Segal-Bargmann Transforms Associated to a Family of Coupled Supersymmetries

TL;DR

This paper develops Segal-Bargmann transforms for a class of coupled supersymmetries, extending the classical transform to new spaces with non-Gaussian weights, revealing novel reproducing kernel Hilbert spaces.

Contribution

It introduces Segal-Bargmann transforms for coupled SUSYs, including non-Gaussian weight functions, expanding the scope of the classical transform.

Findings

Segal-Bargmann spaces for coupled SUSYs differ from classical spaces

Associated weight functions are non-Gaussian and more restrictive

New examples of reproducing kernel Hilbert spaces are provided

Abstract

The Segal-Bargmann transform is a Lie algebra and Hilbert space isomorphism between real and complex representations of the oscillator algebra. The Segal-Bargmann transform is useful in time-frequency analysis as it is closely related to the short-time Fourier transform. The Segal-Bargmann space provides a useful example of a reproducing kernel Hilbert space. Coupled supersymmetries (coupled SUSYs) are generalizations of the quantum harmonic oscillator that have a built-in supersymmetric nature and enjoy similar properties to the quantum harmonic oscillator. In this paper, we will develop Segal-Bargmann transforms for a specific class of coupled SUSYs which includes the quantum harmonic oscillator as a special case. We will show that the associated Segal-Bargmann spaces are distinct from the usual Segal-Bargmann space: their associated weight functions are no longer Gaussian and are spanned by stricter subsets of the holomorphic polynomials. The coupled SUSY Segal-Bargmann spaces provide new examples of reproducing kernel Hilbert spaces.