In this study, we have generalized the fractional action integral by using the Saigo-Maeda fractional operators defined in terms of the Appell hypergeometric function of two variables , F3(a, a′, β, β′; γ; z, ζ) with complex parameters. We have derived the associated Euler-Lagrange equation and we have studied the harmonic oscillator problem. We have proved that a PT -symmetric quantum-mechanical Hamiltonian characterized by real and discrete spectra is obtained although the system is characterized by complex trajectories. The associated thermodynamical properties were discussed and it was revealed the entropy of the quantum system decreases with time toward an asymptotically positive value similar to what is observed in quantum Maxwell demon.

中文翻译：

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Saigo-Maeda 算子涉及 Appell 函数、对称量子哈密顿量的实谱和违反量子阻尼振荡器的热力学第二定律

在本研究中，我们通过使用根据两个变量 F3(a, a', β, β'; γ; z, ζ) 的 Appell 超几何函数定义的 Saigo-Maeda 分数算子推广了分数作用积分复杂的参数。我们导出了相关的欧拉-拉格朗日方程，并且研究了谐振子问题。我们已经证明，尽管系统具有复杂的轨迹特征，但获得了具有实谱和离散谱特征的PT对称量子力学哈密顿量。讨论了相关的热力学性质，并揭示了量子系统的熵随时间向渐近正值减小，类似于在量子麦克斯韦妖中观察到的情况。