Newton's Polyhedron and Weyl's Formula for the Spectrum of the Schrödinger Operator with Polynomial Potential

Abstract

Introduce the notation: V is a nonnegative polynomial, Q(V) is the corresponding Newton polyhedron, L= -△+V is the Schrodinger operator on \({\mathbb{L}}_2 ({\mathbb{R}}^n )\), N(λ,L) is the number of eigenvalues of L that are less than λ. The upper estimate for N(λ,L) is derived in terms of Q(V), and the Weyl asymptotic formula is established under certain assumptions on the geometry of Q(V). The case where the behavior of V(x) is not regular as \(\left| x \right| \to \infty \) is also considered. Bibliography: 6 titles.