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We study a geometric generalization of the time-dependent Schrödinger equation for the harmonic oscillator $$\left(D_t+\frac{1}{2}\Delta +V\right)\psi =0(0.1)$$ where $\Delta$ is the Laplace-Beltrami operator with respect to a “scattering metric” on a compact manifold $M$ with boundary (the class of scattering metrics is a generalization of asymptotically Euclidean metrics on ${ℝ}^n$, radially compactified to the ball) and $V$ is a perturbation of $\frac{1}{2}{\omega }^2{x}^{-2}$, with $x$ a boundary defining function for $M$ (e.g. $x=1/r$ in the compactified Euclidean case). Using...

We discuss recent progress in understanding the effects of certain trapping geometries on cut-off resolvent estimates, and thus on the qualititative behavior of linear evolution equations. We focus on trapping that is unstable, so that strong resolvent estimates hold on the real axis, and large resonance-free regions can be shown to exist beyond it.

We show that the ``radiation field'' introduced by F.G. Friedlander, mapping Cauchy data for the wave equation to the rescaled asymptotic behavior of the wave, is a Fourier integral operator on any non-trapping asymptotically hyperbolic or asymptotically conic manifold. The underlying canonical relation is associated to a ``sojourn time'' or ``Busemann function'' for geodesics. As a consequence we obtain some information about the high frequency behavior of the scattering...