q-deformed Fourier Theory

TL;DR

This paper develops a family of Fourier transformations for the q-deformed Heisenberg algebra, utilizing self-adjoint extensions and q-deformed functions, revealing a connection to elliptic functions.

Contribution

It introduces a novel family of Fourier transforms based on q-deformed algebra and elliptic functions, extending classical Fourier analysis to quantum algebra contexts.

Findings

Existence of a one-parameter family of Fourier transformations.

Construction of transformations using q-deformed trigonometric functions.

Characterization of the family by an elliptic function.

Abstract

We solve the problem of Fourier transformation for the one-dimensional $q$-deformed Heisenberg algebra. Starting from a matrix representation of this algebra we observe that momentum and position are unbounded operators in the Hilbert space. Therefore, in order to diagonalise the position operator in a momentum eigenbasis we have to study self-adjoint extensions of these operators. It turns out that there exist a whole family of such extensions for the position operator. This leads, correspondingly, to a one-parametric family of Fourier transformations. These transformations, which are related to continued fractions, are constructed in terms of $q$-deformed trigonometric functions. The entire family of the Fourier transformations turns out to be characterised by an elliptic function.