Weighted Mourre's commutator theory, application to Schrödinger operators with oscillating potential (Q2857164)

Let \(\mathcal H\) be a complex Hilbert space and \(H,A\) be self-adjoint operators in \(\mathcal H\). Let \(\mathcal {I, J}\) be open intervals of \(\mathbb R\). Given \(k\in\mathbb N\), we say that \(H\in C^k_{\mathcal J}\) if, for all \(\chi\in C_0^\infty({\mathbb R})\) with support in \(\mathcal J\) and for all \(f \in\mathcal H\), the map \({\mathbb R}\ni t\to e^{itA} \chi(H)a^{-itH}f\in {\mathcal H}\) has the usual \(C^k\) regularity. Denote by \(E_{\mathcal I} (H)\) the spectral measure of \(H\) on the bounded interval \(\mathcal I\). It is a classical result due to \textit{E. Mourre} [Commun. Math. Phys. 78, 391--408 (1981; Zbl 0489.47010)] that, for \(H\in C^2_{\mathcal J}(A)\), \({\mathcal I}\subset {\mathcal J}\), the commutator estimate NEWLINE\[NEWLINEE_{\mathcal I}[H,iA] E_{\mathcal I} \geq c_0 E_{\mathcal I}NEWLINE\]NEWLINE implies the limiting absorption principle NEWLINE\[NEWLINE\sup\limits_{\text{Re} z\in {\mathcal I}, \text{Im}z\neq 0}\left\|(1+A^2)^{-s/2}(H-z)^{-1} (1+A^2)^{-s/2} \right\|< \infty,\quad s>\frac{1}{2}.NEWLINE\]NEWLINE In the paper under review, the authors consider the following ``weighted Mourre estimate'' NEWLINE\[NEWLINEE_{\mathcal I}[H,i\varphi(A)] E_{\mathcal I} \geq c_1 E_{\mathcal I}(1+A^2)^{-\frac{1+\varepsilon}{2}} E_{\mathcal I},NEWLINE\]NEWLINE where \(\varepsilon =2s-1>0\) and \(\varphi\) is some appropriate non-negative, bounded, smooth function on \(\mathbb R\). It is proved that, for \(H\in C^1_{\mathcal J}(A)\), this estimate implies the limiting absorption principle.NEWLINENEWLINE The authors apply this result to the proof of the limiting absorption principle for Schrödinger operators \(H_1\) with a perturbed Wigner-von Neumann potential \(W(x)=q\frac{\sin k| x|}{| x|}\) at suitable energies. A long range perturbation of the Wigner-von Neumann potential is also under consideration. The authors show that \(H_1\) does not have the required regularity to apply the usual Mourre theory, based on differential inequalities and on the generator of dilation.