Cauchy problem for nonlinear Schrodinger equations with inverse-square potentials

The wellposedness of nonlinear Schrodinger equations (NLS) with inverse-square potentials is discussed in this article. The usual (NLS) is regarded as the potential-free case. The wellposedness of the usual (NLS) is well-known for a long time. In fact, several methods have been developed up to now. Among others, the Strichartz estimates seem to be essential in addition to the restriction on the nonlinear term caused by the Gagliardo-Nirengerg inequality. However, a parallel argument is not available when we apply such estimates to (NLS) with inverse-square potentials. Thus, we shall give only partial answer to the question in this article.