Low-energy sum rules for two-particle scattering

We consider the low-energy expansion of the operators describing a two-particle scattering system and derive recursion relations for the expansion coefficients. The particles of the system are allowed to interact via general (not necessarily spherically symmetric) potentials and can also form zero-energy bound states and/or resonances. Using these expansions, we prove low-energy sum rules for such a system. They relate the negative-energy moments of the trace of the resolvent difference to the bound-state eigenvalues. As an application, we first discuss the low-energy expansion of the two-body total cross section. In particular, we derive a closed expression for the expansion coefficients. We also find that there is always a finite characteristic length in the problem, even in the presence of zero-energy resonances. Secondly, we study the low-temperature behavior of the second virial coefficient.