Lower bounds on the eigenvalue sums of the Schr\"odinger operator and the spectral conservation law

Abstract

In the first part of the paper we consider the Schr\"odinger operator $ -\Delta-V(x),\quad 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 $H_-=-\Delta-V$.