The past and future wave operators on the singular spectrum (Q690577)

Let \(H_1\) and \(H_2\) be complex Hilbert spaces, \(U_i\) be a unitary operator on \(H_i\) \((i=1,2)\), and \(A:H_1\to H_2\) be a bounded operator. The past and the future wave operators \(W_{-}\) and \(W_{+}\) are defined as the limits (if they exist) of the sequence \(\{ U_{2}^{-n}AU_{1}^{n}\}_{n\in {\mathbb Z}}\) as \(n\to -\infty\) and \(n \to +\infty\), respectively. If the limits exist in the strong (weak) operator topology, then it is said that the strong (weak) wave operators \(W_{\pm}\) exist. The well known Kato-Rosenblum-Pearson Theorem asserts that, in the case when the spectral measures of \(U_1\) and \(U_2\) are absolutely continuous with respect to Lebesgue measure and \(AU_1-U_2A\) is a trace class operator, then the strong wave operators exist. The assumption of absolute continuity is essential. However, if the convergence of the arithmetical means \(\tfrac{1}{n+1}\sum_{k=0}^{n}U_{2}^{-k}AU_{1}^{k}\) is considered, the so-called averaged wave operators exist even for some unitary operators that have a non-trivial singular component of the spectral measure. In this paper, a slightly more general situation is studied. Let \(\{ p_{r,n}\}_{r\in E, n\in {\mathbb Z}_+}\), where \(E\) is a totally ordered set, be a family of positive numbers which determines an \(s\)-regular summation method. Let \[ W_{-}(r)=\sum_{n=0}^{+\infty}p_{r,n}U_{2}^{n}AU_{1}^{-n}\quad\text{ and}\quad W_{+}(r)=\sum_{n=0}^{+\infty}p_{r,n}U_{2}^{-n}AU_{1}^{n}. \] The past weak averaged wave operator is the limit \(W_{-}=\lim_{r}W_{-}(r)\) and the future weak averaged wave operator is \(W_{+}=\lim_{r}W_{+}(r)\). The main theorem of the paper says that, if the spectral measures of \(U_1\) and \(U_2\) are singular with respect to Lebesgue measure and \(AU_1-U_2A\) is of rank at most two, then the weak limit \(\lim_r(W_{+}(r)-W_{-}(r))\) exists and equals zero. It follows that the weak averaged wave operators exist or do not exist simultaneously and, in the case of existence, they are equal. Some other interesting related results are discussed as well.