The Trajectory-Coherent Approximation and the System of Moments for the Hartree-Type Equation

The general construction of quasi-classically concentrated solutions to the Hartree-type equation, based on the complex WKB-Maslov method, is presented. The formal solutions of the Cauchy problem for this equation, asymptotic in small parameter \h (\h\to0), are constructed with a power accuracy of O(\h^{N/2}), where N is any natural number. In constructing the quasi-classically concentrated solutions, a set of Hamilton-Ehrenfest equations (equations for middle or centered moments) is essentially used. The nonlinear superposition principle has been formulated for the class of quasi-classically concentrated solutions of the Hartree-type equations. The results obtained are exemplified by the one-dimensional equation Hartree-type with a Gaussian potential.Comments: 6 pages, 4 figures, LaTeX Report no: Subj-class: Accelerator Physics


Introduction
The nonlinear Schrödinger equation {−i∂ t +Ĥ(t, |Ψ| 2 )}Ψ = 0, (0.1) whereĤ(t, |Ψ| 2 ) is a nonlinear operator, arises in describing a broad spectrum of physical phenomena. In statistical physics and quantum field theory, the generalized model of the evolution of bosons is described in terms of the second quantization formalism by the Schrödinger equation [1] which, in Hartree's approximation, leads to the classical multidimensional Schrödinger equation with a nonlocal nonlinearity for one-particle functions, i.e., a Hartree type equation.
The quantum effects associated with the propagation of an optical pulse in a nonlinear medium are also described in the second quantization formalism by the one-dimensional Schrödinger equation with a delta-shaped interaction potential. In this case, the Hartree approximation results in the classical nonlinear Schrödinger equation [2,3] which is integrated by the Inverse Scattering Transform (IST) method and has soliton solutions [4]. Solitons are localized wave packets propagating without distortion and interacting elastically in mutual collisions. The soliton theory has found wide application in various fields of nonlinear physics [5,6,7,8].
Investigations of the statistical properties of optical fields have led to the concept of compressed states of a field in which quantum fluctuations are minimized and the highest possible accuracy of optical measurements is achieved. The important problem of the correspondence between the stressed states describing the quantum properties of a radiation and the optical solitons is analyzed in [2,3].
The Hartree type equation is nonintegrable by the IST method. Nevertheless, approximate solutions showing some properties characteristic of solitons can be constructed. Solutions of this type are referred to as solitary waves or "quasi-solitons" to differentiate them from the solitons (in the strict sense) arising in IST integrable models.
For the functions of noncommuting variables, we use the Weyl ordering [27,28]. In this case, we can write, for instance, for the operatorĤ: i ( x − y), p H p, x + y 2 , t Ψ( y, t, ), (1.6) where H(z, t) = H( p, x, t) is the Weyl symbol of the operatorĤ(t) and ., . is the Euclidean scalar product of the vectors p, x = n j=1 p j x j , p, x ∈ R n , z, w = 2n j=1 z j w j , z, w ∈ R 2n .
We here are interested in localized solutions of equation (1.1) for each fixed ∈ [0, 1[ and t ∈ R, belonging to the Schwartz space with respect to the variable x ∈ R n . For the operatorsĤ(t) andV (t, Ψ) to be at work in this space, it is sufficient that their Weyl symbols H(z, t) and V (z, w, t) be smooth functions 1 and grow, together with their derivatives, with |z| → ∞ and |w| → ∞ no more rapidly as the polynomial and uniformly in t ∈ R 1 . Therefore, we believe that the following conditions for the functions H(z) and V (z, w, t) are satisfied: Supposition 1 For any multi-indices α, β, µ, and ν there exist constant C α β (T ) and C αµ βν (T ), such that the inequalities z α w µ ∂ |β+ν| V (z, w, t) ∂z β ∂w ν ≤ C αµ βν (T ), z, w ∈ R 2n 0 ≤ t ≤ T are fulfilled.
We are coming now to the description of the class of functions for which we shall find asymptotical solutions to equation (1.1).

The class of trajectory-concentrated functions
Let us introduce a class of functions singularly depending on a small parameter , which is a generalization of the notion of a solitary wave. It appears that asymptotical solutions to equation (1.1) can be constructed based on functions of this class, which depend on the phase trajectory z = Z(t, ), the real function S(t, ) (analogous to the classical action at κ = 0 in the linear case), and the parameter . For → 0 the functions of this class are concentrated in the neighborhood of a point moving along a given phase curve z = Z(t, 0). Functions of this type are well known in quantum mechanics. In particular, among these are coherent and "compressed" states of quantum systems with a quadric Hamiltonian [24,25,29,30,31,32,33,34,35,36]. Note that the soliton solution localized only with respect to spatial (but not momentum) variables does not belong to this class.
Let us denote this class of functions as P t (Z(t, ), S(t, )) and define it as where the function ϕ( ξ, t, ) belongs to the Schwartz space S in variable ξ ∈ R n and depends smoothly on t and regularly on √ for → 0. Here, ∆ x = x − X(t, ), and the real function S(t, ) and the 2n-dimensional vector function Z(t, ) = ( P (t, ), X(t, )), which characterize the class P t (Z(t, ), S(t, )), depend regularly on √ in the neighborhood of = 0 and are to be determined. In the cases where this does not give rise to ambiguity, we use a shorthand symbol of P t for P t (Z(t, ), S(t, )).
The functions of the class P t are normalized to Φ(t) 2 = Φ(t)|Φ (t) in the space L 2 (R n x ) with the scalar product In the subsequent manipulation, the argument t in the expression for the norm may be omitted: In constructing asymptotical solutions, it is useful to define, along with the class of functions P t (Z(t, ), S(t, )), the following class of functions where the functions ϕ, as distinct from (2.1), are independent of . At any fixed point in time t ∈ R 1 , the functions belonging to the class P t are concentrated, in the limit of → 0, in the neighborhood of a point lying on the phase curve z = Z(t, 0), t ∈ R 1 (the sense of this property is established exactly in theorems 2.1-2.3 below). Therefore, it is natural to refer to the functions of the class P t as trajectory-concentrated functions. The definition of the class of trajectory-concentrated functions includes the phase trajectory Z(t, ) and the scalar function S(t, ) as free "parameters". It appears that these "parameters" are determined unambiguously from the Hamilton-Ehrenfest equations (see Sec. 3) fitting the nonlinear (κ = 0) Hamiltonian of equation (1.1). Note that for a linear Schrödinger equation, in the limiting case of κ = 0, the principal term of the series in → 0 determines the phase trajectory of the Hamilton system with the Hamiltonian H( p, x, t), and the function S(t, 0) is the classical action along this trajectory. In particular, in this case, the class P t includes the well-known dynamic (compressed) coherent states of quantum systems with quadric Hamiltonians when the amplitude of ϕ in (2.1) is taken as a Gaussian exponential: where Q(t) is a complex symmetrical matrix with a positive imaginary part, and the time factor is given by (see for details [21]). Let us consider the principal properties of the functions of the class P t (Z(t, ), S(t, )), which are also valid for those of the class C t (Z(t, ), S(t, )).
After the change of variables and taking into consideration the implicit form of the functions belonging to the class P t (Z(t, ), S(t, )), we find Since ϕ( ξ, t, ) depends on √ regularly and M 0 (t, ) > 0, we get and thus the theorem is proved. Let us denote by the symbolÔ( ν ) an operatorF , such that for any function Φ belonging to the space P t (z(t, ), S(t, )) the asymptotical estimate Theorem 2.2 For the functions belonging to P t (Z(t, ), S(t, )), the following asymptotical estimates are valid: Proof is similar to that of relation (2.3).
Proof. Let us consider an arbitrary function φ( x) ∈ S. Then for any function Φ( x, t, ) ∈ P t the integral Let us pass in the last equality to the limit of → 0, and, in view of and a regular dependence of the function ϕ( ξ, t, ) on √ , we arrive at the required statement. The proof of relation (2.9) is similar to the previous one if we notice that the Fourier transform of the function Φ( x, t, ) ∈ P t can be represented as Denote by L (t) the mean value of the operatorL(t), t ∈ R 1 , self-conjugate in L 2 (R n x ), calculated from the function Φ( x, t, ) ∈ P t . Then the following corollary is valid: For any function Φ( x, t, ) ∈ P t (Z(t, ), S(t, )) and any operatorÂ(t, ) whose Weyl symbol A(z, t, ) satisfies Supposition 1, the equality Proof is similar to that of relations (2.8) and (2.9). Following [21], we introduce For any function Φ ∈ P t (Z(t, ), S(t, )), the representation where Φ (k) ( x, t, ) ∈ C t (Z(t, ), S(t, )), is valid. Representation (2.11) naturally induces the expansion of the space P t (Z(t, ), S(t, )) in a direct sum of subspaces Here, the functions Φ ∈ P t (Z(t, ), S(t, ), l) ⊂ P t (Z(t, ), S(t, )), according to (2.2), have estimates by the norm where the function µ(t) is independent of and continuously differentiable with respect to t. Similar to the proof of the estimates (2.5) and (2.6), it can be shown that the operators do not disrupt the structure of the expansion (2.11), (2.12), and {∆ẑ} α : P t (Z(t, ), S(t, ), l) → P t (Z(t, ), S(t, ), l + |α|), ,˙ X(t, ) + Ż (t, ), J∆ẑ } : (2.14) Remark 2.2 From Corollary 2.3.2 it follows that the solution Ψ( x, t, ) of equation (1.1), belonging to the class P t ,, is quasi-classically concentrated.
The limiting character of the conditions (2.8) and (2.9) and the asymptotical character of the estimates (2.3)-(2.6) valid for the class of trajectory-concentrated functions make it possible to construct quasi-classically concentrated solutions to the Hartree type equation not exactly, but approximately. In this case, the L 2 norm of the error has an order of α , α > 1 for → 0 on any finite time interval [0, T ]. Denote such an approximate solution as Ψ as = Ψ as ( x, t, ). This solution satisfies the following problem: where O( α ) denotes the function g (α) ( x, t, ), the "residual" of equation (1.1). For the residual, the following estimate is valid: max Below we refer to the function Ψ as ( x, t, ) satisfying the problem (2.15)-(2.17) as a quasi-classically concentrated solution (mod α , → 0) of the Hartree type equation (1.1).
Thus, the quasi-classically concentrated solutions Ψ (N ) ( x, t, ) of the Hartree type equation describe approximately the evolution of the initial state Ψ 0 ( x, ) if the latter has been taken from a class of trajectoryconcentrated functions P 0 . The operatorsĤ(t) andV (t, Ψ) entering in the Hartree type equation (1.1) leave the class P t invariant on a finite time interval 0 ≤ t ≤ T since their symbols satisfy Supposition 1. Therefore, in constructing quasi-classically concentrated solutions to the Cauchy problem, the initial conditions can be taken in the form Ψ( x, t, )| t=0 = Ψ 0 ( x, ), Ψ 0 ∈ P 0 (z 0 , S 0 ). (2.18) The functions from the class P 0 have the following form: where Z 0 ( ) = ( P 0 ( ), X 0 ( )) is an arbitrary point of the phase space R 2n px , and the constant S 0 ( ) can be put equal to zero without loss of generality.
Important particular cases of the initial conditions of type (2.19) are where the real n×n matrix A is positive definite and symmetrical. Then relationship (2.19) defines the Gaussian packet; where the complex n×n matrix Q is symmetrical and has a positive definite imaginary part Im Q and H ν ( η) and η ∈ R n are multidimensional Hermite polynomials of multi-index ν = (ν 1 , . . . , ν n ) [42].

The set of Hamilton-Ehrenfest equations
In view of Supposition 1 for the symbols H(z, t) and V (z, w, t), the operator H(ẑ, t) (1.2) is self-conjugate to the scalar product Ψ|Φ in the space L 2 (R n x ) and the operator V (ẑ,ŵ, t) (1.3) is self-conjugate to the scalar product L 2 (R 2n xy ): Therefore, for the exact solutions of equation (1.1) we have and for the mean values of the operatorÂ(t) = A(ẑ, t), calculated for these solutions, the equality Using the rules of composition for Weyl symbols [27], we find for the symbol of the operatorĈ =ÂB Here, the index over an operator symbol specifies the turn of its action. We suppose that for the Hartree type equation . After cumbersome, but not complicated calculations similar to the calculations that were performed for the linear case with κ = 0 (see for details [21]), we then obtain, restricting ourselves to the moments of order N , the following set of ordinary differential equations:ż Here,κ = κ Ψ 0 ( x, ) 2 and Ψ 0 ( x, ) is the initial function from (2.18), By analogy with the linear theory (κ = 0) [21], we refer to equations (3.4) as Hamilton-Ehrenfest equations of order N . In view of the estimates (2.3), these equations are equivalent, for the class P t , to the nonlinear Hartree type equation (1.1) accurate to O( (N +1)/2 ).
For the case of N = 2, the Hamilton-Ehrenfest equations take the form where Equations (3.7) can be written in the equivalent form if in the second equation we put and then it becomesȦ

Linearization of the Hartree type equation
Let us now construct a quasi-classically concentrated (for → 0) solution to equation (1.1), satisfying the initial condition (2.18). Designate by the solution of the Hamilton-Ehrenfest equations of order N (3.4) with the initial data y (N ) (0, ) (3.5) determined by the initial function Ψ 0 ( x, ) (2.18), i.e., the mean values Z(0, ) and ∆ α (0, ) are calculated from the function Ψ 0 ( x, ). Let us expand the "kernel" of the operatorV (t, Ψ) in a Taylor power series for the operators ∆ŵ =ŵ − Z(t, ): Substituting this series into equation (1.1), we obtain for the functions Ψ ∈ P t where In view of the asymptotical estimates (2.3), the functions z(t, ) and ∆ α (t, ) can be determined with any degree of accuracy from the Hamilton-Ehrenfest equations (3.4) as  The following statement is valid: (4.6), satisfying the initial condition Φ| t=0 = Ψ 0 , the function

Now we expand the operators
in a Taylor power series for the operator ∆ẑ and present the operator −i ∂ t in the form Here, the group of terms in braces containing −i ∂ t , in view of (2.14), has an order ofÔ( ). Other terms can be estimated, in view of (2.6), by the parameter . Substitute the obtained expansions into (4.6). Take (accurate to O( N/2 )) the real function S(t, ) entering in the definition of the class P t (Z(t, ), S(t, )) in the form As a result, equation (4.6) will not contain operators of multiplication by functions depending only on t and .
In view of the estimates (2.5) and (2.6) valid for the class P t (Z(t, ), S(t, )), we obtain for (4.3) with the following notations: Here, k = 1, N and the functions Z (k) (t) are the coefficients of the expansion of the projection Z(t, ) of the solution y (N ) (t, ) of the Hamilton-Ehrenfest equations on the phase space R 2n in a power series of the regular perturbation theory for √ : From the Hamilton-Ehrenfest equations, in view of the fact that the first-order moments are zero (∆ α (t, , N ) = 0 for |α| = 1), it follows that the coefficientŻ (1) (t) is equal to zero.

Remark 4.1
The solutions of the set of Hamilton-Ehrenfest equations depend on the index N that denotes the highest order of the centered moments ∆ α , α ∈ Z 2n + . We shall omit the index N if this does not give rise no ambiguity.

The trajectory-coherent solutions of the Hartree type equation
The solution of the Schrödinger equation with a quadric Hamiltonian is well known [24,25]. For our purposes, it is convenient to take quasi-classical trajectory-coherent states (TCS's) [21] as a basis of solutions to equation (4.14). We shall refer to the solution of the nonlinear Hartree type equation, which coincides with the TCS at the time zero, as a trajectory-coherent solution of the Hartree type equation. Now we pass to constructing solutions like this.
Let us write the symmetry operatorsâ(t, Ψ 0 ) of equation (4.14), linear with respect to the operators ∆ẑ, in the formâ where N a is a constant and b(t) is a 2n-space vector. From the equation which determines the operatorsâ(t), in view of the explicit form of the operatorĤ 0 (t, Ψ 0 ) (4.10), we obtain Taking into account the commutative relations Then we obtain for the determination of the 2n-space vector a(t) from (5.3) We call the set of equations (5.4), by analogy with the linear case [18], a set of equations in variations.
is a symmetry operator for equation (4.14) if the vector a(t) = a(t, Ψ 0 ) is a solution of the equations in variations (5.4). For each given solution Z(t, ) of the Hamilton-Ehrenfest equations (3.4), we can find 2n linearly independent solutions a k (t) ∈ C 2n to the equations in variations (5.4). Since to each 2n-space vector a k (t) corresponds an operatorâ k (t, Ψ 0 ), we obtain 2n operators n of which commutate with one another and form a complete set of symmetry operators for equation (4.14).
Now we turn to constructing the basis of solutions to equation (4.14) with the help of the operatorsâ k (t, Ψ 0 ). Equation (4.14) is a (linear) Schrödinger equation with a quadric Hamiltonian and admits solutions in the form of Gaussian wave packets where the real phase S(t, ) is defined in (4.7), N is a normalized constant, and the real functions φ 0 (t) and φ 1 (t) and the complex n × n matrix Q(t) are to be determined.
Remark 5.1 Asymptotical solutions in the form of Gaussian packets (5.6) for equations with an integral nonlinearity of more general form than (1.1) were constructed in [44]. In this case, the Hamilton-Ehrenfest equations depend substantially on the initial condition for the starting nonlinear equation.
Substitution of (5.6) into (4.14) yields Equating the coefficients at the terms with the same powers of the parameter and the operator ∆ x, we obtain As a result we have The matrix Q(t) is determined from the Riccati type equatioṅ Thus, the construction of a solution to equation (4.14) in the form of the Gaussian packet (5.6) is reduced to solving the set of ordinary differential equations (5.9). Let us now construct the Fock basis of solutions to the (linear) Hartree equation in the trajectory-coherent approximation (4.14). This is the first step in constructing the solution to recurrent equations (4.14)-(4.16).
Consider the properties of the symmetry operatorsâ k (t) (5.5) of the zero-order associated Schrödinger equation (4.14), which are necessary to construct the Fock basis. Statement 5.1 Let a 1 (t) and a 2 (t) be two solutions of the equations in variations andâ 1 (t) andâ 2 (t) be the respective symmetry operators of equation (4.14), defined in (5.5). Then the equality is valid.
Actually, upon direct checking we are convinced that Here, we have used the rules of commutation for the operators ∆ẑ. The skew scalar product holds and, hence, the statement is proved.

Remark 5.2 If the initial conditions for the equations in variations are taken such that
and N k = 1/ √ d k , then the following canonical commutation relations for the boson operators of "creation" (â + k (t)) and "annihilation" (â k (t)) are valid: The simplest example of initial data satisfying the conditions (5.11) is  Here, we have d k = 2 Im b k > 0, k = 1, n.
Theorem 5.1 The function 14) where N = [(π ) −n det D 0 ] 1/4 is a "vacuum" state for the operatorsâ j (t), such that a j (t)|0, t = 0, j = 1, n. Proof. Actually, substituting (5.5) and (5.14) into (5.15), we get Recollect that from the fact that the matrix D 0 is positive definite and diagonal follows det C(t) = 0, and so the matrix Im Q(t) is positive definite as well (see Appendix A).
Let us define the denumerable set of states |ν, t as a result of the action of the "creation" operators upon the "vacuum" state |0, t : By analogy with the linear theory (κ = 0), we call the functions |ν, t (5.16) quasi-classical trajectory-coherent states and consider their simplest properties.
Actually, we have and thus the statement is proved.
Then we calculate In view of (A.8) and the explicit form of the complex phase in (5.14), we have The matrix Im Q(t) is real and positive definite; hence, the matrix Im Q(t) does exist, such that Let us perform in the integral of (5.20) the change and then obtain Im b k . Thus, the functions |ν, t, Ψ 0 (5.16) form the Fock basis of solutions to equation To do this, we construct the Green function of the Cauchy problem for the zero-order associated Schrodinger equation. Although the Green function G (0) ( x, y, t, s) for quadric quantum systems is well known [25,38,39,40], we give for completeness its explicit form, as convenient to us. This function will allow us to demonstrate explicitly the nontrivial dependence of the evolution operator of the associated linear equation on the initial conditions for the starting Hartree type equation.
By definition we have where the operatorĤ 0 is defined in (4.10). We make use of the simplifying assumption that det H pp (s) = 0, det ∂p k (t, z 0 ) ∂p 0j = 0. (6.2) If the condition (6.2) is not valid, the solution of the problem can be found following the work [39,40]. For the problem under consideration, exact solutions of the Schrödinger equation (4.14) are known: these are the functions |ν, t, Ψ 0 (5.16) that form a complete set of functions. Thus we have Details of similar calculations can be found, for instance, in [25]. However, for our purposes the following approach seems to be convenient. Let us carry out an −1 Fourier transform in equation (6.1). For the Fourier transform of the Green functioñ Here,ˆ p = p andˆ x = i ∂ ∂ p and the symbols of the operators of equations (6.1) and (6.5) coincide: Equation (6.5) coincides to notations with (4.14) and, hence, admits solutions of type (5.6) G (0) ( p, y, t, s, Ψ 0 ) = exp − i S 0 (t, s, y) + G(t, s, y), ∆ p + 1 2 ∆ p,Q(t, s, y)∆ p , (6.6) where ∆ p = p − P (t, ). Here, the functions S 0 (t, s, y) = S 0 (t), G(t, s, y) = G(t) andQ(t, s, y) =Q(t) are to be determined and, according to (6.5), satisfy the initial conditions lim t→sQ (t, s, y) = 0, lim t→s G(t, s, y) = y, lim t→s S 0 (t, s, y) = p 0 , y . (6.7) Substituting (6.6) into (6.5), we writẽ Equating the terms with the same powers of ∆ p, we obtain the following set of equations: In view of (6.2), the solution to the Cauchy problem (A.2), (6.11) will then have the form where the matrices λ t 3 (t) and λ t 4 (t) are defined in (A.16). The matrix will then satisfy equation (6.8) with the initial conditions (6.7). Provided that (6.2) is valid, from (A.12) and (A.10) follows ). (6.14) In a similar manner, we obtain for In view of (A.12) and Liouville's lemma A.3, we obtain To calculate the last integral in (6.16), we use relationship (A.15) and, in view of (6.15), we get where the matrix λ t 2 (t) is defined in (A.16). Hence, we have Substituting (6.19), (6.15), and (6.13) into (6.6), we obtain the well-known expression (see, e.g., [39]) Now we substitute (6.19) into (6.4) and make use of the relationship We then obtain Here, we used the relationships It follows that for short times we have (see, e.g., [41]) Thus we have proved Theorem 6.1 Let the symbols of the operatorsĤ(t) andV (t, Ψ) satisfy the conditions of Supposition 1. Then the function 0 ( x, ) ∈ C t (z 0 , S 0 ). Then for the recurrent associated linear equations (4.14)-(4.16) we arrive at a Cauchy problem with initial data: The solution to these recurrent equations can readily be constructed as its expansion over the complete set of orthonormalized Fock functions |ν, t (5.15). As a result we obtain Denote byF (N ) (t, Ψ 0 ) the operator defined by the relationship whereÛ 0 (t, τ ) is the evolution operator of the associated Schrödinger equation (4.14) and the following notation has been made:Ĥ Thus we have proved the following statement: Theorem 7.1 Let the symbols of the operatorsĤ(t) andV (t, Ψ) satisfy the conditions of Supposition 1. Then the function ., (7.7) where N ≥ 2, is an asymptotical, accurate to O( (N +1)/2 ), solution of equation (1.1) and satisfies the initial condition (2.8).

The Green function and the nonlinear superposition principle
Let us show that in the class of trajectory-concentrated functions for the Hartree type equation (1.1) we can construct, with any given accuracy in 1/2 , the kernel of the evolution operator or the Green function of the Cauchy problem for equation (1.1). The explicit form of the quasi-classical asymptotics Ψ (N ) ( x, t, ) (7.7) makes it possible to obtain an expression for the Green function G (N ) ( x, y, t, s, Ψ 0 ) valid on finite time intervals t ∈ [0, T ]. Actually, according to (7.7), for any function ϕ( x, ) ∈ P 0 , the solution of the Cauchy problem with the initial condition for the associated linear Schrödinger equation (4.8) has the form 2) and the function G (0) ( x, y, t, s, Ψ 0 ) is defined in (6.21). It follows that SinceR (N ) (0, Ψ 0 ) = 1, we have for an arbitrary s < t being the Green function of the Cauchy problem (8.1) with s = 0. Obviously, for the functions G (N ) ( x, y, t, s, Ψ 0 ) the following composition rule is valid: Denoting byÛ (N ) (t, 0, Ψ 0 ) the approximate evolution operator of the linear equation (4.8) we obtain it from (8.3) in the form of the T-ordered Dyson expansion Here, we have used the following notations [41]: the domain of integration is an open hypertriangle the operatorĤ 1 (τ, t, Ψ 0 ) is a perturbation operator in the representation of the interaction It follows that operator (8.5) is an approximate evolution operator for the Hartree type equation (1.1) in the class of trajectory-concentrated functions. For the constructed asymptotical solutions, from expression (8.7) immediately follows [45,46] Theorem 8.1 (nonlinear superposition principle) Let Ψ 1 ( x, t, , y 3 (t, )) + c 2 Ψ 2 ( x, t, , y 3 (t, )).

The one-dimensional Hartree type equation with a Gaussian potential
Let us illustrate the above scheme for constructing asymptotical solutions by the example of a nonlinear interaction with a Gaussian potential [47]. By this example we shall show in an explicit form how the procedure of constructing quasi-classically concentrated solutions to the Hartree type equation necessarily leads to Hamilton-Ehrenfest equations. Moreover, it becomes possible to elucidate the "nonlinearity" of the generalized superposition principle for the Hartree type equation. We write equation (1.1) with a Gaussian potential for the one-dimensional case as In this case, for the class of functions P t (S(t, ), Z(t)) in which we shall find solutions to equation (9.1), in accordance with (2.1), we find Here, the function ϕ(ξ, t, ) ∈ S (Schwartz space) with respect to the variable ξ = ∆x √ and depends regularly on with ∆x = x − x(t). The functions S(t, ) and Z(t) = (P (t), X(t)) are real, depend regularly on , and are to be determined. Let us expand the exponential in equation (9.1) in a Taylor series for ∆x = x − X(t), ∆y = y − X(t) and restrict ourselves to the terms of the order of not above four in ∆x and ∆y. In view of the estimates (2.7), equation (9.1) will then take the form where σ xx (t, ) = α is the variance. The function Φ(x, t, ) belongs to the class P t S (t, ), Z(t) , wherẽ and satisfies the equation Φ (t, )) + Here, we have made use of α (k) We shall seek the approximate (mod 5/2 ) solution Φ(x, t, ) to equation (9.5) in the form , Z(t)). In equation (9.5) we equate the terms having the same estimate in √ in the sense of (2.7). Denote byL 0 the operator Earlier we have shown thatL 0 =Ô( ). We then have The function is a solution of equation (9.6). Here, we have used the fact that X(t) and P (t) are solutions of the ordinary differential equations and C(t) denotes the complex function satisfying the equations Similarly, equations (9.7) are Hamiltonian equations for a harmonic oscillator with frequency and their solution is In the linear case (κ = 0), the frequency Ω = 0 and equations (9.8) become equations in variations for equations (9.7). In view of (9.8), we find the variance of the coordinate x in an explicit form: Then we get It can readily be noticed that α Φ (0) (t, ). Hence, from (9.9) and (9.10) it can be inferred that for κV 0 < 0 the variance α  If C(t) and B(t) are solutions of equations (9.8), the operatorâ(t) commutates with the operatorL 0 . So the function Φ will also be a solution of the Schrödinger equation (9.5). Commuting the operatorsâ + (t) with the function Φ (0) 0 (x, t, ), we obtain the Fock basis of solutions for linear equation (9.11) where H n (ξ) are Hermite polinomials. Determining N a from the condition [â(t),â + (t)] = 1 and representing the solution of the equations in variations as we get Using the properties of Hermite polinomials, we can readily be convinced that the mean α Then we have Similarly, we find and for functions Ψ l (x, t) (9.14) the condition α is not a solution of equation (9.6) and, hence, the linear superposition principle is invalid for the functions (9.13) even in the class of asymptotical solutions P t S(t, ), P (t), X(t) accurate to O( 3/2 ). Thus, the presence of the term α (1) Φ (0) (t, ) in equation (9.6) violates the linear superposition principle (9.14).
We seek the solution to equation (9.1) in the class P t S(t, ), Z(t, ) , i.e., localize the solution asymptotically in the neighborhood of the trajectory z = Z(t, ) depending explicitly on parameter . With that, the estimates (2.7) remain valid. Let us take the dependence of Z(t, ) on the parameter → 0 such that the equation for the function Φ (0) (x, t, ) be linear. For doing this, we subject the functions X(t, ) and P (t, ) to the equations 15) and the functions C(t) and B(t) to the equations The function Φ (0) (x, t, ) will then satisfy the equation Unlike equations (9.6)-(9.8), equations (9.15)-(9.17) are dependent. Note that, within the accuracy under consideration, the principal term of the asymptotic will not change if equations (9.15) and (9.16) are solved accurate to O( 3/2 ) and O( ), respectively. Then equations (9.15) become k (x, t, ): Here, Φ k (x, t) is determined by expression (9.12) where X(t) and P (t) ought to be replaced by X(t, ) and P (t, ), respectively. Substitute (9.19) into (9.9). In view of the properties of Hermite polinomials Equations (9.18) will then take the form Integration of the obtained equations yields As a result, the principal term of the asymptotic can be represented in the form It follows that function (9.21) depends on Θ 1 and Θ 2 as on parameters: Here, Θ 1 and Θ 2 are determined by the sets of equations (9.20) and (9.22), respectively. Let us consider the Cauchy problem for equation (9.1): where G k = const . Denote by Ψ k (x, t, , Θ k 1 , Θ k 2 ) the principal term of the asymptotic solution to equation (9.1), satisfying the initial conditions (9.23). Then from the explicit form of function (9.13) follows Relationship (9.24) represents the nonlinear superposition principle for the asymptotical solutions of equation (9.1) in the class P t S(t, ), Z(t, ) .
The work was supported in part by the Russian Foundation for Basic Research (Grant No. 00-01-00087).
The set of equations (A.1) is called a set of equations in variations in vector form. Denote by B(t) and C(t) the n × n matrices composed of the "momentum" and "coordinate" parts of the solution of the equations in variations: The matrices B(t) and C(t) satisfy the set of equations which is called a set of equations in variations (5.4) in matrix form. Let us consider some properties of the solutions of this set of equations, which determine the explicit form of the asymptotical solution of the Hartree type equation and its approximate evolution operator. The complex number {a 1 , a 2 } = a 1 , Ja 2 is called a skew-scalar product of the vectors a 1 and a 2 , a k ∈ C 2n . Obviously, the skew-scalar product is antisymmetric: This statement can be checked immediately by differentiating the skew-scalar product {a 1 (t), a 2 (t)} with respect to t: d dt {a 1 (t), a 2 (t)} = ȧ 1 (t), Ja 2 (t) + a 1 (t), Jȧ 2 (t) = = JH zz (t)a 1 (t), Ja 2 (t) + a 1 (t), JJH zz (t)a 2 (t) = = a 1 (t), H zz (t)a 2 (t) − a 2 (t), H zz (t)a 2 (t) = 0.
Here  The relation of the matrices B(t) and C(t) to the matrix Q(t) and, in view of (5.6), to the function φ 1 (t) yields Statement A.3 Let the n × n-matrices B(t) and C(t) be solutions to equations in variations (5.4). Then, if det C(t) = 0, t ∈ [0, T ], the matrix Q(t) = B(t)C −1 (t) satisfies the Riccati matrix equation (5.9).
and since from C −1 (t)C(t) = I followsĊ −1 (t)C(t) + C −1 (t)Ċ(t) = 0, we havė A similar property is also valid for the matrix Q −1 (t): Here, A t denotes the transpose to the matrix A.
Actually, from (5.9) followsQ Hence, the matrix Q t (t) satisfies equation (5.9) with the same initial conditions as the matrix Q(t) since, as agreed, the matrix Q(0) is symmetrical. The validity of the statement follows from the uniqueness of the solution to the Cauchy problem.
Statement A.5 The imaginary parts of the matrices Q(t) and Q −1 (t) can be presented in the form Here, the matrix D 0 is defined by relationship (A.5).
Actually, by definition we have Q.E.D. Similarly, relationship (A.9) can be proved. The proof is similar to that of Statement A.6.
Statement A.7 If the matrix D 0 (A.5) is positive definite, the relation is valid.
As agreed, the matrix D 0 is positive definite, and, hence, the above equality holds only for | k| = 0. The obtained contradiction proves the lemma.