On the eigenvalue counting function for Schrödinger operator: some upper bounds

Abstract

The aim of this work is to provide an upper bound on the eigenvalues counting function N(Rn, −∆+V, e) of a Schr¨odinger operator −∆+V on R^n corresponding to a potential V ∈ L^(n/2 +ε) (Rn, dx), in terms of the sum of the eigenvalues counting function of the Dirichlet integral D with Dirichlet boundary conditions on the subpotential domain {V < e}, endowed with weighted Lebesgue measure (V − e)− · dx and the eigenvalues counting function of the absorption-to-reflection operator on the equipotential surface {V = e}.

Keywords

• Schr¨odinger operators
• eigenvalues counting function
• Dirichlet-to-Neumann operator