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havior originating from the harmonic trap. If the interac- tion is neglected, the recovery would be exact. However, due to the interaction, ρ ( r $_{1}$, τ ) is not exactly equal to ρ ( r $_{1}$, τ + 2 π ). With the above parameters, the deviation is very small. Say, when r 1 = a , we have ρ ( a , 0) = 0 . 2553 and ρ ( a , 2 π ) = 0 . 2544. Obviously, in the period (0 , 2 π ) the system undergoes a pair of collisions. When the time goes on, a series of head-on collisions occur repeatedly in the trap. Although the effect of interaction on a round of collision is weak, the effect of many rounds might be accumulated and therefore might become strong. This is shown below. GLYPH<s32> 

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In Fig. 2 ρ ( a , τ ) is given at τ = 2 kπ , where k is an integer. It implies that ρ is observed at the initial posi- tion of the incident particle repeatedly. If the interaction is removed, all the symbols in the figure would lie along a horizontal line (implying an exact periodicity with a period 2 π ). However, the interaction causes a deviation. The deviation would become larger if the interaction is stronger (the black circles to be compared with the trian- gles) and/or if τ is larger. It is found that, when the time goes on, ρ ( a , 2 kπ ) against increasing k will first arrive at a minimum, then arrive at the second maximum which is a little lower than the first maximum at τ = 0, then again a minimum, and afterward arrive at the third maximum with a height close to the first maximum. This behavior will repeat again and again. E.g., for V$_{0}$ = 20 (black cir- cles), the first, second and third maxima appear at τ = 0, 54 π , and 110 π , respectively. Whereas for V$_{0}$ = 10 (red diamonds), they appear at 0, 104 π and 210 π . 

On the other hand, making use of the expansion Eq. (6), we calculate the overlap |〈 Ψ(0) | Ψ(2 kπ ) 〉| which varies with k . When k leads to a minimum (maximum) of ρ ( a , 2 kπ ), the overlap is small (close to one). For exam- ples, with the parameters for Fig. 1, when τ = 48 π , 104 π and 210 π associated with the first minimum, the sec- ond maximum, and the third maximum, respectively, we have |〈 Ψ(0) | Ψ( τ ) 〉| = 0 . 103, 0 . 918 and 0 . 994. It is further 

noted that the imaginary part of 〈 Ψ(0) | Ψ(210 π ) 〉 is very small. Thus, Ψ(210 π ) is extremely close to the initial state. Therefore, from the time-dependent Schr¨ odinger equation, we know that what happens during the inter- val (0 , 210 π ) will nearly exactly repeat again in the next interval (210 π, 420 π ), and so on. It implies the existence of another nearly periodic behavior. Thus, there are two distinct periodic behaviors. One has a period 2 π , and the other one has a much longer period (say, for the above case, the period is 210 π ). GLYPH<s32> 

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<img src='content_image/422384.jpg'>

