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 $N({\mathbb{R}}^n,-\Delta +V,e)$ of a Schödinger operator $-\Delta +V$ on ${\mathbb{R}}^n$ corresponding to a potential $V\in L^{\frac{n}{2}+\epsilon }({\mathbb{R}}^n,dx)$, in terms of the sum of the eigenvalues counting function of the Dirichlet integral $\mathcal{D}$ with Dirichlet boundary conditions on the subpotential domain $\{V<e\}$, endowed with weighted Lebesgue measure $(V-e{)}_{-}\cdot dx$ and the eigenvalues counting function of the absorption-to-reflection operator on the equipotential surface $\{V=e\}$.