A formula relating localisation observables to the variation of energy in Hamiltonian dynamics
aa r X i v : . [ m a t h - ph ] J a n A formula relating localisation observables to the variation ofenergy in Hamiltonian dynamics
A. Gournay ∗ and R. Tiedra de Aldecoa † Universit´e de Neuchˆatel, Rue E.-Argand 11, CH-2000 Neuchˆatel, Switzerland. Facultad de Matem´aticas, Pontificia Universidad Cat´olica de Chile,Av. Vicu˜na Mackenna 4860, Santiago, Chile
E-mails: [email protected], [email protected]
Abstract
We consider on a symplectic manifold M with Poisson bracket { · , · } an Hamiltonian H with completeflow and a family Φ ≡ (Φ , . . . , Φ d ) of observables satisfying the condition {{ Φ j , H } , H } = 0 for each j . Under these assumptions, we prove a new formula relating the time evolution of localisation observablesdefined in terms of Φ to the variation of energy along classical orbits. The correspondence between this formulaand a formula established recently in the framework of quantum mechanics is put into evidence.Among other examples, our theory applies to Stark Hamiltonians, homogeneous Hamiltonians, purely ki-netic Hamiltonians, the repulsive harmonic potential, the simple pendulum, central force systems, the Poincar´eball model, covering manifolds, the wave equation, the nonlinear Schr¨odinger equation, the Korteweg-de Vriesequation and quantum Hamiltonians defined via expectation values. ∗ Work made while on leave from the Max Planck Institute f¨ur Mathematik. † Supported by the Fondecyt Grant 1090008 and by the Iniciativa Cientifica Milenio ICM P07-027-F “Mathematical Theory of Quantumand Classical Magnetic Systems”. ontents ∇ H = g ( H ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 H = h ( p ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 The assumption {{ Φ j , H } , H } = 0 as a differential equation . . . . . . . . . . . . . . . . . 134.4 Passing to a covering manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.5 Infinite dimensional Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.5.1 Classical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.5.2 Quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Introduction and main results
The purpose of the present paper is to put into evidence a new formula in Hamiltonian dynamics, both simpleand general, relating the time evolution of localisation observables to the variation of energy along classicalorbits.Our result is the following. Let M be a (finite or infinite-dimensional) symplectic manifold with symplectic -form ω and Poisson bracket { · , · } . Let H ∈ C ∞ ( M ) be an Hamiltonian on M with complete flow { ϕ t } t ∈ R .Let Φ ≡ (Φ , . . . , Φ d ) ∈ C ∞ ( M ; R d ) be a family of observables satisfying the condition (cid:8) { Φ j , H } , H (cid:9) = 0 (1.1)for each j ∈ { , . . . , d } . Then we have (see Theorem 3.3, Corollary 3.4 and Lemma 3.6 for a precise statement): Theorem 1.1.
Let H and Φ be as above. Let f : R d → C be such that f = 1 on a neighbourhood of , f = 0 at infinity, and f ( x ) = f ( − x ) for each x ∈ R d . Then there exist a closed subset Crit ( H, Φ) ⊂ M and anobservable T f ∈ C ∞ (cid:0) M \ Crit ( H, Φ) (cid:1) satisfying { T f , H } = 1 on M \ Crit ( H, Φ) such that lim r →∞ Z ∞ d t (cid:2)(cid:0) f (Φ /r ) ◦ ϕ − t (cid:1) ( m ) − (cid:0) f (Φ /r ) ◦ ϕ t (cid:1) ( m ) (cid:3) = T f ( m ) (1.2) for each m ∈ M \ Crit ( H, Φ) . The observable T f admits a very simple expression given in terms of the Poisson brackets ∂ j H := { Φ j , H } and the vector ∇ H := ( ∂ H, . . . , ∂ d H ) , namely, T f := − Φ · ( ∇ R f )( ∇ H ) , (1.3)where ∇ R f : R d → C d is some explicit function (see Section 2).In order to give an interpretation of Formula (1.2), consider for a moment the situation where M := T ∗ R n ≃ R n is the standard symplectic manifold with canonical coordinates ( q, p ) and -form ω := P nj =1 d q j ∧ d p j . Furthermore, let H ( q, p ) := h ( p ) be a purely kinetic energy Hamiltonian and let Φ( q, p ) := q be the stan-dard family of position observables. In such a case, the condition (1.1) is readily verified, the vector ∇ H reducesto the usual velocity observable ∇ h associated to H , and the l.h.s. of Formula (1.2) has the following meaning:For r > and m ∈ M \ Crit ( H, Φ) fixed, it is equal to the difference of times spent by the classical orbit { ϕ t ( m ) } t ∈ R in the past (first term) and in the future (second term) within the region Σ r := supp[ f (Φ /r )] ⊂ M defined by the localisation observable f (Φ /r ) . Moreover, if we interpret the map dd H := { T f , · } as a derivationon C ∞ (cid:0) M \ Crit ( H, Φ) (cid:1) , then T f on the r.h.s. of (1.2) can be seen as an observable “derivative with respect tothe energy H ” on M \ Crit ( H, Φ) , since dd H ( H ) = { T f , H } = 1 on M \ Crit ( H, Φ) . Therefore, Formula (1.2)provides a new relation between sojourn times and variation of energy along classical orbits. Evidently, thisinterpretation remains valid in the general case provided that we consider the observables Φ j as the componentsof an abstract position observable Φ (see Remark 3.7).Our interest in this issue has been aroused by a recent paper [31], where the authors establish a similarformula in the framework of quantum (Hilbertian) theory. In that reference, H is a selfadjoint operator in aHilbert space H , Φ ≡ (Φ , . . . , Φ d ) is a family of mutually commuting selfadjoint operators in H , (1.1) is asuitable version of the commutation relation (cid:2) [Φ j , H ] , H (cid:3) = 0 , and T f is a time operator for H ( i.e. a symmetricoperator satisfying the canonical commutation relation [ T f , H ] = i ). So, apart from its intrinsic interest, thepresent paper provides also a new example of result valid both in quantum and classical mechanics. Points ofthe symplectic manifold correspond to vectors of the Hilbert space, complete Hamiltonian flows correspond toone-parameter unitary groups, Poisson brackets correspond to commutators of operators, etc. (see [1, Sec. 5.4]and [24] for the interconnections between classical and quantum mechanics). Accordingly, we try put intolight throughout all of the paper the relation between both theories. For instance, we link in Remark 3.5 theconfinement (resp. the non-periodicity) of the classical orbits { ϕ t ( m ) } t ∈ R , m ∈ M , to the affiliation of thecorresponding quantum orbits { e itH ψ } t ∈ R , ψ ∈ H , to the singular (resp. absolutely continuous) subspace of H . Moreover, we show in Section 4.5.2 that the Hilbertian space theory of [31] can be recast into the presentframework of symplectic geometry by using expectation values.3e also mention that Formula (1.2), with r.h.s. defined by (1.3), provides a crucial preliminary step for theproof of the existence of classical time delay for abstract scattering pairs { H, H + V } (see [10], [14, Sec. 4.1],and [35, Sec. 3.4] for an account on classical time delay). If V is an appropriate perturbation of H and S is theassociated scattering map, then the classical time delay τ ( m ) for m ∈ M defined in terms of the localisationoperators f (Φ /r ) should be reexpressed as follows: it is equal to the l.h.s. of (1.2) minus the same quantity with m replaced by S ( m ) . Therefore, if m and S ( m ) are elements of M \ Crit ( H, Φ) , then the classical time delayfor the scattering pair { H, H + V } should satisfy the equation τ ( m ) = ( T f − T f ◦ S )( m ) . Now, the property { T f , H } ( m ) = 1 implies that T f ( m ) = ( T f ◦ ϕ t )( m ) − t for each t ∈ R . Since S commuteswith ϕ t , this would imply that τ ( m ) = (cid:2) ( T f − T f ◦ S ) ◦ ϕ t (cid:3) ( m ) for all t ∈ R , meaning that the classical time delay is a first integral of the free motion. This property correspondsin the quantum case to the fact that the time delay operator is decomposable in the spectral representation of thefree Hamiltonian (see [32, Rem. 4.4]).Let us now describe more precisely the content of this paper. In Section 2 we recall some definitionsin relation with the function f that appear in Theorem 1.1. The function R f is introduced and some of itsproperties are presented. Then we prove various versions of Formula (1.2) in the particular case where thefunctions Φ ◦ ϕ ± t : M → R d are fixed vectors x ± ty , x, y ∈ R d (see Proposition 2.3, Lemma 2.4 and Corollary2.6). In Section 3.1, we introduce the Hamiltonian system ( M, ω, H ) and the abstract position observable Φ .Then we define the (closed) set of critical points Crit ( H, Φ) associated to H and Φ as (see [31, Def. 2.5] for thequantum analogue): Crit ( H, Φ) := (cid:8) m ∈ M | ( ∇ H )( m ) = 0 (cid:9) . When H ( q, p ) = h ( p ) and Φ( q, p ) := q on M = R n , Crit ( H, Φ) coincides with the usual set Crit ( H ) ofcritical points of the Hamiltonian vector field X H , i.e. Crit ( H ) ≡ (cid:8) m ∈ M | X H ( m ) = 0 (cid:9) = (cid:8) ( q, p ) ∈ R n | ( ∇ h )( p ) = 0 (cid:9) = Crit ( H, Φ) . But, in general, we simply have the inclusion
Crit ( H ) ⊂ Crit ( H, Φ) .In Section 3.2, we prove the main results of this paper. Namely, we show Formula (1.2) when the localisa-tion function f is regular (Theorem 3.3) or equal to a characteristic function (Corollary 3.4). We also establish inTheorem 3.8 a discrete-time version of Formula (1.2). The interpretation of these results is discussed in Remarks3.5 and 3.7.In Section 4, we show that our results apply to many Hamiltonian systems ( M, ω, H ) appearing in lit-erature. In the case of finite-dimensional manifolds, we treat, among other examples, Stark Hamiltonians, ho-mogeneous Hamiltonians, purely kinetic Hamiltonians, the repulsive harmonic potential, the simple pendulum,central force systems, the Poincar´e ball model and covering manifolds. In the case of infinite-dimensional man-ifolds, we discuss separately classical and quantum Hamiltonians systems. In the classical case, we treat thewave equation, the nonlinear Schr¨odinger equation and the Korteweg-de Vries equation. In the quantum case,we explain how to recast into our framework the (Hilbertian) examples of [31, Sec. 7], and we also treat anexample of Laplacian on trees and complete Fock spaces. In all these cases, we are able to exhibit a familyof position observables Φ satisfying our assumptions. The diversity of the examples covered by our theory,together with the existence of a quantum analogue [31], make us strongly believe that Formula (1.2) is of nat-ural character. Moreover it also suggests that the existence of time delay is a very common feature of classicalscattering theory. In this section, we prove an integral formula and a summation formula for functions on R d . For this, we startby recalling some properties of a class of averaged localisation functions which appears naturally when dealing4ith quantum scattering theory. These functions, which are denoted R f , are constructed in terms of functions f ∈ L ∞ ( R d ) of localisation around the origin ∈ R d . They were already used, in one form or another, in[17, 31, 32, 37, 38]. We use the notation h x i := p | x | for any x ∈ R d . Assumption 2.1.
The function f ∈ L ∞ ( R d ) satisfies the following conditions:(i) There exists ρ > such that | f ( x ) | ≤ Const . h x i − ρ for almost every x ∈ R d .(ii) f = 1 on a neighbourhood of . It is clear that lim r →∞ f ( x/r ) = 1 for each x ∈ R d if f satisfies Assumption 2.1. Furthermore, one hasfor each x ∈ R d \ { } (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ d µµ (cid:2) f ( µx ) − χ [0 , ( µ ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z d µµ | f ( µx ) − | + Const . Z + ∞ d µ µ − (1+ ρ ) < ∞ , where χ [0 , denotes the characteristic function for the interval [0 , . Therefore the function R f : R d \ { } → C given by R f ( x ) := Z + ∞ d µµ (cid:2) f ( µx ) − χ [0 , ( µ ) (cid:3) is well-defined.In the next lemma we recall some differentiability and homogeneity properties of R f . We also give theexplicit form of R f when f is a radial function. The reader is referred to [38, Sec. 2] for proofs and details. Thesymbol S ( R d ) stands for the Schwartz space on R d . Lemma 2.2.
Let f satisfy Assumption 2.1.(a) Assume that ∂f∂x j ( x ) exists for all j ∈ { , . . . , d } and x ∈ R d , and suppose that there exists some ρ > such that (cid:12)(cid:12) ∂f∂x j ( x ) (cid:12)(cid:12) ≤ Const . h x i − (1+ ρ ) for each x ∈ R d . Then R f is differentiable on R d \ { } , and itsgradient is given by ( ∇ R f )( x ) = Z ∞ d µ ( ∇ f )( µx ) . In particular, if f ∈ S ( R d ) then R f belongs to C ∞ ( R d \ { } ) .(b) Assume that R f belongs to C m ( R d \ { } ) for some m ≥ . Then one has for each x ∈ R d \ { } and t > the homogeneity properties x · ( ∇ R f )( x ) = − , (2.1) t | α | ( ∂ α R f )( tx ) = ( ∂ α R f )( x ) , where α ∈ N d is a multi-index with ≤ | α | ≤ m .(c) Assume that f is radial, i.e. there exists f ∈ L ∞ ( R ) such that f ( x ) = f ( | x | ) for almost every x ∈ R d .Then R f belongs to C ∞ ( R d \ { } ) , and ( ∇ R f )( x ) = − x − x . In the sequel, we say that a function f : R d → C is even if f ( x ) = f ( − x ) for almost every x ∈ R d . Proposition 2.3.
Let f : R d → C be an even function as in Lemma 2.2.(a). Then we have for each x ∈ R d andeach y ∈ R d \ { } lim r →∞ Z ∞ d t h f (cid:16) x − tyr (cid:17) − f (cid:16) x + tyr (cid:17)i = − x · ( ∇ R f )( y ) . (2.2) In particular, if f is radial, the l.h.s. is independent of f and equal to ( x · y ) /y . roof. The change of variables µ := t/r , ν := 1 /r , and the fact that f is even, gives lim r →∞ Z ∞ d t (cid:2) f (cid:0) x − tyr (cid:1) − f (cid:0) x + tyr (cid:1)(cid:3) = lim ν ց Z ∞ d µν (cid:2) f ( νx − µy ) − f ( νx + µy ) (cid:3) = lim ν ց Z ∞ d µ (cid:8) ν (cid:2) f ( νx − µy ) − f ( − µy ) (cid:3) − ν (cid:2) f ( νx + µy ) − f ( µy ) (cid:3)(cid:9) . (2.3)By using the mean value theorem and the assumptions of Lemma 2.2.(a), one obtains that ν (cid:12)(cid:12) f ( νx ± µy ) − f ( ± µy ) (cid:12)(cid:12) ≤ Const . sup ξ ∈ [0 , (cid:10) ξνx ± µy (cid:11) − (1+ ρ ) for some ρ > . Therefore, if µ is big enough, the integrant in (2.3) is bounded by Const . (cid:10) µ | y | − | x | (cid:11) − (1+ ρ ) . for all ν ∈ (0 , . This implies that the integrant in (2.3) is bounded uniformly in ν ∈ (0 , by a functionbelonging to L (cid:0) [0 , ∞ ) , d µ (cid:1) . So, we can apply Lebesgue’s dominated convergence theorem to interchange thelimit on ν with the integration over µ in (2.3). This, together with the fact that ( ∇ f )( − x ) = − ( ∇ f )( x ) , leadsto the desired result: lim r →∞ Z ∞ d t (cid:2) f (cid:0) x − tyr (cid:1) − f (cid:0) x + tyr (cid:1)(cid:3) = Z ∞ d µ (cid:2) x · ( ∇ f )( − µy ) − x · ( ∇ f )( µy ) (cid:3) = − Z ∞ d µ x · ( ∇ f )( µy )= − x · ( ∇ R f )( y ) . The result of Proposition 2.3 can be extended to less regular functions f : R d → C . The interested readercan check that the result holds for functions f admitting a weak derivative f ′ such that, for every real line L ⊂ R d , f ′ is of class L on L (see [41, Thm. 2.1.6]). We only present here the case (of particular interest forthe theory of classical time delay) where f is the characteristic function χ for the unit ball B := { x ∈ R d || x | ≤ } . Lemma 2.4.
One has for each x ∈ R d and each y ∈ R d \ { } lim r →∞ Z ∞ d t h χ (cid:16) x − tyr (cid:17) − χ (cid:16) x + tyr (cid:17)i = x · yy . Proof.
Direct calculations and the change of variables µ := t/r , ν := 1 /r , give Z ∞ d t χ (cid:0) x ± tyr (cid:1) = R ∞ µν χ [0 , (cid:0) | νx ± µy | (cid:1) = Z ∞ d µ χ [0 ,y − ] (cid:16) ν x y ± νµx · yy + µ (cid:17) = Z ∞ d µν χ [0 ,y − ] (cid:16)(cid:0) µ ± νx · yy (cid:1) + ν y (cid:0) x y − ( x · y ) (cid:1)(cid:17) = Z ∞ d µν χ [ − a ( ν,x,y ) ,y − − a ( ν,x,y )] (cid:16)(cid:0) µ ± νx · yy (cid:1) (cid:17) , with a ( ν, x, y ) := ν y (cid:0) x y − ( x · y ) (cid:1) . Now, a ( ν, x, y ) ≥ , and y − − a ( ν, x, y ) ≥ if ν > is small enough.So, the last expression is equal to Z ∞ d µν χ [0 ,y − − a ( ν,x,y )] (cid:16)(cid:0) µ ± νx · yy (cid:1) (cid:17) = Z ∞ d µν χ (cid:2) − √ y − − a ( ν,x,y ) ∓ νx · yy , √ y − − a ( ν,x,y ) ∓ νx · yy (cid:3) ( µ )= 1 ν p y − − a ( ν, x, y ) ∓ x · yy ν is small enough. This implies that lim r →∞ Z ∞ d t (cid:2) χ (cid:0) x − tyr (cid:1) − χ (cid:0) x + tyr (cid:1)(cid:3) = lim ν ց (cid:16) ν p y − − a ( ν, x, y ) + x · yy − ν p y − − a ( ν, x, y ) + x · yy (cid:17) = x · yy . For the next corollary, we need the following version of the Poisson summation formula (see [15, Thm. 5]or [39, Thm. 45]).
Lemma 2.5.
Let g : (0 , ∞ ) → C be a continuous function of bounded variation in (0 , ∞ ) . Suppose that lim t →∞ g ( t ) = 0 and that the improper Riemann integral R ∞ d t g ( t ) exists. Then we have the identity g (0) + X n ≥ g ( n ) = Z ∞ d t g ( t ) + 2 X n ≥ Z ∞ d t cos(2 πnt ) g ( t ) . Corollary 2.6.
Let f : R d → C be an even function such that(i) f = 1 on a neighbourhood of .(ii) For each α ∈ N d with | α | ≤ , the derivative ∂ α f exists and satisfies | ( ∂ α f )( x ) | ≤ Const . h x i − (1+ ρ ) for some ρ > and all x ∈ R d .Then we have for each x ∈ R d and each y ∈ R d \ { } lim r →∞ X n ≥ h f (cid:16) x − nyr (cid:17) − f (cid:16) x + nyr (cid:17)i = − x · ( ∇ R f )( y ) . (2.4) In particular, if f is radial, the l.h.s. is independent of f and equal to ( x · y ) /y .Proof. For r > given, the function g r : (0 , ∞ ) → C , t g r ( t ) := f (cid:0) x − tyr (cid:1) − f (cid:0) x + tyr (cid:1) , satisfies all the hypotheses of Lemma 2.5. Thus lim r →∞ X n ≥ (cid:2) f (cid:0) x − nyr (cid:1) − f (cid:0) x + nyr (cid:1)(cid:3) = lim r →∞ Z ∞ d t g r ( t ) + lim r →∞ X n ≥ Z ∞ d t cos(2 πnt ) g r ( t ) . The first term is equal to − x · ( ∇ R f )( y ) due to Proposition 2.3. For the second term, the change of variables µ := t/r , ν := 1 /r , and two integrations by parts give lim r →∞ X n ≥ Z ∞ d t cos(2 πnt ) g r ( t )= lim ν ց X n ≥ Z ∞ d µν cos(2 πnµ/ν ) (cid:2) f ( νx − µy ) − f ( νx + µy ) (cid:3) = X j y j lim ν ց X n ≥ Z ∞ d µ sin(2 πnµ/ν )2 πn (cid:18) ∂f∂x j ( νx − µy ) + ∂f∂x j ( νx + µy ) (cid:19) = X j y j lim ν ց X n ≥ ν (2 πn ) ∂f∂x j (cid:0) νx (cid:1) − X j,k y j y k lim ν ց X n ≥ Z ∞ d µ ν cos(2 πnµ/ν )(2 πn ) (cid:18) ∂ f∂x k ∂x j ( νx − µy ) + ∂ f∂x k ∂x j ( νx + µy ) (cid:19) . P n ≥ /n < ∞ , one sees directly that the first term is equal to zero. Using the fact that (cid:12)(cid:12) ∂ f∂x k ∂x j ( x ) (cid:12)(cid:12) ≤ Const . h x i − (1+ ρ ) for some ρ > and all x ∈ R d , one also obtains that the second term is equal to zero.Therefore, lim r →∞ X n ≥ (cid:2) f (cid:0) x − nyr (cid:1) − f (cid:0) x + nyr (cid:1)(cid:3) = − x · ( ∇ R f )( y ) , and the claim is proved. In the sequel, we require the presence of a symplectic structure in order to speak of Hamiltonian dynamics.However our results still hold if one is only given a Poisson structure. A lack of examples and some complica-tions in infinite dimension regarding the identification of vector fields with derivations have led us to restrict thediscussion to the symplectic case for the sake of clarity.
Let M be a symplectic manifold, i.e. a smooth manifold endowed with a closed two-form ω such that themorphism T M ∋ X ω ♭ ( X ) := ι X ω is an isomorphism. In infinite dimension, such a manifold is said tobe a strong symplectic manifold (in opposition to a weak symplectic manifold, when the above map is onlyinjective; see [2, Sec. 8.1]). When the dimension is finite, the dimension must be even, say equal to n , andthe n -form ω n must be a volume form. The Poisson bracket is defined as follows: for each f ∈ C ∞ ( M ) wedefine the vector field X f := ( ω ♭ ) − (d f ) , i.e. d f ( · ) = ω ( X f , · ) , and set { f, g } := ω ( X f , X g ) for each f, g ∈ C ∞ ( M ) .In the sequel, the function H ∈ C ∞ ( M ) is an Hamiltonian with complete vector field X H . So, the flow { ϕ t } associated to H is defined for all t ∈ R , it preserves the Poisson bracket: (cid:8) f ◦ ϕ t , g ◦ ϕ t (cid:9) = { f, g } ◦ ϕ t , t ∈ R , and satisfies the usual evolution equation: dd t f ◦ ϕ t = { f, H } ◦ ϕ t , t ∈ R . (3.1)In particular, the Hamiltonian H is preserved along its flow, i.e. H ◦ ϕ t = H for all t ∈ R . We also consider anabstract family Φ ≡ (Φ , . . . , Φ d ) ∈ C ∞ ( M ; R d ) of observables , and define the associated functions ∂ j H := { Φ j , H } ∈ C ∞ ( M ) and ∇ H := ( ∂ H, . . . , ∂ d H ) ∈ C ∞ ( M ; R d ) . Then, one can introduce a natural set of critical points:
Definition 3.1 (Critical points) . The set
Crit ( H, Φ) := ( ∇ H ) − ( { } ) ⊂ M is called the set of critical points associated to H and Φ . The set
Crit ( H, Φ) is closed in M since ∇ H is continuous. Furthermore, since { Φ j , H } = dΦ j ( X H ) , theset Crit ( H ) := (cid:8) m ∈ M | X H ( m ) = 0 (cid:9) ≡ (cid:8) m ∈ M | d H m = 0 (cid:9) of usual critical points of H satisfies the inclusion Crit ( H ) ⊂ Crit ( H, Φ) .Our main assumption is the following: If need be, the results of this article can be extended to the case where H and Φ j are functions of class C with { Φ j , H } also C . ssumption 3.2. One has (cid:8) { Φ j , H } , H (cid:9) = 0 for each j ∈ { , . . . , d } . Assumption 3.2 imposes that all the brackets { Φ j , H } are first integrals of the motion given by H . When M is a symplectic manifold of dimension n , these first integrals are functions of k ∈ { , , . . . , n − } independent first integrals J ≡ H, J , . . . , J k ( J , . . . , J k are independent in the sense that their differential arelinearly independent at each point of M ) . So, one should have { Φ j , H } = g j ( J , . . . , J k ) for some functions g j ∈ C ∞ ( R n ; R ) . Using the properties of { · , H } as a derivation, one infers that (cid:8) g j ( J , . . . , J k ) − Φ j , H (cid:9) = 1 outside g j ( J , . . . , J k ) − ( { } ) . Thus, if k first integrals as J , . . . , J k are known, finding functions Φ j satisfyingAssumption 3.2 is to some extent equivalent to finding functions Φ solving { Φ , H } = 1 (the equivalence isnot complete because these functions Φ are in general not C ∞ since { · , H } is necessarily on Crit ( H ) ).For further use, we define the C ∞ -function T f : M \ Crit ( H, Φ) → R by T f := − Φ · ( ∇ R f )( ∇ H ) . When f is radial, T f is independent of f and equal to T := Φ · ∇ H ( ∇ H ) , due to Lemma 2.2.(c). In fact, the closed subset T − ( { } ) of M \ Crit ( H, Φ) admits an interesting interpretation:If we consider the observables Φ j as the components of an abstract position observable Φ , then ∇ H can be seenas the velocity vector for the Hamiltonian H , and the condition T ( m ) = 0 ⇐⇒ Φ( m ) · ( ∇ H )( m ) = 0 (3.2)means that the position and velocity vectors are orthogonal at m ∈ T − ( { } ) . Alternatively, one has T ( m ) = 0 if and only if the vector fields X | Φ | and X H are ω -orthogonal at m , that is, ω m (cid:0) X | Φ | ( m ) , X H ( m ) (cid:1) = 0 .The simplest example illustrating the condition (3.2) is when Φ( q, p ) := q and H ( q, p ) := | p | are the usualposition and kinetic energy on ( M, ω ) := (cid:0) R n , P nj =1 d q j ∧ d p j (cid:1) . In such a case, (3.2) reduces to q · p = 0 . Next Theorem is our main result. We refer to Remark 3.7 below for its interpretation.
Theorem 3.3.
Let H and Φ satisfy Assumption 3.2. Let f : R d → C be an even function as in Lemma 2.2.(a).Then we have for each point m ∈ M \ Crit ( H, Φ)lim r →∞ Z ∞ d t (cid:2)(cid:0) f (Φ /r ) ◦ ϕ − t (cid:1) ( m ) − (cid:0) f (Φ /r ) ◦ ϕ t (cid:1) ( m ) (cid:3) = T f ( m ) . (3.3) In particular, if f is radial, the l.h.s. is independent of f and equal to Φ( m ) · ( ∇ H )( m )( ∇ H )( m ) .Proof. Equation (3.1) implies that dd t Φ j ◦ ϕ t = { Φ j , H } ◦ ϕ t for each t ∈ R . Similarly, using Assumption 3.2, one gets that dd t { Φ j , H } ◦ ϕ t = (cid:8) { Φ j , H } , H (cid:9) ◦ ϕ t = 0 . So, Φ j varies linearly in t along the flow of X H , and one gets for any m ∈ M (Φ j ◦ ϕ t )( m ) = (Φ j ◦ ϕ )( m ) + t (cid:16) dd t (Φ j ◦ ϕ t )( m ) (cid:12)(cid:12)(cid:12) t =0 (cid:17) = Φ j ( m ) + t ( ∂ j H )( m ) . In the setup of Liouville’s theorem [5, Sec. 49], we have k = n and the first integrals are mutually in involution. Furthermore, on theconnected components of submanifolds given by fixing the values of these n integrals in involution, the flow is conjugate to a translationflow on cylinders R n − ℓ × T ℓ (see [1, Thm. 5.2.24]). lim r →∞ Z ∞ d t (cid:2)(cid:0) f (Φ /r ) ◦ ϕ − t (cid:1) ( m ) − (cid:0) f (Φ /r ) ◦ ϕ t (cid:1) ( m ) (cid:3) = lim r →∞ Z ∞ d t h f (cid:16) Φ( m ) − t ( ∇ H )( m ) r (cid:17) − f (cid:16) Φ( m )+ t ( ∇ H )( m ) r (cid:17)i = T f ( m ) . Due to Lemma 2.4, the proof of Theorem 3.3 also works in the case f = χ . So, we have the followingcorollary. Corollary 3.4.
Let H and Φ satisfy Assumption 3.2. Then we have for each point m ∈ M \ Crit ( H, Φ)lim r →∞ Z ∞ d t (cid:2)(cid:0) χ (Φ /r ) ◦ ϕ − t (cid:1) ( m ) − (cid:0) χ (Φ /r ) ◦ ϕ t (cid:1) ( m ) (cid:3) = Φ( m ) · ( ∇ H )( m )( ∇ H )( m ) . (3.4)We know from the proof of Theorem 3.3 that (Φ j ◦ ϕ t )( m ) = Φ j ( m ) + t ( ∂ j H )( m ) for all t ∈ R and all m ∈ M . (3.5)Therefore, the l.h.s. of (3.3) and (3.4) are zero if m ∈ Crit ( H, Φ) .For the next remark, we recall that any selfadjoint operator A in a Hilbert space H , with spectral measure E A ( · ) , is reduced by an orthogonal decomposition [40, Sec. 7.4] H = H ac ( A ) ⊕ H p ( A ) ⊕ H sc ( A ) ≡ H ac ( A ) ⊕ H s ( A ) , where H ac ( A ) , H p ( A ) , H sc ( A ) and H s ( A ) are respectively the absolutely continuous, the pure point, the sin-gular continuous and the singular subspaces of A . Furthermore, a vector ϕ ∈ H is said to have spectral supportwith respect to A in a set J ⊂ R if ϕ = E A ( J ) ϕ . Remark 3.5. If m ∈ Crit ( H, Φ) , then one must have ϕ t ( m ) ∈ Crit ( H, Φ) for all t ∈ R , since (3.5) implies ( ∂ j H )( ϕ t ( m )) = ( ∂ j H )( m ) for all t ∈ R . Conversely, if m ∈ M \ Crit ( H, Φ) , then one must have ϕ t ( m ) = m for all t = 0 , since Φ cannot take two different values at a same point. So, under Assumption 3.2, each orbit { ϕ t ( m ) } t ∈ R either stays in Crit ( H, Φ) if m ∈ Crit ( H, Φ) , or stays outside Crit ( H, Φ) and is not periodic if m / ∈ Crit ( H, Φ) .In the corresponding Hilbertian framework [31], the Hamiltonian H and the functions Φ j are selfadjointoperators in a Hilbert space H , and the critical set κ associated to H and Φ is a closed subset of the spectrum of H . Outside κ , the spectrum of H is purely absolutely continuous [31, Thm. 3.6.(a)]. Therefore, vectors ψ ∈ H having spectral support with respect to H in κ belong to the singular subspace H s ( H ) of H , and thus leadto orbits { e itH ψ } t ∈ R confined in H s ( H ) (for instance, e itH ψ stays in a one-dimensional subspace of H if ψ is an eigenvector of H ). Conversely, vectors ψ ∈ H having spectral support outside κ belong to the absolutecontinuous subspace H ac ( H ) of H , and thus lead to orbits { e itH ψ } t ∈ R contained in H ac ( H ) (see [3, Prop. 5.7]for the escape properties of such orbits). These properties are the quantum counterparts of the confinement to Crit ( H, Φ) (when m ∈ Crit ( H, Φ) ) and the non-periodicity outside Crit ( H, Φ) (when m / ∈ Crit ( H, Φ) ) of theclassical orbits { ϕ t ( m ) } t ∈ R . Lemma 3.6. If H , Φ and f satisfy the assumptions of Theorem 3.3, then we have { T f , H } ◦ ϕ t ≡ dd t ( T f ◦ ϕ t ) = 1 (3.6) on M \ Crit ( H, Φ) . In particular, one has T f ◦ ϕ t = T f + t on M \ Crit ( H, Φ) . If we interpret the map dd H := { T f , · } as a derivation on C ∞ (cid:0) M \ Crit ( H, Φ) (cid:1) , this implies that T f canbe seen as an observable “derivative with respect to the energy H ” on M \ Crit ( H, Φ) , since dd H ( H ) = { T f , H } = 1 on each orbit { ϕ t ( m ) } t ∈ R , with m ∈ M \ Crit ( H, Φ) .10 roof of Lemma 3.6. The first equality in (3.6) follows from (3.1). For the second one, we use successively thefact that ϕ t leaves invariant H and the Poisson bracket, Assumption 3.2, and Equation (2.1). Doing so, we geton M \ Crit ( H, Φ) the following equalities dd t ( T f ◦ ϕ t ) = − dd t (Φ ◦ ϕ t ) · ( ∇ R f ) (cid:0) { Φ ◦ ϕ t , H } (cid:1) = − dd t (cid:0) Φ + t ( ∇ H ) (cid:1) · ( ∇ R f ) (cid:0) { Φ + t ( ∇ H ) , H } (cid:1) = − dd t (cid:0) Φ + t ( ∇ H ) (cid:1) · ( ∇ R f )( ∇ H )= − ( ∇ H ) · ( ∇ R f )( ∇ H )= 1 . Remark 3.7.
Theorem 3.3 relates the sojourn times of classical orbits within expanding regions of M to theobservable T f . If we consider the observables Φ j as the components of an abstract position observable Φ , thenthe l.h.s. of Formula (3.3) has the following meaning: For r > and m ∈ M \ Crit ( H, Φ) fixed, it can beinterpreted as the difference of times spent by the classical orbit { ϕ t ( m ) } t ∈ R in the past (first term) and inthe future (second term) within the region Σ r := supp[ f (Φ /r )] ⊂ M defined by the localisation observable f (Φ /r ) . Thus, Formula (3.3) shows that this difference of times tends as r → ∞ to the value of the observable T f at m . Since T f can be interpreted as an observable derivative with respect to the energy H , Formula (3.3) provides a new relation between sojourn times and variation of energy along classical orbits. As a final result, we give a discrete-time counterpart of Theorem 3.3, which could be of some interest inthe context of approximation of symplectomorphisms by time- maps of Hamiltonians flows (see e.g. [7], [18,Appendix B], [23] and references therein). Theorem 3.8.
Let H and Φ satisfy Assumption 3.2. Let f : R d → C be an even function such that(i) f = 1 on a neighbourhood of .(ii) For each α ∈ N d with | α | ≤ , the derivative ∂ α f exists and satisfies | ( ∂ α f )( x ) | ≤ Const . h x i − (1+ ρ ) for some ρ > and all x ∈ R d .Then we have for each point m ∈ M \ Crit ( H, Φ)lim r →∞ X n ≥ (cid:2)(cid:0) f (Φ /r ) ◦ ϕ − n (cid:1) ( m ) − (cid:0) f (Φ /r ) ◦ ϕ n (cid:1) ( m ) (cid:3) = T f ( m ) . In particular, if f is radial, the l.h.s. is independent of f and equal to Φ( m ) · ( ∇ H )( m )( ∇ H )( m ) .Proof. Let m ∈ M \ Crit ( H, Φ) . Then we have by Equation (3.5) lim r →∞ X n ≥ (cid:2)(cid:0) f (Φ /r ) ◦ ϕ − n (cid:1) ( m ) − (cid:0) f (Φ /r ) ◦ ϕ n (cid:1) ( m ) (cid:3) = lim ν ց X n ≥ h f (cid:16) Φ( m ) − n ( ∇ H )( m ) r (cid:17) − f (cid:16) Φ( m )+ n ( ∇ H )( m ) r (cid:17)i , and the claim follows by Formula (2.4). In this section we show that Assumption 3.2 is satisfied in various situations. In these situations all the resultsof Section 3 such as Theorem 3.3 or Formula (3.6) hold. Some of the examples presented here are the classicalcounterparts of examples discussed in [31, Sec. 7] in the context of Hilbertian theory.The configuration space of the system under consideration will sometimes be R n , and the correspondingsymplectic manifold M = T ∗ R n ≃ R n . In that case, we use the notation ( q, p ) , with q ≡ ( q , . . . , q n ) and p ≡ ( p , . . . , p n ) , for the canonical coordinates on M , and set ω := P nj =1 d q j ∧ d p j for the canonicalsymplectic form. We always assume that f = χ or that f satisfies the hypotheses of Theorem 3.3.11 .1 ∇ H = g ( H ) Suppose that there exists a function g ≡ ( g , . . . , g d ) ∈ C ∞ ( R ; R d ) such that ∇ H = g ( H ) . Then H and Φ satisfy Assumption 3.2 since { g j ( H ) , H } = 0 for each j . Furthermore, one has Crit ( H, Φ) = ( g ◦ H ) − ( { } ) ,and T f = − Φ · ( ∇ R f ) (cid:0) g ( H ) (cid:1) on M \ Crit ( H, Φ) . We distinguish various cases:(A) Suppose that g is constant, i.e. g = v ∈ R d \ { } . Then Crit ( H ) = Crit ( H, Φ) = ∅ , and we have theequality T f = − Φ · ( ∇ R f )( v ) on the whole of M .Typical examples of functions H and Φ fitting into this construction are Friedrichs-type Hamiltonians andposition functions. For illustration, we mention the case (with d = n ) of H ( q, p ) := v · p + V ( q ) and Φ( q, p ) := q on M := R n , with v ∈ R n \ { } and V ∈ C ∞ ( R n ; R ) . In such a case, one has ∇ H = v and ϕ t ( q, p ) = (cid:0) vt + q, p − R t d s ( ∇ V )( vs + q ) (cid:1) . Stark-type Hamiltonians and momentum functions also fit into the construction, i.e. H ( q, p ) := h ( p )+ v · q and Φ( q, p ) := p on M := R n , with v ∈ R n \ { } and h ∈ C ∞ ( R n ; R ) . In such a case, one has ∇ H = − v and ϕ t ( q, p ) = (cid:0) q + R t d s ( ∇ h )( p − vs ) , p − vt (cid:1) . Note that these two examples are interesting since the Hamiltonians H contain not only a kinetic part, butalso a potential perturbation.(B) Suppose that Φ has only one component ( d = 1 ), and assume that g ( λ ) = λ for all λ ∈ R (in the Hilbertianframework, one says in such a case that H is Φ -homogeneous [9]). Then Crit ( H, Φ) = H − ( { } ) andwe have the equality T f = − Φ( ∇ R f )( H ) on M \ H − ( { } ) . We present a general class of pairs ( H, Φ) satisfying these assumptions:The Hamiltonian flow of the function D ( q, p ) := q · p on R n is given by ϕ Dt ( q, p ) = (e t q, e − t p ) . So, D is the generator of a dilations group on R n (in the Hilbertian framework, the corresponding operator is theusual generator of dilations on L ( R n ) , see e.g. [4, Sec. 1.2]). Therefore, the relation { D, H } ∝ H holdsfor a large class of homogeneous functions H on R n , due to Euler’s homogeneous function theorem. Letus consider an explicit situation. Take α > and let M be some open subset of ( R n \ { } ) × R n . Defineon M the function Φ := α D and the Hamiltonian H ( q, p ) := h ( p ) + V ( q ) , where h ∈ C ∞ ( R n ; R ) ispositive homogeneous of degree α and V ∈ C ∞ ( R n \ { } ; R ) is positive homogeneous of degree − α .Then one has ∇ H ≡ { Φ , H } = H on M , and Crit ( H ) = (cid:8) ( q, p ) ∈ M | ( ∇ h )( p ) = ( ∇ V )( q ) = 0 (cid:9) ⊂ (cid:8) ( q, p ) ∈ M | p · ( ∇ h )( p ) = q · ( ∇ V )( q ) = 0 (cid:9) = (cid:8) ( q, p ) ∈ M | H ( q, p ) = 0 (cid:9) = Crit ( H, Φ) . Furthermore, if the functions h and V and the subset M are well chosen, the Hamiltonian vector field X H of H is complete. For instance,(i) If V ≡ , then one can take M = R n , and one has ϕ t ( q, p ) = (cid:0) q + t ( ∇ h )( p ) , p (cid:1) and Crit ( H ) = (cid:8) ( q, p ) ∈ M | ( ∇ h )( p ) = 0 (cid:9) ⊂ (cid:8) ( q, p ) ∈ M | p · ( ∇ h )( p ) = 0 (cid:9) = Crit ( H, Φ) (when h ( p ) = | p | is the classical kinetic energy, one has Crit ( H ) = Crit ( H, Φ) = R n × { } ).(ii) Let K > . Then the Hamiltonian given by H ( q, p ) := ( | p | + K | q | − ) on M := R n \ { } × R n has a complete Hamiltonian vector field X H . To see it, we use the push-forward of X H by thediffeomorphism ι : R n \ { } × R n → R n \ { } × R n , ( q, p ) (cid:0) q | q | − , p (cid:1) ≡ ( r, p ) , namely, [ ι ∗ ( X H )]( r, p ) = X j (cid:18)(cid:0) | r | p j − p · r ) r j (cid:1) ∂∂r j (cid:12)(cid:12)(cid:12) ( r,p ) + Kr j | r | ∂∂p j (cid:12)(cid:12)(cid:12) ( r,p ) (cid:19) . ι ∗ ( X H ) is complete by using the criterion [1, Prop. 2.1.20] with the properfunction g : R n \ { } × R n → [0 , ∞ ) given by g ( r, p ) := | p | + K | r | . Since ι is a diffeomorphism,this implies that X H is also complete (see [19, Lemma 1.6.4]).(C) Many other examples with ∇ H = g ( H ) can be obtained using homogeneous Hamiltonians functions.For instance, consider H ( q, p ) := q /q + q /q and Φ( q, p ) := p q + p q on M := ( R \ { } ) × R .Then one has ∇ H = H − , ϕ t ( q, p ) = (cid:0) q, p − t ∂H∂q ( q, p ) (cid:1) and Crit ( H ) = Crit ( H, Φ) = (cid:8) q ∈ R \ { } | q = ± q (cid:9) × R . H = h ( p ) Consider on M := R n a purely kinetic Hamiltonian H ( q, p ) := h ( p ) with h ∈ C ∞ ( R n ; R ) , and take the usualposition functions Φ( q, p ) := q with d = n . Then ϕ t ( q, p ) = (cid:0) q + t ( ∇ h )( p ) , p (cid:1) , ∇ H = ∇ h , and Assumption3.2 is satisfied: (cid:8) { Φ j , H } , H (cid:9) = (cid:8) ( ∂ j h )( p ) , h ( p ) (cid:9) = 0 . In this example, we have
Crit ( H ) = Crit ( H, Φ) = R n × ( ∇ h ) − ( { } ) . {{ Φ j , H } , H } = 0 as a differential equation Consider on M := R n an Hamiltonian function H with partial derivatives H p k := ∂H/∂p k and H q k := ∂H/∂q k . Then, finding the functions Φ j of Assumption 3.2 amounts to solving for Φ the second-order linearequation (cid:8) { Φ , H } , H (cid:9) ≡ (cid:18) n X ℓ =1 (cid:0) H p ℓ ∂ q ℓ − H q ℓ ∂ p ℓ (cid:1)(cid:19) Φ = 0 . As observed in Section 3.1, this is essentially equivalent (when k independent first integrals J ≡ H, J , . . . , J k are known) to find the solutions Φ to { Φ , H } = n X ℓ =1 (cid:0) H p ℓ ∂ q ℓ − H q ℓ ∂ p ℓ (cid:1) Φ = g ( J , . . . , J k ) . (4.1)The case g ≡ is sufficient, though trying to solve { Φ , H } = 1 can at best provide solutions which are C ∞ outside the set Crit ( H ) . A way to remove these singularities could be to multiply the solutions by a function g ( H ) that vanishes and is infinitely flat on Crit ( H ) . For instance, if H (cid:0) Crit ( H ) (cid:1) consists of a finite number ofvalues c , . . . , c s ∈ R , one could take g ( H ) = Q sj =1 e − ( H − c j ) − . Another possibility is to restrict the study toa submanifold M ′ of M (typically an open subset of the same dimension). However, problems can arise as thesame (induced) symplectic structure (or Poisson bracket) must be used for the dynamic to remain unchanged;in particular, it must checked that the Hamiltonian flow preserves M ′ .(A) Repulsive harmonic potential. In this example we first solve the equation { Φ , H } = 1 , and then correctthe functions Φ to make them C ∞ . So, let us consider for K = 0 the Hamiltonian H ( q, p ) := (cid:0) | p | − K | q | (cid:1) on M := R n . One can check that Crit ( H ) = { } and that ϕ t ( q, p ) = (cid:0) Kq + p K e Kt + Kq − p K e − Kt , Kq + p e Kt − Kq − p e − Kt (cid:1) . For j ∈ { , . . . , n } , take Φ j ( q, p ) := K tanh − ( Kq j /p j ) , where tanh − ( z ) ≡ ln (cid:12)(cid:12) z − z (cid:12)(cid:12) is C ∞ on R \ {± } . Whenever p j = ± Kq j , the Φ j are not well-defined, but outside these regions, they satisfy { Φ j , H } = 1 . It is possible in this case to get rid of the singular regions. Indeed, the functions H j ( q, p ) := (cid:0) p j − K ( q j ) (cid:1) are first integrals of the motion and the singular regions correspond to the level sets H − j ( { } ) . Therefore, the functions Φ ′ j := e − H − j Φ j are well-defined and satisfy Assumption 3.2: (cid:8) { Φ ′ j , H } , H (cid:9) = (cid:8) e − H − j , H (cid:9) = 0 . In this example, one has { } = Crit ( H ) ( Crit ( H, Φ ′ ) = T j H − j ( { } ) .13B) Simple pendulum. In this example we first consider the dynamics on a manifold and then restrict it to anappropriate submanifold. For K > , take H ( q, p ) := (cid:0) p + K (1 − cos q ) (cid:1) on M := R . One has Crit ( H ) = π Z × { } (the values q ∈ π Z correspond to minima, while q ∈ π Z + π correspond toinflexion points). Then, consider the open subset M ′ of M defined by the relation H > K , i.e. M ′ := (cid:8) ( q, p ) ∈ R | p / − K cos ( q/ > (cid:9) . One verifies easily that M ′ is preserved by the Hamiltonianflow, that M ′ ∩ Crit ( H ) = ∅ and that M ′ corresponds to the region where the values of q along an orbitcover all of R . Define also Φ( q, p ) := s H ( q, p ) F (cid:0) q/ (cid:12)(cid:12)p K/H ( q, p ) (cid:1) ≡ √ Z q/ d ϑ q H ( q, p ) − K sin ( ϑ ) , where F ( · | · ) denotes the incomplete elliptic integral of the first kind. Then one verifies that the function Φ is well-defined on M ′ and a direct calculation gives { Φ , H } ( q, p ) = p/ | p | for each ( q, p ) ∈ M ′ . Now, p/ | p | = 1 on one connected component of M ′ and p/ | p | = − on the other one. Thus Assumption 3.2 isverified on M ′ and Crit ( H, Φ) = ∅ .(C) Unbounded trajectories of central force systems. Once again, we first consider the dynamics on a manifoldand then restrict it to an appropriate submanifold. For K ∈ R \ { } , take H ( q, p ) := (cid:0) | p | − K | q | − (cid:1) on M := ( R n \ { } ) × R n , with n > if K > and n ≥ if K < . One has Crit ( H ) = ∅ .When K > (and n > ), we must restrict our attention to the case where the Hamiltonian function H is positive (to avoid periodic orbits), and where at least one of the two-dimensional angular momenta L ij ( q, p ) := q i p j − q j p i is nonzero (to avoid collisions, i.e. orbits whose flow is not defined for all t ∈ R ,see [29]). Therefore, the open set M ′ := (cid:8) ( q, p ) ∈ M | H ( p, q ) > , P ni,j =1 | L ij ( q, p ) | = 0 (cid:9) is anappropriate submanifold of M when K > .Consider now the real valued functions on M (resp. M ′ ) when K < (resp. K > and n > ) given by Φ ± ( q, p ) := p · q H ( q, p ) ∓ K (cid:0) H ( q, p ) (cid:1) / ln (cid:16) | q | (cid:0) H ( q, p ) + | p | (cid:1) ± p H ( q, p ) p · q (cid:17) . Since | p | < H ( q, p ) (resp. | p | > H ( q, p ) ), then (cid:0)p H ( q, p ) − | p | (cid:1) > ⇒ H ( q, p ) | p | ± p H ( q, p ) p · q | q | > ⇐⇒ | q | (cid:0) H ( q, p ) + | p | (cid:1) ± p H ( q, p ) p · q > . So, Φ ± are well-defined, and further calculations show that { Φ ± , H } = 1 on M (resp. M ′ ). As before, Crit ( H ) = Crit ( H, Φ ± ) = ∅ . Note that Φ ± ( q, p ) = p · q/ | p | when K = 0 , which is coherent with thecanonical function Φ for the purely kinetic Hamiltonian H ( q, p ) = | p | .One can construct a more intuitive function Φ in terms of Φ ± , namely, Φ ( q, p ) := (Φ + + Φ − )( q, p ) = p · q H ( q, p ) − K (cid:0) H ( q, p ) (cid:1) / tanh − (cid:18) p H ( q, p ) p · q | q | (cid:0) H ( q, p ) + | p | (cid:1) (cid:19) , which also satisfies { Φ , H } = 1 . Since the functions satisfying Assumption 3.2 are linear in t , one canregard them as inverse functions for the flow. The appearance of the inverse hyperbolic function tanh − in Φ is related to the fact that unbounded trajectories of the central force system given by H > arehyperbolas.(D) Poincar´e ball model. Consider B := (cid:8) q ∈ R n | | q | < (cid:9) endowed with the Riemannian metric g givenby g q ( X q , Y q ) := 4(1 − | q | ) ( X q · Y q ) , q ∈ B , X q , Y q ∈ T q B ≃ R n . M := T ∗ B ≃ (cid:8) ( q, p ) ∈ B × R n (cid:9) be the cotangent bundle on B with symplectic form ω := P nj =1 d q j ∧ d p j , and let H : M → R , ( q, p ) n X j,k =1 g jk ( q ) p j p k = | p | (cid:0) − | q | (cid:1) be the kinetic energy Hamiltonian. It is known that the integral curves of the vector field X H correspondto the geodesics curves of ( B , g ) (see [19, Thm. 1.6.3] or [11, Sec. 6.4]). Since, ( B , g ) is geodesicallycomplete (see Proposition 3.5 and Exercice 6.5 of [25]), this implies that X H is complete. There remainsonly to find a function Φ satisfying Assumption 3.2 in order to apply the theory.Some calculations using spherical-type coordinates suggest the function Φ : M → R , ( q, p ) e − /H ( q,p ) tanh − ( p · q )(1 − | q | ) p H ( q, p )(1 + | q | ) ! . Indeed, since (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( p · q )(1 − | q | ) p H ( q, p )(1 + | q | ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) p · q ) | p | (1 + | q | ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | q | | q | < , the function Φ is well-defined. Furthermore, direct calculations show that Φ is C ∞ and that { Φ , H } =e − /H √ H . Therefore, Assumption 3.2 is verified and one has Crit ( H ) = Crit ( H, Φ) = B × { } .In one dimension, q ( t ) := tanh( t ) is (up to speed and direction) the only geodesic curve, and Φ( q, p ) = e − /H ( q,p ) tanh − (cid:18) pq | p | (1 + q ) (cid:19) = 2 e − /H ( q,p ) p | p | tanh − ( q ) . So, apart from the smoothing factor − /H , our Φ coincides in one dimension with the inverse functionof the flow. In this subsection we briefly discuss a way of avoiding the obstruction of periodic orbits: Given M a symplecticmanifold with symplectic form ω and Hamiltonian H , we let π : f M → M \ Crit ( H ) be C ∞ -covering manifold.In order to preserve the dynamics, we endow the manifold f M with the pullback e ω := π ∗ ω of the symplecticform ω and with the pullback e H := π ∗ H of the Hamiltonian H . Here are two simple examples of finite-dimensional symplectic covering manifolds.(A) Consider on the sphere M := S (as seen in R and with its standard symplectic structure) the Hamil-tonian H given by the projection onto the z -coordinate. Outside the polar critical points, all the orbitsare periodic: the flow corresponds to rotations around the z -axis. In this case, one can use the covering of S \ { (0 , , ± } given by f M := (cid:8) ( ϑ, z ) | ϑ ∈ R , z ∈ ( − , } and the covering map π : f M → M \ Crit ( H ) ≡ S \ { (0 , , ± } , ( ϑ, z ) (cid:0)p − z cos( ϑ ) , p − z sin( ϑ ) , z (cid:1) . Consequently, e H : f M → ( − , is the projection onto the z -coordinate and e ω = d ϑ ∧ d z . One can alsocheck that ϕ t ( ϑ, z ) := ( ϑ + t, z ) is the flow of e H and that (cid:8) Φ , e H (cid:9) = 1 for Φ( ϑ, z ) := ϑ . So, Assumption3.2 is verified on f M and Crit ( e H ) = Crit ( e H, Φ) = ∅ . If one wants to consider only a Poisson manifold M , a Poisson structure can also be defined on f M given that π is C ∞ . Indeed, for U ⊂ M \ Crit ( H ) a sufficiently small open set ( i.e. such that π − ( U ) is a disjoint union of diffeomorphic copies), connected componentsof π − ( U ) are diffeomorphic to U and the Poisson structure can be induced by this diffeomorphism. M := R n (with its standard symplectic structure) the Hamiltoniangiven by H ( q, p ) := (cid:0) | p | + K | q | (cid:1) , where K ∈ R \{ } . Define f M := (cid:8) ( r, ϑ ) | r ∈ (0 , ∞ ) n , ϑ ∈ R n (cid:9) and π : f M → M \ Crit ( H ) ≡ R n \ { } , with π ( r, ϑ ) := (cid:0) K − r cos( ϑ ) , . . . , K − r n cos( ϑ n ) , r sin( ϑ ) , . . . , r n sin( ϑ n ) (cid:1) . Then e H ( r, ϑ ) = | r | , e ω = K − P nj =1 r j d r j ∧ d θ j , and ϕ t ( r, ϑ ) = ( r, ϑ − Kt ) is the flow of e H .Furthermore, one has (cid:8) Φ j , e H (cid:9) = − K for each function Φ j ( r, ϑ ) := ϑ j . Therefore, Assumption 3.2 isverified on f M with Φ ≡ (Φ , . . . , Φ n ) and Crit ( e H ) = Crit ( e H, Φ) = ∅ . In the following examples, the infinite dimensional manifold M is either L ( R ) or L ( R ) ⊕ L ( R ) (equivalenceclasses of real valued square integrable functions) . The atlas of M consists in only one chart, the tangent space T u M at a point u ∈ M is isomorphic to M , and the Riemannian metric on M is flat ( i.e. independent of thebase point in M ) and given by the usual scalar product h · , · i on L ( R ) or L ( R ) ⊕ L ( R ) .To define the symplectic form on M in terms of the metric h · , · i we let H s , s ∈ R , denote the real Sobolevspace H s ( R ) or H s ( R ) ⊕ H s ( R ) (see [4, Sec. 4.1] for the definition in the complex case) and we let S denotethe real Schwartz space S ( R ) or S ( R ) ⊕ S ( R ) . Then we consider an operator J : S → S (which can beinterpreted by continuity as an endomorphism of the tangent spaces T u M ≃ M ) satisfying the following:(i) There exists a number d J ≥ , called the order of J , such that for each s ∈ R the operator J extends toan isomorphism H s → H s − d J (which we denote by the same symbol).(ii) J is antisymmetric on S , i.e. h Jf, g i = −h f, Jg i for all f, g ∈ S .It is known [22, Lemma 1.1] that the operator ¯ J := − J − : M → H d J (of order − d J ) is bounded andanti-selfadjoint in M . In consequence, for each s ≥ the map ω : H s × H s → R given by ω ( f, g ) := − (cid:10) ¯ Jf, g (cid:11) defines a symplectic form on H s .The functions on the phase space (such as H or Φ j ) are infinitely Fr´echet differentiable mappings from O s H (a subset of H s H for some s H ≥ ) to R , i.e. elements of C ∞ ( O s H ; R ) . The Hamiltonian function H ∈ C ∞ ( O s H ; R ) is defined as follows: for some h ∈ C ∞ ( R k +1 ; R ) (or h ∈ C ∞ ( R k +1) ; R ) if M = L ( R ) ⊕ L ( R ) ), one has for each u ∈ O s H H ( u ) := Z R d x h ( u , u , . . . , u s H ) , where u j := d j u d x j . Since H ∈ C ∞ ( O s H ; R ) , the differential of H at u ∈ O s H on a tangent vector f ∈ S ⊂ M ≃ T u M is given by d H u ( f ) = lim t → t (cid:2) H ( u + tf ) − H ( u ) (cid:3) = Z R d x s H X j =0 ∂h∂u j d j f d x j = s H X j =0 Z R d x ( − j f d j d x j ∂h∂u j , where the second equality is obtained using integrations by parts (with vanishing boundary contributions).The (Riemannian) gradient vector field grad H associated to the linear functional d H satisfies by definition (cid:10) (grad H )( u ) , f (cid:11) = d H u ( f ) for all u ∈ O s H and f ∈ S (here (grad H )( u ) a priori only belongs to the In the case of the wave and the Schr¨odinger equations below, one can easily extend the results to the situation where L ( R ) is replacedby L ( R n ) . We restrict ourselves to the case n = 1 for the sake of notational simplicity. S ∗ of S , which means that h · , · i denotes a priori the duality map between S ∗ and S ). So, (grad H )( u ) is given by (grad H )( u ) = s H X j =0 ( − j d j d x j ∂h∂u j . (4.2)Then, the Hamiltonian vector field X H is the map O s H → S ∗ satisfying (cid:10) ¯ Jf, X H ( u ) (cid:11) = − ω (cid:0) f, X H ( u ) (cid:1) = d H u ( f ) = (cid:10) f, (grad H )( u ) (cid:11) for all u ∈ O s H and f ∈ S . Since ¯ J is anti-selfadjoint, this implies that ¯ JX H ( u ) = − (grad H )( u ) in S ∗ , which is equivalent to X H ( u ) = J (grad H )( u ) in S ∗ . So, the equation of motion with Hamiltonian H has the form dd t u = J (grad H )( u ) , and { Φ , H } = dΦ( X H ) = (cid:10) gradΦ , J (grad H ) (cid:11) for all functions Φ , H ∈ C ∞ ( O s H ; R ) with appropriate gradient.Before passing to concrete examples, we refer to [20] for standard results on the local existence in time ofHamiltonian flows (global existence is specific to the system considered).(A) The wave equation. We refer to [1, Ex. 5.5.1], [2, Ex. 8.1.12], [13, Sec. 2.1] and [30, Sec. X.13] for adescription of the model. The existence of the flow for all times depends on the nonlinear term in theHamiltonian (see for instance [30, Thm. X.74] and the corollary that follows).In this example, the scale {H s } s ≥ is given by H s := H s ( R ) ⊕ H s ( R ) . The metric on M := L ( R ) ⊕ L ( R ) is given for each ( p, q ) , ( e p, e q ) ∈ M by (cid:10) ( p, q ) , ( e p, e q ) (cid:11) := R R d x ( p e p + q e q ) , and the operator J isgiven by J : M → M, ( p, q ) ( − q, p ) . It is an isomorphism of degree with ¯ J = J . Given m ≥ and F ∈ C ∞ ( R ; R ) , one can find a subset O ⊂ H (depending on F ) such that the Hamiltonian function H : O → R , ( p, q ) Z R d x h ( p, q, ∂ x q ) ≡ Z R d x (cid:8) p + ( ∂ x q ) + m q + 2 F ( q ) (cid:9) , is well-defined and C ∞ . In fact, we assume that O is chosen such that (i) all the functions on the phasespace appearing below are elements of C ∞ ( O ; R ) , and (ii) integrations by parts involving these functionscome vanishing boundary contributions. Then one checks that (grad H )( p, q ) = (cid:0) p, m q + F ′ ( q ) − ∂ x q (cid:1) due to (4.2), and that X H ( p, q ) is trivial if and only if p = 0 and m q + F ′ ( q ) − ∂ x q = 0 . The constrainton q depends on the choice of F . For example, when F ( q ) = 0 , q or q , the solution q of the differentialequation does not decay as | x | → ∞ . In consequence, the corresponding pairs ( p, q ) cannot belong to M ,and Crit ( H ) = { (0 , } . The equation of motion dd t ( p, q ) = J (grad H )( p, q ) (4.3)coincides with the usual the wave equation since the combination of dd t p = ∂ x q − m q − F ′ ( q ) and dd t q = p gives d d t q = ∂ x q − m q − F ′ ( q ) . When m = 0 , this equation is called the Klein-Gordon equation, and F is usually assumed to be anonlinear term of the form F ( q ) = q λ for some λ ∈ R . A first relevant observation is that the function C ∈ C ∞ ( O ; R ) given by C ( p, q ) := R R d x p ( ∂ x q ) is a first integral of the motion. Furthermore, thefunction Φ ∈ C ∞ ( O ; R ) given by Φ ( p, q ) := R R d x id R h ( p, q, ∂ x q ) has gradient (gradΦ )( p, q ) = (cid:0) id R p, id R m q + id R F ′ ( q ) − ∂ x (id R ∂ x q ) (cid:1) . Therefore, { Φ , H } ( p, q ) = (cid:10) (gradΦ )( p, q ) , J (grad H )( p, q ) (cid:11) = Z R d x p (cid:8) id R ∂ x q − ∂ x (id R ∂ x q ) (cid:9) = − C ( p, q ) , Φ satisfies Assumption 3.2. Here, we clearly have Crit ( H, Φ ) = C − ( { } ) = (cid:8) ( p, q ) ∈ O | R R p ( ∂ x q ) d x = 0 (cid:9) ) { (0 , } = Crit ( H ) . If we assume further that F ≡ , then the equation of motion (4.3) is linear. Therefore any pair ( ∂ jx p, ∂ jx q ) , j ≥ , with ( p, q ) a solution of (4.3), also satisfies (4.3). Consequently, if the subsets O j ⊂ H j haveproperties similar to the ones of O , then the functions C j ∈ C ∞ ( O j +1 ; R ) and H j ∈ C ∞ ( O j +1 ; R ) given by C j ( p, q ) := R R d x (cid:0) ∂ jx p (cid:1)(cid:0) ∂ j +1 x q (cid:1) and H j ( p, q ) := R R d x h (cid:0) ∂ jx p, ∂ jx q, ∂ j +1 x q (cid:1) are first integralsof the motion. Accordingly, one deduces that the functions Φ j ∈ C ∞ ( O j +1 ; R ) given by Φ j ( p, q ) := R R d x id R h (cid:0) ∂ jx p, ∂ jx q, ∂ j +1 x q (cid:1) satisfy { Φ j , H } = − C j on O j +1 . So, if F ≡ , there is an infinitefamily of functions Φ j satisfying Assumption 3.2, and one has again Crit ( H, Φ j ) ) Crit ( H ) , with ∂ jx : Crit ( H, Φ j ) → Crit ( H, Φ ) an isomorphism.Finally, when F ≡ and m = 0 one can check that the function e Φ ∈ C ∞ ( O ; R ) given by e Φ ( p, q ) := R R d x id R p ( ∂ x q ) has gradient (cid:0) grad e Φ (cid:1) ( p, q ) = (id R ∂ x q, − id R ∂ x p − p ) . Then, (cid:8)e Φ , H (cid:9) ( p, q ) = Z R d x (cid:0) id R ( ∂ x q )( ∂ x q ) − id R p ∂ x p − p (cid:1) = − Z R d x (cid:0) ( ∂ x q ) + p (cid:1) = − H ( p, q ) , where the third equality is obtained using integrations by parts (with vanishing boundary contributions).Thus e Φ satisfies Assumption 3.2. Furthermore, since { e Φ , H } ( p, q ) = 0 implies R R d x (cid:8) ( ∂ x q ) + p (cid:9) =0 , one has Crit ( H, e Φ ) = Crit ( H ) = { (0 , } . As before, any derivative of a solution of the equa-tion of motion is still a solution of the equation of motion. So, it can be checked that the functions e Φ j ∈ C ∞ ( O j +1 ; R ) given by e Φ j ( p, q ) := R R d x id R (cid:0) ∂ jx p (cid:1)(cid:0) ∂ j +1 x q (cid:1) satisfy { e Φ j , H } = − H j on O j +1 .Therefore, one has once again Crit ( H, e Φ j ) = Crit ( H ) = { (0 , } and the e Φ j ’s constitutes a secondinfinite family of functions satisfying Assumption 3.2.(B) The nonlinear Schr¨odinger equation. We refer to [22, Ex. 1.3, p. 3 & 5] for a description of the model.The existence of the flow for all times depends on the nonlinear term in the Hamiltonian (see for instance[8, Sec. I.2] and [33, Sec. 3.2.2-3.2.3]).The setting is the same as that of the previous example, except that the Hamiltonian function H ∈ C ∞ ( O ; R ) is given by H ( p, q ) := Z R d x (cid:8) ( ∂ x p ) + ( ∂ x q ) + V · ( p + q ) + F ( p + q ) (cid:9) , where V, F ∈ C ∞ ( R ; R ) . Using (4.2), one checks that the gradient of H at ( p, q ) ∈ O is (grad H )( p, q ) = (cid:0) − ∂ x p + V p + pF ′ ( p + q ) , − ∂ x q + V q + qF ′ ( p + q ) (cid:1) . So, the equation of motion dd t ( p, q ) = J (grad H )( p, q ) is equivalent to the nonlinear Schr¨odinger equation dd t u = i (cid:0) − ∂ x u + V u + uF ′ ( | u | ) (cid:1) , (4.4)with u := p + iq . Without additional assumptions on F or V , it is hardly possible to determine the set Crit ( H ) of functions u for which the r.h.s. of (4.4) vanishes. However, it is known that in general Crit ( H ) is not trivial, as in the case of elliptic stationary nonlinear Schr¨odinger equations (see Theorem 1.1 andProposition 1.1 of [6]).Now, assume that V ≡ F ≡ and for each j ≥ let O j ⊂ H j be a subset having properties similarto the ones of O . Then the functions H j ∈ C ∞ ( O j ; R ) and C j ∈ C ∞ ( O j +1 ; R ) given by H j ( p, q ) := R R d x (cid:8) ( ∂ jx q ) + ( ∂ jx p ) (cid:9) ≡ R R d x h j ( p, q ) and C j ( p, q ) := R R d x (cid:8) ( ∂ jx q )( ∂ j +1 x p ) − ( ∂ j +1 x q )( ∂ jx p ) (cid:9) ≡ R R d x c j ( p, q ) are first integrals of the motion. Furthermore, the functions Φ j ∈ C ∞ ( O j ; R ) and e Φ j ∈ C ∞ ( O j +1 ; R ) given by Φ j ( p, q ) := R R d x id R h j ( p, q ) and e Φ j ( p, q ) := R R d x id R c j ( p, q ) satisfy { Φ j , H } = j and { e Φ j , H } = 4 H j +1 on O j +1 . So, the Φ j ’s and the e Φ j ’s constitute two infinite families of func-tions satisfying Assumption 3.2. Note that the sets Crit ( H, Φ j ) = C − j ( { } ) = (cid:8) ( p, q ) ∈ O j +1 | R R d x ( ∂ jx q )( ∂ j +1 x p ) = 0 (cid:9) (with isomorphisms ∂ jx : Crit ( H, Φ j ) → Crit ( H, Φ ) ) are rather large,whereas Crit ( H, e Φ j ) = Crit ( H ) = { (0 , } .Some of the above functions still work when V and F are not trivial. For instance, the identity { Φ , H } = C on O remains valid for all V and F . Furthermore, if V = Const . , then { C , H } = 0 on O . Conse-quently, Φ satisfies Assumption 3.2 for all F and for V = Const . , and one has Crit ( H, Φ ) ) Crit ( H ) .This last example is interesting since it applies to a large class of nonlinear Schr¨odinger equations.(C) The Korteweg-de Vries equation. Among many other possible references, we mention [1, Ex. 5.5.7] and[22, Ex. 1.4, p. 3 & 5]. For the global existence of the flow, we refer the reader to [12, Sec. 1] and referencestherein.In this example, the scale {H s } s ≥ is given by H s := H s ( R ) and the sets O j , j ∈ N , are appropriatesubsets of H j . The Hamiltonian function H ∈ C ∞ ( O ; R ) is given by H ( u ) := Z R d x (cid:0) ( ∂ x u ) + u (cid:1) , and the isomorphism J := ∂ x is of order .The gradient of H at u ∈ O is − ∂ x u + 3 u . So, the elements of Crit ( H ) are functions u satis-fying − ∂ x u + 3 u = 0 ; these are Weierstrass ℘ -functions [21, Sec. 134.F], that is, functions withmany singularities and no decay at infinity. Thus, Crit ( H ) = { } . Furthermore, the equation of motion dd t u = J (grad H )( u ) coincides with the KdV equation dd t u = ∂ x (cid:0) − ∂ x u + 3 u (cid:1) .There exists an infinite number of first integrals of the motion with polynomial density, that is, of the form H j := R R d x h j , where h j is a finite polynomial in u and its derivatives (see [27, Sec. 3]). For example,when h ( u ) = u , h ( u ) = u , h ( u ) = ( ∂ x u ) + u , or h ( u ) = ( ∂ x u ) + 10 u ( ∂ x u ) + 5 u . So, let Φ ∈ C ∞ ( O ; R ) be given by Φ ( u ) := R R d x id R u . Then the gradient of Φ at u is id R , and { Φ , H } = − H on O . Since H is a first integral of the motion, this implies that Φ satisfies Assumption 3.2.Furthermore, the fact that H ( u ) = k u k L ( R ) implies that Crit ( H, Φ ) = { } = Crit ( H ) .Looking for others Φ of the form Φ( u ) = R R d x g ( x ) G ( u, ∂ x u, . . . , ∂ kx u ) , with G a polynomial and g a C ∞ function, is unnecessay. Indeed, both { Φ , H } and Υ( t ) := Φ − t { Φ , H } are first integrals of themotion with density C ∞ in x and polynomial in u and its derivatives (and t -linear in the case of Υ ). Thus,we know from [34, Thm. 1 & Rem. 3] that they are completely characterised, up to the usual equivalenceof conservation laws [28, Sec. 4.3]. Therefore, the functions Φ are also completely characterised. Notehowever, that it is not excluded that functions Φ with an integrand G involving fractional derivatives,an infinite number of derivatives, or of class C ∞ might work. Non-polynomial conserved densities areknown to exist in the periodic case (see [27, Sec. 5]). Let H be a complex Hilbert space, with scalar product h · , · i antilinear in the left entry. Define on H the usualquantum-mechanical symplectic form ω : H × H → R , ( ψ , ψ ) Im h ψ , ψ i . The pair ( H , ω ) has the structure of an (infinite-dimensional) symplectic vector space. Now, define for anybounded selfadjoint operator H op ∈ B ( H ) the expectation value Hamiltonian function H : H → R , ψ
7→ h H op i ( ψ ) := h ψ, H op ψ i . Then, it is known [26, Cor. 2.5.2] that the vector field and the flow associated to H are X H = − iH op and ϕ t ( ψ ) = e − itH op ψ . Therefore, the Poisson bracket of two such Hamiltonian functions H, K satisfies for each ψ ∈ H { K, H } ( ψ ) = ω (cid:0) X K ( ψ ) , X H ( ψ ) (cid:1) = − ω ( K op ψ, H op ψ ) = (cid:10) ψ, i [ K op , H op ] ψ (cid:11) . H ≡ h H op i and Φ j ≡ h (Φ j ) op i satisfying the commutation relation (cid:2) [(Φ j ) op , H op ] , H op (cid:3) = 0 . (4.5)In concrete examples, the operators H op and (Φ j ) op are usually unbounded. Therefore, the preceding calcula-tions can only be justified (using the theory of sesquilinear forms) on subspaces of H where all the operatorsare well-defined. We do not present here the whole theory since much of it, examples included, is similar to thatof [31]. We prefer to present a new example inspired by [16], where all the calculations can be easily justified.Let U be an isometry in H admitting a number operator, that is, a selfadjoint operator N such that U N U ∗ = N − . Define on H the bounded selfadjoint operators ∆ := Re ( U ) ≡ ( U + U ∗ ) and S := Im ( U ) ≡ i ( U − U ∗ ) . Then we know from [16, Sec. 3.1] that any polynomial in U and U ∗ leaves invariant the domain D ( N ) ⊂ H of N . In particular, the operator A := ( SN + N S ) , D ( A ) := D ( N ) , is well-defined and symmetric. In fact, it is shown that A admits a selfadjoint extension A with domain D ( A ) = D ( N S ) . Furthermore, one has on D ( N ) the identity i [ A, ∆] = ∆ − . So, if we define the Hamiltonianfunctions H : H → R , ψ
7→ h ∆ i ( ψ ) and Φ : D ( N ) → R , ψ
7→ h A i ( ψ ) , we obtain for each ψ ∈ D ( N )( ∇ H )( ψ ) = { Φ , H } ( ψ ) = h i [ A, H ] i ( ψ ) = h ∆ − i ( ψ ) , and Assumption 3.2 is verified for each ψ ∈ D ( N ) : (cid:8) { Φ , H } , H (cid:9) ( ψ ) = ω (cid:0) X h ∆ − i ( ψ ) , X h ∆ i ( ψ ) (cid:1) = (cid:10) i [∆ − , ∆] (cid:11) ( ψ ) = 0 . Now, since the spectrum of ∆ is [ − , , the operator − ∆ is positive, so we have the equivalences h ∆ − i ( ψ ) = 0 ⇐⇒ (cid:13)(cid:13) (1 − ∆ ) / ψ (cid:13)(cid:13) = 0 ⇐⇒ ψ ∈ E ∆ ( {± } ) . Thus,
Crit ( H, Φ) ≡ ( ∇ H ) − ( { } ) = (cid:8) ψ ∈ D ( N ) | h ∆ − i ( ψ ) = 0 (cid:9) = D ( N ) ∩ E ∆ ( {± } ) . On the other hand, the elements ψ ∈ Crit ( H ) satisfy the condition X H ( ψ ) = − i ∆ ψ ⇐⇒ ψ ∈ E ∆ ( { } ) . This implies that
Crit ( H ) = { } , since the spectrum of ∆ is purely absolutely continuous outside the points ± [16, Prop. 3.2]. Finally, the function T f is given by T f = −h A i · ( ∇ R f ) (cid:0) h ∆ − i (cid:1) on D ( N ) \ Crit ( H, Φ) .Typical examples of operators ∆ and N of the preceding type are Laplacians and number operators ontrees or complete Fock spaces (see [16] for details). Acknowledgements
Part of this work was done while R.T.d.A was visiting the Max Planck Institute for Mathematics in Bonn. Hewould like to thank Professor Dr. Don Zagier for his kind hospitality. R.T.d.A also thanks Professor M. Mussofor a useful conversation on the stationary nonlinear Schr¨odinger equation.20 eferences [1] R. Abraham and J. E. Marsden.
Foundations of mechanics . Benjamin/Cummings Publishing Co. Inc.Advanced Book Program, Reading, Mass., 1978. Second edition, revised and enlarged, With the assistanceof Tudor Rat¸iu and Richard Cushman.[2] R. Abraham, J. E. Marsden and T. Rat¸iu.
Manifolds, tensor analysis, and applications . Applied Mathe-matical Sciences, 75. Springer-Verlag, New York, 1988. Second edition.[3] W. O. Amrein.
Hilbert space methods in quantum mechanics . Fundamental Sciences. EPFL Press, Lau-sanne, 2009.[4] W. O. Amrein, A. Boutet de Monvel, and V. Georgescu. C -groups, commutator methods and spectraltheory of N -body Hamiltonians , volume 135 of Progress in Math.
Birkh¨auser, Basel, 1996.[5] V. I. Arnold.
Mathematical methods of classical mechanics , volume 60 of
Graduate Texts in Mathematics .Springer-Verlag, New York, second edition, 1989. Translated from the Russian by K. Vogtmann and A.Weinstein.[6] A. Bahri and Y. Y. Li. On a min-max procedure for the existence of a positive solution for certain scalarfield equations in R N . Rev. Mat. Iberoamericana , 6(1-2):1–15, 1990.[7] G. Benettin and A. Giorgilli. On the Hamiltonian interpolation of near-to-the-identity symplectic mappingswith application to symplectic integration algorithms.
J. Statist. Phys. , 74(5-6):1117–1143, 1994.[8] J. Bourgain.
Global solutions of nonlinear Schr¨odinger equations , volume 46 of
American MathematicalSociety Colloquium Publications . American Mathematical Society, Providence, RI, 1999.[9] A. Boutet de Monvel and V. Georgescu. The method of differential inequalities. In
Recent developmentsin quantum mechanics (Poiana Bras¸ov, 1989) , volume 12 of
Math. Phys. Stud. , pages 279–298. KluwerAcad. Publ., Dordrecht, 1991.[10] V. Buslaev and A. Pushnitski. The scattering matrix and associated formulas in Hamiltonian mechanics.
Comm. Math. Phys. , 293(2):563–588, 2010.[11] O. Calin and D.-C. Chang.
Geometric mechanics on Riemannian manifolds . Applied and Numerical Har-monic Analysis. Birkh¨auser Boston Inc., Boston, MA, 2005. Applications to partial differential equations.[12] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao Sharp global well-posedness for KdV andmodified KdV on R and T . J. Amer. Math. Soc. 16 (2003), no. 3, 705–749 (electronic).[13] P. R. Chernoff and J. E. Marsden. Properties of infinite dimensional Hamiltonian systems , volume 425 of
Lecture Notes in Mathematics . Springer-Verlag, Berlin-New York, 1974.[14] C. A. A. de Carvalho and H. M. Nussenzveig. Time delay.
Phys. Rep. , 364(2):83–174, 2002.[15] L. Dixon and W. L. Ferrar. On the summation formulae of Vorono¨ı and Poisson.
The Quarterly Journalof Mathematics , 8(1):66–74, 1937.[16] G. Georgescu and S. Gol´enia. Isometries, Fock spaces, and spectral analysis of Schr¨odinger operators ontrees.
J. Funct. Anal. , 227(2):389–429, 2005.[17] C. G´erard and R. Tiedra de Aldecoa. Generalized definition of time delay in scattering theory.
J. Math.Phys. , 48(12):122101, 15, 2007.[18] `A. Haro. The primitive function of an exact symplectomorphism.
Nonlinearity , 13(5):1483–1500, 2000.[19] J. Jost.
Riemannian geometry and geometric analysis . Universitext. Springer-Verlag, Berlin, fourth edition,2005. 2120] T. Kato. Quasi-linear equations of evolution, with applications to partial differential equations.
Spectraltheory and differential equations , pp. 25–70. Springer-Verlag, Lecture Notes in Mathematics 448, 1975.[21] Kiyosi It¯o, editor.
Encyclopedic dictionary of mathematics. Vol. I–IV . MIT Press, Cambridge, MA, secondedition, 1987. Translated from the Japanese.[22] S. Kuksin.
Nearly integrable infinite-dimensional Hamiltonian systems.
Springer-Verlag, Lecture notes inmathematics 1556, 1993.[23] S. Kuksin and J. P¨oschel
On the inclusion of analytic symplectic maps in analytic Hamiltonian flowsand its applications.
Seminar on Dynamical Systems (St. Petersburg, 1991), 96–116, Progr. NonlinearDifferential Equations Appl., 12, Birkh¨auser, Basel, 1994.[24] N. P. Landsman.
Mathematical topics between classical and quantum mechanics . Springer Monographsin Mathematics. Springer-Verlag, New York, 1998.[25] J. M. Lee.
Riemannian manifolds , volume 176 of
Graduate Texts in Mathematics . Springer-Verlag, NewYork, 1997. An introduction to curvature.[26] J. E. Marsden and T. S. Ratiu.
Introduction to mechanics and symmetry , volume 17 of
Texts in AppliedMathematics . Springer-Verlag, New York, second edition, 1999. A basic exposition of classical mechanicalsystems.[27] R. M. Miura, C. S. Gardner, and M. D. Kruskal. Korteweg-de Vries equation and generalizations. II.Existence of conservation laws and constants of motion.
J. Mathematical Phys. , 9:1204–1209, 1968.[28] P. J. Olver.
Applications of Lie groups to differential equations , volume 107 of
Graduate Texts in Mathe-matics . Springer-Verlag, New York, second edition, 1993.[29] M. ¨Onder and A. Verc¸in. Orbits of the n -dimensional Kepler-Coulomb problem and universality of theKepler laws. European J. Phys. , 27(1):49–55, 2006.[30] M. Reed and B. Simon.
Methods of modern mathematical physics. Volume II. Fourier analysis, self-adjointness.
Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.[31] S. Richard and R. Tiedra de Aldecoa. A new formula relating localisation operators to time operators.preprint on http://arxiv.org/abs/0908.2826 .[32] S. Richard and R. Tiedra de Aldecoa. Time delay is a common feature of quantum scattering theory.preprint on http://arxiv.org/abs/1008.3433 .[33] C. Sulem and P.-L. Sulem.
The nonlinear Schr¨odinger equation , volume 139 of
Applied MathematicalSciences . Springer-Verlag, New York, 1999. Self-focusing and wave collapse.[34] J. A. Sanders and J. P. Wang. Classification of conservation laws for KdV-like equations.
Math. Comput.Simulation , 44(5):471–481, 1997.[35] W. Thirring.
Classical mathematical physics . Springer-Verlag, New York, third edition, 1997. Dynamicalsystems and field theories, Translated from the German by Evans M. Harrell, II.[36] R. Tiedra de Aldecoa. Time delay and short-range scattering in quantum waveguides.
Ann. Henri Poincar´e ,7(1):105–124, 2006.[37] R. Tiedra de Aldecoa. Anisotropic Lavine’s formula and symmetrised time delay in scattering theory.
Math. Phys. Anal. Geom. , 11(2):155–173, 2008.[38] R. Tiedra de Aldecoa. Time delay for dispersive systems in quantum scattering theory.
Rev. Math. Phys. ,21(5):675–708, 2009. 2239] E. C. Titchmarsh.
Introduction to the theory of Fourier integrals . Oxford University Press, London, secondedition, 1948.[40] J. Weidmann.
Linear operators in Hilbert spaces , volume 68 of
Graduate Texts in Mathematics . Springer-Verlag, New York, 1980. Translated from the German by Joseph Sz¨ucs.[41] W. P. Ziemer,