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To control the size of the space, a number N$_{0}$ is intro- duced and 2 n + 2 N + l + L ≤ N$_{0}$ is required for all the basis functions. There are two choices to obtain symmetrized eigenstates of H$_{in}$ . In the first choice, the set Φ$_{Jm}$$_{Π}$$_{q}$( r , R ) is firstly symmetrized and orthonormal- ized before carrying on the diagonalization. However, this procedure is complicated. Therefore we make the second choice, in which the set Φ$_{Jm}$$_{Π}$$_{q}$( r , R ) is simply used without symmetrization but with the requirement that l must be even and all the basis functions satisfying 2 n +2 N + l + L ≤ N$_{0}$ are included without missing. This requirement assures that the space is close under per- mutations, and no basis functions that will contribute to the symmetrized eigenstates would be missed, unless they are too high to have 2 n +2 N + l + L > N$_{0}$ . However, in this choice, a number of unphysical eigenstates with confused symmetry will emerge together with those with correct symmetry. Therefore a discrimination is needed as shown below. 

Let an eigenstate be denoted as Ψ$_{Jm}$$_{Π}$$_{i}$ where i is a serial number of the Jm Π − series. Expanding in terms of the basis functions, 

where the l in q must be even, and the coefficients B Jm Π iq can be directly known from the diagonalization of H$_{in}$ . If Ψ$_{Jm}$$_{Π}$$_{i}$ is correctly symmetrized, the coefficients would obey 

and 

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In the summations of Eqs. (14) and (15), l (in q ) is re- stricted to be even. With the help of Eqs. (14) and (15), the states with confused symmetry can be discriminated and dropped, and all the symmetrized eigenstates under the restriction caused by N$_{0}$ can be extracted without missing. They are one-to-one identical to those obtained via the first choice if the same N$_{0}$ are used. In what follows Ψ$_{Jm}$$_{Π}$$_{i}$ denotes only the symmetrized eigenstate, and the associated energy is denoted by E$_{J}$$_{Π}$$_{i}$ . 

## IV. EVOLUTION AND THE REPEATING 3-BODY COLLISIONS 

With the eigenstates it is straight forward to obtain the time-dependent solution of H$_{evol}$ as 

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where 

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and τ = ωt . Obviously, Eqs. (16) to (18) give only an approximate solution because the set Ψ$_{Jm}$$_{Π}$$_{i}$ obtained via diagonalization each would deviate more or less from the corresponding exact eigenstate, and because only finite number of Ψ$_{Jm}$$_{Π}$$_{i}$ are used in the expansion. However, it is believed that, when the number of basis functions becomes larger and larger, the above Ψ( t ) would be closer and closer to the exact solution. The crucial point is the convergency. In this paper the interaction is assumed to be weak. It turns out that, in this case, the convergency is satisfying as shown below. 

We shall demonstrate that the evolution is a repeating 3-body collisions, and the effect of interaction will be also shown. For these purposes, we extract the following quantities from Ψ( t ). 

It was found that the c.m. is distributed very close to the origin, therefore R is approximately proportional to r $_{3}$. Therefore, the behavior of the particle 3 can be roughly understood via ζ ( R , t ). Incidentally, the behav- iors of all the particles are the same due to the sym- metrization. The analytical expression of these densities are given in the appendix. 

In order to have numerical results, as an example, the interaction is assumed to be a repulsive core as V ( r ) = V$_{0}$ if r ≤ 0 . 4, or zero if r > 0 . 4, where V$_{0}$ is a constant to be given. The other parameters are chosen as η = 1 . 5, | a | = 1 . 5, and N$_{0}$ = 20. To show the initial localization of the particles, we define the one-body density of the initial state as ρ$_{1}$ ( r $_{1}$) ≡ ∫ d r $_{2}$d r $_{3}$Ψ ∗ $_{I}$Ψ$_{I}$ . This density is plotted in Fig.1. Starting from Ψ$_{I}$, the details of evolution are given as follows. 

ρ$_{R}$ ( R, t ), ρ$_{r}$ ( r, t ), and ρ$_{θ}$ ( θ, t ) against t in the earliest stage of evolution is plotted in Fig2a to 2c. Where V$_{0}$ = 0 . 08 is assumed, and ωt is from 0 to 2 π (say, if ω = 

