ρ . The semi-norms defined in (14) give a topology to the space of ( k, ρ ) -symbols. For σ a ( k, ρ )-symbol, we define the following function on T ∗ Z : T k σ : ( x, ξ, h ) (cid:55)→ (2 πh ) − k ( d +1) (cid:90) i =1 ...k e ih [ (cid:80) (cid:104) x i − x,ξ i − ξ (cid:105) ] σ ( x, ξ ; ( x , ξ ) , . . . , ( x k , ξ k )) dx i dξ i , where the integration has been taken over the universal cover T ∗ ( H d +1 ) k . Remark 2. T k σ is well defined. Indeed, if we first perform the integration in the ξ i variables, we obtain Fourier transforms of symbols. Those are distributions whosesingular support is reduced to { } and are decreasing faster than any power at infinity.Using compact support in y ,...,k — depending on y — we see that such distributionscan be integrated against . Recall the notation ∇ x . ∇ ξ = ∂ y ∂ Y + (cid:88) i =1 ...d ∂ θ i ∂ J i Proposition 1.17. Let σ be some ( k, ρ ) -symbol, of order β . Then T k σ is in S | β | ρ ( Z ) and we have the following expansion : T k σ ( x, ξ, h ) ∼ (cid:88) α ∈ N k ( ih ) | α | α ! (cid:34) k (cid:89) ( ∇ x i . ∇ ξ i ) α i (cid:35) σ ( x, ξ ; x , ξ , . . . , x k , ξ k ) | ( x i ,ξ i )=( x,ξ ) . In addition, T k is continuous from the space of ( k, ρ ) -symbols to S ρ . ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 17 Remark 3. All the terms in the expansion are in the right symbol class : if σ is a ( k, ρ ) -symbol of order β , the terms with coefficient h | α | are in finite number, and aresymbols in S | β |−| α | ρ .We will only use this proposition for k = 1 and k = 2 Proof. We prove the result by induction on k . First, if k = 0, this is obvious, since T σ = σ . Now, if we assume it is true for k , let σ be some ( k + 1 , ρ ) symbol. Then, wecan consider that σ is a ( k, ρ ) symbol in its k first coordinates, with the last coordinatesas parameters. Applying T k on those first coordinates, we obtain that T k σ is a (1 , ρ )symbol by the induction hypothesis (here, the continuity of T k σ is important). Then,we remark that T k +1 σ = T T k σ .Hence, proving the announced property for T is sufficient. Assume for now that : Lemma 1.18. If σ is a (1 , ρ ) symbol of order ( k , k ) with k < , for some constant | T σ ( x, ξ, h ) | ≤ C (cid:104) yξ (cid:105) k + k . Let σ be some (1 , ρ ) symbol of order ( k , k ). Changing variables ( v, V ) = ( x , ξ ) − ( x, ξ ) in T σ ,(15) T σ : ( x, ξ, h ) (cid:55)→ (2 πh ) − ( d +1) (cid:90) e ih (cid:104) v,V (cid:105) σ ( x, ξ ; ( x, ξ ) + ( v, V )) dvdV. Hence, the following identities hold : X y T σ = T ( y ∂ y σ ) + T ( y ∂ y σ ) X θ T σ = T ( y ∂ θ σ ) + T ( y ∂ θ σ ) where θ i is the θ coordinate for x i X Y T σ = T (cid:18) y ( ∂ Y σ + ∂ Y σ ) (cid:19) X J T σ = T (cid:18) y ( ∂ J σ + ∂ J σ ) (cid:19) . We deduce then from 1.18 that T σ is in S ρ with the correct order whenever k < V variable : σ ( x, ξ ; x + v, ξ + V ) = n (cid:88) s =0 s ! d sξ σ ( x, ξ ; ( x + v, ξ )) .V ⊗ s + (cid:90) (1 − t ) n n ! ( d n +1 ξ σ )( x, ξ ; x + v, ξ + tV ) .V ⊗ n +1 dt. Plugging this in the formula for T σ , integrating by parts in the v variable, we obtaina sum(16) n (cid:88) s =0 ( ih ) s s ! ( ∇ x . ∇ ξ ) s ( σ ( x, ξ ; x , ξ )) | ( x ,ξ )=( x,ξ ) and a remainder term(2 πh ) − ( d +1) ( ih ) n +1 (cid:90) (1 − t ) n n ! (cid:90) e ih (cid:104) v,V (cid:105) (cid:2) ( ∇ x . ∇ ξ ) n +1 σ (cid:3) ( x, ξ ; x + v, ξ + tV ) dvdV dt Actually, after rescaling V by a factor t , this remainder term is seen to be( ih ) n +1 (cid:90) (1 − t ) n n ! T σ ∗ ( x, ξ, th ) dt where σ ∗ is of order ( k , k − n − 1) and depends continuously on σ (we only tooka finite number of derivatives). If we take n ≥ k , we already know that T σ ∗ is asymbol, so that the remainder is O ( h n +1 ) is S nρ , with constants depending on a finitenumber of derivatives of σ . Together with remark 3, this is enough to conclude. (cid:3) Now, let us prove lemma 1.18. Proof. We rescale variable V in (15) to W = V /h , absorbing the h − d − constant. Let χ ∈ C ∞ c ( R d +1 ) equal 1 near 0, and break the integral into two parts with 1 = ( χ + (1 − χ ))( h ρ yW ). In the part with 1 − χ , we can also introduce 1 = ( h ρ yW ) N / ( h ρ yW ) N for some N big enough, and get (cid:90) e i (cid:104) v,W (cid:105) (cid:20) χ ( h ρ yW ) + (1 − χ ( h ρ yW )) ( h ρ yW ) N ( h ρ yW ) N (cid:21) σ ( x, ξ ; x + v, ξ + hW ) dvdW If we integrate the second term by parts in the v variable 2 N times, we get rid of the( h ρ yW ) N on top. We see that for both terms we obtain an expression of the form (cid:90) e i (cid:104) v,W (cid:105) ψ ( h ρ yW ) σ ∗ ( x, ξ ; x + v, ξ + hW ) dvdW where either ( σ ∗ , ψ ) = ( σ, χ ) or ( σ ∗ , ψ ) = ( h ρN y N ( ∂ y + ∂ θ ) N σ, (1 − χ ( x )) /x N ). Inboth cases, σ ∗ has the same properties as σ (including support, bounds, and order),and ψ is some symbol on R d +1 in the usual sense, of order − N . We apply the sametrick in the v variable now, introducing1 = χ ( h − ρ v/y ) + (1 − χ )( h − ρ v/y ) ( h − ρ v/y ) N ( h − ρ v/y ) N and integrating by parts in the W variable for the second term. When differentiating ψ , the powers of h compensate; when differentiating σ ∗ , we gain a positive power h − ρ .In the end, we get new expressions of the form (cid:90) e i (cid:104) v,W (cid:105) ψ ( h ρ yW ) ˜ ψ ( h − ρ v/y ) σ ∗ ( x, ξ ; x + v, ξ + hW ) dvdW where σ ∗ still has the same properties as σ , and ψ, ˜ ψ are some symbols on R d +1 inthe usual sense, of order − N . We can take the L norm of the integrand, and it is ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 19 bounded by : C (cid:90) R d +1) (cid:104) h ρ yW (cid:105) − N (cid:104) h − ρ v/y (cid:105) − N (cid:104) y ( ξ + hW ) (cid:105) k (cid:104) yξ (cid:105) k dvdW. rescaling both v and W , this is bounded by C (cid:104) yξ (cid:105) k (cid:90) R d +1) (cid:104) v (cid:105) − N (cid:104) W (cid:105) − N (cid:104) yξ + h − ρ W (cid:105) k dvdW ≤ C (cid:104) yξ (cid:105) k (cid:90) R d +1 (cid:104) W (cid:105) − N (cid:104) yξ + h − ρ W (cid:105) k dW for N > d .We break the integral into two parts : {| W | > ε | yξ |} and {| W | ≤ ε | yξ |} . Since k < (cid:104) yξ + hW (cid:105) k ≤ 1, and the first part is bounded by C (cid:104) yξ (cid:105) k (cid:90) | W | >ε | yξ | (cid:104) W (cid:105) − N = O ( (cid:104) yξ (cid:105) k + d − N +1 ) = O ( (cid:104) yξ (cid:105) k + k ) when N ≥ k + d + 1.The second part is bounded by C (cid:104) yξ (cid:105) k + k × (cid:90) R (cid:104) W (cid:105) − N = O ( (cid:104) yξ (cid:105) k + k ) . (cid:3) Symbolic calculus consequences. We start this section by proving that theclass of operator Ψ ρ is stable by composition. Proposition 1.19. Let a ∈ S mρ ( Z ) and b ∈ S nρ ( Z ) . Then, there is a symbol c ∈ S m + nρ ( Z ) and a negligible family of operators R h ∈ O ( h ∞ )Ψ −∞ such that Op( a ) Op( b ) = Op( c ) + R h where (17) c ( x, ξ ) ∼ (cid:88) α ∈ N ( − α ( ih ) | α | | α | α ! ( ∇ x . ∇ ξ ) α ( ∇ x . ∇ ξ ) α a ( x , ξ ) b ( x , ξ ) | x = x = x,ξ = ξ = ξ Proof. First, we choose a truncation η ∈ C ∞ c ( R ) equal to 1 around the origin. Then1 − η is a symbol in S ( R , 0) vanishing around 0, so we can apply proposition 1.15, andreplace Op( a ) and Op( b ) by respectively Op( a ) η and Op( b ) η . Recalling corollary 1.14,there exists R h ∈ O ( h ∞ )Ψ −∞ such thatOp( a ) Op( b ) = Op( a ) η Op( b ) η + R h . If K wσ is the kernel of Op wh ( σ ) on H d +1 as in (6), we have σ ( x, ξ ) = (cid:90) e ih (cid:104) u,ξ (cid:105) K wσ (cid:16) x − u , x + u (cid:17) du. Since both Op wh ( a ) η and Op wh ( b ) η act on H d +1 , the product also, and its kernel on H d +1 is K w ( x, ˜ x ) = (cid:90) K wa ( x, x (cid:48) ) K wb ( x (cid:48) , ˜ x ) η (cid:18) y x (cid:48) y x − y x y x (cid:48) (cid:19) η (cid:18) y ˜ x y x (cid:48) − y x (cid:48) y ˜ x (cid:19) dx (cid:48) Hence, the solution to our problem is (formally) the function c defined by c ( x, ξ ) = h − d − (cid:90) e ih φ σ ( u, x (cid:48) , ξ , ξ ) χ (cid:18) y + y u / y (cid:48) , y − y u / y (cid:48) (cid:19) dudξ dx (cid:48) dξ integrating over ( T ∗ R d +1 ) , where φ = (cid:104) u, ξ (cid:105) + (cid:104) x − u/ − x (cid:48) , ξ (cid:105) + (cid:104) x (cid:48) − x − u/ , ξ (cid:105) σ ( u, x (cid:48) , ξ , ξ ) = a (cid:18) x − u/ x (cid:48) , ξ (cid:19) b (cid:18) x + u/ x (cid:48) , ξ (cid:19) and χ is a smooth function on R supported in a rectangle { ( τ, κ ) ∈ R | < (cid:15) ≤ τ ≤ /(cid:15) < (cid:15) ≤ κ ≤ /(cid:15) } After a change of variables, we will be able to use our stationary phase lemma. Let x = 12 ( x + u/ x (cid:48) ) x = 12 ( x − u/ x (cid:48) )we get to write c in the suitable fashion c ( x, ξ ) = (cid:18) h (cid:19) − d − (cid:90) e ih ( (cid:104) x − x ,ξ − ξ (cid:105) + (cid:104) x − x ,ξ − ξ (cid:105) ) σ χ = T ( σ χ )( x, ξ, h/ . where σ = a ( x , ξ ) b ( x , ξ ) χ = χ (cid:18) y − y + yy + y − y , y − y + yy + y − y (cid:19) . An elementary computation shows that χ is supported in some { (cid:15) (cid:48) y ≤ y , ≤ y/(cid:15) (cid:48) } ; hence, it is a smooth function of y , /y and it will have a good behavior w.r.tvector fields y∂ y . Function σ χ is supported in Ω ,(cid:15) (cid:48) . In addition, since the weights( (cid:104) y i ξ j (cid:105) ) i =1 , are equivalent to (cid:104) y ξ j (cid:105) in Ω ,(cid:15) , σ satisfies the desired estimates in thatregion, and σ χ is a (2 , ρ )-symbol. From proposition 1.17, we conclude directly that c is in S α + βρ ( Z ). (cid:3) Proposition 1.20. The adjoint of Op h ( a ) for the L inner product is Op h ( a ) , so thatreal symbols yield self-adjoint operators, which is a key feature of the Weyl quantization. ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 21 Proof. Taking L : f ∈ L ( Z ) → y − ( d +1) / f ∈ L ( dydθ ), we see thatOp( a ) = L ∗ Op wh ( a ) L . Since the usual Weyl quantization on R n has the property announced for Op, we deducethe first part of the proposition: Op h ( a ) ∗ = Op h ( a ).Now, we use proposition 8.5 in appendix A in Taylor [Tay11]. It suffices to provethat when a is real, Op h ( a ) ± i is surjective. Since a is real, a ± i never vanishes, andwe can find a symbol σ ± N such thatOp( a ± i ) Op( σ ± N ) = 1 + O ( h N Ψ − N ) . for h small enough, the operator on the LHS is invertible. In particular it is surjective,and so is Op h ( a ± i ). (cid:3) Proposition 1.21. Let a and b be in S ρ ( Z ) . Then, with R h ∈ O ( h ∞ )Ψ −∞ , [Op( a ) , Op( b )] = Op( c ) + R h where c is a semi-classical symbol with an asymptotic expansion with only odd powersof h , such that : c ( x, ξ ) = hi { a, b } + O ( h − ρ ) S n + m − ρ ) where { ., . } is the Poisson bracket defined with the symplectic form dξ ∧ dx : { f, g } = ∇ ξ f. ∇ x g − ∇ x f. ∇ ξ g. Proof. Since in the asymptotic expansion (17) the terms in odd powers of h are sym-metric in a and b , this other key feature of Weyl quantization is now trivial. (cid:3) Proposition 1.22. Let a ∈ S ρ ( Z ) . Then Op( a ) is bounded on L , with norm (cid:107) a (cid:107) ∞ + o h → (1) .Proof. We have all the ingredients to make the classical proof work. Consider (cid:107) Op h ( a ) (cid:107) L = (cid:107) Op h ( a ) ∗ Op h ( a ) (cid:107) L Op h ( a ) ∗ Op h ( a ) = Op h ( | a | ) + O ( hS n − ρ ) for a ∈ S nρ ( Z )When a has negative order, | a | has a more negative order. Since Ψ − d − operatorsare bounded with norm O ( h − ρd − ), one can prove by induction that for any (cid:15) , (cid:15) (cid:48) ,Ψ − (cid:15) operators are bounded on L whenever (cid:15) > 0, with norm O ( h − (cid:15) (cid:48) ). Now, take a ∈ S ρ ( Z ). Let M = (cid:107) a (cid:107) ∞ . We just have to prove that M − Op h ( a ) ∗ Op h ( a ) + o (1) isa positive operator. Take κ > 0. Consider M + κ − Op h ( a ) ∗ Op h ( a ) = Op h ( M + κ − | a | ) + O ( h − ρ S − ρ ) . But M + κ + | a | > κ so b = (cid:112) M + κ + | a | is in S ρ ( Z ) and real, so that Op( b ) isself-adjoint, and M + κ + Op h ( a ) ∗ Op h ( a ) = Op h ( b ) + O ( h − ρ S − ρ ) ≥ − C.h − ρ − (cid:15) (cid:48) for any (cid:15) (cid:48) given . We deduce that M − Op h ( a ) ∗ Op h ( a ) ≥ − Co (1). (cid:3) Proposition 1.23. Let f ∈ S ( R , n ) . With P = − h ∆ / , we define f ( P ) by thespectral theorem. Recall that p is the symbol of P . Then there is a symbol σ such that σ = f ◦ p + O ( hS n − ) and f ( P ) = Op( σ ) + R where R is asymptotically negligible and commutes with ∂ θ .Proof. If f has positive order n , consider f ( P ) = ( P + i ) n +1 f ( x + i ) n +1 ( P ) . Since ( P + i ) n +1 is a pseudor, we only need to consider cases when f has negative order.Following the method in lemma 1.11, we get symbols q N ( z ) and r N ( z ) such that( P + z ) Op( q N ( z )) = + Op( h N r N ( z )) . What is more, the symbol norms of q N and r N are bounded by a power |(cid:61) z | − L N with L N → ∞ when N → ∞ . This establishes that ( P + z ) − is a pseudor up to anasymptotically negligible remainder, for fixed z . Now, using a quasi-analytic extensionof f as in p.358 in [Zwo12] (theorem 14.8 therein), and the bounds on q N and r N , wesee that the same can be said about f ( P ).To conclude, observe that Op( σ ) commutes with ∂ θ whenever σ does not depend on θ , which is the case for q N and r N . (cid:3) To prove a trace formula, it is convenient to be able to change quantizations. Lemma 1.24. On R d +1 , we can define a family of quantization as usual by Op th ( σ ) f ( x ) = 1(2 πh ) d +1 (cid:90) e i (cid:104) x − x (cid:48) ,ξ (cid:105) /h σ ( tx + (1 − t ) x (cid:48) , ξ ) f ( x (cid:48) ) dξdx (cid:48) and then define Op th ( σ ) := L ∗ Op th ( σ ) L , for σ ∈ S ρ ( Z ) — so that Op / = Op .If a ∈ S nρ , there is a family a t of symbols so that for all t ∈ [0 , , Op t ( a t ) = Op( a ) + O ( h ∞ Ψ −∞ ) . What is more a t = a + O ( h − ρ S n − ρ ) . ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 23 Proof. Following the scheme of proof of the composition lemma, we truncate the kernelof Op / around y = y (cid:48) with a η compactly supported, and we want to solveOp t ( a t ) η = Op( a ) η . If K t is the kernel of Op th ( a t ), we have a t ( x, ξ ) = (cid:90) e i (cid:104) u,ξ (cid:105) /h K t ( x + ( t − u, x + tu ) du. so it is legitimate to consider b t ( x, ξ ) := (2 πh ) − d − (cid:90) e i (cid:104) u,ξ − ξ (cid:48) (cid:105) /h a ( x + ( t − / u, ξ (cid:48) ) χ (cid:18) y + ty u y + ( t − y u (cid:19) dudξ (cid:48) = T ( σ t )( x, ξ, h )with σ t ( x, ξ ; x , ξ ) = a ((1 / t ) x + (1 / − t ) x , ξ ) χ (cid:18) y (1 + t ) − ty ty + y (1 − t ) (cid:19) and χ ( x ) = η ( x − /x ) . One can check that σ t is a (1 , ρ )-symbol. We deduce then from proposition 1.17 thatOp( a ) = Op t ( b t ) + O ( h ∞ Ψ −∞ ) . and b t = a + O ( h − ρ S n − ρ ) ∈ S nρ . (cid:3) Before we turn to a trace formula, observe that when one imposes Dirichlet condi-tions at y = y and considers the Laplacian on L ( Z, { y > y } ), one finds that it hascontinuous spectrum [ d / , + ∞ ). We cannot expect our operators to be trace class, ifthey are not even compact. This is why we introduce the following.Let Π ∗ is the orthogonal projection (in L ( Z )) on the non-zero Fourier modes in the θ direction. Also let Λ (cid:48) be the dual lattice to Λ and Λ (cid:48)∗ = Λ (cid:48) \ { } . Lemma 1.25. Let (cid:15) > and κ > . Let χ ∈ C ∞ b ( Z ) be supported in { y > κ } . When a ∈ S − ( d +1) / − (cid:15)ρ is supported in { y > κ } , both Op ( a )Π ∗ and Π ∗ Op ( a ) are Hilbert-Schmidt. As a consequence, for any A ∈ Ψ − ( d +1) / − (cid:15)ρ , both Π ∗ χAχ and χAχ Π ∗ areHilbert-Schmidt — this is also true if A is only asymptotically negligible.Proof. The Hilbert-Schmidt (HS) norm (on L ( Z )) of an operator A with kernel K w.r.t to the Lebesgue measure on the cylinder is (cid:107) A (cid:107) HS = (cid:90) Z × Z | K ( x, x (cid:48) ) | (cid:18) y (cid:48) y (cid:19) d +1 dydθdy (cid:48) dθ (cid:48) . Recall the Poisson formula (the covolume of Λ is 1) (cid:88) (cid:36) ∈ Λ e i (cid:104) (cid:36),W (cid:105) = (cid:88) W i ∈ Λ (cid:48) δ W i ( W ) . Using (8), we deduce that the kernel of Op ( a ) is K a ( x, x (cid:48) ) = (2 π ) − d − (cid:18) yy (cid:48) (cid:19) d +12 (cid:88) J ∈ Λ (cid:48) (cid:90) e i (cid:104) x − x (cid:48) ,ξ (cid:105) a ( x, hξ ) dY, where ξ = ( Y, J ) . Seing this as a Fourier transform in the x (cid:48) variabla, by Parseval : (cid:107) Op ( a )Π ∗ (cid:107) HS = 1(2 π ) d (cid:88) J ∈ Λ (cid:48)∗ (cid:90) Z × R | a ( y, θ, hY, hJ ) | dydθdY Since a ∈ S − ( d +1) / − (cid:15)ρ is supported in { y ≥ κ } , this is less than C (cid:88) J ∈ Λ (cid:48)∗ (cid:90) ∞ (cid:90) R (cid:104) h y ( Y + J ) (cid:105) − d − − (cid:15) dY dy ≤ C N (cid:88) J ∈ Λ (cid:48)∗ (cid:90) ∞ κ dy hy (1 + ( hy | J | ) ) d/ (cid:15) . After some change of variables, this is seen to be finite for fixed h . Observe that ifwe did not remove J = 0, it would not be the case; there would be a logarithmicdivergence. Hence Op ( a )Π ∗ is HS. Taking adjoints, we see that that Π ∗ Op ( a ) is alsoHS.Then, we write for some N > P + 1) − ( d +1) / − (cid:15) = (1 + Op ( h N r N )) − Op ( q N ) = Op ( q N )(1 + Op ( h N r N )) − We deduce that bothΠ ∗ χ ( P + 1) − ( d +1) / − (cid:15) Π ∗ ( P + 1) − ( d +1) / − (cid:15) χ are HS.Take R asymptotically negligible, supported in { y > κ } . Then for any N , ( P + 1) N R is bounded on L , so that we can write Π ∗ R = Π ∗ ˜ χ ( P + 1) − N ˜ R where ˜ χ is supportedin some { y > κ (cid:48) > } , equal to 1 on the support of R , and ˜ R is a bounded operatorfor h small enough. We deduce that Π ∗ R is HS for h small enough, and similarly for R Π ∗ .Now, if A ∈ Ψ − ( d +1) / − (cid:15)ρ , it writes as A = Op ( a ) + O ( h ∞ Ψ −∞ ), and χAχ Π ∗ = Op ( χa ) χ Π ∗ + χ O ( h ∞ Ψ −∞ ) χ Π ∗ so that χAχ Π ∗ is HS. (cid:3) Proposition 1.26. Take any (cid:15) > . Let A ∈ Ψ − d − − (cid:15)ρ with principal symbol a . Then χAχ Π ∗ is trace class, and T r [ χAχ Π ∗ ] = 1 h d +1 (cid:20)(cid:90) T ∗ Z χ ( x ) a ( x, hξ ) + O ( h + h d | log h | ) (cid:21) . ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 25 If A = Op ( a ) , then the remainder is improved to O ( h + h d | log h | ) . Observe that in the case of surfaces, the remainder is not as good as for compactmanifolds. Proof. First, if R ∈ O ( h ∞ )Ψ −∞ is supported in { y > κ } , then R Π ∗ and Π ∗ R are traceclass since, for example, we can writeΠ ∗ R = Π ∗ χ ( P + 1) − N Π ∗ ( P + 1) N R and this is the product of two HS operators.Now, if A ∈ Ψ − d − − (cid:15)ρ , we can write A = Op (˜ a ) + O (Ψ −∞ )with ˜ a = a + O ( hS − d − − (cid:15)ρ ). Observe χ Op (˜ a ) χ Π ∗ = (cid:2) χ Op (˜ a ) √ χ ( P + 1) ( d +1) / (cid:15)/ Π ∗ (cid:3) (cid:2) ( P + 1) − ( d +1) / − (cid:15)/ Π ∗ √ χ (cid:3) In the RHS, we have shown that the second term of the product is HS. Using propo-sition 1.23, we can write the first term as Op b ) + R where b ∈ S − ( d +1) / − (cid:15)/ ρ and R is asymptotically negligible. It is thus also HS, and the product is trace class.Writing the trace as the integral of the kernel along the diagonal, we obtain T rχAχ Π ∗ = 1(2 π ) d +1 (cid:88) J ∈ Λ (cid:48)∗ (cid:90) Z × R χ ˜ a ( x, hξ ) dxdY. To end the proof, we use: Lemma 1.27. If a ∈ S − d − − (cid:15)ρ is supported in some { y > κ > } , (cid:88) J ∈ Λ (cid:48)∗ (cid:90) Z × R a ( x, hξ ) dxdY = 1 h d +1 (cid:20)(cid:90) T ∗ Z a + O ( h + h d | log h | ) (cid:21) . and both sums converge absolutely. (cid:3) The proof of lemma 1.27: Proof. Let D (cid:48) be fundamental domain for the action of Λ (cid:48) on R d . Assume D (cid:48) to besymmetric around 0, and of bounded diameter. Its volume is 1. Then, for (cid:36) ∈ Λ (cid:48)∗ , (cid:12)(cid:12)(cid:12)(cid:12) f ( h(cid:36) ) − h d (cid:90) h(cid:36) + hD (cid:48) f ( J ) dJ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C h h d (cid:90) h(cid:36) + hD (cid:48) (cid:107) d J f ( J ) (cid:107) . hence the difference between the two main terms in lemma 1.27 is bounded up to someconstant by h − d − (cid:90) ( x,ξ ) ∈ T ∗ Z,J ∈ hD (cid:48) | a ( x, ξ ) | + h − d (cid:90) ( x,ξ ) ∈ T ∗ Z,J / ∈ hD (cid:48) (cid:107) d J a (cid:107) . For the first term, first integrate variable θ (losing a constant vol( D )) and thenvariable Y after rescaling. Obtain a bound by h − d − (cid:90) J ∈ hD (cid:48) ,y>κ y (1 + y J ) − d − (cid:15) Rescaling the y variable, this is bounded by (note the use of polar coordinates in J ), h − d − (cid:90) Ch r d − dr (cid:90) + ∞ κr dyy (1 + y ) d + (cid:15) . This is O ( h − | log h | ). Likewise for the second term, it is bounded by : h − d (cid:90) ∞ Ch r d − dr r (cid:90) + ∞ κr ydy (1 + y ) d + (cid:15) This is O ( h − d ) if d > d = 1, O ( h − ), and when d = 2, O ( h − | log h | ). (cid:3) Applications Now we will present some applications of the cusp-quantization.2.1. Cusp manifolds. Quantization. As we said in the introduction, cusp manifolds are described asa compact manifold with boundary to which is glued a finite number of cusps. Here,we give a formal definition that will simplify the construction of the quantization : Definition 2.1. Let ( M, g ) be a complete ( d + 1) -dimensional riemannian manifold. M is said to be a cusp manifold if it is endowed with a cusp atlas F , that is • a finite collection ( U i , U (cid:48) i , γ i ) i of R d +1 -charts, that is, diffeomorphisms γ i : U i ⊂ M → U (cid:48) i ⊂ R n , with U i relatively compact. • a finite collection ( Z j , Z (cid:48) j , γ cj ) j of cusp-charts, that is, diffeomorphisms γ cj : Z j ⊂ M → Z (cid:48) j ⊂ Z Λ j such that γ cj is an isometry, and Z (cid:48) j is of the form { y > a j } .We require that • No two X j ’s intersect. • The coordinate changes between two ( U (cid:48) i ) ’s or Z j and U i are diffeomorphisms. • The lattices Λ i have covolume . This is a convention, and there is only onechoice of height function y i that is coherent with that choice. ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 27 Definition 2.2. In R d +1 , we define the Kohn-Nirenberg symbols of order n , in theusual way, as in [Zwo12, p.207] : σ ∈ S nρ ( R d +1 ) whenever for all k, k (cid:48) ≥ there is aconstant C k,k (cid:48) > , | d k (cid:48) x d kξ σ | ≤ C k,k (cid:48) (cid:104) ξ (cid:105) n − k , for x, ξ ∈ R d +1 The class S nρ ( M ) of hyperbolic symbols of order ( n, ρ ) is composed of the functions σ on T ∗ M such that for any chart ( U, V ⊂ N, γ ) in the atlas, the function σ U ( x, ξ ) := γ ∗ σ (cid:2) = σ ( γ − ( x ) , dγ ( γ − ( x )) ∗ ξ ) (cid:3) is the restriction to T ∗ V of some element of S nρ ( N ) (with N = R d +1 or N = Z Λ i ).The invariance by coordinate changes of the Kohn-Nirenberg class [Zwo12, theorem9.4, p.207] implies that this is well defined — it does not depend on the choice of theatlas. To define a quantization on cusp manifolds that enjoys all usual properties, we followthe procedure in p. 347 through to p. 352 in [Zwo12]. A pseudo-differential operatoron M is defined as an operator C ∞ c ( M ) → C ∞ ( M ) such that restricted to any chart,it is pseudo-differential — in the case of a cusp-chart, this means that it is in someΨ ρ ( Z Λ i ). We also require that they are pseudo-local — that is, when we truncate theirkernel at a fixed distance of the diagonal, we obtain negligible operators.Lemma 1.8 proves that it suffices to check the above properties for the finite set ofcharts of some cusp-atlas. Lemmas 1.8 and 1.15 ensure that the class of pseudors is notreduced to compactly supported operators, because pull backs of elements of Ψ( Z Λ i ) are pseudo-local.We can define the semi-classical principal symbol σ ( A ) of a pseudor A as for pseu-dors on compact manifolds — once again thanks to lemma 1.8 — and according tothe definition, σ ( A ) is in some S nρ ( M ). The class of A such that σ ( A ) ∈ S nρ ( M ) isdenoted Ψ nρ . We let Ψ −∞ ρ ( M ) be the class of smoothing operators in the Sobolev sense— as in definition 1.13. Let Ψ ρ ( M ) = ∪ n ≥−∞ Ψ nρ . When we omit the ρ , we refer to thecase ρ = 0.Using charts, our quantization Op in Z Λ i and the usual Weyl quantization on R d +1 ,we are able to build a quantization procedure Op on M , that is a section to thesymbol map. Using classical results, and the first part of the article, we see that Ψ ρ gives bounded operators on L ( M ), whose norm is the L ∞ norm of the symbol, up toa o (1) term as h → a bigger than all the a j , we define Π ∗ a as the projection on non zeroFourier modes in { y > a } : Π ∗ a f := f − ( y > a ) (cid:90) f dθ. The following hold : Proposition 2.3. Let A ∈ Ψ − ρ ( M ) . Then Π ∗ a A is compact on L .Let A ∈ Ψ − nρ ( M ) with n > d + 1 . Then Π ∗ A Π ∗ is Hilbert-Schmidt.Let A ∈ Ψ − nρ ( M ) with n > d + 1 . Then Π ∗ A Π ∗ is trace class, and T r Π ∗ a A Π ∗ a = 1 h d +1 (cid:20)(cid:90) T ∗ M σ ( A ) + O ( h + h d log h ) (cid:21) Proof. In the compact part of M , these are classical results — see theorem 4.28 p. 89,and remark (C.3.6) p. 412 in [Zwo12]; see also proposition 9.2 and theorem 9.5 p.112and following in [DS99]. So we only need to prove this when A is only supported inthe cusps, and for negligible operators. For the Hilbert-Schmidt and the trace-classproperty, this are the contents of lemma 1.25 and proposition 1.26 when A is supportedonly in the cusps.As to negligible operators, they can always be written down as the product of anothernegligible operator with some power of ( P + 1) − , and the arguments we used in theproof of lemma 1.25 and proposition 1.26 will carry on, so that what we really need toprove is that Π ∗ a A is compact on L ( M ) when A ∈ Ψ − ρ .But that is a consequence of the fact that Π ∗ a H ( M ) is compactly injected in L ( M ).Once again, as this is always true for compact manifolds with boundary (Rellich’stheorem), it suffices to prove it for for the cusps. More precisely, we need to showthat { y>a f | f ∈ H ( Z Λ ) , Π ∗ f = f } is compactly injected in L ( Z Λ ). We recall theproof from [LP76] — see pp. 206 and following. Consider the fact that f ∈ H ( Z Λ ) (cid:55)→ a In this section, we give an Ehrenfest timeEgorov lemma, which was the original motivation for what we have done so far. TheLevi-Civita connexion on M is associated to a splitting of T T ∗ M = V ⊕ H . V and H are subbundles that can be identified with respectively T ∗ M and T M . The onlymetric on T ∗ M that renders V orthogonal to H and that makes those identificationsisometries is called the Sasaki metric. It is in some sense the natural metric to useon T ∗ M for our problem; we recall a few facts on it in appendix C. Now the wehave specified a riemannian metric on T ∗ M , we can define the spaces C k ( T ∗ M ) as inappendix C. Let us introduce a particular class of symbols : Definition 2.4. Let U be some open set of R . For E > , let S EC denote the classof functions σ on U × T ∗ M that are C ∞ ( T ∗ M ) in the second variable, supported in ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 29 ( T ∗ M ) E := { p ≤ E } . Additionally require there are constants C k > such that (cid:107) σ ( h, τ ; . ) (cid:107) C k ( T ∗ M ) ≤ C k e Ck | τ | where ( h, τ ) are the coordinates in R . From proposition C.2, elements of S EC are symbols in S −∞ for fixed t . Additionally,if the open set U is { C | τ | ≤ ρ | log h |} with ρ < / 2, elements of S EC are symbols in S −∞ ρ , and can be quantized. We will assume that U takes this form in the rest of thearticle.Let us point out that when A is in some Ψ n ( M ), and σ ∈ S EC , up to a negligibleoperator R , [ A, Op( σ )] − hi Op( { σ ( A ) , σ } ) = Op(˜ σ ) + R where ˜ σ is O (( he C | τ | ) ) in S EC .Let us introduce Definition 2.5. The maximal Lyapunov exponent of the geodesic flow on ( T ∗ M ) E isdefined as λ max ( E ) := sup ξ ∈ ( T ∗ M ) E lim sup t →∞ | t | log (cid:107) d ξ ϕ t (cid:107) . Using Jacobi fields and Rauch’s comparison theorem — see 1.28 in section 1.10 of[CE75] — one can prove that λ max ( E ) is bounded by Eκ where − κ is the minimum ofthe curvature of M . Observe that proposition B.1 implies that for any λ > λ max ( E ),and any f ∈ C ∞ ( T ∗ M ) supported in ( T ∗ M ) E , f ◦ ϕ t is in S Eλ .Recall that the Schr¨odinger propagator is U ( t ) = e − itP/h We have Theorem 3. Let σ ∈ C ∞ ( T ∗ M ) be supported in ( T ∗ M ) E . Then, for any ρ < / andany λ > λ max ( E ) , there exists a symbol ˜ σ ρ that is in S Eλ , with U = {| τ | ≤ ρ | log h | /λ } .On U , ˜ σ ( t, x, ξ ) = σ ( ϕ t ( x, ξ )) + O ( h | t | e λ | t | ) , and U ( − t ) Op( σ ) U ( t ) = Op(˜ σ ) + O (( | t | he λ | t | ) ∞ ) where the remainder is asymptotically smoothing. Since Beal’s theorem — see theorem 8.3 in [Zwo12] — is not available to us, we canonly prove that the remainder is asymptotically smoothing. Proof. Let us assume that we found an exact solution ˜ σ . Then, we would have :Op(˜ σ ) = e itP/h Op( σ ) e − itP/h i.e Op( σ ) = e − itP/h Op(˜ σ ) e itP/h . Differentiating with t , 0 = e − itP/h (cid:20) Op( ∂ t ˜ σ ) − ih [ P, Op(˜ σ )] (cid:21) e itP/h All along our development, we will follow the proof in [Zwo12] closely. Let us buildby induction a family of operators B n ( t ) = Op( b n ) , E n ( t ) = Op( e n )where b n and e n are in S Eλ , satisfying : hi ∂ t B n = [ P, B n ] + E n + R n .B n (0) = Op( σ )the remainder R n being negligible, and with the estimates : b n − b n − = O S Eλ (( | t | h ) n e nλ | t | ) for n > e n = O S Eλ ( h n | t | n e (2 n +2) λ | t | ) . For n = 1, define B = Op( σ ◦ ϕ t ) . This is O (1) in S Eλ . Then hi ∂ t B = hi Op( { p, σ ◦ ϕ t } )= [ P, B ] + E + R , where R is negligible and E = Op( e ). From the product formula, we get that e isstill supported in ( T ∗ M ) E , and it is O ( h e λ | t | ) in S Eλ .Assume that all the assumptions hold for some n ≥ 0, and let c n +1 = ih (cid:90) t e n ( s ) ◦ ϕ t − s ds.c n +1 is in S Eλ , and it is O (( | t | h ) n +1 e (2 n +2) λ | t | ). One gets hi ∂ t Op( c n +1 ) = hi Op( { p, c n +1 } + ih e n )= [ P, Op( c n +1 )] + E n − E n +1 + R n +1 ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 31 where E n +1 = Op( e n +1 ) with e n +1 = O S Eλ ( h n | t | n +1 e (2 n +4) λ | t | )At last, define B n +1 = B n − C n +1 Such b n ’s and e n ’s satisfy the announced properties. Now, since hi ∂ t [ e − itP/h B n e itP/h ] = e − itP/h ( E n + R n ) e itP/h Duhamel’s formula gives : B n ( t ) = e itP/h Op( σ ) e − itP/h − ih (cid:90) t U ( s − t )[ E n ( s ) + R n ( s )] U ( t − s ) ds. Since the operators U ( t ) are bounded from H s to H s for any s , we deduce that this is O (( | t | he λ | t | ) n +1 h − N ) in H − N → H N operator norm. We can find a symbol ˜ σ ∈ S Eλ such that ˜ σ ∼ b − ∞ (cid:88) c n = σ ◦ ϕ t + O S Eλ ( h | t | e λ | t | ) . Then, ˜ σ satisfies the condition of the theorem. (cid:3) Remark 4. Following the support of the b n ’s, e n ’s, we find that ˜ σ is exactly supportedin ϕ t (supp( σ )) . Actually, the whole operator is microsupported on that set ; if wemultiply our conjugated operator by some Op( η ) such that η vanishes on supp ˜ σ , weobtain a negligible operator (not only asymptotically). Extending a result of Semyon Dyatlov. Spectral theory and Eisenstein functions. The following facts on the spectraltheory of the Laplacian on cusp-manifolds are contained in [M¨ul83]. However, in thatarticle, M¨uller considered cusps where the horizontal slices were arbitrary compact d -dimensional manifolds instead of tori, so that his definition of Riemannian manifoldswith cusps is more general than our cusp-manifolds. However, he also wrote an articlein the case of surfaces [] with the same definition of cusp, which is a good place tostart if one wants to learn about cusp surfaces.The non-negative Laplacian − ∆ acting on C ∞ ( M ) functions has a unique self-adjointextension to L ( M ) and its spectrum consists of(1) Absolutely continuous spectrum σ ac = [ d / , + ∞ ) with multiplicity k (thenumber of cusps). (2) Discrete spectrum σ d = { λ = 0 < λ ≤ · · · ≤ λ i ≤ . . . } , possibly finite,and which may contain eigenvalues embedded in the continuous spectrum. To λ ∈ σ d , we associate a family of orthogonal eigenfunctions that generate itseigenspace ( u iλ ) i =1 ...d λ ∈ L ( M ) ∩ C ∞ ( M ).The generalized eigenfunctions associated to the absolutely continuous spectrum arethe Eisenstein functions, ( E j ( x, s )) i =1 ...k . Each E j is a meromorphic family (in s ) ofsmooth functions on M . Its poles are contained in the open half-plane {(cid:60) s < d/ } orin ( d/ , g E j ( ., s ) = s ( d − s ) E j ( ., s )(2) In the cusp Z i , i = 1 . . . k , the zeroth Fourier coefficient of E j in the θ variableequals δ ij y si + φ ij ( s ) y − si where y i denotes the y coordinate in the cusp Z i and φ ij ( s ) is a meromorphic function of s .Let us recall the construction of the Eisenstein functions. On M we define a function y M that corresponds to y i on X i ∩ { y i ≥ a } , and equals 1 on M . Let χ be a smoothmonotonous function that equals 1 on [3 a, + ∞ [, and vanishes on ] − ∞ , a ]. We let χ i be the function supported in cusp Z i , where it is χ ◦ y i . Now, let ˜ χ ∈ C ∞ c ( R , [0 , χ ≡ − ∞ , ln 4] and ˜ χ ≡ , + ∞ [. For s ∈ R + , let(18) χ s := ˜ χ (cid:16) ln (cid:16) y M a (cid:17) − s (cid:17) . Take E ( s, x ) = y sM . Then consider E i ( s, x ) := χ i E ( s, x ) + ( − ∆ − s ( d − s )) − [∆ , χ i ] E ( s, x ) . Since χ (cid:48) is compactly supported, [∆ , χ i ] E ( s, . ) is compactly supported and in L , sothis is well defined. One can check that( − ∆ − s ( d − s )) E i = − [∆ , χ i ] E + [∆ , χ i ] E = 0 . to see that the E i ’s satisfy the announced properties. Uniqueness is then straightfor-ward. In what follows, we will use the notations : s = d/ i/h + η ( h ) W = h s ( d − s ) = h (cid:20) d h − η − i ηh (cid:21) = 12 (cid:20) − iηh + h (cid:18) d − η (cid:19)(cid:21) . Let us define the measures µ i,η announced in the introduction. For f ∈ C c ( T ∗ M )compactly supported, let µ ± i,η ( f ) := 2 ηa η (cid:90) R × T Λ e − ηt f ◦ ϕ −± t ( a, θ, ± /a, dtdθ ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 33 This defines two Radon measures. We also recall the definition of the Wigner distri-butions (cid:104) µ hi,j ( s ) , σ (cid:105) := (cid:104) Op( σ ) E i ( s ) , E j ( s ) (cid:105) for σ ∈ C ∞ c ( T ∗ M ).We will prove the following theorem Theorem 4. Consider s h = 1 / ± i/h + η ( h ) . All the limits are taken when h → . (1) If η ( h ) → ν > , then ηµ hi,j ( s h ) (cid:42) δ i,j πµ ± i,ν in C ∞ c ( T ∗ M ) (cid:48) . (2) Assume that M has negative curvature. Whenever η → with lim inf η | log h | log | log h | > λ max (cid:18) (cid:19) , then ηµ hi,j ( s h ) (cid:42) δ i,j π L . The case when η → ν > M .It suffices to consider the case (cid:61) s → + ∞ , the other can be deduced thereof.2.2.2. Reduction to a lagrangian expression. We fix an exponent λ > λ max (1 / p of the geodesic flow is 2-homogeneous, ϕ t ( κξ ) = κϕ κt ( ξ ). Consider Φ κ : T ∗ M → T ∗ M the multiplication by κ . Then dϕ t = d Φ κ ◦ dϕ κt ◦ d Φ − κ . If κ ≥ 1, we have (cid:107) d Φ κ (cid:107) = κ and (cid:107) d Φ − κ (cid:107) = 1 (by inspecting the behaviour of d Φ κ onthe vertical and horizontal bundles of T ∗ M ). We deduce that λ max ( κE ) = κλ max ( E ) . It follows that for any (cid:15) > λ > λ max ( E = 1 / (cid:15) ).Let us take T > σ is supported in { y M ≤ ae T } . We aim to replace E i ( s )on the support of σ by a propagated incoming wave. That is why we define :˜ E i ( s, t ) = χ T − ln 3 e ih t ( P − W ) χ T + t χ i E ( s )˜ E i ( s, t ) = χ T − ln 3 e ih t ( P − W ) χ T + t E i ( s ) and prove : Lemma 2.6. When η remains bounded, (cid:107) χ T − ln 3 E i ( s ) − ˜ E i ( s, t ) (cid:107) L = O (cid:18) e − ηt η (cid:19) + O (( | t | he λ | t | ) ∞ ) . Proof. We write χ T − ln 3 E i − ˜ E i = ( χ T − ln 3 E i − ˜ E i ) + ( ˜ E i − ˜ E i ) . Then, we prove successively Lemma 2.7. ˜ E i − ˜ E i = O L (cid:18) e − ηt η (cid:19) . and Lemma 2.8. χ T − ln 3 E i − ˜ E i = O L (( | t | he λ | t | ) ∞ ) . (cid:3) we start with lemma 2.7. Proof. We have˜ E i − ˜ E i = χ T − ln 3 e ith ( P − W ) χ T + t ( − ∆ − s ( d − s )) − [∆ , χ i ] E Thus (cid:107) ˜ E i − ˜ E i (cid:107) L ≤ e −(cid:60) itWh (cid:107) ( − ∆ − s ( d − s )) − (cid:107) L → L (cid:107) [∆ , χ i ] E (cid:107) L . since ∆ is self adjoint, we have (cid:107) ( − ∆ − s ( d − s )) − (cid:107) L → L ≤ h η . What is more, (cid:60) ( itW ) = hηt . Now,[∆ , χ i ] E = (∆ χ i ) E + 2 ys∂ y χ i E = O L (cid:18) h (cid:19) . putting all three inequalities together, we conclude. (cid:3) we go on to lemma 2.8. ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 35 Proof. When t = 0, ˜ E i (0) = χ T − ln 3 E i because χ T − ln 3 χ T = χ T − ln 3 ( s + ln 3 ≤ ln 5 ⇒ s ≤ ln 4). For τ = 0 . . . t , let A ( τ ) = χ T − ln 3 e iτh ( P − W ) χ T + t E i ddτ A = χ T − ln 3 e iτh ( P − W ) ih [ P, χ T + t ] E i . We want to use Egorov’s lemma, first we need to localize the expression. Let (cid:15) > f ∈ C ∞ c ( R ) so that f is supported at distance less than (cid:15) of1 / / 2. Let F = Op( f ◦ p ) .F is a parametrix for f ( P ), but we do not use that fact. We claim that(1 − F )[ P, χ T + t ] E i ( s, . ) = O L ( h ∞ )First, remark that f ◦ p is indeed a symbol in the class S −∞ . By ellipticity, we cansolve 1 − F = Op( r n )( P − / n for all n ∈ N , with symbols r n in S − n . Observe( P − / E i = ( W − / E i = O ( ηh ) E i . and ( P − / n [ P, χ ] = n (cid:88) k =0 (cid:18) nk (cid:19) P [ k +1] [ χ ] ( P − / n − k where P [ k ] [ χ ] = [ P, [ P, . . . , [ P, χ ] . . . ] with k occurences of P . From the proof of lemma2.7, we know that the L norm of E ( s, . ) restricted to any compact set is O (1 /η ). Now,since r n ∈ S − n , Op( r n ) P k [ χ ] — k ≤ n + 1 — is bounded on L with norm h k , and iscompactly supported; the claim follows since η is bounded.Now, we have localized our formulae in the momentum variable :(19) ddl A = χ T − ln 3 e ilh ( P − W ) (cid:18) ih F [ P, χ T + t ] E i + O L ( h ∞ ) (cid:19) . According to the support hypothesis we have made, we can pick a function g ∈ C ∞ c ( R ) such g ◦ y M ≡ ∂ y χ T + t ), and that for all 0 < l < t , ϕ − l (supp( f ◦ p × g ◦ y M )) does not intersect the δ -neighbourhood of supp( χ T − ln 3 ) where δ is some positive number.We can insert 1 = g + 1 − g in (19) between F and [ P, χ T + t ]. Now, remark 4 givesthat for (cid:15) > χ T − ln 3 e ilh ( P − W ) F g = e − ηl O L → L (( | t | he λ | t | ) ∞ ) Since (cid:107) [ P, χ T + t ] E i (cid:107) L is bounded by some finite power of h , we can conclude. (cid:3) We deduce the following lemma : Lemma 2.9. For (cid:15) > small enough, there is a symbol σ (cid:15) that is supported at distance ≤ (cid:15) of the energy shell { p = 1 / } , and coincides with σ on the neighbourhood { / − (cid:15)/ ≤ p ≤ / (cid:15)/ } , such that (cid:104) Op( σ ) E i , E j (cid:105) = (cid:104) Op( σ (cid:15) ) ˜ E i , ˜ E j (cid:105) + O (cid:18) e − ηt η (cid:19) + O (( | t | he λ | t | ) ∞ ) + O ( h ∞ ) . Proof. We claim that the quantity in the LHS of the equation is well defined. We onlyhave to prove that Op δ ( σ ) := y δM Op( σ ) y δM is bounded on L for some δ > y − δ E i ∈ L ( M ). It suffices to prove it in the cusps. A simple computation shows thatin Z Λ , with a ∈ S nρ ( Z Λ ), y δ Op( a ) y δ = Op( y δ a ) ζ with ζ ( x ) = 4 δ ( x + 4) − δ . Since σ iscompactly supported, y δM σ still is a symbol, and Op δ ( σ ) is bounded on L .From the pseudo-locality properties of Op( σ ), and the bound (cid:107) y − (cid:15) E i (cid:107) L = O (1 /η ),we know that (cid:104) Op( σ ) E i , E j (cid:105) = (cid:104) Op( σ ) χ T − ln 3 E i , χ T − ln 3 E j (cid:105) + O ( h ∞ ) . We use the same trick as in the previous proof : we introduce 1 = F + (1 − F ) with F = Op( f ◦ p ), where f is smooth, supported in [1 / − (cid:15), / (cid:15) ], and equals 1 on[1 / − (cid:15)/ , / (cid:15)/ σ ) χ T − ln 3 E i = Op( σ ) F χ T − ln 3 E i + O L ( h ∞ ) . But then Op( σ ) F = Op( σ (cid:15) )+ R where R is a negligible operator and σ (cid:15) is as announced.From there : (cid:12)(cid:12)(cid:12) (cid:104) Op( σ ) E i , E j (cid:105) − (cid:104) Op( σ (cid:15) ) ˜ E i , ˜ E j (cid:105) (cid:12)(cid:12)(cid:12) ≤ (cid:104) Op( σ (cid:15) )( χ T − ln 3 E i − ˜ E i ) , χ T − ln 3 E j (cid:105) + (cid:104) Op( σ (cid:15) ) χ T − ln 3 E i , χ T − ln 3 E j − ˜ E j (cid:105) + (cid:104) Op( σ (cid:15) )( χ T − ln 3 E i − ˜ E i ) , χ T − ln 3 E j − ˜ E j (cid:105) + O ( h ∞ )We can conclude using the previous lemma, and : (cid:107) χ T − ln 3 E i (cid:107) L ≤ C + (cid:107) ( − ∆ − s ( d − s )) − [∆ , χ i ] E (cid:107) L ≤ C (1 + 1 η ) (cid:3) From now on, we choose a small enough (cid:15) > 0. We write : (cid:104) Op( σ (cid:15) ) ˜ E i , ˜ E j (cid:105) = e − itWh + itWh (cid:104) Aχ T + t χ i E , χ T + t χ j E (cid:105) ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 37 where, again with the notation U ( t ) = e − itPh , A = U ( t ) χ T − ln 3 Op( σ (cid:15) ) χ T − ln 3 U ( − t ) . Here again, Egorov’s lemma gives A = Op( σ (cid:15) ◦ ϕ − t ) + O L → L ( h | t | e λ | t | ) . Actually, when i (cid:54) = j , χ T + t χ i E and χ T + t χ j E have a distinct support, so that remark4 implies that when i (cid:54) = j ,(20) η (cid:104) Op( σ ) E i , E j (cid:105) = O (cid:18) e − ηt η (cid:19) + O ( η ( | t | he λ | t | ) ∞ ) + O ( ηh ∞ ) . Now, we assume that i = j , unless specifically stated. We denote σ (cid:15),t = σ (cid:15) ◦ ϕ − t . Weclaim that when η remains bounded and η × t → ∞ , there are constants C and C (depending on T ) such that(21) C η ≤ e − ηt (cid:107) χ T + t χ i E (cid:107) L ≤ C η . Indeed e − ηt (cid:107) χ T + t χ i E (cid:107) L = (cid:90) y>a χ T + t ( y ) χ ( y ) y η e − ηt dyy = (cid:90) s> ln a e ηs ˜ χ ( s − T − t − ln a ) χ ( e s ) e − ηt dse − ηt (cid:107) χ T + t χ i E (cid:107) L ≤ (cid:90) ln(5 a )+ T + t ln 2 a e η ( s − t ) = 12 η [ e η (ln(5 a )+ T ) − e η (ln(2 a ) − t ) ] ≤ C η (1 + o (1)) e − ηt (cid:107) χ T + t χ i E (cid:107) L ≥ (cid:90) ln(4 a )+ T + t ln 3 a e η ( s − t ) = 12 η [ e η (ln(4 a )+ T ) − e η (ln(3 a ) − t ) ] ≥ C η (1 + o (1)) . Hence, when η × t → + ∞ , and λ > λ max , η (cid:104) Op( σ ) E i , E i (cid:105) = ηe − ηt (cid:104) Op( σ t ) χ T + t χ i E , χ T + t χ i E (cid:105) + O (cid:18) e − ηt η + ( h | t | e λ | t | ) ∞ + h | t | e λ | t | (cid:19) . Letting t = t | log h | / (2 λ ), where 0 < t < 1, and assuming η ≥ C λ log | log h || log h | with C λ > λ/t > λ max (1 / η (cid:104) Op( σ ) E i , E i (cid:105) = ηe − ηt (cid:104) Op( σ (cid:15),t ) χ T + t χ i E , χ T + t χ i E (cid:105) + o h → (1) For i (cid:54) = j , equation 20 gives(23) η (cid:104) Op( σ ) E i , E j (cid:105) = o h → (1) . Stationary phase computations. The idea behind the proof here is that χ T + t χ i E is a lagrangian state, thus mapped to another lagrangian state by Op( σ (cid:15),t ) which is apseudo-differential operator. Lemma 2.10. Assume t , λ and η satisfy the above conditions. Then, ηe − ηt (cid:104) Op( σ (cid:15),t ) χ T + t χ i E , χ T + t χ i E (cid:105) = (cid:20) πa η ηe − ηt (cid:90) dθdτ e ητ [ χ T + t χ ] ( ae τ ) σ (cid:15),t − τ ( a, θ, a , (cid:21) + O ( h − t ) Proof. This computation only takes place in cusp Z i , and we forget the dependence in i until the end of the proof of this lemma.First, we can eliminate the integration in the θ (cid:48) and J variable in the LHS becauseof the following fact. When ς is tempered, Λ periodic in the first variable, (cid:90) D (cid:18)(cid:90) R d ς (cid:18) θ + θ (cid:48) , hJ (cid:19) e i (cid:104) θ − θ (cid:48) ,J (cid:105) dθ (cid:48) dJ (cid:19) dθ = (cid:90) D (cid:88) J ∈ Λ (cid:48) ˆ ς ( J/ , hJ/ e i (cid:104) θ,J (cid:105) dθ = (cid:90) D ς ( θ, dθ where ˆ ς was the discrete Fourier transform in the first variable. Hence, the quantityin the LHS in the lemma is the integral over θ ∈ D of the following expression :(24) h − ηe − ηt (cid:90) y s − y (cid:48) s − e i ( y − y (cid:48) ) Y/h [ χ T + t χ ]( y )[ χ T + t χ ]( y (cid:48) ) σ (cid:15),t (cid:18) y + y (cid:48) , θ, Y, (cid:19) dydy (cid:48) dY. We want to use the fact that if ς is a symbol in some S n ( Z ), not depending on θ noron J , then the function ˜ ς ( s, v ) = ς ( e s , e − s v ) is a symbol in the usual Kohn-Nirenbergsense, in S n ( R ) — notation of definition 2.2. Remark that the behavior is not so clearin the θ variable, for which periodicity and rescaling are not compatible.We introduce the following rescalings : y = ae τ , y (cid:48) = y (1 + u ), Y = (1 + v ) /y . Up toa factor h − ηa η e − η ( t − τ ) χ T + t χ ( ae τ ), the expression in (24) is the integral over τ ∈ R of (cid:90) (1 + u ) η − / e i (log(1+ u ) − u (1+ v )) /h [ χ T + t χ ]( ae τ (1 + u )) σ (cid:15),t (cid:18) ae τ (1 + u , θ, vae τ , (cid:19) dudv. Remark that this integral vanishes when τ / ∈ [ln 2 , T + t + ln 5], and write σ (cid:15),t (cid:18) ae τ (1 + u , θ, vae τ , (cid:19) = σ (cid:15),t − τ (cid:18) a (1 + u , θ, a (1 + v ) , (cid:19) . ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 39 Then, introduce a cutoff (cid:37) ( u ), supported around 0, and 1 = (cid:37) + 1 − (cid:37) to separate theintegral into two parts (I) and (II).Let us examine first (II) which is not stationnary, and supported for | u | > δ . Weinsert 1 = u N /u N and integrate by parts in v . We take the L bound, considering that σ (cid:15),t is supported in { p ∈ [1 / − (cid:15), / (cid:15) ] } , and using symbol estimates on σ (cid:15),t . It gives | (II) | ≤ C N h N e Nλ ( t − τ ) (cid:90) (1 − (cid:37) ( u ))(1 + u ) η − / (1 + u/ N u N [ χ T + t χ ]( ae τ (1 + u )) (cid:104) (1 + u/ | v | ≤ √ (cid:15) (cid:105) dudv. after some rescaling in the v variable, and considering[ χ T + t χ ]( ae τ (1 + u )) ≤ ( − ≤ u ≤ + ∞ ) , this is bounded (uniformly in θ and τ ) by Ch N (1 − ρ ) (cid:90) + ∞− du (1 − (cid:37) )(1 + u ) η − / (cid:0) u/ (cid:1) N − u N = O (cid:0) h − t (cid:1) . Part (I) of the integral supported around u = 0 is an oscillatory integral that candirectly be estimated. Indeed, on that domain, the phase function satisfies symbolicestimates and has only one critical point ( u, v ) = (0 , − uv + O ( u , v ).Further consider that the function under the integral is smooth and uniformly com-pactly supported in v . When we differentiate it in v , we lose a O ( e λ | t − τ | ) constant.When differentiating in u , either we differentiate σ (cid:15),t − τ , losing again a O ( e λ | t − τ | ) con-stant, or we differentiate ρ ( u )(1 + u ) η − / χ T + t χ . We chose — recall (18) — the cutoffs χ T + t and χ exactly so that we lose only O (1) constants by doing so.The basic stationnary phase theorem — see theorem 7.7.5 in [H¨or03] — in the planeapplies and we find(I) = 2 πh [ χ T + t χ ]( ae τ ) σ (cid:15),t − τ (cid:18) a, θ, a , (cid:19) + O ( h − t ) , uniformly in variables θ and τ .Recall we are to integrate (I)+(II) in θ and τ with a prefactor h − ηa η e − η ( t − τ ) χ T + t χ ( ae τ ).But (cid:90) dθdτ ηa η e − η ( t − τ ) χ T + t χ ( ae τ ) = O (1) . and that estimate ends the proof. (cid:3) Dynamical properties and conclusion. Recall that whenever ς is a compactlysupported continuous function on T ∗ M , in the coordinates of cusp Z i , µ + i,ν ( ς ) = 2 νa ν (cid:90) t ∈ R e − νt ς ◦ ϕ − t (cid:0) a, θ, a , (cid:1) dθ. When η ( h ) → ν > η remains bounded, and we can directly apply lemma 2.10 andequation (22). Letting h → 0, we find η (cid:104) Op( σ ) E i ( s ) , E i ( s ) (cid:105) → πµ + i,ν ( σ (cid:15) ) . Since µ + i is supported on S ∗ M Actually, when η → 0, slowly enough, lemma 2.10 andequation 22 also imply that | η (cid:104) Op( σ ) E i ( s ) , E i ( s ) (cid:105) − πµ + i,η ( h ) ( σ ) | = O ( h − t ) + O ( (cid:107) σ (cid:107) L ∞ e − η ( h ) t ( h ) )= O ( | log h | − t ) . The proof of theorem 4 will therefore be complete if we can prove Lemma 2.11. Assume M has strictly negative curvature. For all σ ∈ C c ( T ∗ M ) , forall i = 1 . . . k , as ν → + , µ ± i,ν ( σ ) → (cid:90) σd L where L is the normalized Liouville measure on the unit cotangent bundle of M . It is as far as we know an open question as to whether the Liouville measure is aGibbs measure in such a cusp-manifold — that is to say, whether a Ruelle inequalityholds. If it were, we could apply directly theorem 3 in [Bab02]. However, mimicking theproof therein and using the classical Hopf argument, we are able to conclude. Observethat replacing the hypothesis of negative curvature by ergodic , or even mixing , we arenot able to prove that conclusion still holds: we really use the stable and unstablefoliations, and the fact that they are absolutely continuous.In this part of the proof, it is easier to consider only µ − , that is supported onincoming horospheres. Proof. From now on, we work on the unit cotangent sphere since both µ − ν,i and L aresupported on S ∗ M .Take ε > 0. Since σ is compactly supported, we can find a δ > | σ ( ξ ) − σ ( ξ (cid:48) ) | < ε whenever | ξ − ξ (cid:48) | < δ . The measures µ − i,ν are obtained by propagating anincoming horocycle. Following the idea of proof in [Bab02], we want to thicken thehorocycle. Denote by H i,a the incoming horosphere at height a in cusp Z i , and considerthe set Ω i,a,C := (cid:91) ξ ∈ H i,a B ( ξ, C, W s ( ξ )) , ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 41 where B ( ξ, C, W s ( ξ )) is a ball of radius C > ξ ,centered at ξ . For C > i,a,C , H i,a is a local section of the weak-stable foliation of T ∗ , M , with projection π su . From the contraction properties on theweak-stable foliation, for some constant C > 0, on Ω i,a,Cδ , | σ ◦ ϕ t − σ ◦ ϕ t ◦ π | ≤ ε .From theorem 7.6 in [PPS12], there is a locally bounded measurable density ρ suchthat in Ω i,a,Cδ , d L = ρ ( ξ ) d vol W s ( π su ξ ) d vol H i,a The measure on H i,a being dθ of mass 1. We let g vanish out of Ω i,a,Cδ and on Ω i,a,Cδ , g ( ξ ) := 1 ρ ( ξ )vol( B ( π su ξ, C, W s ( π su ξ ))) . Then, g is in L ( S ∗ M ), and (cid:107) g (cid:107) L ( S ∗ M ) = vol( H i,a ) = 1. Hence, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) H i,a σ ◦ ϕ t dθ − (cid:90) Ω i,a,Cδ σ ◦ ϕ t × gd L (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Consider that in the definition of µ + ν,i , since σ is compactly supported, we can inte-grate in t for only t ∈ [ − T, + ∞ [. Additionally, the prefactor a η tends to 1, and thepart t ∈ [ − T, 0] will not contribute, so we write µ − ν,i ( σ ) = O ( νT (cid:107) σ (cid:107) ∞ ) + 2 ν (cid:90) ∞ dte − νt (cid:90) σ ◦ ϕ t × gd L + O ( ε )Using the Hopf argument (as in [Cou07]), and theorem 7.6 from [PPS12] again, onecan see that the geodesic flow is mixing for the Liouville measure. Actually, it sufficesfor it to be ergodic . Indeed,2 ν (cid:90) ∞ dte − νt (cid:90) σ ◦ ϕ t × gd L = (cid:90) R + te − t (cid:90) T ∗ , M g ( ξ ) F (cid:18) t ν , ξ (cid:19) dtd L ( ξ ) , where F ( t, ξ ) is the Birkhoff average of σ for a time t along the trajectory of ξ . Since g is L , and σ is bounded, by dominated convergence and ergodicity, the limit of thiswhen ν → (cid:107) g (cid:107) L L ( σ ). For all ε > 0, we find for any limit value σ of µ − i,ν ( σ ), | σ − L ( σ ) | = O ( ε ) . letting ε → (cid:3) Appendix A. Functionnal spaces in a cusp First, let us recall some definitions on covariant derivatives. If S is a tensor on ariemannian manifold, one defines its covariant derivative in the following way:( ∇ X S )( Y , . . . , Y n ) := X ( S ( Y , . . . , Y n )) − (cid:88) i S ( Y , . . . , ∇ X Y i , . . . , Y n ) In particular, when f is a function on a riemannian manifold N , one defines a familyof tensors ∇ n f in the following way. ∇ f : X (cid:55)→ X ( f ) and ∇ n +1 X ,...,X n f := ( ∇ X ∇ n ) X ,...X n f. We also define it for vectors — which are (0 , 1) tensors: ∇ Z : X (cid:55)→ ∇ X Z and ∇ n +1 X ,...,X n Z := ( ∇ X ∇ n ) X ,...X n Z. This enables us to define, for x ∈ N (cid:107)∇ n f (cid:107) ( x ) = sup X ,...,X n ∈ T x N |∇ n f ( X , . . . , X n ) |(cid:107) X (cid:107) . . . (cid:107) X n (cid:107) Then, the space C n ( N ) is the set of functions on N that are C n , and such that (cid:107) f (cid:107) C n ( N ) := n (cid:88) k =0 sup x ∈ N (cid:107)∇ k f (cid:107) ( x ) < ∞ . Now, we turn to Sobolev spaces. When N is complete, L ( N ) is a Hilbert space.For n ≥ (cid:107) f (cid:107) H n ( Z ) := (cid:88) k ≤ n (cid:13)(cid:13) (cid:107)∇ k f (cid:107) ( x ) (cid:13)(cid:13) L ( dx ) The Sobolev space H n ( N ) of order n is the completion of C ∞ ( N ) for this norm. If N has no boundary, then H − n ( N ) is defined as the dual of H n ( N ).Using the Lax-Milgram theorem, exactly as for the Laplacian on R n , one proves thatfor any (cid:15) > − ∆ + (cid:15) is invertible on H ( N ) with values in H − ( N ). Since it is alsopositive, one can use the spectral theorem to define ( − ∆ + 1) s for any s ∈ R . Oneobserves that (cid:107) . (cid:107) H ( N ) and (cid:107) ( − ∆ + 1) / . (cid:107) are equivalent norms on H ( N ).The cusp Z is complete, so the above apply. Now, one can compute the following : ∇ X y X y = 0 ∇ X y X θ i = 0 ∇ X θi X y = − X θ i ∇ X θi X θ j = δ ij X y (25)From this and the definition of ∇ n , if α is a space-index of length n , we find ∇ n f ( X α ) = X α f + (cid:88) β ± X β f where β are other space-indices, of length < n . Whence by induction on n ≥ (cid:107) f (cid:107) C n ( Z ) is equivalent to (cid:88) | α |≤ n (cid:107) X α f (cid:107) L ∞ ( Z ) . ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 43 and(27) (cid:107) f (cid:107) H n ( Z ) is equivalent to (cid:88) | α |≤ n (cid:107) X α f (cid:107) L ( Z ) . Now, we define, for s a real number (cid:107) f (cid:107) s := (cid:107) ( − ∆ + 1) s f (cid:107) L We want to show that the completion of C ∞ ( Z ) is the sobolev space H s ( Z ) for integer s , and that then, (cid:107) . (cid:107) s is equivalent to (cid:107) . (cid:107) H s ( Z ) . Such a result is deduced of an ellipticestimate similar to that in pp 358 in [Tay11]. Actually, the proof therein adapts to acusp if one defines the slope operators D j,h in the following way D j,h f ( x ) = 1 h ( f ( x + hX j ) − f ( x )) , j = y, θ , . . . , θ d . Then, using P = − h ∆ / 2, we also define the semi-classical Sobolev norms : (cid:107) f (cid:107) s,h := (cid:107) ( P + 1) s/ f (cid:107) L ( Z ) . One gets for some constant C > C h s + (cid:107) f (cid:107) s ≤ (cid:107) f (cid:107) s,h ≤ Ch − s − (cid:107) f (cid:107) s . where s + and s − are the positive and negative part of s .To finish this section, we define the non-integer Sobolev spaces using complex inter-polation — as in pp 321 from [Tay11]. Appendix B. Estimating the derivatives of a flow on a Riemannianmanifold The following proposition should be classical, but for lack of a reference, we enclosea proof. Proposition B.1. Let ϕ t be a flow in a manifold N , such that all the covariantderivatives of the vector field V of the flow and of the curvature tensor of N arebounded. Assume also that the maximal Lyapunov exponent λ of ϕ t — as definedin 2.5 — is finite . Then for all λ > λ , there are constants C n > , such that for f ∈ C n ( N ) , for t ∈ R , (cid:107) f ◦ ϕ t (cid:107) C n ( N ) ≤ C n e nλ | t | (cid:107) f (cid:107) C n ( N ) The proof is inspired by [DG14] ( N is a convex co-compact hyperbolic surface),which itself comes from [BR02] ( N = R n ). In the usual proofs of this type of result,at some point, one uses coordinates to transport the problem to R n . When N iscompact, this is reasonnable because all metrics on N are equivalent. When N is noncompact, it is probably possible to take a similar approach. However, one would have be careful and take coordinate charts with derivatives nicely bounded. We chose toavoid taking coordinates altogether, and give an intrinsic formulation of the proof,hence the appearance of many tensors.The main idea of the proof is to avoid estimating higher derivatives of the flow, andreplace them by higher derivatives of the vector field of the flow. Proof. We want to compute ∇ X ,...X n ( f ◦ ϕ t ) . We are going to compare this with ∇ ϕ t ∗ X ,...ϕ t ∗ X n f. In the first expression, there are a priori , higher derivatives of the flow, while thesecond one only contains first order derivatives that are much easier to estimate. Let z ∈ N , and X , . . . , X n ∈ T z N . Let W nt ( X , . . . X n ) f := (cid:2) ( ϕ t ) ∗ (cid:0) ∇ ϕ t ∗ X ,...,ϕ t ∗ X n (cid:1)(cid:3) f That is : [ W nt ( X , . . . , X n ) f ] ◦ ϕ − t = (cid:0) ∇ ϕ t ∗ X ,...,ϕ t ∗ X n (cid:1) ( f ◦ ϕ − t ) . From the definition, we see that W nt f is a tensor. We observe that(29) W nt = ∇ W n − t + (cid:88) i =2 ,...n W n − t ( X , . . . , ∇ X X i − ( ϕ t ) ∗ ( ∇ ϕ t ∗ X ϕ t ∗ X i ) , . . . , X n ) . One can compute ( ϕ t ) ∗ ( ∇ ϕ t ∗ X ϕ t ∗ X i ). Indeed, consider the fact ∂ t ( ϕ t ) ∗ X ( t ) = ( ϕ t ) ∗ [ V, X ( t )] + ( ϕ t ) ∗ ∂ t X We deduce that ∂ t ( ϕ t ) ∗ ( ∇ ϕ t ∗ X ϕ t ∗ Y ) = ( ϕ t ) ∗ (cid:0) [ V, ∇ ϕ t ∗ X ϕ t ∗ Y ] − ∇ [ V,ϕ t ∗ X ] ϕ t ∗ Y − ∇ ϕ t ∗ X [ V, ϕ t ∗ Y ] (cid:1) . That is ∂ t ( ϕ t ) ∗ ( ∇ ϕ t ∗ X ϕ t ∗ Y ) = ( ϕ t ) ∗ Z ( ϕ t ∗ X, ϕ t ∗ Y )with Z ( ϕ t ∗ X, ϕ t ∗ Y ) = ∇ ϕ t ∗ Xϕ t ∗ Y V + R ∇ ( V, ϕ t ∗ X ) ϕ t ∗ Y, where R ∇ is the curvature tensor. So,(30) ∇ X Y − ( ϕ t ) ∗ ( ∇ ϕ t ∗ X ϕ t ∗ Y ) = − (cid:90) t ( ϕ s ) ∗ Z ( ϕ s ∗ X, ϕ s ∗ Y ) ds ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 45 This equation we found in [DG14], and the rest of the proof is devoted to proving similarformulae for higher order derivatives. Let us call the tensor in the RHS L t ( X, Y ). Wecan already compute explicitly W t f = X f W t f = ∇ X ,X f + L t ( X , X ) f. Now, we introduce a class of vector-valued tensors T . Elements of T depend on twotime-parameters s, t . First, the identity is in T . Second, if T ( s, t ) , . . . , T k ( s, t ) arein T with k ≥ 2, and if R is a smooth (vector-valued) k -tensor with all its covariantderivatives bounded, then(31) T RT ,...,T k ( s, t ) := (cid:90) ts ( ϕ u ) ∗ R ( ϕ u ∗ T ( u, t ) , . . . , ϕ u ∗ T k ( u, t )) du is also in T . We require that T is the vector space generated by the above tensors.For example, L t is T − ZId,Id (0 , t ); we denote L ( s, t ) = T − ZId,Id ( s, t ). Then Lemma B.2. W nt f can be written as a sum of terms (32) ∇ k f ( T (0 , t ) , . . . , T k (0 , t )) where the T i ’s are in T of the correct order.Proof. We have already checked it for n = 1 and n = 2. Actually, we will check thatif A t is an ( n − ∇ A t − (cid:88) i =1 ,...,n A t ( X , . . . , L t ( X , X i ) , . . . , X n )is a sum of such operators (of orders n and n − ∇ (cid:2) ∇ k ( T (0 , t ) , . . . , T k (0 , t )) (cid:3) = ∇ k +1 ( Id, T (0 , t ) , . . . , T k (0 , t )) − (cid:88) i =1 ,...,k ∇ k ( T (0 , t ) , . . . , ∇ T i (0 , t ) , . . . , T k (0 , t )) . We deduce that it suffices to show that when T ∈ T is a k -tensor, T (cid:48) := ∇ X T ( s, t ) + (cid:90) s ( ϕ w ) ∗ Z ( ϕ w ∗ X, ϕ w ∗ T ( s, t )) dw + k (cid:88) i =1 T (0 , t )( X , . . . , L t ( X , X i ) , . . . , X k )is in T . We prove this by induction on k . First, if k = 1, T is the identity, and wefind T (cid:48) ( s, t ) = L ( s, t ).Assume we are done for all k ≤ n . Then, let T be a n + 1 tensor in T . Byconstruction, it is a sum of terms as in (31). Since the property we are trying to prove is stable by taking sums, assume there is only one term in the sum. The T i ’s all are oforder < n + 1, and we can compute, using (30) in the first line ∇ X T ( s, t ) = (cid:90) ts (cid:90) u ( ϕ u − w ) ∗ ( − Z )( ϕ u − w ∗ X, ϕ − w ∗ R ( ϕ u ∗ T ( u, t ) , . . . , ϕ u ∗ T k ( u, t ))) dwdu + (cid:90) ts ( ϕ u ) ∗ ∇ ϕ u ∗ X (cid:0) R ( ϕ u ∗ T ( u, t ) , . . . , ϕ u ∗ T k ( u, t )) (cid:1) du. ∇ X T ( s, t ) = T − ZId,T ( s, t ) + (cid:90) s ( ϕ w ) ∗ ( − Z )( ϕ w ∗ X, ϕ w ∗ T ( s, t )) dw + (cid:90) ts ( ϕ u ) ∗ ( ∇ ϕ u ∗ X R )( ϕ u ∗ T k ( u, t ) , . . . , ϕ u ∗ T k ( u, t )) du. + k (cid:88) i =1 (cid:90) ts ( ϕ u ) ∗ R ( ϕ u ∗ T ( u, t ) , . . . , ∇ ϕ u ∗ X ϕ u ∗ T i ( u, t ) , . . . , ϕ u ∗ T k ( u, t )) du. Hence we find ∇ X T ( s, t )+ (cid:90) s ( ϕ w ) ∗ Z ( ϕ w ∗ X, ϕ w ∗ T ( s, t )) dw = T − ZId,T ( s, t ) + T ∇ RId,T ,...,T k ( s, t )+ k (cid:88) i =1 (cid:90) ts ( ϕ u ) ∗ R ( ϕ u ∗ T ( u, t ) , . . . , ∇ ϕ u ∗ X ϕ u ∗ T i ( u, t ) , . . . , ϕ u ∗ T k ( u, t )) du But we precisely have( ϕ u ) ∗ ∇ ϕ u ∗ X ϕ u ∗ T i ( u, t ) = ∇ X T i ( u, t ) + (cid:90) u ( ϕ w ) ∗ Z ( ϕ w ∗ X, ϕ w ∗ T i ( u, t )) dw. so we can use the induction hypothesis, and conclude. (cid:3) Lemma B.3. When T ∈ T is a n -tensor, there is a constant C > such thatwhenever ≤ s ≤ t , (cid:107) ϕ s ∗ T ( s, t )(( ϕ t ) ∗ X , . . . , ( ϕ t ) ∗ X n ) (cid:107) ≤ Ce nλ ( t − s ) (cid:107) X (cid:107) . . . (cid:107) X n (cid:107) . Proof. We proceed by induction. First, for the identity, this is true because the max-imal lyapunov exponent of the flow is bounded. Now, we assume it is true for all k -tensors in T with k ≤ n , and let T ∈ T be a n + 1 tensor. (cid:107) ϕ s ∗ T ( s, t ) (cid:107) ≤ (cid:90) ts e λ ( u − s ) k (cid:89) i =1 (cid:107) ϕ u ∗ T i ( u, t ) (cid:107) du If we use the induction hypothesis, we get (cid:107) ϕ s ∗ T ( s, t ) (cid:107) ≤ C (cid:107) X (cid:107) . . . (cid:107) X n (cid:107) (cid:90) ts e λ ( u − s )+( n +1) λ ( t − u ) du (cid:124) (cid:123)(cid:122) (cid:125) ≤ Ce λ ( n +1)( t − s ) ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 47 (cid:3) We conclude the proof by observing that ∇ n ( f ◦ ϕ − t )( X , . . . , X n ) = W nt f (( ϕ t ) ∗ X , . . . , ( ϕ t ) ∗ X n ) (cid:3) Appendix C. On the Sasaki metric C.1. The curvature tensor of a Sasaki metric. There is a useful — and easilyaccessible — reference for the Sasaki metric on tangent spaces: [GK02]. We are going torely heavily on it to avoid introducing too much machinery. In the following paragraph,we retain the notations therein. We want to show that proposition B.1 applies to thegeodesic flow cusp surfaces. We prove Proposition C.1. Assume that the curvature tensor of M is bounded, and all itscovariant derivatives also. For R > , let T M R := { v ∈ T M | (cid:107) v (cid:107) ≤ R } be endowedwith the Sasaki metric. Then (1) The curvature tensor of T M R , and all its derivatives are bounded. (2) It is also the case for the vector of the geodesic flow Remark that when the curvature of M is constant, the covariant derivative of thecurvature tensor is just 0, so the above proposition applies to cusp manifolds — andmore generally to any geometrically finite manifold with hyperbolic ends. Proof. We denote ( p, u ) for points of T ∗ M . If X is a vector in T p M , we denote by X h (resp. X v ) its horizontal (resp. vertical) lift, which are vectors in T ( p,u ) T M .Let T be a vector valued tensor on M . From T we can construct a variety ofvector valued on T M . Indeed, first, we can construct tensors on M valued in T T M by taking either the vertical of the horizontal lift of T . Then, we can compose T byeither X v (cid:55)→ X or X h (cid:55)→ X . We consider now the class B of tensors on T M thatare obtained in this way when T and all its derivatives are bounded. We also requirethat 0-tensors u h and u v are in B . Now, B is the smallest class of tensors stable bycomposition and sums that contains B .From the formulae page 16 (prop. 7.5) for the curvature tensor of the Sasaki metric,we see that it is in B since the curvature tensor R of M as well as all its derivativesare bounded. The vector of the geodesic flow also is in B because it is V ( p, u ) = u h .We want to prove that B is stable under covariant derivatives . We work in localcoordinates. Observe that since covariant derivatives behave well with compositionand sums, it suffices to prove that covariant derivatives of elements of B are in B . Let p ∈ M , let U be some small open set containing p where the normal coordinatesat p , exp − p : U → T p M are well defined. Taking an orthonormal basis X , . . . , X n in T p M , we have coordinates x , . . . , x n on U . Then, we can consider coordinates v , . . . , v n on T U as in page 6 of [GK02]. Since we have taken normal coordinates, theChristoffel coefficients vanish at p , and we have (see lemma 4.3 p. 7) ∂ x i ( p ) h = ∂ v i ∂ x i ( p ) v = ∂ v n + i . At a point ( p (cid:48) , u ), we have(33) u = (cid:88) v n + k ∂ x k ( p (cid:48) ) . Since we have taken normal coordinates, the ∇ ∂ x ∂ x i vanish at p . From this and theformulae for covariant derivatives in proposition 7.2 page 15, we find ∇ a h + b v u h = b h + 12 ( R p ( u, b ) u ) h − 12 ( R p ( a, u ) u ) v . and ∇ a h + b v u v = b v + 12 ( R p ( u, u ) a ) h . Now, we take T a tensor on M with all its derivatives bounded, and we just considerthe case when T is a 1 tensor, and T (cid:48) ( a h + b v ) = ( T ( a )) h . This defines an element of B . ( ∇ X h + Y v T (cid:48) )( a h + b v ) = ∇ X h + Y v ( T ( a )) h − T (cid:48) ( ∇ X h + Y v ( a h + b v )) . Using again the formulae for Sasaki covariant derivatives, we can expand this expres-sion. There will be terms containing ∇ X T and terms involving R p , u and T , so theresult will be an element of B .To give a complete proof, we would have to consider all the possibilities that leadto similar computations; we leave this as an exercise for the reader. (cid:3) C.2. The Sasaki metric in a cusp, and symbols. Now, M is a cusp manifold.The Sasaki metric is a priori defined on the tangent space. However, there is a cor-respondance v (cid:55)→ (cid:104) v, . (cid:105) between T M and T ∗ M , and we define the Sasaki metric on T ∗ M by pushing forward the metric on T M . As a consequence, T ∗ M is endowed witha connection ∇ and C k norms. The following fact is the key to proving the Egorovlemma 3. Proposition C.2. Take E > , and consider functions on T ∗ M supported in ( T ∗ M ) E := { p ( ξ ) ≤ E } . For such functions, the C k ( T ∗ M ) norm is equivalent to the norm givenby symbol estimates with k (cid:48) ≤ k derivatives.Proof. The part of ( T ∗ M ) E above the compact part of M is relatively compact, so all C k norms over it are equivalent. We just have to work in the cusps. Let us first startby finding the expression for the Sasaki metric in a cusp Z ; we use again [GK02]. We ONG TIME QUANTUM EVOLUTION OF OBSERVABLES ON CUSP SURFACES 49 have coordinates y, θ , and the coordinates on the tangent space v y , v θ . From (25), wecan compute y ∇ ∂ y ∂ y = − ∂ y y ∇ ∂ y ∂ θ i = − ∂ θ i y ∇ ∂ θi ∂ y = − ∂ θ i y ∇ ∂ θi ∂ θ j = δ ij ∂ y We deduce that the Sasaki metric on T M is g = 1 y (cid:18) dy + dθ + ( dv y + 1 y ( v θ .dθ − v y dy )) + ( dv θ − y ( v θ dy + v y dθ )) (cid:19) Now, v y = y Y , and v θ = y J , so this gives on T ∗ Mg = dy + dθ y + y (cid:18) ( dY + 1 y ( J.dθ − Y dy )) + ( dJ − y ( J dy + Y dθ )) (cid:19) Recall that p = | ξ | / − h ∆ / 2. We get that g ( X y ) = g ( X θ i ) = 1 + 2 p and g ( X Y ) = g ( X J i ) = 1 . and when k (cid:54) = i — (cid:104) ., . (cid:105) being the scalar product, (cid:104) X y , X θ i (cid:105) = (cid:104) X θ k , X θ i (cid:105) = (cid:104) X Y , X J i (cid:105) = (cid:104) X J k , X J i (cid:105) = 0 (cid:104) X y , X Y (cid:105) = − yY, (cid:104) X y , X J i (cid:105) = − yJ i , (cid:104) X θ i , X Y (cid:105) = yJ i , (cid:104) X θ i , X J k (cid:105) = − δ ik yY. If we use the Koszul formula [Pau14] to determine the covariant derivatives of X y,θ,Y,J ,we will find that they are of the type aX y + bX θ + cX Y + dX J , where a, b, c, d areelements of S V — defined in the paragraph after (4). As a consequence, if α is a finitesequence of α j ∈ { y, θ i , Y, J k } of length k , there are symbols f β ∈ S V for all sequences β of the same type, of length k (cid:48) < k , such that f β is of order ≤ k − k (cid:48) , andˆ ∇ kX α ...X αk = X α + (cid:88) β f β X β From this we deduce that on ( T ∗ M ) E , the norms (cid:110) (cid:88) | α |≤ k q n,α (cid:111) n and (cid:88) | α |≤ k sup T ∗ M (cid:107) ˆ ∇ kX α (cid:107) are equivalent. We are left to prove that the latter is equivalent to the C k ( T ∗ M ) norm.It is a priori bounded by it, so we need to prove a lower bound.We have coordinates in each T ξ ( T ∗ Z ) given by T ξ ( T ∗ Z ) (cid:51) X = u y X y + u θ X θ + u Y X Y + u J X J this defines a map u ξ : T ξ T ∗ Z → R d +2 . 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