  We present a pair of conjectural formulas that compute the leading term of
the spectral asymptotics of a Schr\"odinger operator on $L^2(\bR^n)$ with
quasi-homogeneous polynomial magnetic and electric fields. The construction is
based on the orbit method due to Kirillov. It makes sense for any nilpotent Lie
algebra and is related to the geometry of coadjoint orbits, as well as to the
growth properties of certain ``algebraic integrals,'' studied by Nilsson. By
using the direct variational method, we prove that the formulas give the
correct answer not only in the ``regular'' cases where the classical formulas
of Weyl or Colin de Verdi\`ere are applicable but in many ``irregular'' cases,
with different types of degeneration of potentials.
