We study the resolvent for nontrapping obstacles on manifolds with Euclidean ends. It is well known that for such manifolds, the outgoing resolvent satisfies $|chi R(k) chi|_{L^2to L^2}leq C{k}^{-1}$ for ${k}>1$, but the constant $C$ has been little studied. We show that, for high frequencies, the constant is bounded above by $2/pi$ 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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