Oleg Safronov
Lower bounds on the eigenvalue sums of the Schr\"odinger operator and the spectral conservation law.
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ABSTRACT.  We study the Schr\"odinger operator $-\Delta+V$ where $V$ is a real potential. It is known that oscillations of $V$ at the infinity lead to the same effects as a decay. However, it is not understood in which way one is equivalent to the other. In this paper we suggest a method that allows one to compare oscillating potentials with the decaying ones quantitatively. We suggest to formulate conditions on $V$ not in terms of the $L^p$ norm of the potential but in terms of the rate of accumulation of negative eigenvalues to zero.