Segal-Bargmann Transforms Associated to a Family of Coupled Supersymmetries

DOI10.1007/S11785-022-01272-1arXiv2110.08995WikidataQ114221741 ScholiaQ114221741MaRDI QIDQ6380608

Publication date: 17 October 2021

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.

Mathematics Subject Classification ID

Coherent states (81R30) Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) (46E22) Representations of Lie algebras and Lie superalgebras, analytic theory (17B15) Bessel and Airy functions, cylinder functions, ({}_0F_1) (33C10) Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) (47B32) General integral transforms (44A05)

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