  The envelope theory is an easy-to-use approximation method to obtain
eigensolutions for some quantum many-body systems. Even if the solutions are
reliable and an improvement procedure exists, the method can lack of accuracy
for some systems. In a previous work, two hypotheses were proposed to explain
the low precision: the presence of a divergence in the potential and a mix of
variational characters. In the present work, different systems are studied to
test these hypotheses. Theses tests show that the presence of a divergence
causes indeed less accurate results, while the mix of variational characters
reduces the impact of the improvement procedure.

Quantum Physics