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.

In non relativistic quantum mechanics time enters as a parameter in the Schrödinger equation. However, there are various situations where the need arises to view time as a dynamical variable. In this paper we consider the dynamical role of time through the construction of a Lyapunov variable -i.e., a self-adjoint quantum observable whose expectation value varies monotonically as time increases. It is shown, in a constructive way, that a certain class of models admit a Lyapunov variable and that the existence of a Lyapunov variable implies the existence of a transformation mapping the original quantum mechanical problem to an equivalent irreversible representation. In addition, it is proved that in the irreversible representation there exists a natural time ordering observable splitting the Hilbert space at each t > 0 into past and future subspaces.

We prove dynamical upper bounds for discrete one-dimensional Schrödinger operators in terms of various spacing properties of the eigenvalues of finite volume approximations. We demonstrate the applicability of our approach by a study of the Fibonacci Hamiltonian.