The initial value problem (IVP) for the non-isotropic Schrödinger equation posed on the two-dimensional cylinders and Formula: see text is considered. The IVP is shown to be locally well-posed for small initial data in Formula: see text if Formula: see text. For the IVP posed on Formula: see text, given data are considered in the anisotropic Sobolev spaces thereby obtaining the local well-posedness result in Formula: see text, if Formula: see text and Formula: see text. In the purely periodic case, a particular case of the IVP is shown to be locally well-posed for any given initial data in Formula: see text if Formula: see text. In some cases, ill-posedness issues are also considered showing that the IVP posed on Formula: see text, in the focusing case, is ill-posed in the sense that the application data-solution fails to be uniformly continuous for data in Formula: see text if Formula: see text.
Corcho et al. (Wed,) studied this question.