Propagation properties for Schrödinger operators affiliated with certain \(C^*\)-algebras
The paper under review is concerned with propagation properties of scattering states of selfadjoint \(n\)-dimensional Schrödinger operators \(H=-\Delta +V\) with potential \(V\). Remember that the scattering states of \(H\) are defined by the property that they propagate away from each bounded region of the configuration space \({\mathbb R}^n\) as time \(t\to\pm\infty\). If the potential \(V\) tends to zero sufficiently rapidly at infinity, a description of the behaviour of \(\text{ e}^{-\text{ i}tH}f\) for a scattering state \(f\) at large times \(t\) is provided by standard scattering theory. On the other hand, in more complicated situations (e.g., if the asymptotic behaviour of \(V\) is anisotropic), little is known about the propagation of scattering states, and it is expected that certain asymptotic regions of configuration space should be inaccessible to states of certain energies. Using a careful spectral analysis of the Hamiltonians \(H\) under study, detailed results about the propagation and non-propagation of states for one-dimensional Schrödinger operators and multi-dimensional operators periodic in all but one dimension are obtained in [\textit{E. B. Davies} and \textit{B. Simon}, Commun. Math. Phys. 63, 277-301 (1978; Zbl 0393.34015)]. In the paper under review, a different method for obtaining non-propagation properties, based on a recent approach to spectral theory in the framework of \(C^*\)-algebras, is proposed. Along these lines, an abstract theorem on non-propagation and applications to Schrödinger Hamiltonians that are asymptotically periodic in several cones are given.
scientific article; zbMATH DE number 1874656