FBI transforms in Gevrey classes (Q1385423)

The following theorem is proved: Let \({\mathcal O}\) be a Gevrey \(s\) strictly convex obstacle, \(1 \leq s < 3\). Then for every positive \(\varepsilon\) there are only finitely many resonances in the region \(\{ k \in {\mathbb C} \mid\text{Re }k \geq 1, \text{ Im } k \geq -(C_{0, a} - \varepsilon) (\text{Re } k)^{1/3} \}\). Here \(C_{0, a}\) is the constant found in a preceding paper by J. Sjöstrand. To prove such a result they develop a suitable Gevrey class FBI machinery, which will very likely be useful for other future applications.