| Abstract: | The widely cited definition of quantization in terms of square-integrable wave functions does not apply to continuum wave functions, to such phenomena as metastable states, or many-body resonances. A better philosophical foundation for quantum mechanics separates the probabilistic aspects based on square integrable Hilbert space functions from the dynamical aspects based upon the solutions of Shroedinger's (or Dirac's) equation. A Hilbert space may have a non-Hilbert space basis, which may be described by Stieltjes integrals and a spectrum measure. This viewpoint is expounded by reference to a very detailed analysis of a simple model, through which a precise definition of a Bohr–Feshbach resonance can be given. We propose a definition of a “metastable state,” showing that it is consistent with accepted usage, and that it overcomes a series of objections which have been catalogued by Simon. Its rate of decay is given by the Fourier–Stieltjes transform of the spectral density function; it is moreover the longest-lived initially localized state which can be formed from a small span of energy eigenfunctions near its mean energy. |
