Nonlinear Schr(?)dinger Equation With A Harmonic Potential

The Schrodinger equation is the basic mathematic model in the quantum mechanics'2"4'10'43'48'54'83'84). Prom the view point of mathematics, the researching of nonlinear Schrodinger equation has only been for thirty years. Segal pointed out the nonlinear semi-group theory of the equation, and Strauss'72'73' studied the scattering of small solution for the equation. Then, Glassey got some significant results of blowing-up properties for the equation. According with the estimates of Strichartz, Ginibre and Velo established the local well-posedness of the nonlinear Schrodinger equation, which is basic work for further researching. With studying of the classic nonlinear Schrodinger equation, a series of important results have been got. As the representative persons who accomplished these studies, mathematicians such as J. Bourgain, F.Merle, W.A. Strauss, H. Levine, T. Cazenave, P.L. Lions, Y. Tsutsumi, T. Tao, D.Q. Li and B.L. Guo play a symbolic role in the modern advance of PDE and the core mathematics.The nonlinear Schrodinger equation with a harmonic potential models the famous Bose-Einstein condensates (BEC) in the quantum mechanics (see [2-4,10,43,48,54,83,84]). In 1924 Einstein made a famous forecast that was: under some low temperature, a macro-number of atoms would condensate into the lowest energy state of zero momentum, then would form the macro-quantum state. In 1995 the forecast was realized in a experiment, which gave rise to two Nobel physics prizes. Now BEC is one of the most important researching topics in the...