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We consider the Cauchy problem for nonlinear Schrodinger equations in thepresence of a smooth, possibly unbounded, potential. No assumption is made onthe sign of the potential. If the potential grows at most linearly at infinity,we construct solutions in Sobolev spaces (without weight), locally in time.Under some natural assumptions, we prove that the $H^1$-solutions are global intime. On the other hand, if the potential has a super-linear growth, then theSobolev regularity of positive order is lost instantly, not matter how large itis, unless the initial datum decays sufficiently fast at infinity.

On the Cauchy problem in Sobolev spaces for nonlinear Schrödinger equations with potential