Semiclassical resolvent estimates for trapping perturbations
The authors obtain semiclassical estimates for the resolvent \[ R(z) = (L(h)-z)^{-1}, \quad z \in \mathbb{C} \backslash \mathbb{R}, \] of a self-adjoint operator \(L(h)\) in a Hilbert space \(\mathcal{H}\) depending on \(h \in ]0,h_{0}]\) for \quad \(z=\lambda+i \tau \in \mathbb{C}\), where \(\lambda \in ]\mu_{0},\mu_{1}[ \subset \subset \mathbb {R}\) and \(\pm \tau \in ]0,1]\). While the case that \(\lambda\) is a non-trapping energy level has been studied by many authors, in the present paper the more general case of long-range trapping ``black-box'' perturbations is considered. The main result is the estimate \[ \|R(\lambda \pm i \tau) \|_{s,-s} \leq C_{s} \exp(C_{s} h^{-p}), \quad 0<h \leq h_{1}, \] for the resolvent considered as a mapping from \(\mathcal{H}^{0,s}\) to \(\mathcal{H}^{0,-s}\), where \(s>1/2\). The principal assumption for this estimate to hold is the cut-off estimate \[ \|\chi (x) R(\lambda \pm i 0) \chi (x)\|_{\mathcal{H} \rightarrow \mathcal{H}} \leq C \exp(C h^{-p}), \quad \lambda \in ]\mu_{0}-\delta,\mu_{1}+\delta[ , \quad 0<h \leq h_{0}, \] with \(\delta > 0\) and \(p \geq 1\). The involved proof uses microlocal estimates to finally exploit the known estimates for non-trapping energy levels.
scientific article; zbMATH DE number 1529108