Spectral asymptotics for the Schrödinger operator with potential which steadies at infinity

Abstract

We consider the discrete spectrum of the selfadjoint Schrödinger operatorA _h =−h ² Δ+V defined inL ²(ℝ^m) with potentialV which steadies at infinity, i.e.V(x)=g+|x|⁻ f(1+o(1)) as |x|→∞ forα>0 and some homogeneous functionsg andf of order zero. Letℜ _h (λ),λ≧0, be the total multiplicity of the eigenvalues ofA _h smaller thanM−λ, M being the minimum value ofg over the unit sphereS ^m−1 (hence,M coincides with the lower bound of the essential spectrum ofA _h ). We study the asymptotic behaviour ofℜ ₁(λ) asλ↓0, or ofℜ _h (λ) ash↓0, the numberλ≧0 being fixed. We find that these asymptotics depend essentially on the structure of the submanifold ofS ^m−1, where the functiong takes the valueM, and generically are nonclassical, i.e. even as a first approximation (2π)^m ℜ _h (λ) differs from the volume of the set {(x, ξ)∈ℝ^2m:h ²|ξ|²+V(x)<M−λ}.