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High frequency estimates for the Dirichlet-to-Neumann andNeumann-to-Dirichlet operators are obtained for the Helmholtz equation in theexterior of bounded obstacles. These a priori estimates are used to study thescattering of plane waves by an arbitrary bounded obstacle and to prove thatthe total cross section of the scattered wave does not exceed four geometricalcross sections of the obstacle in the limit as the wave number $k\to \infty$.This bound of the total cross section is sharp.

A priori estimates for high frequency scattering by obstacles of arbitrary shape