The quantum Zeno effect -- suppression of decay by frequent measurements --
was believed to occur only when the response of the detector is so quick that
the initial tiny deviation from the exponential decay law is detectable.
However, we show that it can occur even for exactly exponentially decaying
systems, for which this condition is never satisfied, by considering a
realistic case where the detector has a finite energy band of detection. The
conventional theories correspond to the limit of an infinite bandwidth. This
implies that the Zeno effect occurs more widely than expected so far.
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