On the Semigroup Decomposition of the Time Evolution of Quantum Mechanical Resonances

A way of utilizing Lax-Phillips type semigroups for the description of time evolution of resonances for scattering problems involving Hamiltonians with a semibounded spectrum was recently introduced by Y. Strauss. In the proposed framework the evolution is decomposed into a background term and an exponentially decaying resonance term evolving according to a semigroup law given by a Lax-Phillips type semigroup; this is called the semigroup decomposition. However, the proposed framework assumes that the S-matrix in the energy representation is the boundary value on the positive real axis of a bounded analytic function in the upper half-plane. This condition puts strong restrictions on possible applications of this formalism. In this paper it is shown that there is a simple way of weakening the assumptions on the S-matrix analyticity while still obtaining the semigroup decomposition of the evolution of a resonance.