Newton Polyhedron and Spectral Asymptotics of Differential Operators

Let p(ξ) be a nonnegative polynomial in ξ ∈ Rᵈ . The differential operator P(Dx)—the Friedrichs extension of the operator p(Dx) defined on C₀∞(Ω ) —is considered. The Newton polyhedron of the polynomial p is introduced. The asymptotics of eigenvalues is obtained for the case when the polynomial is nondegenerate with respect to its Newton polyhedron. The geometry of the polyhedron shows whether the asymptotics is given by Weyl’s law or by another formula.