Beyond the classical Weyl and Colin de Verdiere's formulas for Schrodinger operators with polynomial magnetic and electric fields

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.