In this article, we present a quantum Painlevé second Lax pair that explicitly involves the Planck constant ħ and an arbitrary field variable v which distinguish it from the existing Flaschka–Newell Painlevé second Lax pair. We show that Flaschka–Newell pair appears as a special case of our quantum Painlevé second Lax pair. It is shown that the compatibility of quantum Painlevé second Lax pair simultaneously yields a quantum Painlevé equation and a quantum commutation relation between field variable v and independent variable z. We also show that the field variable v with different choices yields various analogs of classical Painlevé second equation as matrix Painlevé second equation, derivative matrix Painlevé second equation, and noncommutative Painlevé second equation. Further, we construct the gauge equivalence of quantum Painlevé second Lax pair whose compatibility condition generates a quantum P34 equation involving ħ with power +1 that brings the classical P34 equation close to its quantum analogs as compared to the existing one which carries ħ with power +2.

中文翻译：

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量子 Painlevé 第二个松弛对和经典 Painlevé II 方程的量子（矩阵）类似物

在本文中，我们提出了一个量子 Painlevé 第二 Lax 对，它明确涉及普朗克常数ħ和任意场变量v，这将其与现有的 Flaschka-Newell Painlevé 第二 Lax 对区分开来。我们证明 Flaschka-Newell 对是我们的量子 Painlevé 第二个 Lax 对的特例。结果表明，量子 Painlevé 第二个 Lax 对的相容性同时产生了量子 Painlevé 方程以及场变量v和自变量z之间的量子交换关系。我们还表明，不同选择的场变量v会产生经典 Painlevé 第二方程的各种类似物，如矩阵 Painlevé 第二方程、导数矩阵 Painlevé 第二方程和非交换 Painlevé 第二方程。此外，我们构造了量子 Painlevé 第二个 Lax 对的规范等价，其兼容性条件生成了一个涉及ħ的量子 P34 方程，其幂次为 +1，与现有的ħ次幂为 +1的方程相比，该方程使经典 P34 方程更接近其量子类似物。2.