A new way of defining unstable states

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We define a new unstable state in the Friedrichs model of a two-level atom. This unstable state is a complex eigenstate of the time evolution operator (-iHt) with a restricted test function space, which is obtained from causality conditions. The unstable state shows exact exponential decay for t≥0. Its emitted field is confined inside the future light-cone. In this way the long-standing problem of exponential catastrophe is removed. This is an example of quantum mechanics outside Hilbert space, which consists of generalized eigenstates in a distribution space, and a dual (test function) space.