Regular subspaces of a quaternionic Hilbert space from quaternionic Hermite polynomials and associated coherent states

TL;DR

This paper introduces quaternionic Hermite polynomials, constructs regular and anti-regular subspaces in quaternionic L^2-spaces, and develops associated coherent states, extending complex Hermite polynomial theory to quaternionic settings.

Contribution

It defines quaternionic Hermite polynomials and constructs new regular and anti-regular subspaces with associated coherent states in quaternionic L^2-spaces.

Findings

Orthogonal quaternionic Hermite polynomial families established

Regular and anti-regular subspaces constructed in quaternionic L^2-spaces

Reproducing kernels and coherent states derived for quaternionic setting

Abstract

We define quaternionic Hermite polynomials by analogy with two families of complex Hermite polynomials. As in the complex case, these polynomials consatitute orthogonal families of vectors in ambient quaternionic $L^2$-spaces. Using these polynomials, we then define regular and anti-regular subspaces of these $L^2$-spaces, the associated reproducing kernels and the ensuing quaternionic coherent states.