  When two Landau levels are brought to a close coincidence between them and
with the chemical potential in the Integer Quantum Hall regime, the two Landau
levels can just cross or collapse while the external or pseudospin field that
induces the alignment changes. In this work, all possible crossings are
analyzed theoretically for the particular case of semiconductor trilayer
systems, using a variational Hartree-Fock approximation. The model includes
tunneling between neighboring layers, bias, intra-layer and inter-layer Coulomb
interaction among the electrons. We have found that the general pseudospin
anisotropy classification scheme used in bilayers applies also to the trilayer
situation, with the simple crossing corresponding to an easy-axis ferromagnetic
anisotropy analogy, and the collapse case corresponding to an easy-plane
ferromagnetic analogy. An isotropic case is also possible, with the levels just
crossing or collapsing depending on the filling factor and the quantum numbers
of the two nearby levels. While our results are valid for any integer filling
factor $\nu$ (=1,2,3,...), we have analyzed in detail the crossings at $\nu=3$
and $4$, and we have given clear predictions that will help in their
experimental search. In particular, the present calculations suggest that by
increasing the bias, the trilayer system at these two filling factors can be
driven from an easy-plane anisotropy regime to an easy-axis regime, and then
can be driven back to the easy-plane regime. This kind of reentrant behavior is
an unique feature of the trilayers, compared with the bilayers.
