Quasiparticle solutions for the nonlocal NLSE with an anti-Hermitian term in semiclassical approximation

The nonlinear Schrödinger equation (NLSE) is a common model of collective excitations in nonlinear media. Its simplest variation with a local cubic nonlinearity and external potential is known as the Gross–Pitaevskii (GP) equation [2]. If one considers an open system, the NLSE with an anti-Hermitian term comes into play [3]. Such variation of the NLSE is widely used in modelling the propagation of optical pulses in nonlinear media [4, 5]. When the model includes the source of light and losses [6, 7], it deals with a fundamentally open system. The NLSE with an anti-Hermitian term also plays crucial role in modelling the Bose–Einstein condensate (BEC) within the framework of the GP model. The interaction of the BEC with an environment described by the anti-Hermitian terms leads to nontrivial effects in this model such as formation of the vortex lattice [8]. The anti-Hermitian terms are also necessary for the mathematical description of an atom laser [9].

The form of nonlinear terms in the NLSE varies greatly depending on specific physical model. In most cases, the cubic nonlinearity is considered as the simplest physically motivated nonlinearity. In the GP model, such nonlinearity describes the effective mean field of interpaticle interaction. The optics related models such as, e.g., the Haus equation [10] often account for the Kerr effect through the nonlinear term. However, even the simplest cubic nonlinearity leads to quite complex models from the mathematical point of view when one deals with the nonlocal nonlinearity. Returning to the GP model, the nonlocality is motivated by the consideration of long-range interactions such as the dipole-dipole interaction [11, 12, 13, 14]. In the Haus-like models, where both of the independent variables are associated with time, the nonlocality describes the memory effect of the medium. Besides the nonlocal Kerr effect, the saturation of the laser medium also can be included into the kernel of the nonlocal nonlinearity within the framework of such models [15]. A number of papers are devoted to the mathematical efforts in consideration of various aspects of nonlocality in the NLSE (see, e.g., [16, 17, 18]) as well to the derivation of such nonlocal models [19, 20, 21, 22, 23, 24]. However, it quite hard to obtain exact mathematical results for such complex equations, especially when the nonlocality and the non-Hermiticity are considered at the same time. Hence, many results rely only on numerical calculations [12, 25].

In order to advance in the problem under consideration, we will apply to the semiclassical approximation. A powerful tool that allows one to deals with such problem is the Maslov complex germ method [26, 27]. In [28], it was shown that this method can be applied to the nonlocal NLSE of a quite general form. Asymptotics for various specific nonlocal NLSE were obtained using the ideas of the Maslov complex germ method in the past years (see, e.g., [29, 30, 31, 32]). In [33], we have shown that the approach [28] can be generalized to the nonlocal NLSE with an anti-Hermitian term. That approach allows one to construct the asymptotic solutions to the Cauchy problem localized in a neighbourhood of one point moving along the trajectory determined by ”classical”  equations. The limitation of such approach with respect to the physics is that it really deals only with weakly nonlocal effects since the respective asymptotic solutions have trivial geometry. On the other hand, the attractive feature of the nonlocal models is the possibility to consider long-range interactions that can lead to nontrivial spatial patterns [34, 35]. Thus, we come to the even more complex problem of constructing asymptotic solutions to the nonlocal NLSE with an anti-Hermitian part that effectively depend on the behaviour of the nonlinearity kernel on its whole support rather than the neighbourhood of a center point, i.e. account for the long-range interactions. It turned out to be possible within the framework of the Maslov complex germ based approach if we introduce the so-called semiclassical quasiparticles similar to the ones in [36] where we dealt with the classical problem of a population dynamics.

The concept of quasiparticles is known in a theory of solitons for nonlinear equations that are exactly integrable in terms of the inverse scattering transform (IST) [37, 38]. The approximate solutions based on this conception can be obtained using the perturbation theory for the equations that are close to the exactly solvable ones with IST [39], e.g., the $(1+1)$-dimensional Kortewe-de Vries equation, the sine-Gordon equation, the NLSE with the local cubic nonlinearity, and some others. Such soliton solutions are treated as modes of the field excitation. Although our problem is far from the exactly solvable cases, the semiclassical quasiparticles can be treated in a similar way. Usually, the term ”quasiparticles”  implies the components of solution that are distinguishable in either the momentum space (see, e.g. [40]) or the coordinate space. The latter one is our case. Note that, for the Haus-like models, where there is no the spatial variables, the quasiparticles can be treated as a train of optical pulses. The distinctive feature of our approach is that the interaction of the semiclassical quasiparticles is ruled by the exact ”classical mechanics”  (dynamical system) rather than by the pertubation of the exactly solvable wave equation.

In this work, we construct the asymptotic solutions to the Cauchy problem for the nonlocal NLSE with an anti-Hermitian term that are semiclassically localized in a neighbourhood of few trajectories associated with the dynamics of quasiparticles. The paper is organized as follows. In Section II, we give the mathematical statement of the problem under consideration, and clarify the meaning of the semiclassically concentrated states. In Section III, we introduce the wave functions of quasiparticles that are auxiliary mathematical objects allowing us to construct the approximate solution to the original NLSE. The moments of such wave functions are defined in Section IV. The solutions to the dynamical system describing these moments is a key elements of our approach. In Section V, we pose the Cauchy problem for the equations associated with the NLSE and give its relation to the original problem. Section VI is devoted to the reduction of the original complex nonlinear problem to the linear ones within the framework of our quasiparticle formalism. The approximate evolution operator for the original NLSE is constructed. In Section VII, we provide the general formalism with a physically motivated example. The one-dimensional NLSE with dipole-dipole interaction, optical-lattice potential, and phenomenological damping is considered. It is shown that the non-perturbative interaction of semiclassical quasiparticles plays a crucial role in the behaviour of the solution to the NLSE. In Section VIII, we conclude with some remarks.

We deal with a quite general form of the non-Hermitian nonlocal NLSE that reads as follows:

$$\begin{array}[]{l}\bigg{\{}-i\hbar\partial_{t}+H(\hat{z},t)[\Psi]-i\hbar\Lambda\breve{H}(\hat{z},t)[\Psi]\bigg{\}}\Psi(\vec{x},t)=0,\cr\cr H(\hat{z},t)[\Psi]=V(\hat{z},t)+\varkappa\displaystyle\int\limits_{{\mathbb{R}}^{n}}d\vec{y}\,\Psi^{*}(\vec{y},t)W(\hat{z},\hat{w},t)\Psi(\vec{y},t),\cr\breve{H}(\hat{z},t)[\Psi]=\breve{V}(\hat{z},t)+\varkappa\displaystyle\int\limits_{{\mathbb{R}}^{n}}d\vec{y}\,\Psi^{*}(\vec{y},t)\breve{W}(\hat{z},\hat{w},t)\Psi(\vec{y},t).\end{array}$$

(1)

Here, $\vec{x}\in{\mathbb{R}}^{n}$,$\hat{\vec{p}}_{x}=-i\hbar\partial_{\vec{x}}$, $\hat{z}=(\hat{\vec{p}}_{x},\vec{x})$, and $\hat{w}=(\hat{\vec{p}}_{y},\vec{y})$. As usual for a semiclassical formalism, an operator of the equation is defined in terms of pseudo-differential operators [41, 42]. So, the operators $V(\hat{z},t)$, $\breve{V}(\hat{z},t)$, $W(\hat{z},\hat{w},t)$, $\breve{W}(\hat{z},\hat{w},t)$ belong to the set ${\mathcal{A}}^{t}_{\hbar}$ of pseudo-differential operators with smooth symbols growing not faster than polynomial (some related properties and formal definitions are given in Appendix A). Since the equation (1) is given in the coordinate representation, the operator $\vec{x}$ is the operator of multiplication by $\vec{x}$. Hereinafter, we put a right arrow over and only over the $n$-dimensional vectors. The nonlocal interactions of atoms in BEC usually lead to the kernels $W$ and $\breve{W}$ with ”almost compact support”, while the saturation of the medium in the optics related models yield the step-like contribution to the $\breve{W}$. Due to the wide variety of kernels in physics, the application of our formalism to the specific models can benefit greatly from the generality of our problem statement.

Hereinafter, $s=\overline{1,K}$ stands for $s=1,2,...,K$, where $K\in{\mathbb{N}}$ is interpreted as a total number of quasiparticles. The $2n$-tuple vector $Z_{s}(t)=\left(\vec{P}_{s}(t),\vec{X}_{s}(t)\right)$ can be treated as the phase coordinate of the $s$-th quasiparticle, and $\mu_{s}(t)$ corresponds to a quantity of matter related to the $s$-th quasiparticle (a ”mass”  of the quasiparticle). We term the weight functions $\mu_{s}(t)$ as ”masses”  of quasiparticles. Note that $\mu_{s}(t)$ is not direct analog for a mass of a classical particle since it is not a measure of inertia of the $s$-th particle in a common sense (it will be clear in the specific example in Section VII). However, there are some reasons to draw the analogy with masses for $\mu_{s}(t)$ that are given below.

Let $\mu_{\Psi}(t,\hbar)=||\Psi||^{2}(t,\hbar)$. Note that the following relation holds in the class ${\mathcal{T}}^{t}_{\hbar}\left(\left\{Z_{s}(t),\mu_{s}^{(0)}(t)\right\}_{s=1}^{K}\right)$:

$$\lim\limits_{\hbar\to 0}\mu_{\Psi}(t,\hbar)=\sum_{s=1}^{K}\mu^{(0)}_{s}(t).$$

(3)

This relation is obtained using the substitution of $\hat{A}=1$ into (2). Hereinafter, we will denote the second functional parameters of the class ${\mathcal{T}}^{t}_{\hbar}\left(\left\{Z_{s}(t),\mu_{s}^{(0)}(t)\right\}_{s=1}^{K}\right)$ by $\mu^{(0)}_{s}(t)$ since it corresponds to the zeroth order approximation (with respect of $\hbar$) of the zeroth moments of the quasiparticle wave function, which will be introduces in Section IV.

Let us also consider the mean value of the coordinate $\vec{X}_{\Psi}(t,\hbar)=\displaystyle\frac{1}{\mu_{\Psi}(t,\hbar)}\displaystyle\int\limits_{{\mathbb{R}}^{n}}\vec{x}|\Psi(\vec{x},t)|^{2}$. From (2) one readily gets the following relation:

$$\lim\limits_{\hbar\to 0}\vec{X}_{\Psi}(t,\hbar)=\displaystyle\frac{\sum_{s=1}^{K}\mu^{(0)}_{s}(t)\vec{X}_{s}(t)}{\sum_{s=1}^{K}\mu^{(0)}_{s}(t)}.$$

(4)

We can treat $\lim\limits_{\hbar\to 0}\vec{X}_{\Psi}(t,\hbar)$ as the coordinate of the center of mass of the system within the semiclassical approximation and $\lim\limits_{\hbar\to 0}\mu_{\Psi}(t,\hbar)$ as its mass. Then, the equations (3) and (4) correspond to the well-known laws for the mass of a classical system and the position of its center of mass, respectively, where $\mu^{(0)}_{s}(t)$ act exactly as masses of the quasiparticles that constitute such system.

Now we go on to the derivation of the equations that determine the parameters of the class ${\mathcal{T}}^{t}_{\hbar}\left(\left\{Z_{s}(t),\mu_{s}^{(0)}(t)\right\}_{s=1}^{K}\right)$ on solutions of (1). The equation for $\mu_{\Psi}(t,\hbar)$ can be readily obtained using the direct substitution of $\partial_{t}\Psi(\vec{x},t)$ from the original equation (1) into $\dot{\mu}_{\Psi}(t,\hbar)$ and reads as follows:

$$\begin{gathered}\dot{\mu}_{\Psi}(t,\hbar)=-2\Lambda\displaystyle\int\limits_{{\mathbb{R}}^{n}}d\vec{x}\,\Psi^{*}(\vec{x},t;\hbar)\breve{H}(\hat{z},t)[\Psi]\Psi(\vec{x},t;\hbar)=-2\Lambda\langle\Psi|\breve{H}[\Psi]|\Psi\rangle=\\ =-2\Lambda\left(\langle\Psi|\breve{V}(\hat{z},t)|\Psi\rangle+\varkappa\langle\Psi|\displaystyle\int\limits_{{\mathbb{R}}^{n}}d\vec{y}\,\Psi^{*}(\vec{y},t)\breve{W}(\hat{z},\hat{w},t)\Psi(\vec{y},t)|\Psi\rangle\right).\end{gathered}$$

(5)

Using (2), we also obtain

$$\begin{gathered}\lim\limits_{\hbar\to 0}\langle\Psi|\breve{V}(\hat{z},t)|\Psi\rangle=\sum_{s=1}^{K}\mu^{(0)}_{s}(t)\breve{V}(Z_{s}(t),t),\\ \lim\limits_{\hbar\to 0}\langle\Psi|\displaystyle\int\limits_{{\mathbb{R}}^{n}}d\vec{y}\,\Psi^{*}(\vec{y},t)\breve{W}(\hat{z},\hat{w},t)\Psi(\vec{y},t)|\Psi\rangle=\sum_{s=1}^{K}\sum_{r=1}^{K}\mu^{(0)}_{s}(t)\mu^{(0)}_{r}(t)\breve{W}(Z_{s}(t),Z_{r}(t),t).\end{gathered}$$

(6)

Then, the equation (5), in the limit $\hbar\to 0$, can be written as follows:

$$\sum_{s=1}^{K}\dot{\mu}^{(0)}_{s}(t)=-2\Lambda\sum_{s=1}^{K}\mu^{(0)}_{s}(t)\left(\breve{V}(Z_{s}(t),t)+\varkappa\sum_{r=1}^{K}\mu^{(0)}_{r}(t)\breve{W}(Z_{s}(t),Z_{r}(t),t)\right).$$

(7)

Now, let $A_{\Psi}(t)=\langle\hat{A}\rangle_{\Psi}=\langle\Psi|\hat{A}|\Psi\rangle$. The exact equation for $A_{\Psi}(t)$ is as follows:

	$\displaystyle\displaystyle\frac{\partial}{\partial t}\langle\hat{A}(t)\rangle_{\Psi}=\bigg{\langle}\displaystyle\frac{\partial\hat{A}(t)}{\partial t}\bigg{\rangle}_{\Psi}+\frac{i}{\hbar}\big{\langle}\big{[}H(\hat{z},t)[\Psi],\hat{A}(t)\big{]}\big{\rangle}_{\Psi}-\Lambda\big{\langle}\big{[}\breve{H}(\hat{z},t)[\Psi],\hat{A}(t)\big{]}_{+}\big{\rangle}_{\Psi}=$		(8)
	$\displaystyle=\bigg{\langle}\displaystyle\frac{\partial\hat{A}(t)}{\partial t}\bigg{\rangle}_{\Psi}+\displaystyle\frac{i}{\hbar}\big{\langle}[V(\hat{z},t),A(\hat{z},t)]\big{\rangle}_{\Psi}-\Lambda\big{\langle}[\breve{V}(\hat{z},t),A(\hat{z},t)]_{+}\big{\rangle}_{\Psi}+$		(9)
	$\displaystyle+\varkappa\bigg{\langle}\displaystyle\int\limits_{{\mathbb{R}}^{n}}d\vec{y}\,\Psi^{*}\Big{(}\displaystyle\frac{i}{\hbar}[W(\hat{z},\hat{w},t),A(\hat{z},t)]-\Lambda[\breve{W}(\hat{z},\hat{w},t),A(\hat{z},t)]_{+}\Big{)}\Psi(\vec{y},t)\bigg{\rangle}_{\Psi}.$		(10)

Let us use the property (68) of pseudo-differential operators. Then, for the operator $\hat{A}=A(\hat{z},t)$ with a Weyl symbol $A(z,t)$, in the limit $\hbar\to 0$, we come at the following equation:

$$\begin{gathered}\displaystyle\frac{d}{dt}\sum_{s=1}^{K}\mu^{(0)}_{s}(t)A(Z_{s}(t),t)=\sum_{s=1}^{K}\mu^{(0)}_{s}(t)\Bigg{(}\displaystyle\frac{\partial A(z_{s},t)}{\partial t}-\left\{V(z_{s},t),A(z_{s},t)\right\}-2\Lambda\breve{V}(z_{s},t)A(z_{s},t)+\\ +\varkappa\sum_{r=1}^{K}\mu^{(0)}_{r}(t)\Big{(}-\left\{W(z_{s},w_{r},t),A(z_{s},t)\right\}-2\Lambda\breve{W}(z_{s},w_{r},t)A(z_{s},t)\Big{)}\Bigg{)}\Big{|}_{z_{s}=Z_{s}(t),\,w_{r}=Z_{r}(t)}.\end{gathered}$$

(11)

In particular, for $A(z,t)=z$, we have

$$\begin{gathered}\displaystyle\frac{d}{dt}\sum_{s=1}^{K}\mu^{(0)}_{s}(t)Z_{s}(t)=\sum_{s=1}^{K}\mu^{(0)}_{s}(t)\Bigg{(}JV_{z}(Z_{s}(t),t)-2\Lambda\breve{V}(Z_{s}(t),t)Z_{s}(t)+\\ +\varkappa\sum_{r=1}^{K}\mu^{(0)}_{r}(t)\Big{(}JW_{z}(Z_{s}(t),Z_{r}(t),t)-2\Lambda\breve{W}(Z_{s}(t),Z_{r}(t),t)Z_{s}(t)\Big{)}\Bigg{)}.\end{gathered}$$

(12)

Note that the system of the first order ordinary differential equations (ODEs) (7), (12) is closed.

The system of $(2n+1)$ equations (7), (12) admits particular solutions that satisfy the following system of $K(2n+1)$ equations:

$$\begin{gathered}\dot{\mu}^{(0)}_{s}(t)=-2\Lambda\mu^{(0)}_{s}(t)\left(\breve{V}(Z_{s}(t),t)+\varkappa\sum_{r=1}^{K}\mu^{(0)}_{r}(t)\breve{W}(Z_{s}(t),Z_{r}(t),t)\right),\\ \dot{Z}_{s}(t)=JV_{z}(Z_{s}(t),t)+\varkappa\sum_{r=1}^{K}\mu^{(0)}_{r}(t)JW_{z}(Z_{s}(t),Z_{r}(t)),\quad s=\overline{1,K}.\end{gathered}$$

(13)

We will try an asymptotic solution to the equation (1) in the class ${\mathcal{T}}^{t}_{\hbar}\left(\left\{Z_{s}(t),\mu_{s}^{(0)}(t)\right\}_{s=1}^{K}\right)$ where the functional parameters $Z_{s}(t)$, $\mu^{(0)}_{s}(t)$, $s=\overline{1,K}$, satisfy the system of ”classical”  equations (13). We term the system (13) as the zeroth order $K$-particle Hamilton-Ehrenfest system by analogy with [28].

Let us introduce the family of classes of trajectory concentrated functions ${\mathcal{P}}_{\hbar}^{t}(Z_{s}(t),S_{s}(t,\hbar))$ that reads as follows [28]:

$\displaystyle{\mathcal{P}}_{\hbar}^{t}(Z_{s}(t),S_{s}(t,\hbar))=\bigg{\{}\Phi:\Phi(\vec{x},t,\hbar)=\hbar^{-n/4}\cdot\varphi\Big{(}\frac{\Delta\vec{x}_{s}}{\sqrt{\hbar}},t,\hbar\Big{)}\cdot\exp\Big{[}\frac{i}{\hbar}\left(S_{s}(t,\hbar)+\langle\vec{P}_{s}(t),\Delta\vec{x}_{s}\rangle\right)\Big{]}\bigg{\}}.$

(14)

Here, $\Phi(\vec{x},t,\hbar)$ is a general element of the class ${\mathcal{P}}_{\hbar}^{t}(Z_{s}(t),S_{s}(t,\hbar))$; the real functions $Z_{s}(t)=(\vec{P}_{s}(t),\vec{X}_{s}(t))$ and $S_{s}(t,\hbar)$ are functional parameters of the class ${\mathcal{P}}_{\hbar}^{t}(Z_{s}(t),S_{s}(t,\hbar))$; $\Delta\vec{x}_{s}=\vec{x}-\vec{X}_{s}(t)$; the function $\varphi(\vec{\xi},t,\hbar)$ belongs to the Schwartz space $\mathbb{S}$ with respect to the variables $\vec{\xi}\in{\mathbb{R}}^{n}$; the functions $Z_{s}(t),S_{s}(t,\hbar)$, and $\varphi(\vec{\xi},t,\hbar)$ smoothly depend on $t$ and regularly depend on $\sqrt{\hbar}$ in a neighbourhood of $\hbar=0$. The notations $\langle\cdot,\cdot\rangle$ stands for the Euclidean scalar product of vectors.

The functions $Z_{s}(t)$ and $\mu^{(0)}_{s}(t)$, which determine the trajectory and ”mass”  of the $s$-th quasiparticle, will be subjected to the system of equation (13), and the function $S_{s}(t,\hbar)$ will be defined later.

We will seek for a solution to the equation (1) in the set of functions that can be presented as follows:

$$\Psi(\vec{x},t,\hbar)=\sum_{s=1}^{K}\Psi_{s}(\vec{x},t,\hbar),\quad\Psi_{s}(\vec{x},t,\hbar)\in{\mathcal{P}}_{\hbar}^{t}(Z_{s}(t),S_{s}(t,\hbar)).$$

(15)

Note that the function $\Psi(\vec{x},t,\hbar)$ does not belong to the family of classes ${\mathcal{P}}_{\hbar}^{t}$ in the general case $K>1$. In the specific case $K=1$, the formalism that we propose here can be reduced to the one constructed in [33] under some simplifying assumptions. The function $\Psi$ of the form (15) meets the definition (2). Thus, $\Psi_{s}$ can be termed as semiclassical wave function of the $s$-th quasiparticle.

One readily gets, that, if the function $\Psi_{s}(\vec{x},t,\hbar)$ satisfies the system

$$\begin{array}[]{l}\bigg{\{}-i\hbar\partial_{t}+H(\hat{z},t)[\sum_{r=1}^{K}\Psi_{r}]-i\hbar\Lambda\breve{H}(\hat{z},t)[\sum_{r=1}^{K}\Psi_{r}]\bigg{\}}\Psi_{s}(\vec{x},t)=0,\quad s=\overline{1,K},\end{array}$$

(16)

the function $\Psi(\vec{x},t,\hbar)$ given by (15) obeys the original equation (1). For brevity, we will denote the class ${\mathcal{P}}_{\hbar}^{t}(Z_{s}(t),S_{s}(t,\hbar))$ as ${\mathcal{P}}_{\hbar}^{t}(s)$ where it does not cause the confusion.

It was proved in [28, 43] that, for functions from the class ${\mathcal{P}}_{\hbar}^{t}(s)$ on a finite time interval $t\in[0;T]$, the following asymptotic estimates hold:

	$\displaystyle\{\Delta\hat{z}_{s}\}^{\alpha}=\hat{{\rm O}}(\hbar^{|\alpha|/2}),\quad\Delta\hat{z}_{s}=(\Delta\hat{\vec{p}}_{s},\Delta\vec{x}_{s}),$		(17)
	$\displaystyle\langle\Phi_{s}|\{\Delta\hat{z}_{s}\}^{\alpha}|\Phi_{s}\rangle={\rm O}(\hbar^{|\alpha|/2}),\quad\Phi_{s}\in{\mathcal{P}}_{\hbar}^{t}(s).$		(18)

Here, the following notations are used. The estimate $\hat{A}=\hat{{\rm O}}(\hbar^{m})$, $m\geq 0$, in (17) means that

$\displaystyle\frac{||\hat{A}\Phi||}{||\Phi||}={\rm O}(\hbar^{m}),\quad\Phi\in{\mathcal{P}}_{\hbar}^{t}|_{s};$

(19)

$Z_{s}(t)=(\vec{P}_{s}(t),\vec{X}_{s}(t))$, $\Delta\hat{z}_{s}=\hat{z}-Z_{s}(t)=(\Delta\hat{\vec{p}}_{s},\Delta\vec{x}_{s})$, $\Delta\hat{\vec{p}}_{s}=\hat{\vec{p}}-\vec{P}_{s}(t)$, $\Delta\vec{x}_{s}=\vec{x}-\vec{X}_{s}(t)$; $\{\Delta\hat{z}_{s}\}^{\alpha}$ is the operator determined by the Weyl symbol $(\Delta{z}_{s})^{\alpha}=(z-Z_{s}(t))^{\alpha}$. The multi-index $\alpha\in{\mathbb{Z}}^{2n}_{+}$ ($2n$-tuple) reads $\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{2n})$; $\alpha_{j}\in{\mathbb{Z}}^{1}_{+}$, $j=1,2,\ldots,2n$; $|\alpha|=\alpha_{1}+\alpha_{2}+\ldots+\alpha_{2n}$. For $v=(v_{1},v_{2},\ldots,v_{2n})$, we put $v^{\alpha}=v_{1}^{\alpha_{1}}v_{2}^{\alpha_{2}}\ldots v_{2n}^{\alpha_{2n}}$.

In particular, we have

	$\displaystyle\Delta{x}_{s}=\hat{{\rm O}}(\sqrt{\hbar}),\quad\Delta\hat{p}_{s}=\hat{{\rm O}}(\sqrt{\hbar}),$		(20)
	$\displaystyle-i\hbar\partial/\partial t-\dot{S}_{s}(t,\hbar)+\langle{\vec{P}}_{s}(t),\dot{\vec{X}}_{s}(t)\rangle+\langle\dot{Z}_{s}(t),J{\Delta\hat{z}}_{s}\rangle=\hat{{\rm O}}(\hbar).$		(21)

The functions $\Delta^{(\alpha)}_{\Phi_{s}}(t,\hbar)$ are $|\alpha|$-th order central moments of the function $\Phi_{s}$.

Hereinafter, all calculations and commentaries are given for $t\in[0;T]$ where $T<\infty$.

Let us introduce the following definition for m-th order central moments of the functions $\Psi_{s}(\vec{x},t,\hbar)$:

$$\begin{gathered}\Delta_{s,j_{1}j_{2}...j_{m}}(t,\hbar)[\Psi_{s}]=\langle\Psi_{s}|\Delta\hat{z}_{s,j_{1}}\Delta\hat{z}_{s,j_{2}}...\Delta\hat{z}_{s,j_{m}}|\Psi_{s}\rangle\Big{|}_{\text{symmetrized over }j_{1},...,j_{m}}.\end{gathered}$$

(22)

Indices $j_{k}$, $k=\overline{1,m}$, stand for a number of the element $z\in{\mathbb{R}}^{2n}$. The formula (22) symmetrized over $j_{1},...,j_{m}$ means that $\Delta_{s,j_{1}j_{2}...j_{m}}(t,\hbar)$ is the expectation of the operator with the Weyl symbol $A(z)=\Delta{z}_{s,j_{1}}\Delta{z}_{s,j_{2}}...\Delta{z}_{s,j_{m}}$, $\Delta z_{s}=z-Z_{s}(t)$. Thereinafter, the zeroth order moment of the functions $\Psi_{s}(\vec{x},t,\hbar)$ will be denoted by

$$\mu_{s}(t,\hbar)[\Psi_{s}]=||\Psi_{s}||^{2}.$$

(23)

It is clear that $\Delta_{s,j_{1}j_{2}...j_{m}}(t,\hbar)[\Psi_{s}]={\rm O}(\hbar^{m/2})$ from (18). Let the trajectories $\vec{x}=\vec{X}_{s}(t)$ do not intersect for different $s$, i.e.

$$Z_{r}(t)\neq Z_{s}(t),\quad r\neq s,\quad\forall t\in[0,T].$$

(24)

Under assumption (24), the asymptotic expansion for the operator $H(z,t)[\sum_{r=1}^{K}\Psi_{r}]$ reads

$$\begin{gathered}H(\hat{z},t)\Big{[}\sum_{r=1}^{K}\Psi_{r}\Big{]}=V(\hat{z},t)+\varkappa\sum_{r=1}^{K}\bigg{(}W(\hat{z},w,t)\mu_{r}(t,\hbar)[\Psi_{r}]+\\ +\sum_{m=1}^{M}\displaystyle\frac{1}{m!}W_{w_{j_{1}}w_{j_{2}}...w_{j_{m}}}(\hat{z},w,t)\Delta_{r,j_{1}j_{2}...j_{m}}(t,\hbar)[\Psi_{r}]\bigg{)}\Big{|}_{w=Z_{r}(t)}+\hat{{\rm O}}(\hbar^{(M+1)/2}).\end{gathered}$$

(25)

Hereinafter, the summation over repeated indices $j_{k}$ and $i_{k}$, $k\in{\mathbb{N}}$, is implied. The summation over other repeated indices is not implied until it is explicitly stated. The similar relation holds for $\breve{H}$ up to changes of $H\to\breve{H}$, $V\to\breve{V}$, and $W\to\breve{W}$. Note that we have taken into account in (25) that the following estimate holds under assumption (24):

$$\langle\Psi_{r}|A(\hat{z},t)|\Psi_{s}\rangle={\rm O}(\hbar^{\infty}),\quad r\neq s.$$

(26)

In the class ${\mathcal{P}}_{\hbar}^{t}(s)$, the moments (22), (23) also admit the following expansion:

$$\begin{gathered}\Delta_{s,j_{1}j_{2}...j_{m}}(t,\hbar)[\Psi_{s}]=\sum_{k=m}^{M}\hbar^{k/2}\Delta^{(k)}_{s,j_{1}j_{2}...j_{m}}(t)[\Psi_{s}]+{\rm O}(\hbar^{(M+1)/2}),\\ \mu_{s}(t,\hbar)[\Psi_{s}]=\sum_{k=0}^{M}\hbar^{k/2}\mu^{(k)}_{s}(t)[\Psi_{s}]+{\rm O}(\hbar^{(M+1)/2}).\end{gathered}$$

(27)

Hence, the equations (10) for the moments of up to the $M$-th order with the accuracy of ${\rm O}(\hbar^{(M+1)/2})$ form a closed system of ODEs. We term this system as the $M$-th order $K$-particle Hamilton–Ehrenfest system for $M\geq 1$ following [28]. The Hamilton–Ehrenfest systems for $M\leq 3$ are derived in Appendix B.

Let us denote

$$\hat{L}\Big{[}\sum_{r=1}^{K}\Psi_{r}\Big{]}=-i\hbar\partial_{t}+H(\hat{z},t)\Big{[}\sum_{r=1}^{K}\Psi_{r}\Big{]}-i\hbar\Lambda\breve{H}(\hat{z},t)\Big{[}\sum_{r=1}^{K}\Psi_{r}\Big{]}.$$

(28)

We pose the Cauchy problem for (1) as follows:

$$\Psi(\vec{x},t,\hbar)\Big{|}_{t=0}=\psi(\vec{x},\hbar).$$

(29)

Let the initial condition $\psi(\vec{x},t,\hbar)$ meets the condition (15), i.e. we have

$$\psi(\vec{x},\hbar)=\sum_{s=1}^{K}\psi_{s}(\vec{x},\hbar),\qquad\psi_{s}(\vec{x},\hbar)\in{\mathcal{P}}_{\hbar}^{0}(Z_{s}(0),S_{s}(0,\hbar)).$$

(30)

Then, the solution to the Cauchy problem for (16) with the initial conditions

$$\Psi_{s}(\vec{x},t,\hbar)\Big{|}_{t=0}=\psi_{s}(\vec{x},\hbar),\quad s=\overline{1,K},$$

(31)

constitutes the solution to the Cauchy problem (1), (29), (30) following (15).

Let ${\bf C}$ be set of initial conditions for the Hamilton–Ehrenfest system:

$${\bf C}=\left(\mu_{s}(0,\hbar),\left(\left(\Delta_{s,j_{1}j_{2}...j_{m}}(0,\hbar)\right)_{j_{k}=1}^{2n}\right)_{m=1}^{\infty}\right)_{s=1}^{K},\quad k=\overline{1,m}.$$

(32)

The parameters ${\bf C}$ enumerate the parametric family of the solutions to the Cauchy problem for the $M$-th order Hamilton–Ehrenfest system. We need those solutions from this family that correspond to the initial condition (31):

$${\bf C}\Big{[}\big{(}\psi_{s}\big{)}_{s=1}^{K}\Big{]}=\left(\mu_{s}(0,\hbar)[\psi_{s}],\left(\left(\Delta_{s,j_{1}j_{2}...j_{m}}(0,\hbar)[\psi_{s}]\right)_{j_{k}=1}^{2n}\right)_{m=1}^{\infty}\right)_{s=1}^{K},\quad k=\overline{1,m}.$$

(33)

Then, if we replace the moments in (25) with the solutions to the Hamilton–Ehrenfest system corresponding to the initial conditions ${\bf C}$, the resulting operators $H(\hat{z},t,{\bf C})$ and $\breve{H}(\hat{z},t,{\bf C})$ can be treated as linear ones parameterized by ${\bf C}$. The linear operators $H(\hat{z},t,{\bf C})$ and $\breve{H}(\hat{z},t,{\bf C})$ act on the function $\Psi_{s}$ in the same way as $H(\hat{z},t)\Big{[}\sum_{r=1}^{K}\Psi_{r}\Big{]}$ and $\breve{H}(\hat{z},t)\Big{[}\sum_{r=1}^{K}\Psi_{r}\Big{]}$ if the parameters ${\bf C}$ obey the following algebraic condition:

$${\bf C}={\bf C}\Big{[}\big{(}\psi_{s}\big{)}_{s=1}^{K}\Big{]}.$$

(34)

Note that, within the accuracy of ${\rm O}(\hbar^{(M+1)/2})$, we need to determine only a finite set of elements of ${\bf C}$, i.e. (34) is equal to the finite system of algebraic equations.

In the class ${\mathcal{P}}_{\hbar}^{t}(s)$, in view of (13), the estimate (21) yields:

$$\begin{gathered}-i\hbar\partial_{t}-\dot{S}_{s}(t)+\langle\vec{P}_{s}(t),\dot{\vec{X}}_{s}(t)\rangle+\\ +\big{\langle}V_{z}(Z_{s}(t),t)+\varkappa\sum_{r=1}^{K}\mu^{(0)}_{r}(t)W_{z}(Z_{s}(t),Z_{r}(t),t),\Delta\hat{z}_{s}\big{\rangle}=\hat{{\rm O}}(\hbar).\end{gathered}$$

(35)

Expanding operators $H(\hat{z},t,{\bf C})$ and $\breve{H}(\hat{z},t,{\bf C})$ into an asymptotic series in a neighbourhood of the trajectory $z=Z_{s}(t)$, we obtain the following estimate in the class ${\mathcal{P}}_{\hbar}^{t}(s)$:

$$\begin{gathered}H(\hat{z},t,{\bf C})=\sum_{m=0}^{M}\displaystyle\frac{1}{k!}H_{z_{j_{1}}...z_{j_{m}}}(z,t,{\bf C})\Big{|}_{z=Z_{s}(t)}\Delta\hat{z}_{s,j_{1}}...\Delta\hat{z}_{s,j_{m}}+\hat{{\rm O}}(\hbar^{(M+1)/2}).\end{gathered}$$

(36)

Using the estimates (36),(25), and (35) for $M=1$, in the class ${\mathcal{P}}_{\hbar}^{t}(s)$, one readily gets that the operator (28) can be written as

$$\hat{L}\Big{[}\sum_{r=1}^{K}\Psi_{r}\Big{]}=\sum_{m=2}^{M}\hat{L}_{s}^{(m)}\Big{[}\sum_{r=1}^{K}\Psi_{r}\Big{]}+\hat{{\rm O}}(\hbar^{(M+1)/2}),\quad\hat{L}_{s}^{(m)}\Big{[}\sum_{r=1}^{K}\Psi_{r}\Big{]}=\hat{{\rm O}}(\hbar^{m/2}),$$

(37)

if we put the functional parameter $S_{s}(t)$ of the class ${\mathcal{P}}_{\hbar}^{t}(s)$ as follows:

$$\begin{gathered}\dot{S}_{s}(t,\hbar)=\langle\vec{P}_{s}(t),\dot{\vec{X}}_{s}(t)\rangle-\Bigg{(}V(z,t)+\varkappa\sum_{r=1}^{K}\bigg{(}\mu_{r}(t,\hbar)W(z,w)+\\ +\Delta_{r,j}(t,\hbar)W_{w_{j}}(z,w,t)\bigg{)}\Big{|}_{w=Z_{r}(t)}\Bigg{)}\bigg{|}_{z=Z_{s}(t)},\end{gathered}$$

(38)

where summation over repeated index $j$ is implied, and the moments $\mu_{r}(t,\hbar)$, $\Delta_{r,j}(t,\hbar)$ correspond to the initial conditions (34). Note that we replace the operators $\hat{L}_{s}^{(m)}\Big{[}\sum_{r=1}^{K}\Psi_{r}\Big{]}$ with the linear operators $\hat{L}_{s}^{(m)}({\bf C})$ under the algebraic condition (34) in view of (28). In (38) and thereinafter, we will omit the argument ${\bf C}$ if it does not cause the confusion.

Let us give the explicit form of the operators $\hat{L}_{s}^{(m)}$ in (37) for $m=2$ and $m=3$:

$$\begin{gathered}\hat{L}_{s}^{(2)}=-i\hbar\partial_{t}-\dot{S}_{s}(t,\hbar)+\vec{P}_{s,j_{1}}(t)\dot{\vec{X}}_{s,j_{1}}(t)+\displaystyle\frac{\varkappa}{2}\sum_{r=1}^{K}\Delta_{r,j_{1}j_{2}}(t,\hbar)W_{|j_{1}j_{2}}(Z_{s}(t),Z_{r}(t),t)-\\ -i\hbar\Lambda\breve{V}(Z_{s}(t),t)-i\hbar\Lambda\varkappa\sum_{r=1}^{K}\mu_{r}(t,\hbar)\breve{W}_{|}(Z_{s}(t),Z_{r}(t),t)+\\ +V_{j_{1}}(Z_{s}(t),t)\Delta\hat{z}_{s,j_{1}}+\varkappa\sum_{r=1}^{K}\mu_{r}(t,\hbar)W_{i_{1}|}(Z_{s}(t),Z_{r}(t),t)\Delta\hat{z}_{s,i_{1}}+\\ +\varkappa\sum_{r=1}^{K}\Delta_{r,j_{1}}(t,\hbar)W_{i_{1}|j_{1}}(Z_{s}(t),Z_{r}(t),t)\Delta\hat{z}_{s,i_{1}}+\displaystyle\frac{1}{2}V_{j_{1}j_{2}}(Z_{s}(t),t)\Delta\hat{z}_{s,j_{1}}\Delta\hat{z}_{s,j_{2}}+\\ +\displaystyle\frac{\varkappa}{2}\sum_{r=1}^{K}\mu_{r}(t,\hbar)W_{i_{1}i_{2}|}(Z_{s}(t),Z_{r}(t),t)\Delta\hat{z}_{s,i_{1}}\Delta\hat{z}_{s,i_{2}},\end{gathered}$$

(39)