Scattering Resonances and theRFunction

The relationship between scattering resonances and $R$-function states is explored by investigation of the separate properties of the "hard-sphere" and "$R$-function" portions of the phase shift. The $R$-function states are found to be good models for bound states and for some resonances below potential barriers, but they have no direct relation to scattering resonances above barriers. An $R$-function proof and explication are given for Levinson's theorem, according to which the number of scattering resonances is in general finite and small even though the spectrum of $R$-function states is infinite. Implications of the lack of high-energy resonances for the many-body problem are mentioned.