Oleg Safronov
Lower bounds on the eigenvalue sums of the Schr\"odinger operator and the spectral conservation law
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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_-,\qquad 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 $H_-=-\Delta-V$.