The initial value problem for cubic semilinear Schrodinger equations

We present local and global existence theorems for cubic semilinear Schrodinger equations. Our new results are the improvement of our previous ones ([2],[3],[4]). The idea of the proof consists of the energy and the decay estimates. These equations do not allow the classical energy estimates. To avoid this difficulty, we make strong use of S. Doi's method for linear Schrodinger type equations. Combining cubic nonlinearity and S. Doi's method, we obtain the improved results.