Rigged Hilbert space formalism as an extended mathematical formalism for quantum systems. I. General theory

Roberts' proposal of a rigged Hilbert space $\mathrm{\Phi \subset }\mathcal{G}\mathrm{\subset \Phi \times }$ for a certain class of quantum systems is reinvestigated and developed in order to exhibit various properties of this kind of rigged Hilbert spaces which might be of interest for the application of this formalism to special quantum systems. It is shown that on the basis of this proposal one also obtains a satisfactory solution for a rigged Hilbert space for composite systems. Another part is concerned with topological properties of the so‐called eigenoperators γ(t) belonging to an A‐eigen‐integral decomposition of Φ with respect to a self‐adjoint operator A on $\mathcal{G}$ . We derive a representation of γ(t) in terms of the generalized eigenvectors of A and in the same context give a rough topological characterization of these eigenvectors.