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If very frequent periodic measurements ascertain whether a quantum system isstill in its initial state, its evolution is hindered. This peculiar phenomenonis called quantum Zeno effect. We investigate the large-time limit of thesurvival probability as the total observation time scales as a power of themeasurement frequency, $t\propto N^\alpha$. The limit survival probabilityexhibits a sudden jump from 1 to 0 at $\alpha=1/2$, the threshold between thequantum Zeno effect and a diffusive behavior. Moreover, we show that for$\alpha\geq 1$ the limit probability becomes sensitive to the spectralproperties of the initial state and to arithmetic properties of the measurementperiods.

Large-time limit of the quantum Zeno effect