Decay of quantum states in some exactly soluble models

Abstract

To illustrate the decay of quantum states, we compute exactly the elements of the evolution operator, exp(-i tH), and solve the energy-eigenvalue problem for some simple model hamiltonian systems with discrete spectra. Then, the behavior of these solutions in limits for which continuum parts appear in the energy spectra is studied. We find that in such limits the similarity transformation diagonalizing the hamiltonian becomes ill-defined, the evolution operator becomes non-unitary and the complicated quasi-periodic time-evolution characteristic of discrete-spectrum systems is displaced to the infinite future; the remaining decays of states for the limiting systems are sometimes, but not always, complete.