  We start with some methodic remarks referring to purely bosonic quantum
systems and then explain how corrections to the leading--order quasiclassical
result for the fermion--graded partition function Tr{(-1)^F exp(-\beta H)} can
be calculated at small \beta. We perform such calculation for certain
supersymmetric quantum mechanical systems where such corrections are expected
to appear. We consider in particular supersymmetric Yang-Mills theory reduced
to (0+1) dimensions and were surprised to find that the correction of order
\beta^2 vanishes in this case.
  We discuss also a nonstandard N =2 supersymmetric sigma model defined on S^3
and other 3--dimensional conformally flat manifolds and show that the
quasiclassical expansion breaks down for this system.
