The Existence and Completeness of Generalized Wave Operators for Spherically Symmetric, Asymptotically Coulomb Potentials

By applying a theorem of Kuroda, we prove the existence and continuous completeness of the generalized wave operators W_±(H₂, H₁), W_±(H₁, H₂), where W_±(H_j, H_i) = s — lime^iHj^te^−iHi^t × P_i, in the Hilbert space L²(R³). P_i is the projection operator on the subspace of absolute continuity of H_i. H₁ is the self‐adjoint Hamiltonian for a particle in a pure Coulomb potential V_c = ze²/|x|, and H₂ is the self‐adjoint Hamiltonian for a system described by the potential function V_c + V, where V is a real‐valued, measurable function of x ε R³, spherically symmetric [V(x) = V(r = |x|)], satisfies the condition $${\displaystyle {\int }_0^R}{r}^2{\mathrm{|V(r)|}}^2\mathrm{dr+}{\displaystyle {\int }_0^{\infty }}{\text{(1+r)}}^{\delta }{\mathrm{|V(r)|}}^i\mathrm{dr<\infty }$$ for some R(0 ≤ R < ∞), some 0 < δ < 1 with i = 1, 2, and is continuous except at r = 0. In conjunction with our result, we obtain a bound for the radial Coulomb Green's function. Dropping the continuity assumption on V, we have absolutely continuous completeness of the wave operators.