Model of the effective separable potential in the problem of three one-dimensional quantum particles

The goal of this paper is to construct an effective model for studying the asymptotic solution of the scattering problem of three one-dimensional quantum particles with finite (short-range) attractive pair potentials. The asymptotic nature of the solution is defined by the rapid decrease in its discrepancy in the Schrödinger equation. We consider the scattering problem of three one-dimensional quantum particles of equal masses ($m_{i}=\frac{1}{2},\ i=1,2,3$). The dynamics of the system is described by the Schrödinger equation

where $z_{j}$ are the coordinates of particles in the laboratory frame of reference,

Here the indices $(i,j,k)$ form an even permutation.

Let us pass to the reference system associated with the center of mass of the three-particle system

and introduce three systems of Jacobi coordinates $(x_{j},y_{j}),\ j=1,2,3$ on the resulting hyperplane $S=\{{\bf z}\in\mathbb{R}^{3}:\ z_{1}+z_{2}+z_{3}=0\}$; each of the systems is associated with one of the pair subsystems. All three coordinate systems are equivalent and related to each other by rotation transformations. Namely,

We also note the relations connecting the pair Jacobi coordinates with the chosen coordinate system

In the new notation, we arrive at the following Schrödinger equation for $S$:

We assume here that the operator $\Delta$ on the hyperplane $S$ is the Laplace–Beltrami operator

invariant with respect to the choice of a specific Jacobi coordinate system. We assume that all pair potentials $v_{j}(x_{j}),\ j=1,2,3$ are identical, finite, and are attractive potentials supporting one bound state. We also assume that the pair potential is a smooth function $v_{i}\in C_{\mathbb{R}},\ i=1,2,3$, although this condition can be weakened.

We note that the restrictions associated with the equality of particle masses, as well as with the equality of pair potentials, are not fundamental and are introduced only to technically simplify the solution of the problem. The solution of the problem allows a direct generalization to the case of arbitrary masses and arbitrary finite pair potentials (both attractive and repulsive) in different pairs.

We note that the scattering problem of three one-dimensional quantum particles $3\rightarrow 3$ (all three particles are free in the initial and final states of the system) in the situation of repulsive pair potentials was considered within the framework of the diffraction approach in [1]–[4]. At the same time, the asymptotic eigenfunctions of the Schrödinger operator were constructed. The case of $N$-particle scattering [5] was also considered separately. Later, in [6], the limiting values of the resolvent kernel of the Schrödinger operator were constructed on the continuous spectrum within the framework of the alternating Schwartz method. This, in turn, allowed isolating the asymptotics of the three-particle eigenfunctions of the absolutely continuous spectrum [7], which confirmed the results obtained earlier within the diffraction approach.

We emphasize that the analysis of scattering in the system of three particles on a straight line both is the first step towards studying the problem of three particles in three-dimensional space, and is interesting in itself. This is confirmed by the fact that the system of three (neutral or charged) particles on a line has been intensively studied for many years (see, for example, [8]–[23]). Currently, interest in such systems has increased since they have been implemented experimentally (see [24]–[27]). Therefore, mathematically correct and logically clear numerical procedures for constructing the states of the continuous spectrum of the Schrödinger operator of three-particle systems in the one-dimensional case are obviously in demand. The proposed approach allows a natural generalization to the case of slowly decreasing pair potentials in one dimension and in higher dimensions. A generalization to the case of a larger number of particles is also possible in principle.

We note that an attempt was undertaken in [28] to take into account the presence in the system of attractive pair potentials supporting the bound states, within the framework of the diffraction approach, for the $2\rightarrow 2(3)$ scattering problem (in the initial state, the system contains a free particle and a two-particle cluster in a bound state; in the final state, the system may consist of both a free particle and a cluster in a bound state, and of three free particles) in the case of three one-dimensional quantum particles. Although the ideological line of the paper was correct, certain weaknesses appeared in its implementation. In this paper, the outline proposed in [28] is corrected due to a more accurate description of the ansatz of the three-particle solution while preserving the main ideas proposed in [28].

Before we pass to the description of the results obtained in this study, let us recall the optical model that is well known in nuclear physics, proposed by Feshbach, Porter, and Weisskopf in 1954 [29] to describe the averaged behavior of cross sections. The model got its name from the analogy between the scattering of particles on a kernel and the passage of light through a semitransparent sphere. In the optical model, it is assumed that the kernel can be described by a complex potential well using the so-called optical potential

where the imaginary part $W_{opt}(r)$ describes the absorption of particles of the incident ray. The imaginary part of the optical potential is proportional to the average probability of transition from a single-particle state (defined by the initial state of the incident particle) to a more complex state of the compound system (a target kernel and a particle). In other words, such a phenomenological structure of the potential in the optical model takes into account the transition of the system to a new scattering channel (if there is one) and is energy-dependent.

In this paper, we reduce the solution of the scattering problem of a particle on a two-particle cluster to the solution of some auxiliary boundary value problem for the Schrödinger operator $\hat{H}$ in a circle of large radius with a radiation condition at the boundary. This boundary condition corresponds to the breakup of the system into three free particles. The potential in the Schrödinger operator in this case is equal to the sum of the original three-particle potential $V$ and some additional complex separable term $V_{sep}$, depending on the scattering energy. All pair amplitudes of rearrangement of the two-particle clusters are expressed in terms of the functionals of the solution $\Phi$ of the resulting boundary value problem. The amplitude of the breakup of the original system into three free particles is also constructed using $\Phi$. The complete (numerical) solution of the original scattering problem is also constructed in terms of the solution of the boundary value problem. In view of the above, the additional separable potential $V_{sep}$ constructed in our model makes sense and is similar (in structure and content) to the optical potential mentioned above. Namely, on one hand, the constructed model boundary value problem contains an asymptotic boundary condition reflecting the breakup of the system into three free particles, that is, the $2\rightarrow 3$ scattering channel. On the other hand, the structure of the operator contains an additional energy-dependent potential that takes into account the possibility of reclustering, that is, it takes into account the presence of the asymptotic scattering channels $2\rightarrow 2$. In this sense, the constructed separable potential arises as a consequence of eliminating the cluster scattering channels $2\rightarrow 2$ while preserving the scattering channel $2\rightarrow 3$.

1. In this paper, we propose a method for calculating the asymptotic eigenfunctions of the absolutely continuous spectrum of the Schrödinger operator for the $2\rightarrow 2(3)$ scattering problem of three one-dimensional quantum particles with finite pair potentials supporting the bound states.

The proposed method allows reducing the asymptotic solution of the original scattering problem to the construction of some model inhomogeneous boundary value problem for the ‘‘extended’’ Schrödinger operator with a potential equal to the sum of the total initial potential $V$ and some additional term. This additional term is a complex energy-dependent separable potential of finite rank, similar in content to the so-called optical potential. Thus, the asymptotic solution of the original scattering problem is completely defined in terms of Green’s function of the constructed model inhomogeneous boundary value problem.

2. In this paper, we propose a method for solving the constructed model boundary value problem. The method is based on the implementation of the alternating Schwartz method [30], which is one of the variants of the Faddeev equation method.

3. To justify the applicability of the alternating method, we give estimates for the second iteration of the reflection operators $\Gamma_{i}\Gamma_{j},\ \ i\neq j$, where $\Gamma_{i}=V^{(i)}_{sep}R_{i}$. Here $V^{(i)}_{sep}$ is a separable potential of rank one, $R_{i}=(-\Delta+V+V^{(i)}_{sep}-E)^{-1}$.

4. We give an estimate for the rate of decrease of the $n$th iteration of reflection operators.

5. In this paper, we present the results of the numerical implementation of the proposed method (see Sec. 7) for the technically simplest situation of particles of equal masses and identical finite attractive pair potentials supporting one bound state in each pair, for two various values of the total energy of the system $E=0.5$ and $E=2$. For each of the energies, we calculate the reclustering amplitudes in each pair scattering channel and the breakup amplitude in the three-particle channel, and construct a numerically complete asymptotic solution of the original scattering problem. When constructing Green’s function of the model boundary value problem, we take into account the contributions up to and including the second iteration of reflection operators.

6. We note that the method proposed in this paper for an asymptotic solution of the $2\rightarrow 2(3)$ scattering problem can be easily generalized to the case of short-range pair potentials, as well as to the $3\rightarrow 2(3)$ scattering problem. In principle, a generalization to the case of a larger number of particles is possible. We also note that the method proposed within the diffraction approach for solving the three-body problem in the presence of a discrete spectrum in pair subsystems complements the results, including the numerical ones, obtained earlier in [32]–[33] for the case of one-dimensional repulsive pair potentials.

We consider here the $2\rightarrow 2(3)$ scattering process within the framework of the diffraction approach. We assume that the pair potentials $v_{i},\ \ i=1,2,3$ are finite, continuously differentiable (the smoothness requirement can be relaxed), even, nonpositive, and supporting one bound state. We rely here on the Calogero criterion [31] (and its generalization for the potentials set on the axis), which defines the number of bound states in the two-body system. We also assume that the pair $(x,y)$ is a pair of Jacobi coordinates corresponding to the three-body system. Here $x\in\mathbb{R},\ \ y\in\mathbb{R}$. We assume that the masses of particles and the pair potentials are identical.

We will study the scattering problem $2\rightarrow 2(3)$ of three particles on an axis, that is, the coordinate of each particle is characterized by a real number. More precisely, we will study the scattering of a bound pair on a third particle, using the formalism of the diffraction approach, described in detail in [2], [3], [4]. Within the framework of this formalism, the configuration space of the problem after separating the dynamics of the center of mass of the entire system is the plane $S$, each of the three pairs of Jacobi coordinates $(x_{j},y_{j}),\ j=1,2,3$ forms an oriented coordinate system on $S$, these coordinate systems (as well as the pairs of Jacobi coordinates themselves, corresponding to two arbitrary various pair subsystems) are related by a rotation transformation. The complete support of potential (the union of three pair supports of potential) is a family of three rays-‘‘screens’’ $\ l_{j},j=1,2,3$ intersecting at one point with the vicinities. In this case of finite pair potentials, the support of the total potential is the union of three oriented strips on the plane, and the width of each strip in this case is defined by the support of the corresponding pair potential. Each of the ‘‘screens’’ with index $j,\ j=1,2,3$ defines a region in the configuration space such that the particles in the pair $j$ coincide, that is, the equality $x_{j}=0$ holds. Thus, along the ‘‘screen’’ with index $j$, the Jacobi coordinate $y_{j}$ changes, and orthogonally to the screen, with a fixed orientation of the coordinate system, the Jacobi coordinate $x_{j}$ changes. The sign of the coordinate $x_{j}$ is defined by the parity of the permutation of particles in the pair $j$, and the sign of the coordinate $y_{j}$ is determined by the parity of the permutation of particle $j$ and the center of mass of the pair of particles $k$ and $l$. We assume here that the triple of indices $(j,k,l)$ is formed by the permutation of numbers $(1,2,3)$.

We also assume that the asymptotics of the solution of the Schrödinger equation

which almost everywhere satisfies the radiation conditions at infinity in the configuration space

is arranged as follows:

Here the differentiation operator $\frac{\partial}{\partial n}$ denotes the differentiation operator along the normal to the boundary of the region, a circle of large radius $R$. We also use the notation

The index $\tau_{j}=\pm$ defines the sign of the half-screen $l_{j}$ on which the function $\psi_{j}^{\tau_{j}}$ is defined, or, in other words, it defines the sign of the variable $y_{j}$. The notation $\varphi$ stands for the pair bound state.

We also use the notation

Here the symbols $k\in\mathbb{R}$ and $p\in\mathbb{R}$ stand for the description of the moments that are conjugate in a Fourier sense to the Jacobi coordinates $x$ and $y$.

We assume that the particles of the pair $j=1$ in the initial state are in a bound state $\varphi_{1}^{-}$ with energy $\kappa_{1}<0$. The functions $\varphi_{j}^{\tau_{j}}$ satisfy the normalization condition

In this case, the cluster pair solution $\varphi_{j}^{\tau_{j}}$ is an even function on the support of the finite pair potential $v(x_{j})$ and decays exponentially outside the support of potential as the value of $|x_{j}|$ grows.

We note that the first term in the expression (6) corresponds to the incident wave. In this case, the following relations are satisfied:

The second term in the expression (6) corresponds to the superposition of outgoing cluster waves (the $2\rightarrow 2$ processes) with amplitudes $a_{j}^{\tau_{j}}$. Here the index $j=1,2,3$ denotes the number of the pair subsystem, and the index $\tau\in\{+,-\}$ defines the parity of permutation of the coordinate of the particle $j$ and the center of mass of the subsystem with index $j$. In other words, the index $\tau_{j}$ corresponds to the sign of the Jacobi coordinate $y_{j}$ and therefore defines the ‘‘half-screen’’ $\ l_{j}^{\tau_{j}}$. For the outgoing cluster waves, the following relations hold:

We note that each diverging wave with amplitude $a_{j}^{\tau_{j}}$ is defined only on the half-screen that corresponds to the index $\tau_{j}$. On the half-screen with index $-\tau_{j}$, it continues by zero. Finally, the third term in the expression (6) corresponds to the $2\rightarrow 3$ decay process and describes the diverging circular wave with amplitude $A({\hat{\bf X}},{\bf P})$.

Let us now rearrange the solution of the original scattering problem in a bounded, but sufficiently large for the asymptotics of the solution to occur, region. The purpose of such a rearrangement, or “deformation”, of the solution of the original problem is to construct a functional connection between the asymptotic scattering channels $2\rightarrow 2$ and $2\rightarrow 3$. Establishing such a connection allows effective eliminating of the cluster scattering channels and formulating a new model boundary value problem for the ‘‘deformed’’ part of the solution in the $2\rightarrow 3$ scattering channel. In terms of the solution of such a model problem, we reconstruct the complete asymptotic solution of the original scattering problem $2\rightarrow 2(3)$. Let us now describe the procedure for the rearrangement of the solution more precisely. We are going to construct a set of equations relating the amplitudes $a_{j}^{\tau_{j}},\ j=1,2,3;\ \tau_{j}=\pm$ of the $2\rightarrow 2$ scattering processes and the ‘‘deformed’’ in a bounded region part of the solution corresponding to the breakup process $2\rightarrow 3$. In this case, we know the solution of the Schrödinger equation only in the asymptotic region of the configuration space for $X\gg 1$. Let us introduce a smooth cut-off function that “cuts off” the    solution for limiting and small values of $X$. Multiplying the exact solution of the scattering problem (or some part of it) by such a cut-off function, we obtain a new function that remains an exact (up to the terms of the next order of smallness) solution of the Schrödinger equation for large $X$, and for limiting and small values of $X$, although it will no longer be an exact solution of the Schrödinger equation, will generate a known nonzero discrepancy in a bounded region of the configuration space.

This fact allows implementing the following outline for solving the scattering problem. At the first stage, we use Green’s second formula on the plane and find, albeit in terms of some cut-off function, the connection between the scattering amplitudes of the $2\rightarrow 2$ processes and the functionals of the ‘‘deformed’’ in a bounded region of the hyperplane $S$ part of the solution, corresponding to the process $2\rightarrow 3$. In other words, we connect the cluster solutions of the scattering problem (corresponding to the $2\rightarrow 2$ processes) with a ‘‘deformed’’ diverging circular wave (corresponding to the $2\rightarrow 3$ processes). We hence exclude the two-particle scattering channels from consideration, using their connection with a three-particle channel. Such effective elimination of the interaction channels in a multichannel system always leads to the appearance of some so-called optical potential, which is what we observe in this problem.

At the second stage of the solution of the scattering problem, we construct an inhomogeneous boundary value problem for the part of the solution which complements the deformed set of cluster solutions to the complete asymptotic solution of the original scattering problem. The asymptotics of this unknown part of the solution at large distances behaves as a diverging circular wave with a smooth amplitude and satisfies the radiation conditions almost everywhere at infinity.

We begin the construction of the boundary value problem by describing a convenient representation for an exact solution of the scattering problem. Since the asymptotics of the solution at large distances is known and has the form (6), we will seek a solution in a circle of large radius $R\gg 1$ in the form of the sum of an incident wave, smoothly ‘‘cut off’’ for limiting and small values of the hyperradius $X$, cluster waves, also smoothly ‘‘cut off’’ for limiting and small values of the hyperradius $X$, and the unknown function $\Phi$. In this case, the function $\Phi$ is defined everywhere in a circle of radius $R$ and, as follows from the asymptotics (6), behaves at large values of $X$ as a diverging wave with a smooth amplitude. Looking ahead, we will say that it is for the function $\Phi$ that the boundary value problem will be constructed.

Let us introduce the radial cut-off function $\zeta(X)\in C^{2}_{[0,\infty)}$ as follows:

We seek a solution of the scattering problem in the following form:

Let us also introduce the notation

Let us now use Green’s second formula to construct a connection between the cluster amplitude $a_{j}^{\tau_{j}}$ and the functional of $\Phi$. We need a couple of equations

where the symbol $*$ denotes complex conjugation. We note that the discrepancy $Q_{j}^{\tau_{j}}$ of the function $\tilde{\psi}_{j}^{\tau_{j}}$ in the Schrödinger equation obviously differs from zero where the cut-off function smoothly varies from 0 to 1, and vanishes where the cut-off function is a constant. Since $\tilde{\psi}_{j}^{\tau_{j}}$, according to (12) and (7), contains a pair solution $\varphi_{j}^{\tau_{j}}(x_{j})$, exponentially decreasing in $|x_{j}|$ outside the support of the pair potential, the residual $Q_{j}^{\tau_{j}}$ is localized in a curvilinear strip. By choosing the values of $R_{1},\ R_{2}$ sufficiently large, we can approximate the given strip arbitrary well to the rectangular strip

the region of variation of $|x_{j}|$ is defined by the rate of exponential decrease of the function $\varphi_{j}^{\tau_{j}}$. For definiteness, we assume that

The sign of the variable $y_{j}$ is defined by the value of the index $\tau_{j}$, or, in other words, by the choice of the half-screen on which the corresponding cluster solution is defined.

Let us multiply the first of the equations of system (13) by $\tilde{\psi}_{j}^{\tau_{j}}$, and the second by $\Psi^{*}$. Subtract the second equation from the first and integrate the result in a circle $B_{R}$ of large radius $R$.

Applying Green’s second formula, we arrive at the equation

This equation relates the energy flows carried by the distorted diverging wave $\Phi$ and the distorted cluster wave $\tilde{\psi}_{j}^{\tau_{j}}$.

We begin by considering the case $\{j,\tau_{j}\}\neq\{1,-\}$.

Let us pass to the construction of the boundary value problem for the unknown function $\Phi$. For this purpose, let us substitute the expressions obtained in (19) and (21) for the cluster amplitudes $a_{j}^{\tau_{j}}$ into the representation (10) of the solution of the scattering problem $2\rightarrow 2(3)$,

Let us act on Eq. (22) from the left and from the right with the operator $H-E$. As a result, we arrive at the equation

The definition of the residual $Q_{j}^{\tau_{j}}$ was given above in Eq. (13). We also use the notation $Q_{1}^{in}$,

Let us now formulate the boundary value problem for the function $\Phi$ in a circle $B_{R}$ of large radius $R$ with the radiation condition on the boundary ${\partial B_{R}}$,

We use here the notation

The Schrödinger operator $H$ was defined in Eq. (5), and the separable potential of rank six $V_{sep}$ is defined according to Eq. (23),

We note here that all functions $Q_{b},\ Q_{j}^{\tau_{j}}$, localized in the regions $\Pi_{j}^{\tau_{j}}$ (14) on the hyperplane $S$, are smooth in both variables with the exception of the boundaries of the regions $\Pi_{j}^{\tau_{j}}$, on which, however, continuity is preserved.

The solution of the $2\rightarrow 2(3)$ scattering problem is reconstructed in terms of the solution of the boundary value problem (24) using Eq. (22).

The construction itself of the solution of the boundary value problem formulated in Eq. (24) to describe the $2\rightarrow 2(3)$ scattering processes in the system of three one-dimensional quantum particles with finite attractive pair potentials within the framework of the diffraction approach is the next question to be discussed.