Lower bounds on the eigenvalue sums of the Schrödinger operator and the spectral conservation law

Abstract

In the first part of the paper we consider the Schrödinger operator $-\Delta -V(x),V>0.$ We discuss the relation between the behavior of $V$ at the infinity and the properties of the negative spectrum of $H$. After that, we consider the case when $V$ changes its sign: $V=V_{+}-V_{-}$, $2V_{\pm }=|V|\pm V.$ In this case, we treat $V$ and $-V$ symmetrically and study the relation between the behavior of $V$ at the infinity and the negative spectra of the operators $H_{+}=-\Delta +V$ and $H_{-}=-\Delta -V$.