The author studies the initial value problem for the nonlinear Schrödinger equation \(\partial_t u= i(\Delta u- F(u))\), \(t\geq 0\), \(x\in \mathbb{R}^m\), under the assumption that \(F\in C^1(\mathbb{C}, \mathbb{C})\), \(F(0)= 0\), \(DF(\xi)= O(|\xi |^{k- 1})\) for some \(k\geq 1\) as \(|\xi |\to \infty\). The author proves uniqueness in \(L^\infty((0, T); L^2(\mathbb{R}^m))\cap L^r((0, T); L^q(\mathbb{R}^m))\), where \(r\) and \(q\) depend on \(m\) and \(k\). After this, a local existence theorem is proved for \(H^s\)-solutions using Lebesgue-type spaces instead of Besov-type spaces (used by T. Cazenaze and F. B. Weissler). Moreover, the existence of global \(H^s\)-solutions is shown under the assumption that \(F(\xi)= O^{\{s\}}(|\xi|^h)\) as \(|\xi|\to 0\), where \(h= 1+ 4/m\).