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

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

2010-01-30

arxiv.org/abs/1002.0080v2

Mathematics

Spectral Theory

Scientific paper

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 $H_-=-\Delta-V$.

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