Resonance free domains for non globally analytic potentials
In the last years many papers are devoted to the study of the location of quantum resonances. The main purpose of the paper is to study the absence of resonance in some fixed complex neighborhood of any non trapping energy level for a non globally analytic function on \(\mathbb{R}^n\). In the paper under review the \(n\)-dimensional semiclassical Schrödinger operator \[ P(h)=-h^2\Delta+V(x), \] where \(h>0\) is small enough and the potential \(V\in C^{\infty}(\mathbb{R}^n)\) is real valued such function \(V\) is on \(\mathbb{R}^n\setminus K\), where \(K\) is some compact set, and satisfies some conditions. The main results of the paper are the following: for some \(\delta> 0\) and for any \(C>0\), \(P(h)\) admits no resonance in the domain 1. \(\Omega=[{\l}_0-\delta,{\l}_0+\delta]-i[0,Ch\log(h^{-1})]\) if \(V\) is \(C^{\infty}\), 2. \(\Omega=[{\l}_0-\delta,{\l}_0+\delta]-i[0,\delta h^{1-1/s}]\) if \(V\) is Gevrey with index \(s\). Here \(\delta>0\) does not depend on \(h\) and the results are uniform with respect to \(h>0\) small enough. In the proof the author uses an almost analytic extension of the potential, as defined in [\textit{A. Melin} and \textit{J. Sjöstrand}, Lect. Notes Math. 459, 120-223 (1974; Zbl 0306.42007)] and proves for it an equivalent of the exponential microlocal weight estimates introduced in [\textit{A. Lahmer-Benbernou} and \textit{A. Martinez}, Int. Math. Res. Notes 2002, 697-717 (2002; Zbl 1027.35171)]. The obtained results generalize the result of \textit{S. H. Tang} and \textit{M. Zworski} [Commun.Pure Appl. Math. 53, 1305-1334 (2000; Zbl 1032.35148)] and the author's method gives an alternative proof of the result of \textit{M. Rouleux} [Hokkaido Math. J. 30, 475-517 (2001; Zbl 0991.35043)].
scientific article; zbMATH DE number 1890549
scientific article