The inverse decay problem

Let U_{script K}(t) be a one-parameter operator family of positive type in a Hilbert space script K and U(t) its minimal unitary dilation with infinitesimal generator H. If U_{script K}(t) is a contractive semigroup, then H is not positive. If in addition U_{script K}(t)→0 for t→∞, then there exists a state φ ∈ script K on which H is not defined. We interpret these and other results in the context of the quantum-mechanical theory of unstable particles and the scattering theory of Lax and Phillips.