Alternatively, the effects of deep dimers can be taken into account indirectly by using an analytic continuation of the three-body parameter into the complex plane as introduced in Ref. [@Braaten:2002sr]. Then, the partial rate $\beta^{\rm
  deep}_i$ can be obtained by using the relation in Eq. .

D’Incao and Esry have calculated the scattering length dependence of ultracold three-body collisions near overlapping Feshbach resonances for a variety of cases [@DIncao:2009]. For the relaxation rate into the shallow dimer $\beta^{\rm sh}_{i\to j}$, they find that it scales like $a_{j}^2/a_{i}$ times a log-periodic function of $a_{j}$ if $a_{i} \gg a_{j}$ while $a_{k}$ is non-resonant. The rate into deep dimers $\beta^{\rm deep}_i$ scales like $a_{j}^2/a_{i}$ times a prefactor that is a constant for positive $a_{j}$ and a log-periodic function of $a_{j}$ for negative $a_{j}$.

STM equation {#sec:STMeq}
============

The Skorniakov–Ter-Martirosian (STM) equation [@STM57] is an integral equation that describes three-atom scattering interacting through zero-range interactions. In this section, we discuss the STM equation and relate the T-matrix element ${\cal T}^{\rm
  AD}_{ij}$ to the amplitude ${\cal A}_{ij}$ that is the solution to the STM equation.

We consider only the S-wave contribution and assume that higher partial wave contributions are suppressed at low temperature. For the 