Tests of the envelope theory for three-body forces

Many-body forces, and specially three-body forces, are sometimes a relevant
ingredient in various fields, such as atomic, nuclear or hadronic physics. As
their precise structure is generally difficult to uncover or to implement,
phenomenological effective forces are often used in practice. A form commonly
used for a many-body variable is the square-root of the sum of two-body
variables. Even in this case, the problem can be very difficult to treat
numerically. But this kind of many-body forces can be handled at the same level
of difficulty than two-body forces by the envelope theory. The envelope theory
is a very efficient technique to compute approximate, but reliable, solutions
of many-body systems, specially for identical particles. The quality of this
technique is tested here for various three-body forces with non-relativistic
systems composed of three identical particles. The energies, the
eigenfunctions, and some observables are compared with the corresponding
accurate results computed with a numerical variational method.