Quantum Physics and Signal Processing in Rigged Hilbert Spaces by means of Special Functions, Lie Algebras and Fourier and Fourier-like Transforms

TL;DR

This paper explores the integration of quantum physics and signal processing through advanced mathematical frameworks like rigged Hilbert spaces, special functions, and Lie algebras, introducing new bases and transforms for improved analysis.

Contribution

It introduces a novel discrete basis in R, extends Fourier-like transforms to R^+ and R^n, and connects operators to the universal enveloping algebra of io(2).

Findings

Established a rigged Hilbert space framework.

Developed a new discrete basis in R.

Extended Fourier-like transforms to R^+ and R^n.

Abstract

Quantum Mechanics and Signal Processing in the line R, are strictly related to Fourier Transform and Weyl-Heisenberg algebra. We discuss here the addition of a new discrete variable that measures the degree of the Hermite functions and allows to obtain the projective algebra io(2). A Rigged Hilbert space is found and a new discrete basis in R obtained. The operators {O[R]} defined on R are shown to belong to the Universal Enveloping Algebra UEA[io(2)] allowing, in this way, their algebraic discussion. Introducing in the half-line a Fourier-like Transform, the procedure is extended to R^+ and can be easily generalized to R^n and to spherical reference systems.