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

In the first part of the paper we consider the Schrödinger operator $-Δ-V(x),\quad V>0.$ 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 $H_+=-Δ+V$ and $H_-=-Δ-V$.

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