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

The aim of this work is to provide an upper bound on the eigenvalues countingfunctionN(Rn,−∆ +V, e)of a Schr ̈odinger operator−∆ +V on R^n corresponding to a potentialV∈Ln2+ε(R^n, dx), in terms of the sum of the eigenvalues counting function of the DirichletintegralDwith Dirichlet boundary conditions on the subpotential domain{V < e}, endowedwith weighted Lebesgue measure(V−e)−·dxand the eigenvalues counting function of theabsorption-to-reflection operator on the equipotential surface{V=e}.