The Inverse Decay Problem

Abstract

Let UK(t) be a one-parameter operator family of positive type in a Hilbert space K and U(t) its minimal unitary dilation with infinitesimal generator H. If UK(t) is a contractive semigroup, then H is not positive. If in addition UK(t)→0 for t → ∞, then there exists a state φ∈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.