Abstract

We study the resolvent for nontrapping obstacles on manifolds with Euclidean ends. It is well known that for such manifolds the outgoing resolvent satisfies $\parallel \chi R\left(k\right)\chi {\parallel }_{L^2\to L^2}\le C{k}^{-1}$ for $k>1$, but the constant $C$ has been little studied. We show that, for high frequencies, the constant is bounded above by $\frac{2}{\pi }$ times the length of the longest generalized bicharacteristic of $|\xi {|}_g^2-1$ remaining in the support of $\chi$. We show that this estimate is optimal in the case of manifolds without boundary. We then explore the implications of this result for the numerical analysis of the Helmholtz equation.

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Mathematical Subject Classification 2010

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