The authors consider the Schrödinger operator \(H = H_0 + V(Q)\), \(H_0=-\Delta\), where \(V(Q)\) is the multiplication operator by a real valued function \(V\) on \({\mathbb R}^d\) such that \(V=V_{sr} + v\nabla \tilde{V}_{sr} + V_{lr} + V_c +W_{\alpha\beta}\). \(V_c\) is compactly supported and \(H_0\)-compact; \(V_{sr},\tilde{V}_{sr}\) and \(V_{lr}\) are short-range and long-range components, \(W=w(1-\kappa(|x|))|x|^{-\beta}\sin(k|x|^\alpha)\), \(\kappa\in\mathrm{C}^\infty_c({\mathbb R}; {\mathbb R} )\), is an oscillating part. The main result is the following limiting absorption principle (LAP) \[ \sup_{{\mathrm{Re}} z\in{\mathcal{I}} ,\atop {\mathrm{Im}} z\neq 0}\left\|\langle Q\rangle^{-s}(H-z)^{-1}\Pi^\perp\langle Q\rangle^{-s}\right\|\;< +\infty, \] where \(\Pi\) is the orthogonal projection onto the pure point spectral subspace of \(H\), and \(\Pi^\perp = 1-\Pi\) (if \(\mathcal I\) does not intersect the point spectrum, one can remove \(\Pi^\perp\)). The absence of positive eigenvalues is also proved in some cases.
At present time one of the most popular methods of proving LAP is the Mourre commutator method. But this method (and the weighted Mourre commutator method) cannot be applied to recover LAP for arbitrary pairs \((\alpha, \beta)\), since the Hamiltonian \(H\) is not regular enough w.r.t. the generator of dilations. To avoid these difficulties, the authors use the localised Putnam theory developed by S. Golénia and T. Jecko [Complex Anal. Oper. Theory 1, No. 3, 399–422 (2007; Zbl 1167.47010)]. This is what allowed the authors to improve known results.