0 M ay 2 00 6 The L p boundedness of wave operators for Schrödinger operators with threshold singularities II . Even dimensional case

Let H = −∆ + V (x) be a Schrödinger operator on R, m ≥ 1, with real potential V (x) such that |V (x)| ≤ C〈x〉, 〈x〉 = (1+ |x|2) 1 2 , for some δ > 2. Then, H with domain D(H) = H(R), the Sobolev space of order 2, is selfadjoint in the Hilbert space H = L(R) and C 0 (R) is a core. The spectrum σ(H) ofH consists of absolutely continuous part [0,∞) and a finite number of non-positive eigenvalues {λj} of finite multiplicities. The singular continuous spectrum and positive eigenvalues are absent from H . We denote the point, the continuous and the absolutely continuous subspaces for H by Hp, Hc and Hac respectively, and the orthogonal projections onto the respective subspaces by Pp, Pc and Pac. We have Hac = Hc and Pac = Pc; H0 = −∆ is the free Schrödinger operator. The wave operators W± are defined by the following strong limits