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

AUTOR(ES)

Boyarchenko, Mitya

FONTE

National Academy of Sciences

RESUMO

The classical Weyl formula expresses the leading term of the asymptotics of the counting function N(λ, 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 λ. In particular, it is applicable to Schrödinger operators with electric potentials growing at infinity. The Weyl formula is formulated in an invariant form; however, it gives +∞ for magnetic Schrödinger operators with magnetic tensors growing at infinity. For these operators, Colin de Verdière'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 Verdière's formulas for wide classes of Schrödinger 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.

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