We prove an estimate of the kind $\parallel q_1-q_2{\parallel }_{L^{\infty }}\le C\varphi (\parallel A_{q_1}-A_{q_2}{\parallel }_{R,3/2-1/2})$, where $A_{q_i}(\omega ,\theta )$, $i=1,2$ is the scattering amplitude related to the compactly supported potential $q_i\left(x\right)$ at a fixed energy level $k=$ const., $\varphi \left(t\right)=(-lnt{)}^{-\delta }$, $0\<\delta \<1$ and $\parallel ·{\parallel }_{R,3/2-1/2}$ is a suitably defined norm.

We prove a Strichartz inequality for a system of orthonormal functions, with an optimal behavior of the constant in the limit of a large number of functions. The estimate generalizes the usual Strichartz inequality, in the same fashion as the Lieb-Thirring inequality generalizes the Sobolev inequality. As an application, we consider the Schrödinger equation in a time-dependent potential and we show the existence of the wave operator in Schatten spaces.