4 initially), the above qualitative features remain. In par- ticular, for a specific interaction, the longer period is not changed with b . Thus we conclude that the periodic phe- nomenon is common to trapping 2-body scatterings. It is emphasized that this phenomenon is sensitive to the in- teractions. A stronger repulsive (attractive) force would lead to a shorter (longer) long-period. The accuracy of the above numerical results depends on K$_{max}$ . As an example selected values of ρ ( a , τ ) are listed in Tab. I to show the dependence. The periodic behaviors can be shown in more detail by observing directly the densities. Let r 1 = x i , where i is a unit vector along the X − axis. The distribution of ρ ( x i , τ ) along the X − axis is plotted in Figs. 3 and 4, where τ is given at a number of values. Fig. 3a describes the evolution in the interval (0 , π ), where the curve ”1” is for the initial state. From ”1” to ”5” τ goes from 0 to π with a step π/ 4. One can see how the two particles undergo a round of collision. In fact, Fig. 1 and Fig. 3a describe the same thing except that ρ is plotted on the X − Z plane in the former but only along the X − axis in the latter. When τ goes from π to 2 π , the process occurring in (0 , π ) will repeat again but in reverse direc- tion. Thereby the cycle with the 2 π period is completed. Fig. 3b describes the evolution in the interval (48 π, 49 π ) associated with the first minimum of the diamonds in Fig. 2. The peaks in 3b are much lower than those of 3a, and these peaks are located at different places. Therefore the system behaves differently in the two intervals, and the previous clear picture of a head-on collision becomes ambiguous. Fig. 4a is associated with the second max- imum. The collision can be roughly seen but is not as clear as in Fig. 3a. Fig. 4b is associated with the third maximum, and is nearly identical to 3a. Therefore the cycle with the longer period is completed. The convergency appears to be satisfying. Thus the numerical results obtained by using K$_{max}$ = 20 are accu- rate enough in qualitative sense. In this paper the traditional 2-body scattering is con- sidered under a new environment, namely, in a trap. Due to the trap, the two particles collide with each other re- peatedly. Therefore, even the interaction is weak; the effect of interaction can be accumulated via the repeated collisions. Besides, comparing with the case of charged particles, the initial scattering states of neutral particles are more difficult to control. This disadvantage can be overcome by using optical traps. Furthermore, two peri- odic phenomena are found. They are essentially caused by the trap and by the interaction, respectively. The pe- riod of the latter is much longer and is sensitive to the in- teraction. Instead of measuring the differential cross sec- tion as usually does, the observation of the longer period and the details of the time-dependent density might be a valid source of information on weak interactions among neutral particles. The above approach can be easily generalized to the cases with various interactions, and/or to the case of Fermion systems. It is reasonable to expect that the above trapped scattering could be experimentally real- ized via the progress of techniques in trapping neutral particles by optical traps. In Fig. 5, the strength of the attractive tail has been given at three values. In each case a slightly lower peak followed by a higher peak (the third maximum) appears again. With the three choices of c$_{6}$ , the third maximum appears at τ = 210 π , 236 π , and 282 π , respectively. At these three instants, |〈 Ψ(0) | Ψ( τ ) 〉| are all equal to 0 . 994 and the associated imaginary parts are very small. It implies a nearly exact recovery. Thus the nearly peri- odic behavior with the much longer period appears again. When b = 0 (i.e., the target particle is not at the center The support from the NSFC under the grant 10874249 is appreciated. GLYPH<s48> GLYPH<s52>GLYPH<s48> GLYPH<s56>GLYPH<s48> GLYPH<s49>GLYPH<s50>GLYPH<s48> GLYPH<s49>GLYPH<s54>GLYPH<s48> GLYPH<s50>GLYPH<s48>GLYPH<s48> GLYPH<s50>GLYPH<s52>GLYPH<s48> GLYPH<s48>GLYPH<s46> GLYPH<s48>GLYPH<s53> GLYPH<s48>GLYPH<s46> GLYPH<s49>GLYPH<s48> GLYPH<s48>GLYPH<s46> GLYPH<s49>GLYPH<s53> GLYPH<s48>GLYPH<s46> GLYPH<s50>GLYPH<s48> GLYPH<s48>GLYPH<s46> GLYPH<s50>GLYPH<s53> GLYPH<s32> GLYPH<s32> GLYPH<s32>GLYPH<s47>GLYPH<s32> GLYPH<s40> GLYPH<s97> GLYPH<s44> GLYPH<s41> FIG. 5: (Color online) Similar to Fig. 2, but V 0 is fixed at 10 and c 6 has three choices: − 0 . 3 r 6 0 , − 1 . 5 r 6 0 and − 3 r 6 0 . The values of ρ ( a , τ ) are, respectively, marked by black circles, red diamonds, and triangles. The other parameters are the same as in Fig. 1. This figure has a longer range than that in Fig. 2, and only the values of ρ with τ = 8 kπ are shown. TABLE I: ρ ( a , τ ) with three choices of K max . The parameters involved are the same as those for Fig. 1. K max 12 16 20 ρ ( a , 0) 0.253 0.255 0.255 ρ ( a , 50 π ) 0.067 0.068 0.068 ρ ( a , 100 π ) 0.229 0.230 0.229 Acknowledgments 345 (1998). [1] J. Stenger, S. Inouye, D. M. Stamper-Kurn, H.-J. Miesner, A. P. Chikkatur, and W. Ketterle, Nature (London) 396 ,