Wave asymptotics for waveguides and manifolds with infinite cylindrical ends

We describe wave decay rates associated to embedded resonances and spectral thresholds for waveguides and manifolds with infinite cylindrical ends. We show that if the cut-off resolvent is polynomially bounded at high energies, as is the case in certain favorable geometries, then there is an associated asymptotic expansion, up to a $O(t^{-k_0})$ remainder, of solutions of the wave equation on compact sets as $t \to \infty$. In the most general such case we have $k_0=1$, and under an additional assumption on the infinite ends we have $k_0 = \infty$. If we localize the solutions to the wave equation in frequency as well as in space, then our results hold for quite general waveguides and manifolds with infinite cylindrical ends. To treat problems with and without boundary in a unified way, we introduce a black box framework analogous to the Euclidean one of Sj\"ostrand and Zworski. We study the resolvent, generalized eigenfunctions, spectral measure, and spectral thresholds in this framework, providing a new approach to some mostly well-known results in the scattering theory of manifolds with cylindrical ends.Comment: In this revision we work in a more general black box setting than in the first version of the paper. In particular, we allow a boundary extending to infinity. The changes to the proofs of the main theorems are minor, but the presentation of the needed basic material from scattering theory is substantially expanded. New examples are included, both for the main results and for the black box settin

Resonances for Schr\"odinger operators on infinite cylinders and other products

We study the resonances of Schr\"odinger operators on the infinite product $X=\mathbb{R}^d\times \mathbb{S}^1$, where $d$ is odd, $\mathbb{S}^1$ is the unit circle, and the potential $V\in L^\infty_c(X)$. This paper shows that at high energy, resonances of the Schr\"odinger operator $-\Delta +V$ on $X=\mathbb{R}^d\times \mathbb{S}^1$ which are near the continuous spectrum are approximated by the resonances of $-\Delta +V_0$ on $X$, where the potential $V_0$ given by averaging $V$ over the unit circle. These resonances are, in turn, given in terms of the resonances of a Schr\"odinger operator on $\mathbb{R}^d$ which lie in a bounded set. If the potential is smooth, we obtain improved localization of the resonances, particularly in the case of simple, rank one poles of the corresponding scattering resolvent on $\mathbb{R}^d$. In that case, we obtain the leading order correction for the location of the corresponding high energy resonances. In addition to direct results about the location of resonances, we show that at high energies away from the resonances, the resolvent of the model operator $-\Delta+V_0$ on $X$ approximates that of $-\Delta+V$ on $X$. If $d=1$, in certain cases this implies the existence of an asymptotic expansion of solutions of the wave equation. Again for the special case of $d=1$, we obtain a resonant rigidity type result for the zero potential among all real-valued potentials.Comment: 46 pages; v. 2 is attempt to fix uploading erro