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Segal–Bargmann Transforms Associated to a Family of Coupled Supersymmetries

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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.

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Acknowledgements

The author would like to thank Dr. John R. Klauder for the suggestion for this avenue of research and the late Dr. Donald J. Kouri for his mentorship in this project.

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  1. Department of Mathematics, Embry-Riddle Aeronautical University, 3700 Willow Creek Road, Prescott, AZ, 86301, USA

    Cameron L. Williams

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Communicated by Daniel Alpay.

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Williams, C.L. Segal–Bargmann Transforms Associated to a Family of Coupled Supersymmetries. Complex Anal. Oper. Theory 16, 94 (2022). https://doi.org/10.1007/s11785-022-01272-1

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