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

For a bounded real-valued function $V$ on ${\mathbb{R}}^d$, we consider two Schrödinger operators $H_{+}=-\mathrm{\Delta }+V$ and $H_{-}=-\mathrm{\Delta }-V$. 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 $[0,\mathrm{\infty })$.