Well-posedeness for the non-isotropic Schrödinger equations on cylinders and periodic domains

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].