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terior of a single star-shaped obstacle and for any solution which satisfies the boundary condition u = 0 with Cauchy data a t / = 0 vanishing outside a sphere S, that the energy contained in sphere D is less than const. E/t, where E is the total energy and the value of the constant depends only on the radii of S and D. We wish to point out that such a quantitative result about energy decay can hold only if the obstacle satisfies the following geometric condition; No two boundary points of it form a segment exterior to the obstacle and perpendicular to it a t both endpoints. For a narrow high frequency beam directed along such a segment would be reflected back and forth for a length of time proportional to the reciprocal of wave length. If one regards the presence of the obstacle as a perturbation of free space and applies the usual scattering theory formalism, then it follows from Huygens , principle in free space that the so-called wave operators exist, and it follows further from Theorem II that they are unitary. This proves

Addendum to: “On the eigenvalue of positive operators”