The Rate of Accumulation of Negative Eigenvalues to Zero and the Absolutely Continuous Spectrum

Abstract

For a bounded real-valued function V on ℝd, we consider two Schrödinger operators H+ = −Δ+V and H− = −Δ − 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,∞).