Another definition for time delay

Time delay is defined by geometrical considerations which work in classical as well as in quantum mechanics, and its connection with the $S$ matrix and the virial is proven for potentials with $V\left(\overrightarrow{\mathrm{x}}\right)$ and $\overrightarrow{\mathrm{x}}·\overrightarrow{\nabla }V\left(x\right)\mathrm{}$ vanishing as $r^{-1-\epsilon }$ for $r\to \infty$.