On resonances generated by conic diffraction

Abstract

We describe the resonances closest to the real axis generated by diffraction of waves among cone points on a manifold with Euclidean ends. These resonances lie asymptotically evenly spaced along a curve of the form $$\frac{\Im \lambda }{\log \left|\Re \lambda \right|}=-\nu ;$$ here $\nu =(n-1)/2L_0$ where $n$ is the dimension and $L_0$ is the length of the longest geodesic connecting two cone points. Moreover there are asymptotically no resonances below this curve and above the curve $$\frac{\Im \lambda }{\log \left|\Re \lambda \right|}=-\Lambda$$ for a fixed $\Lambda >\nu .$

Slight correction to main theorem: finitely many different values of the constants $C_{\Re }$ and $C_{\Im }$ may be possible if there is more than one maximal diffracted closed orbit. Final version to appear in Ann. Inst. Fourier