Spherical harmonics and rigged Hilbert spaces

TL;DR

This paper explores the mathematical structure of spaces supporting SO(3) and SO(3,2) representations, focusing on spherical harmonics, rigged Hilbert spaces, and the properties of continuous and discrete bases.

Contribution

It demonstrates how discrete and continuous bases coexist in rigged Hilbert spaces and analyzes the continuity of operators within these frameworks.

Findings

Discrete and continuous bases coexist in rigged Hilbert spaces.

Operators are shown to be continuous under appropriate topologies.

Properties of functionals forming the continuous basis are characterized.

Abstract

This paper is devoted to study discrete and continuous bases for spaces supporting representations of SO(3) and SO(3,2) where the spherical harmonics are involved. We show how discrete and continuous bases coexist on appropriate choices of rigged Hilbert spaces. We prove the continuity of relevant operators and the operators in the algebras spanned by them using appropriate topologies on our spaces. Finally, we discuss the properties of the functionals that form the continuous basis.