The effect of geometric focusing on dispersive estimates for the Schr\"odinger and wave equations

Abstract

We classify the long-time decay rate in dispersive estimates for the Schr\"odinger and wave equations on non-trapping asymptotically conic manifolds and exact metric cones in terms of the intensity of geometric focusing. Letting $X_0$ be a metric cone, one of our main results demonstrates that each multiplicity of conjugate points within distance $\pi$ on $Y=\partial X_0$ leads to a $|t|^{1/2}$-loss in the long-time decay order and a half-order shift in the regularity index in the dispersive estimate for the Schr\"odinger equation. Unexpectedly, conjugate point pairs on $Y$ at distance $\pi$ do not cause loss when the Legendre submanifold carrying the wave propagation satisfies a natural admissible condition that we propose. In sum, we give a robust framework for proving dispersive estimates that is stable under geometric perturbations and also accommodates perturbations by potentials.