The generalized scattering problems: ergodic type theorems
The paper uses the results of the paper [the author, Complex Anal. Oper. Theory 12, No. 3, 607--613 (2018; Zbl 1390.34237)]. Schrödinger radial equation with Coulomb-type potential (see the abstract of the mentioned paper for definition and conditions on the potential) is considered. The author proves an ergodicity-type theorem stating that the generalized dynamical scattering operator and the generalized stationary scattering operator coincide. The generalized dynamical scattering operator \(S_{\mathrm{dyn}}\) describes the behaviour of the system in the limit when time goes to infinity. On the other hand, the generalized stationary scattering operator \(S_{\mathrm{st}}\) describes the behaviour when the space coordinate goes to infinity (for definition see the abstract of the above mentioned paper). The relation \(S_{\mathrm{dyn}} = S_{\mathrm{st}}\) is proven under the additional conditions that the integrals \(\int_r^\infty \phi^2(u)\,\mathrm{d}u\), \(\int_r^\infty |\phi'(u)|\,\mathrm{d}u\) and \(\int_r^\infty |q(u)|\,\mathrm{d}u\) are all in \(L^2(a,\infty)\).
scientific article; zbMATH DE number 6860222