  We study Efimov physics for three identical bosons interacting via a pairwise
square-well potential, analyze the validity of the separable approximation as a
function of the interaction strength, and investigate what is needed to improve
this approximation. We find separable approximations to be accurate for
potentials with just one (nearly) bound dimer state. For potentials with more
bound or almost bound dimer states, these states need to be included for an
accurate determination of the Efimov spectrum and the corresponding three-body
observables. We also show that a separable approximation is insufficient to
accurately compute the trimer states for energies larger than the finite-range
energy even when the two-body T matrix is highly separable in this energy
regime. Additionally, we have analyzed three distinct expansion methods for the
full potential that give exact results and thus improve on the separable
approximation. With these methods, we demonstrate the necessity to include
higher partial-wave components of the off-shell two-body T matrix in the
three-body calculations. Moreover, we analyze the behavior of the Efimov states
near the atom-dimer threshold and observe the formation of non-Efimovian trimer
states as the potential depth is increased. Our results can help to elaborate
simpler theoretical models that are capable of reproducing the correct
three-body physics in atomic systems.
