shown in Fig. 2a (see the caption for the parameters a, b and x0). The potential consists of a shallow Gaussian well at the center and a pair of deeper and narrower wells next to the boundaries. The four lowest-energy single particle wavefunctions of potential (2) are also shown in Fig. 2a. The two lower energy states are nearly degenerate and the corresponding wavefunctions have sizable amplitudes at the deep wells near the boundaries, while the two higher energy states have much larger energy separation, with the amplitudes of the corresponding wavefunctions being large in the shallow well in the middle. Accordingly, the effective exchange interactions satisfy J1=J3 and <<1. The dependence of the ratio J2/J1 on the parameters V0 and U in equation (2) are shown in Fig. 2b,c.