Statistical Measures and Complexity of Supersymmetric Polynomials in Quantum Mechanics

We study information-theoretic and complexity measures for the Dunkl-supersymmetric harmonic oscillator to identify the effect of supersymmetry on these quantities. Using the Rakhmanov probability density of the Dunkl-SUSY functions, we analyze the Shannon entropy, spreading measures (Heller, Rényi, and Fisher lengths), and several statistical and dynamical complexities. The Shannon entropy is obtained both asymptotically and in closed analytic form, showing that supersymmetry does not affect the leading large-n scaling. In contrast, spreading measures reveal enhanced localization of the SUSY eigenstates relative to the standard harmonic oscillator. Finally, we find that LMC and Fisher–Shannon complexities are higher in the supersymmetric case.