Traditional Quantum Scattering theory is built upon a Hilbert space formalism. The nature of the Hilbert space has led to a number of ideas about the nature of scattering systems, chief among them being the understanding that the decay of unstable quan- tum systems cannot be purely exponential and that there can only be an approximate relationship between the width Γ of a scattering resonance and the lifetime τ of the resonant state whose formation gives rise to it. That such physical conclusions are de- rived from purely mathematical properties of the underlying topological vector space inform the notion that, in choosing a particular space, one is choosing the mathemat- ical representation of the physical universe for the system. With this notion in mind, purely physical characteristics of quantum systems are used to answer the question: “What is the proper mathematical space for quantum mechanical calculations?” In the particular case of quantum scattering, it is shown that this approach casts the choice of the Hilbert space into doubt and motivates instead the use of rigged Hilbert spaces employing Hardy spaces; in these Hardy rigged Hilbert spaces, exponential decay is permissible and the time evolution of decaying states is governed by semigroups. The original theories concerning deviation from exponential decay in the Hilbert space are revisited to reveal how their conclusions flow from a mathematical property whose physical interpretation is problematic. Finally, an application of a Hardy rigged Hilbert space formalism is presented: Time Asymmetric Quantum Mechanics, as developed by A. Bohm, M. Gadella, and S. Wickramasekara.