Remarks on the Complete Integrability of Quantum and Classical Dynamical

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The problem of integration of Hamiltonian systems and complete integrability has already been discussed in works of Euler, Bernoulli, Lagrange, and Kovalevskaya on the motion of a rigid body in mechanics. Liouville’s theorem states that if a Hamiltonian system with n degrees of freedom has n independent integrals in involution, then it can be integrated in quadratures, see [1]. Such a system is called completely Liouville integrable; for it there is a canonical change of variables that reduces the Hamilton equations to the action-angle form. An important modern development in studying of integrable systems started with the celebrated work of Gardner, Green, Kruskal and Miura [2] where the inverse scattering method for solving nonlinear Korteweg-de Vries equations was developed. A complete integrability of the corresponding Hamiltonian system was shown in [3]. A generalization to the relativistic sin-Gordon equation was obtained in [4]. A study of similar quantum systems showed that the remarkable properties of the classical scattering in such models - the absence of multiparticles production and conservation of sets of momenta of asymptotic states, which determined the scattering operator (S-matrix) - are preserved in the relativistic sin-Gordon quantum field theory [5]. Earlier the complete integrability of some non-relativistic quantum systems was studied by Bethe, Yang, Baxter and others. Completely integrable classical and quantum systems are considered in numerous works, in particular in [6, 7, 8, 9, 10, 11, 12, 13, 14]. Various notions of integrability have been used. In particular, the Lax representation [7], the Yang-Baxter equation [8], an explicit computation of the S-matrix [5], local and non-local conservation laws [9, 10, 11, 12]. The complete integrability of the Schrödinger equation in Cn with a non-degenerate spectrum is proved in [14]. In this paper, we note that the Schrödinger equation iψ˙ = Hψ for any quantum system with an arbitrary self-adjoint Hamiltonian H in a separable Hilbert space is unitary equivalent to a set of classical non-interacting one-dimensional harmonicH oscillators and, in this sense, is completely integrable. According to the spectral theorem there exists a unitary map W : L2(X,µ), where X is a measurable space with a measure µ such that the SchrödingerH → equation is unitary equivalent to a sim- 2 pler equation iϕ˙ x = ωxϕx on the space L (X,µ), ω being a function ω : X R and x X. The last equation, after decomplexification, describes a family of→ classical non-interacting∈ harmonic oscillators. By using this isomorphism we construct higher order integrals of motion for the original Schrödinger equation. Of course, the task of how to construct explicitly the Hilbert space L2(X,µ) and the unitary transformation W is a difficult separate problem. Also, that does not mean that we can explicitly compute the time dependence for expectation value of any quantum observable A in L2(X,µ), (Aϕ(t),ϕ(t)). Then we discuss applications of scattering theory (not the inverse scattering transform method but rather the direct method of the wave operators) to study the integrability of various classical and quantum dynamical systems and the construction of integrals of motion. Let H and H0 be self-adjoint operators on Hilbert space and itH −itH0 exist the limits limt→±∞ e e = Ω± which are called the wave operators. One

2 considers only absolute continuous parts of the Hamiltonians. The wave operators Ω± obey the intertwining property HΩ± = Ω±H0. If H0 is a simple "free" operator then the existence of the intertwining relation can be used to claim the integrability of the system described by the more complicated Hamiltonian H. Even if it is so, still there is a problem of how to compute explicitly the wave operators or the scattering operator ∗ (S-matrix) S = Ω+Ω− which is a task of primary interest. Examples from classical and quantum mechanics, and also from non-linear partial differential equations and quantum field theory are discussed. Higher order integrals of motion for the multi-dimensional nonlinear Klein-Gordon and Schrödinger equations are proposed. Note that there is also another direction in studying properties of quantum systems when one takes the large time limit together with small coupling or small density limit, see [15, 16, 17]. In the next section the finite-dimensional case is considered. In Sect. 3 the complete integrability of arbitrary quantum dynamics, i.e. the Schrödinger equation with an arbitrary self-adjoint Hamiltonian is proved. It follows from the spectral theorem. Integrals of motion are found. Also the complete integrability of classical dynamical systems in the Koopman formalism is mentioned (Remark 4). Applications of direct methods of scattering theory and wave operators for investigation of integrability of classical and quantum mechanics are presented in Sect. 4 and Sect. 5, respectively. Wave operators and integrability for nonlinear partial differential equations such as nonlinear Klein-Gordon and Schrödinger equations are discussed in Sect.6. Finally, in Sect.7 we make some remarks about quantum field theory. Note that scattering theory on p-adic field is considered in [18], that can be used in p-adic mathematical physics [19, 20]. This preprint is an extended version of the paper published in [21]. .

2 Finite-dimensional Case

We first consider the case of a finite-dimensional Hilbert space = Cn. Theorem 1. The Schrödinger equation iψ˙ = Hψ, where H isH a Hermitian operator in Cn, ψ = ψ(t) Cn, regarded as a classical Hamiltonian system, with a symplectic structure obtained∈ by decomplexification the Hilbert space, has n independent integrals in involution. Proof. Let ωj, j = 1, ..., n be the eigenvalues of the operator H. By diagonalizing the matrix H using a unitary transformation, we find that the original Schrödinger equation is unitary equivalent to the system of equations iϕ˙ j = ωjϕj, where ϕj = ϕ (t) C, j =1, ..., n.. j ∈ Passing to the real and imaginary parts of the function ϕj(t)=(qj(t)+ ipj (t))/√2, this system of equations is rewritten in the form of Hamiltonian equations of the family of classical harmonic oscillators

q˙ = ω p , p˙ = ω q . j j j j − j j

3 with the Hamiltonian n 1 H = ω (p2 + q2)=(ψ,Hψ). osc 2 j j j Xj=1 Deﬁne 1 I (ϕ)= ϕ 2 = (p2 + q2), j =1, ..., n. j | j| 2 j j The functions Ij are integrals of motion, independent and in involution, Ij,Im = 0, j, m =1, ..., n.. The corresponding level manifold has the form of an n-dimensional{ } torus. Note that the Hamiltonian is a linear combination of the integrals of motion: n

Hosc = ωjIj. Xj=1 The theorem is proved. Remark 1. We give another, equivalent way of proving the theorem. Let e1, ..., en be an orthonormal basis of the eigenvectors of the Hermitian operator H in Cn, i.e.

Hej = ωjej. Then, writing the function ψ = ψ(t) in the form ψ = j ψjej, where ψj = ψj(t) C, we get that the Schrödinger equation iψ˙ = Hψ Ptakes the form ˙ ∈ iψj = ωjψj. Passing to the real and imaginary parts of the function ψj as done above, ψj = (ξj + iηj)/√2 we get a set of harmonic oscillators with the Hamiltonian Hosc = 2 2 ωj(ηj +ξj )/2. Note that Hosc is equal to the mean value of the quantum Hamiltonian, 2 PHosc = (ψ,Hψ). The integrals of motion are defined as follows: Jk(ψ) = ψk = 2 2 | | (ηk + ξk)/2, k =1, ..., n. We also add that the solution of the Schrödinger equation has the well known form n −iωj t ψ(t)= ψj(0)e ej. Xj=1

3 Complete integrability of the Schr¨odinger equation

Let us prove the complete integrability of the Schrödinger equation with an arbitrary Hamiltonian. The Schrödinger equation is completely integrable in the sense that its solutions are unitary equivalent to the complexified solutions of the Hamilton equations for a family of classical harmonic non-interacting oscillators. The Schrödinger equation after the unitary transformation is rewritten in the form of equations for the family of classical harmonic oscillators. This is a consequence of the spectral theorem. Theorem 2. Let be a separable Hilbert space and H a self-adjoint operator with a dense domain DH(H) . Then the Schrödinger equation ⊂ H ∂ i ψ(t)= Hψ(t), ψ(t) D(H), t R ∂t ∈ ∈ completely integrable in the sense that this equation is unitary equivalent to the complexified system of equations for a family of classical non-interacting harmonic oscillators. There exists a set of non-trivial integrals of motion for the arbitrary Schrödinger equation.

4 Proof. The solution of the Cauchy problem for the Schr¨odinger equation ∂ i ψ(t)= Hψ(t), ψ(0) = ψ D(H) ∂t 0 ∈

−itH by the Stone theorem has the form ψ(t) = Utψ0, where Ut = e is the group of unitary operators, t R. Then, by the spectral theorem [26, 27], there exists a measurable space (X, Σ)∈with σ-finite measure µ, and a measurable finite a.e. function ω : X R such that there is a unitary transformation W : L2(X,µ) such that →∗ (0) (0) −itMω H → Ut = W Ut W where Ut = e , where Mω is the operator of multiplication by ∗ the function ω. On the corresponding domain one has H = W MωW . The initial Schrödinger equation in goes over under the unitary transformation of W into the Schrödinger equation in LH2(X,µ) of the form ∂ i ϕ (t)= ω ϕ (t), x X ∂t x x x ∈ Passing to the real and imaginary parts of the function 1 ϕx(t)= (qx(t)+ ipx(t)), √2 this Schrodinger equation is rewritten in the form of equations of the family of classical harmonic oscillators q˙ = ω p , p˙ = ω q , x X. x x x x − x x ∈ These equations can be obtained from the Hamiltonian 1

1 2 2 Hosc = Hxdµ, Hx = ωx(px + qx), Zx 2 by using the relation

δHosc = δHxdµ = (q ˙xδpx p˙xδqx)dµ. ZX ZX −

Note also that one has Hosc =(ωϕ,ϕ)L2 =(Hψ), ψ)H. Integrals of motion. Lemma 1. Let I : L2(X) R be an integral of motion for the dynamics → U (0), I(U (0)ϕ)= I(ϕ), ϕ L2(X). Then J : R, where J(ψ)= I(W ψ), ψ , is t t ∈ H→ ∈ H the integral of motion for dynamics Ut, J(Utψ)= J(ψ). Proof of Lemma 1 goes as follows:

∗ (0) J(Utψ)= I(W Utψ)= I(W UtW W ψ)= I(Ut W ψ)= I(W ψ)= J(ψ).

(0) Now let us consider the following integrals of motion for the dynamics Ut : 1I am grateful to B. O. Volkov for discussion. For considerations of inﬁnite-dimensional Hamiltonian systems see [22, 23, 24] and refs therein.

5 2 1 2 2 Iγ = Iγ(ϕ)= γx ϕx dµ = γx (px + qx)dµ, ZX | | ZX 2 for any γ L∞(X). Then, by Lemma 1, one gets an integral of motion J for dynamics ∈ γ Ut. The theorem is proved. Remark 2. Consider an example when X = R and the Hilbert space is L2(R,dµ), where µ is a Borel measure on the line. Any measure µ on R admits a unique expansion in the sum of three measures µ = µpp + µac + µs, where µpp is purely pointwise, µac is absolutely continuous with respect to the Lebesgue measure and µs is continuous and singular with respect to Lebesgue measure, and we have respectively

L2(R, dµ)= L2(R, dµ ) L2(R, dµ ) L2(R, dµ ). pp ⊕ ac ⊕ s

A purely point measure has the form µpp = cjδsj , where cj > 0, the sum over j contains a ﬁnite or inﬁnite number of terms,Psj real numbers and δsj Dirac delta function. Then we have ϕ 2 dµ = c ϕ 2. Z | x| pp j| sj | Xj The integrals of motion in this case have the form 1 I (ϕ)= ϕ 2 = (p2 + q2 ), j | sj | 2 sj sj and the Hamiltonian of the system of oscillators is

ωs H = j (p2 + q2 ) osc 2 sj sj Xj with ordinary Poisson brackets, where sequences are considered for which the series converges. Similarly, an absolutely continuous measure has the form dµac = f dx, where f 0 is a locally integrable function and dx is a Lebesgue measure. Then we have ≥

ϕ 2 dµ = ϕ 2f dx. Z | x| ac Z | x| x The integrals of motion in this case have the form 1 I = γ ϕ 2f dx = γ (p2 + q2)f dx, γ Z x| x| x Z x 2 x x x where γ L∞. ∈ One would write formally for the density of the integrals 1 I = ϕ 2 = (p2 + q2), x suppf, x | x| 2 x x ∈

6 and the Hamiltonian of the system of oscillators is ω H = x (p2 + q2) f dx osc Z 2 x x x with ordinary Poisson brackets, where functions are considered for which the integral converges. Remark 3. It is known that another version of the spectral theorem, using the direct integral of Hilbert spaces, allows us to determine the invariants of the Hamiltonian with respect to unitary transformations. The Hilbert space is unitary equivalent to H the direct integral of the Hilbert spaces λ, λ σ(H) , where σ(H) spectrum of H, with respect to some measure ν on σ(H){H, ∈ } ⊕ λdν(λ), Zσ(H) H the operator H in λ corresponds to multiplication by λ. The invariants of H are the spectrum σ(H), theH measure ν (modulo absolute continuity) and the dimension of the Hilbert spaces λ. Remark 4.H The complete integrability of the Liouville equation in Koopman’s approach to classical mechanics and in general any dynamical system is proved in a similar way. Let (M, Σ,α,τt) be a dynamical system, where (M, Σ) measurable space with measure α and τt, t R group of measure-preserving transformations M. Then ∈ 2 the Koopman transform defines a group of unitary operators Ut in L (M,α), (U f)(m)= f(τ (m)), f L2(M,α) t t ∈ Repeating the proof of Theorem 2, we see that the group Ut is unitary equivalent to the family of harmonic oscillators and, in this sense, any dynamical system is completely integrable. Remark 5. Solution of the Cauchy problem for the Schrödinger equation in L2(X,µ), ∂ i ϕ (t)= ω ϕ (t), ϕ (0) = ϕ , x X ∂t x x x x 0x ∈ has the form ϕ (t)= e−itωx ϕ , x X. x 0x ∈ One can expect that quantum chaos [25] is related with ergodicity of classical harmonic oscillators for irrational frequencies ωx. Remark 6. The above interpretation of the spectral theorem as complete integrability in the sense of reducing dynamics to a set of harmonic oscillators does not mean that such a task is easy to perform for specific systems. It is interesting to compare the effectiveness (or complexity) of complete integrability in the above sense of the spectral theorem with the efficiency/complexity of constructing solutions of Hamiltonian systems that are completely integrable in the sense of Liouville. Apparently, the change of variables in the Liouville theorem, which reduces the initial dynamics to action- angle variables on a level manifold, is an analogue of the unitary transformation in the spectral theorem, which reduces the original dynamics to a set of harmonic oscillators.

7 4 Direct methods of scattering theory and integrability of classical dynamical systems

The inverse scattering problem method, based on the representation of certain nonlinear equations in the Lax form, is used, as is known, to prove the complete integrability of such equations, usually on a straight line or on a plane, see, for example, [7]. Here we look at the use of direct scattering theory methods to prove the complete integrability and the construction of integrals of motion in the theory of classical and quantum dynamical systems. This will provide a constructive example to the above general result on the complete integrability of arbitrary quantum and classical systems. Note that a general method was proposed in [9] for constructing conserved currents for a large class of (multidimensional) nonlinear equations by using a dual linear equation.

4.1 Wave operators and integrals of motion

Let φ(t) and φ0(t) be a pair of groups of automorphisms of the phase space in the classical case or groups of unitary transformations in quantum case, t R. Suppose ∈ there exist the limits, called wave operators, see [28, 29, 30, 31],

lim φ( t)φ0(t) = Ω± t→±∞ − The wave operators have the properties of intertwining operators; on the appropriate domain, we have φ(t)Ω± = Ω±φ0(t)

Thus, the interacting dynamics φ(t) is reduced to "free" φ0(t). The following form of Lemma 1 holds Lemma 1a. Let I0(z) be the integral of motion for the dynamics of φ0(t), i.e. I0(φ0(t)z) = I0(z), here z is a phase space point or a Hilbert space vector. Then −1 I(z)= I0(Ω± z) is the integral of motion for the dynamics of φ(t). Indeed, we have

−1 −1 −1 −1 −1 I(φ(t)z)= I0(Ω± φ(t)z)= I0(Ω± φ(t)Ω±Ω± z)= I0(φ0(t)Ω± z)= I0(Ω± z)= I(z).

4.2 Particle in a potential field Let us consider in more detail the motion of a particle in a potential field. Let Γ = ′ M M phase (symplectic) space, where M = Rn is the configuration space and ′ M ×= Rn is the dual space of momenta. Hamiltonian has the form 1 H(x, ξ)= ξ2 + V (x), 2

8 where ξ M ′ and the function V : M R is bounded and the force F (x)= V (x) is locally∈ Lipschitz. Then the solution→(x(t, y, η),ξ(t, y, η)) of the equations of−▽ motion

x˙(t)= ξ(t), ξ˙(t)= F (x(t)) with initial data

x(0,y,η)= y, ξ(0,y,η)= η, y M, η M ′ ∈ ∈ exists and is unique for all t R. We will denote φ(t)(y, η)=(x(t, y, η),ξ(t, y, η)). The ∈ 2 free Hamiltonian is H0(x, ξ) = ξ /2, the solution of the equations of motion has the form φ0(t)(y, η)=(y + tη, η). Let the following conditions also be satisﬁed ∞ lim V (x)=0, sup F (x) dr < |x|→∞ Z0 |x|≥r | | ∞ Then for any (y, η) M M ′ there is a limit [29] ∈ × x(t, y, η) lim = ξ+(y, η), t→∞ t moreover, if ξ (y, η) =0, then there is a limit + 6

lim ξ(t, y, η)= ξ+(y, η), t→∞

−1 ′ and the inverse image of = ξ+ (M 0 ) is the open set of all paths with positive energy unbounded for t D . \{ } →∞ Proposition 1 (see Theorem 2.6.3 in [29]). Let the force satisfy the conditions

∞ α 2 1/2 sup ∂x F (x) (1 + r ) dr < , α =0, 1. Z0 |x|≥r | | ∞ | | Then there exists the limit lim φ( t)φ0(t) = Ω+ t→∞ − ′ ′ uniformly on compact sets in M (M 0 ). The mapping Ω+ : M (M 0 ) is symplectic, continuous and one-to-one.× \{ There} are relations × \{ } →D

H Ω+ = H0, φ(t) Ω+ = Ω+ φ0(t). Theorem 3. The equations of motion of a particle in a potential ﬁeld satisfying the above conditions determine an integrable system in the sense that the dynamics of φ(t) is symplectically equivalent to the free dynamics of φ0(t) on the domain in 2n - dimensional phase space, Ω−1φ(t) Ω = φ (t). On there are n independentD integrals + + 0 D of motion in involution. Proof follows from Proposition 1 and Lemma 1a. Let I : R, j = 1, 2, ..., n 0,j D → be the integral of motion for free dynamics φ0(t), of the form I0,j(y, η) = ηj. Then −1 Ij(z)= I0,j(Ω+ z) is the integral of motion for the dynamics of φ(t).

9 5 Wave operators and integrability in quantum mechanics

Consider the Schrödinger equation in L2(Rn) of the form ∂ψ i = Hψ, (1) ∂t where H = H +V (x), H = - Laplace operator and the potential V is short-range, 0 0 −△ i.e. it satisfIes V (x) C(1 + x )−ν, x Rn, ν> 1. | | ≤ | | ∈ Then H defines a self-adjoint operator and there exist the wave operators Ω±, which are complete and diagonalize H, HΩ± = Ω±H0 [30, 31]. So, one can construct higher order integrals of motion according to the scheme described above. Note that if ϕ = ϕ(t, x) is a smooth solution to the free Schrödinger equation α iϕ˙ + ϕ = 0 then its derivatives ∂x ϕ also are solutions of this equation. Hence, one △ α can get higher integrals of motion just by replacing ϕ by ∂x ϕ in the known integral 2 k Rn I0(ϕ)= Rn ϕ dx. If ϕ is in the Sobolev space H ( ) then one gets a set of integrals | | 2 α 2 of motion:R I0 = ϕ dx, Iα = ∂x ϕ dx, α k. Now, according to Lemma 1a, we | | ± | | |−1| ≤ obtain integrals ofR motion J0,JαR(ψ)= Iα(Ω± ψ) for the original Schrödinger equation (1) with the potential V (x). Remark also that using the Fourier transform Fϕ =ϕ ˜ one can write the free dynam- (0) itk2 2 ics in the form Ut ϕ˜(k)= e ϕ˜(k) with integrals of motion Iγ (ϕ)= Rn γ(k) ϕ˜(k) dk, where γ L∞(Rn). R | | ∈ 6 Wave operators and integrability for nonlinear partial differential equations

Let us consider a nonlinear Klein - Gordon equation of the form u¨ a2 u + m2u + f(u)=0, t R, x Rn,, (2) − △ ∈ ∈ where a> 0, m 0. ≥ Eq. (2) is a Hamiltonian system with the Hamiltonian

1 2 1 2 1 2 2 H = ( )p(x) + ( u) + m u + V (u(x))dx = H0 + V (3) ZRn 2 2 ∇ 2 where V ′ = f and the the Poisson brackets are p(x),u(y) = δ(x y) One can take f(u) = λ u 2u, λ 0 and n {= 3, more} general− case is considered in numerous papers, see, for| instance| ≥ [32, 33, 34] and ref’s therein. If the initial data 1 n 2 n (u(0), u˙ t(0)) H (R ) L (R ) then there exists a global solution u(t) of Eq (2) with these initial data.∈ Here×H1(Rn) is the Sobolev space. For any solution of (2) there exists a unique pair (v+, v−) of solutions for the free Klein-Gordon equation v¨ a2 v + m2v =0, (4) − △ 10 such that lim (u(t), u˙(t)) (v±(t), v˙±(t)) H1×L2 =0. t→±∞ k − k

Moreover, the correspondence (u(0), u˙(0)) (v±(0), v˙±(0)) deﬁnes homeomorphisms 1 2 7→ (wave operators) Ω± on H L . Integrals of motion. × α If v = v(t, x) is a smooth solution of Eq. (4) then its partial derivatives ∂x v is also the solution of Eq. (4). Therefore to get higher order integrals of motion from the 2 2 2 2 2 α energy integral E(v) = (v ˙ + a ( v) + m v ) dx/2 one can just replace v by ∂x v. ∇ α We get higher integralsR of motion Iα(v) = E(∂x v) for Eq. (4). One expects that if |α| ± v H then Jα (u) = Iα(Ω±u) will be integrals of motion for Eq. (2). To construct higher∈ order integrals of motion for the nonlinear equation one can use also the Fourier transform as in the previous section. Similar approach to construct higher order integrals of motion is applied also for the nonlinear Schr¨odinger equation

iu˙ + u + f(u)=0. △ It would be interesting to develop for nonlinear PDE an analogue of Koopman’s approach to classical mechanics. To this end we need a measure in an infinite-dimensional space which is invariant under the Hamilton flow and symplectic transformations. A shift-invariant finitely additive measure on a Hilbert space is constructed in [35].

7 Wave operators and integrability in quantum ﬁeld theory

Quantum field theory deals with operator-valued solutions of partial differential equations [36, 37, 38]. For example, the scalar quantum field uˆ =u ˆ(t, x), t R, x Rn is an operator-valued function which should obey Eq. (2) ∈ ∈

uˆ a2 uˆ + m2uˆ + f(ˆu)=0, tt − △ with initial data satisfying the canonical commutation relation:

[ˆu(0, x), uˆ (0,y)] = iδ(x y) t − A formal solution of these operator equations is given by the expression uˆ(t, x) = eitHˆ uˆ(0, x)e−itHˆ where Hˆ is an operator obtained by quantization of Eq. (3)

1 1 1 Hˆ = ( pˆ(x)2 + ( qˆ)2 + m2qˆ2 + V (ˆq(x))dx (5) ZRn 2 2 ∇ 2 and it is assumed that uˆ(0, x)=ˆq(x), uˆ (0, x)=ˆp(x), [ˆq(x), pˆ(y)] = iδ(x y). t − After an appropriate regularization, one can construct the quantum field uˆ as an operator-valued distribution in a Fock Hilbert space = L2(Rn)⊗sk. F ⊕k 11 The formal scattering theory in quantum field theory is constructed analogously to the previous section. There exists a pair (ˆv+, vˆ−) of solutions (in and out fields) for the free Klein-Gordon equation

vˆ a2 vˆ + m2vˆ =0, (6) tt − △ with initial data satisfying the relation

[ˆv(0, x), vˆ (0,y)] = iδ(x y) t − such that the weak limit lim (ˆu(t) vˆ±(t)=0. t→±∞ −

Moreover, the correspondence (ˆu(0), uˆt(0)) (ˆv±(0), vˆ±t(0)) deﬁnes the wave operators Ω on . 7→ ± F 8 Conclusions

In this paper, it is noted that the Schr¨odinger equation with any self-adjoint Hamil- tonian is unitarily equivalent to the set of classical harmonic oscillators and in this sense is completely integrable. This fact follows immediately from the spectral theorem which provides the existence of the unitary transformation W but not its explicit construction. The problem of how to construct the mapping W is considered by using scattering theory and the wave operators. Higher order integrals of motion are indicated for a number of quantum and classical dynamical systems. Further study and clariﬁcation of the notion of complete integrability is desirable.

9 Acknowledgements

This work was initiated by talks with V.V. Kozlov, who shared with the author his ideas about quadratic integrals for the Schr¨odinger equation. The author is very grateful to I.Ya. Aref’eva, A.K. Gushchin, O. V. Inozemcev, V.V. Kozlov, B.S. Mityagin, V.Zh. Sakbaev, A.S. Trushechkin, B.O. Volkov and V.A. Zagrebnov for fruitful discussions.

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