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    Area of Science:

    • Quantum mechanics
    • Signal processing
    • Mathematical physics

    Background:

    • Linear canonical transforms (LCTs) are crucial in signal processing and optical systems.
    • The Bargmann transform, a complex LCT, was recently normalized by Pei and Huang for better delimitation.
    • The SU(2) finite harmonic oscillator model provides a framework for exploring quantum systems.

    Purpose of the Study:

    • To introduce the discrete normalized Bargmann transform using the SU(2) harmonic oscillator model.
    • To compare this new transform with the existing discrete Bargmann transform.
    • To investigate the properties and applications of the discrete normalized Bargmann transform.

    Main Methods:

    • Following the Pei-Huang algorithm for normalization.
    • Utilizing the relationship between Bargmann and gyrator transforms within the SU(2) model.
    • Comparing discrete Bargmann transforms based on coherent states and the new normalized version.

    Main Results:

    • The discrete normalized Bargmann transform is shown to be invertible and unitary.
    • It converts Hermite-Kravchuk functions into Laguerre-Kravchuk functions, mirroring continuous analogs.
    • Discrete su(1,1) repulsive oscillator functions exhibit self-reproduction with minimal error under this transform.
    • The transform commutes with the fractional Fourier-Kravchuk transform in the SU(2) harmonic oscillator space.

    Conclusions:

    • The discrete normalized Bargmann transform offers a valuable tool for signal processing and quantum mechanics.
    • Its properties, including function conversion and commutation relations, are established.
    • This work extends the utility of Bargmann transforms to discrete quantum systems.