Quantum mechanics gives us information about the spectra of dynamical variables and transition rates including scattering cross sections. They can be exhibited as spectral information in analytically continued spaces and their duals. Quantum mechanics formulated in these generalized spaces is used to study scattering and time evolution. It is shown that the usual asymptotic condition is inadequate to deal with the scattering of composite or unstable particles. Scattering theory needs an amendment when the interacting system is not isospectral with the free Hamiltonian; the amendment is formulated. Perturbation theory in generalized spaces is developed and used to study the deletion and augmentation of the spectrum of the Hamiltonian. A complete set of algebraically independent constants for an interacting system is obtained. The question of the breaking of time symmetry is discussed.