Abstract
In this paper we consider Schrödinger equations with nonlinearities of odd order 2σ + 1 on T^d. We prove that for σd≥2, they are strongly illposed in the Sobolev space H^s for any s < 0, exhibiting norm-inflation with infinite loss of regularity. In the case of the one-dimensional cubic nonlinear Schrödinger equation and its renormalized version we prove such a result for H^s with s < −2/3.