The rate of accumulation of negative eigenvalues to zero and the absolutely continuous spectrum

TL;DR

This paper investigates how the negative eigenvalues of certain Schrödinger operators approach zero and demonstrates that under specific conditions, the absolutely continuous spectrum spans the entire positive real axis.

Contribution

It establishes a link between the rate of negative eigenvalues approaching zero and the coverage of the positive spectrum for Schrödinger operators.

Findings

Negative eigenvalues tend to zero rapidly under certain conditions.

Absolutely continuous spectrum covers [0, ∞) when negative eigenvalues decay sufficiently fast.

Results apply to Schrödinger operators with bounded real-valued potentials.

Abstract

For a bounded real-valued function $V$ on ${\Bbb R}^d$, we consider two Schr\"odinger operators $H_+=-\Delta+V$ and $H_-=-\Delta-V$. We prove that if the negative spectra $H_+$ and $H_-$ are discrete and the negative eigenvalues of $H_+$ and $H_-$ tend to zero sufficiently fast, then the absolutely continuous spectra cover the positive half-line $[0,\infty)$.