Wave Operators for —Δ in a Domain with Non-Finite Boundary

Let ncJ£, ?z > 2 be a domain (open connected set) exterior to obstacles such that the obstacles, not necessarily finite in number, form a closed set enclosed in a cylinder SrQ = {x = (x19 ,xn) = (x,x^ 0}. The complement of SrQ is, therefore, contained in fl. We consider the differential operator —A on CVXfl),' which will be denoted by A. It is easy to see that A is a welldefined, non -negative definite operator in the Hilbert space L2(O), so that it has at least one self-adjoint extension. Let H be any such extension. We are to compare H with the operator H0 in L2(R } defined as follows: D(HQ^= {u^L2(R } : |£|zK|)eL2(JO}, (flo«) (?) = i?i^(lr) f° u^D(H0}, where u denotes the Fourier transform of u, i.e.,