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

Abstract

This paper serves as an extension of our work in defining the inverse momentum operator in a rigged Hilbert space [SPP-2024-PC-25], which is the triplet Φ ⊂ H ⊂ Φ^×. We construct the square of the inverse momentum as a rigged Hilbert space operator, p⁻² : Φ^× → Φ^×, and its Hilbert space reduction, p⁻²|_D : D → H. The particular rigging used is the Schwartz test function space. We show that D is dense in H. Moreover, we show that the RHS reduction p⁻²|_D is unique in the Hilbert space and essentially self-adjoint.