Inverse momentum operator in a rigged Hilbert space (RHS) and its Hilbert space reduction

Abstract

We construct several representations in position basis of the inverse momentum operator in a RHS $\Phi\subset\mathcal{H}\subset{\Phi}^{\times}$ by means of contour integral representations of an otherwise divergent kernel. In the RHS, all representations acting in composition with the momentum operator resolve to the identity operator. However, there exist representations where the eigenvalue equation cannot be constructed; in the case that it does: the eigenvalues are complex, the set of generalized eigenfunctions is approximately complete, and the total probability can be less than or greater than one. In the Hilbert space, all representations are equivalent with the representative element being a densely defined, self-adjoint operator. The existence of a densely defined, Hilbert space reduction implies that the constructed representations are RHS operators.