  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
$\exp(-iHt)$ with a restricted test function space, which is obtained from
causality conditions. The unstable state shows exact exponential decay for
$t\ge0$. 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.
