Generalizations of the classical Weyl and Colin de Verdiere's formulas and the orbit method.

The classical Weyl formula expresses the leading term of the asymptotics of the counting function N(lambda, H) of the spectrum of a self-adjoint operator H in an invariant form: one can "hear" the volume of the subset of the cotangent bundle where the symbol of the operator H is less than lambda. In particular, it is applicable to Schrodinger operators with electric potentials growing at infinity. The Weyl formula is formulated in an invariant form; however, it gives +infinity for magnetic Schrodinger operators with magnetic tensors growing at infinity. For these operators, Colin de Verdiere's formula is known, but the form of the latter is not invariant. In this article, we suggest an invariant generalization of both Weyl's and Colin de Verdiere's formulas for wide classes of Schrodinger operators with polynomial electric and magnetic fields. The construction is based on the orbit method due to Kirillov, and it allows one to hear the geometry of coadjoint orbits.