Addendum to: “On the eigenvalue of positive operators”
Gian‐Carlo Rota · Bulletin of the American Mathematical Society · 1962
terior of a single star-shaped obstacle and for any solution which satisfies the boundary condition u = 0 with Cauchy data at / = 0 vanishing outside a sphere S, that the energy contained in sphere D is less than 7 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 at 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, 8 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 THEOREM III.The perturbed and unperturbed problems are unitarily equivalent', in particular, the spectrum of the wave equation generator is unaffected by the presence of obstacles.