Spectral sum rules for the three-body problem

This paper derives a number of sum rules for nonrelativistic three-body scattering. These rules are valid for any finite region $\Sigma$ in the six-dimensional coordinate space. They relate energy moments of the trace of the on-shell time-delay operator to the energy-weighted probability for finding the three-body bound-state wave functions in the region $\Sigma$. If $\Sigma$ is all of the six-dimensional space, the global form of the sum rules is obtained. In this form the rules constitute higher-order Levinson's theorems for the three-body problem. Finally, the sum rules are extended to allow the energy moments to have complex powers.