a definition, or Section \[Sec-Admissibility\] in our context), but with an estimate on the observability time which is worse than the one we prove here in . Indeed, the technique in [@je3], based on a resolvent characterization of observability due to [@BurqZworski; @Mil05], does not yield explicit estimates on the time of observability. A variant of this strategy was developed in [@je9] for time-discrete approximations of Schrödinger equations in a geometric setting in which the corresponding wave equation is observable, where it was proved that the time-discrete approximations of Schrödinger equations are uniformly observable in any time $T>0$. In that case indeed, the resolvent estimates behave much better at high-frequencies, see e.g. [@TWbook], and the low frequency components of the solutions can be handled following the arguments of [@HarauxTime]. These resolvent estimates were also used to derive observability estimates for space semi-discrete conservative systems in filtered classes of initial data, see [@je8; @je7; @Miller2012], but still with non-explicit estimates on the time of observability.

The new ingredient introduced in this article that allows us to improve the results in [@je3] is the representation formula and careful estimates on the kernel $\rho_\tau$ in . This will be done using classical techniques 