Absolutely continuous spectrum of Schrödinger operators with slowly decaying and oscillating potentials (Q1770247)

Traditionally absolute continuity of spectra of higher dimensional Schrödinger operators has been shown by scattering theory. This, however, requires a relatively fast decay of the potentials. Here the authors introduce a class of slowly decaying potentials, \( V \in \mathcal L^4 ( \mathbb R^d)\), \( d\geq3 \), for which the absolutely continuous spectrum is essentially supported by \([0, \infty)\). This requires in addition some oscillation with respect to one of the variables. They express this in terms of the partial Fourier transform of \(V\). The proof is rather technical and uses the reduction of the Schrödinger equation to the corresponding equation with the radial variable, but with an operator potential. The additional ingredients are trace inequalities and eigenvalue estimates.