An introduction to generalised functions in periodic quantum theory

TL;DR

This paper demonstrates that minimum uncertainty states in periodic quantum systems exist within a Colombeau algebra of generalized functions, extending the mathematical framework beyond traditional Hilbert spaces and Schwartz distributions.

Contribution

It introduces a Colombeau algebra framework for periodic quantum systems, enabling the existence of states and eigen-decompositions not possible in standard Hilbert space.

Findings

Minimum uncertainty states exist in Colombeau algebra but not in Hilbert space.

Generalized eigen-decompositions are proven for operators with periodic spectra.

Colombeau algebra extends the mathematical tools for quantum theory.

Abstract

A proof that minimum uncertainty states of the simplest periodic quantum system exist in a state space that is represented by a Colombeau algebra of generalised functions but not in Hilbert space or in the space of Schwartz distributions is given. There are two significant generalisations of Hilbert space that lead to the special Colombeau algebra. The first step is to a vector space of Schwartz distributions that contains the eigenstates of operators with continuous spectra, the second is to a Colombeau algebra that provides a solution for a differential equation with non-constant coefficients. From the perspective of Colombeau algebra a rigged Hilbert space of Schwartz distributions can be understood as a representation of the linear component of the algebra. The utility of this representation is illustrated by a proof that generalised eigen-decompositions exist for linear operators with periodic continuous spectra.