  In this article, we study the unconditional uniqueness of $\dot H^s$, $0<s<
1$, solutions for the nonlinear Schr\"odinger equation $i\partial_t u +\Delta
u+ c |u|^\alpha u=0$ in ${\mathbb R}^n$. We give a unified proof of the
previously known results in the subcritical cases and critical cases, and we
also extend these results to some previously unsettled cases. Our proof uses in
particular negative order Sobolev spaces (or Besov spaces), general Strichartz
estimates, and the improved regularity property for the difference of two
solutions.
