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

TL;DR

This paper investigates lower bounds on eigenvalue sums of Schrödinger operators, analyzing how the potential's behavior at infinity influences the negative spectrum, including cases where the potential changes sign, and explores spectral conservation laws.

Contribution

It provides new bounds on eigenvalue sums for Schrödinger operators with positive and sign-changing potentials, linking potential decay at infinity to spectral properties.

Findings

Derived bounds for negative eigenvalues based on potential decay

Analyzed spectral properties for sign-changing potentials

Established relations between potential behavior and spectral conservation

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 $H_-=-\Delta-V$.