Analysis and PDE Seminar: Resolvent estimates and wave asymptotics for manifolds with cylindrical ends

Seminar | September 18 | 4:10-5 p.m. | 740 Evans Hall

 Kiril Datchev, Purdue University

 Department of Mathematics

Wave oscillation and decay rates on a manifold \(M\) are well known to be related to the geometry of \(M\) and to dynamical properties of its geodesic flow. When \(M\) is a closed system, such as a bounded Euclidean domain or a compact manifold, there is no decay and the connection is made via the eigenvalues of the Laplacian. When \(M\) is an open system, such as the complement of a bounded Euclidean domain, or a suitable more general manifold with large infinite ends, then the spectrum of the Laplacian is continuous and we look instead at resonances, which give rates of both oscillation and decay.

An interesting intermediate situation is a manifold with infinite cylindrical ends, which we call a mixed system. In this case the continuous spectrum has increasing multiplicity as energy grows, and in general it can have embedded resonances and eigenvalues accumulating at infinity, making wave asymptotics more mysterious. However, we prove that if geodesic trapping is sufficiently mild, then such an accumulation is ruled out, and moreover lossless high-energy resolvent bounds hold. We deduce from this the existence of resonance free regions and compute asymptotic expansions for solutions of the wave equation in terms of eigenvalues, resonances, and spectral thresholds. This is joint work with Tanya Christiansen.