Spectral convergence of self-adjoint operators and generalized wave operators (Q1102511)

The continuity of wave operators \(W_{\pm}(B,A)\) in operator parameters A and B is established in the framework of the trace-class theory. Namely, suppose that \(A_ n\to A\), \(B_ n\to B\) strongly and \(B_ n- A_ n\to B-A\) in the trace-norm as \(n\to \infty\). Let, moreover, \(d(E_{A_ n}(\lambda)x,y)/d\lambda \to d(E_ A(\lambda)x,y)/d\lambda\), \(n\to \infty\), in \(L_ 1\) for arbitrary vectors x,y with \(E_ A(\cdot)\) being the spectral family of A. Then \(W_{\pm}(B_ n,A_ n)\to W_{\pm}(B,A)\) strongly as \(n\to \infty\).